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Add scripts and inp files.

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James Grogan 2024-05-13 20:50:21 +01:00
parent ad937f2602
commit e19f869a1e
390 changed files with 6580687 additions and 10 deletions
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234
Unpublished/XFEM/1D_Solver/2DLevelSet.m Normal file
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function [] = 2DLevelSet()
clear all
% Define Main Solution Mesh
NumX=8;
NumY=8;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
centx=4.;
centy=4.;
rad=2.1;
for i=1:numNodes;
dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
lSet(i)=dist-rad;
end
%for i=1:numNodes;
% dist=Node(i,1)-0.1;
% lSet(i)=dist;
%end
% Plot initial level set
[X Y]=meshgrid(0:1.:8);
Z=zeros(9);
for i=1:81
Z(i)=lSet(i);
end
surf(X,Y,Z)
% LS Algorithm Parameters
lSet'
bandwidth=10.;
% Loop through timesteps
for tstep=1:100
% Identify Narrow Band Elements
NBindex=0;
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,2);
NBelem(NBindex,3)=Element(i,3);
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,3);
NBelem(NBindex,3)=Element(i,4);
end
end
% Velocity BC
F=zeros(numNodes,1);
for i=1:NBindex
if sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,2)))||sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,3)))
F(NBelem(i,1))= 1.;
F(NBelem(i,2))= 1.;
F(NBelem(i,3))= 1.;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(numNodes);
for i=1:NBindex
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
% Apply BCs
RHS=zeros(numNodes,1);
Sub=A*F;
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:numNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(numNodes);
mMatGLS=zeros(numNodes);
f1=zeros(numNodes,1);
f2=zeros(numNodes,1);
f3=zeros(numNodes,1);
h2=0.00001;
visc=0.0005;
for i=1:NBindex
mMatL=zeros(3);
mMatGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
mMatL=mMatL+(phi'*phi)/3.;
mMatGLSL=mMatGLSL+((delphi'*(delset/norm(delset)))*Floc*(h2/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(h2/abs(Floc))*Floc*norm(delset)/3.;
vs=h2*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+h2));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
mMat(NBelem(i,j),NBelem(i,k))=mMat(NBelem(i,j),NBelem(i,k))+mMatL(j,k);
mMatGLS(NBelem(i,j),NBelem(i,k))=mMatGLS(NBelem(i,j),NBelem(i,k))+mMatGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
dt=0.01;
lSet=lSet-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))';
end
lSet'
%[X Y]=meshgrid(0:1.:8);
%Z=zeros(9);
%for i=1:81
% Z(i)=lSet(i);
%end
%surf(X,Y,Z)

234
Unpublished/XFEM/1D_Solver/2DLevelSetFMM.m Normal file
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function [] = 2DLevelSetFMM()
clear all
% Define Main Solution Mesh
NumX=3;
NumY=1;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
%centx=4.;
%centy=4.;
%rad=2.1;
%for i=1:numNodes;
% dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
% lSet(i)=dist-rad;
%end
for i=1:numNodes;
dist=Node(i,1)-0.1;
lSet(i)=dist;
end
% Plot initial level set
%[X Y]=meshgrid(0:1.:8);
%Z=zeros(9);
%for i=1:81
% Z(i)=lSet(i);
%end
%surf(X,Y,Z)
% LS Algorithm Parameters
lSet'
bandwidth=10.;
% Loop through timesteps
for tstep=1:1
% Identify Narrow Band Elements
NBindex=0;
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,2);
NBelem(NBindex,3)=Element(i,3);
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,3);
NBelem(NBindex,3)=Element(i,4);
end
end
% Velocity BC
F=zeros(numNodes,1);
for i=1:NBindex
if sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,2)))||sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,3)))
F(NBelem(i,1))= 1.;
F(NBelem(i,2))= 1.;
F(NBelem(i,3))= 1.;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(numNodes);
for i=1:NBindex
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
% Apply BCs
RHS=zeros(numNodes,1);
Sub=A*F;
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:numNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(numNodes);
mMatGLS=zeros(numNodes);
f1=zeros(numNodes,1);
f2=zeros(numNodes,1);
f3=zeros(numNodes,1);
h2=0.00001;
visc=0.0005;
for i=1:NBindex
mMatL=zeros(3);
mMatGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
mMatL=mMatL+(phi'*phi)/3.;
mMatGLSL=mMatGLSL+((delphi'*(delset/norm(delset)))*Floc*(h2/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(h2/abs(Floc))*Floc*norm(delset)/3.;
vs=h2*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+h2));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
mMat(NBelem(i,j),NBelem(i,k))=mMat(NBelem(i,j),NBelem(i,k))+mMatL(j,k);
mMatGLS(NBelem(i,j),NBelem(i,k))=mMatGLS(NBelem(i,j),NBelem(i,k))+mMatGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
dt=0.01;
lSet=lSet-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))';
end
lSet'
%[X Y]=meshgrid(0:1.:8);
%Z=zeros(9);
%for i=1:81
% Z(i)=lSet(i);
%end
%surf(X,Y,Z)

322
Unpublished/XFEM/1D_Solver/F2DLevelSetFMM.asv Normal file
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function [] = F2DLevelSetFMM()
clear all
% Define Main Solution Mesh
NumX=5;
NumY=1;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
%centx=4.;
%centy=4.;
%rad=2.1;
%for i=1:numNodes;
% dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
% lSet(i)=dist-rad;
%end
for i=1:numNodes;
dist=Node(i,1)-0.1;
lSet(i)=dist;
end
% Plot initial level set
%[X Y]=meshgrid(0:1.:8);
%Z=zeros(9);
%for i=1:81
% Z(i)=lSet(i);
%end
%surf(X,Y,Z)
% LS Algorithm Parameters
lSet'
bandwidth=3.;
% Loop through timesteps
for tstep=1:1
% Identify Narrow Band Elements
NBElems=0;
NBNodes=0;
NGlobal=zeros(NumNodes);
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
for j=1:4
if NewNums(Element(i,j))==0
NBNodes=NBNodes+1
NGlobal(Element(i,j))=NBNodes;
NLocal(NBNodes)=Element(i,j);
end
end
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,2));
NBelem(NBElems,3)=NGlobal(Element(i,3));
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,3));
NBelem(NBElems,3)=NGlobal(Element(i,4));
end
end
% Get local Level Set
for i=1:NBNodes
lSetLocal(i)=lSet(NLocal(i));
end
% Velocity BC
F=zeros(NBNodes,1);
for i=1:NBElems
L1=sign(lSetLocal(NBelem(i,1)));
L2=sign(lSetLocal(NBelem(i,2)));
L3=sign(lSetLocal(NBelem(i,3)));
if L1 ~= L2 || L1 ~= L3
F(NBelem(i,1))= 1.;
F(NBelem(i,2))= 1.;
F(NBelem(i,3))= 1.;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(NBNodes);
for i=1:NBElems
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSetLocal(NBelem(i,1));lSetLocal(NBelem(i,2));lSetLocal(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
% Apply BCs
RHS=zeros(NBNodes,1);
Sub=A*F;
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:NBNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(NBNodes);
mMatGLS=zeros(NBNodes);
f1=zeros(NBNodes,1);
f2=zeros(NBNodes,1);
f3=zeros(NBNodes,1);
h2=0.00001;
visc=0.0005;
for i=1:NBElems
mMatL=zeros(3);
mMatGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSetLocal(NBelem(i,1));lSetLocal(NBelem(i,2));lSetLocal(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
mMatL=mMatL+(phi'*phi)/3.;
mMatGLSL=mMatGLSL+((delphi'*(delset/norm(delset)))*Floc*(h2/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(h2/abs(Floc))*Floc*norm(delset)/3.;
vs=h2*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+h2));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
mMat(NBelem(i,j),NBelem(i,k))=mMat(NBelem(i,j),NBelem(i,k))+mMatL(j,k);
mMatGLS(NBelem(i,j),NBelem(i,k))=mMatGLS(NBelem(i,j),NBelem(i,k))+mMatGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
dt=0.01;
lSetLocal=lSetLocal-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))';
newlSet=lSetLocal;
% Reinitialize LS
nstat=zeros(NBNodes,1);
for i=1:NBelems
L1=sign(lSetLocal(NBelem(i,1)));
L2=sign(lSetLocal(NBelem(i,2)));
L3=sign(lSetLocal(NBelem(i,3)));
if L1 ~= L2 || L1 ~= L3
for j=1:3
nstat(NBelem(i,j))=1;
end
end
end
maincheck=0;
while(maincheck==0)
lmin=1000.;
avlmin=1000.;
eindex=0;
nindex=0;
maincheck=1;
for i=1:NBindex
if nstat(NBelem(i,1))+nstat(NBelem(i,2))+nstat(NBelem(i,3))==2
maincheck=0;
check=0;
ltot=0.;
for j=1:3
if nstat(NBelem(i,j))==0
if abs(lSet(NBelem(i,j)))<=lmin
check=1;
tempindex=j;
end
end
ltot=ltot+abs(lSet(NBelem(i,j)));
end
if check==1 & ltot/3.<=avlmin
eindex=i;
nindex=tempindex;
lmin=lSet(NBelem(eindex,nindex));
avlmin=ltot/3.;
end
end
end
if maincheck==0
% Find New LS for point
xp=Node(NBelem(eindex,nindex),1);
yp=Node(NBelem(eindex,nindex),2);
count=0;
for i=1:3
if i~=nindex
count=count+1;
x(count)=Node(NBelem(eindex,i),1);
y(count)=Node(NBelem(eindex,i),2);
lloc(count)=newlSet(NBelem(eindex,i));
end
end
delxa=x(1)-xp;
delya=y(1)-yp;
delxb=x(2)-xp;
delyb=y(2)-yp;
N=[delxa delya; delxb delyb];
M=N^-1;
A=(M(1)*M(1)+M(2)*M(2));
B=(M(3)*M(3)+M(4)*M(4));
C=2.*(M(1)*M(3)+M(2)*M(4));
a=A+B+C;
b=-2.*lloc(1)*A-2.*lloc(2)*B-C*(lloc(1)+lloc(2));
c=lloc(1)*lloc(1)*A+lloc(2)*lloc(2)*B+lloc(1)*lloc(2)*C-1.;
templ1=(-b+sqrt(b*b-4.*a*c))/(2.*a);
templ2=(-b-sqrt(b*b-4.*a*c))/(2.*a);
if abs(templ1)>abs(templ2)
newlSet(NBelem(eindex,nindex))=templ1;
else
newlSet(NBelem(eindex,nindex))=templ2;
end
nstat(NBelem(eindex,nindex))=1;
end
end
% lSet=newlSet;
end
lSet'

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function [] = F2DLevelSetFMM()
clear all
% Define Main Solution Mesh
NumX=32;
NumY=32;
delX=0.25;
delY=0.25;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
centx=4.;
centy=4.;
rad=2.1;
for i=1:numNodes;
dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
lSet(i)=dist-rad;
end
%for i=1:numNodes;
% dist=Node(i,1)-0.1;
% lSet(i)=dist;
%end
% Plot initial level set
[X Y]=meshgrid(0:0.25:8);
Z=zeros(33);
for i=1:1089
Z(i)=lSet(i);
end
surf(X,Y,Z)
% LS Algorithm Parameters
lSet'
bandwidth=10;
% Loop through timesteps
for tstep=1:10
% Identify Narrow Band Elements
NBElems=0;
NBNodes=0;
NGlobal=zeros(numNodes);
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
for j=1:4
if NGlobal(Element(i,j))==0
NBNodes=NBNodes+1;
NGlobal(Element(i,j))=NBNodes;
NLocal(NBNodes)=Element(i,j);
end
end
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,2));
NBelem(NBElems,3)=NGlobal(Element(i,3));
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,3));
NBelem(NBElems,3)=NGlobal(Element(i,4));
end
end
% Get local Level Set
for i=1:NBNodes
lSetLocal(i)=lSet(NLocal(i));
end
% Velocity BC
F=zeros(NBNodes,1);
for i=1:NBElems
L1=sign(lSetLocal(NBelem(i,1)));
L2=sign(lSetLocal(NBelem(i,2)));
L3=sign(lSetLocal(NBelem(i,3)));
if L1 ~= L2 || L1 ~= L3
F(NBelem(i,1))= 1.;
F(NBelem(i,2))= 1.;
F(NBelem(i,3))= 1.;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(NBNodes);
for i=1:NBElems
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSetLocal(NBelem(i,1));lSetLocal(NBelem(i,2));lSetLocal(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
% Apply BCs
RHS=zeros(NBNodes,1);
Sub=A*F;
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:NBNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(NBNodes);
mMatGLS=zeros(NBNodes);
f1=zeros(NBNodes,1);
f2=zeros(NBNodes,1);
f3=zeros(NBNodes,1);
h2=0.00001;
visc=0.0005;
for i=1:NBElems
mMatL=zeros(3);
mMatGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSetLocal(NBelem(i,1));lSetLocal(NBelem(i,2));lSetLocal(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
mMatL=mMatL+(phi'*phi)/3.;
mMatGLSL=mMatGLSL+((delphi'*(delset/norm(delset)))*Floc*(h2/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(h2/abs(Floc))*Floc*norm(delset)/3.;
vs=h2*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+h2));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
mMat(NBelem(i,j),NBelem(i,k))=mMat(NBelem(i,j),NBelem(i,k))+mMatL(j,k);
mMatGLS(NBelem(i,j),NBelem(i,k))=mMatGLS(NBelem(i,j),NBelem(i,k))+mMatGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
dt=0.01;
lSetLocal=lSetLocal-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))';
newlSet=lSetLocal;
% Reinitialize LS
nstat=zeros(NBNodes,1);
for i=1:NBElems
L1=sign(lSetLocal(NBelem(i,1)));
L2=sign(lSetLocal(NBelem(i,2)));
L3=sign(lSetLocal(NBelem(i,3)));
if L1 ~= L2 || L1 ~= L3
for j=1:3
nstat(NBelem(i,j))=1;
end
end
end
maincheck=0;
while(maincheck==0)
lmin=1000.;
avlmin=1000.;
eindex=0;
nindex=0;
maincheck=1;
for i=1:NBElems
if nstat(NBelem(i,1))+nstat(NBelem(i,2))+nstat(NBelem(i,3))==2
maincheck=0;
check=0;
ltot=0.;
for j=1:3
if nstat(NBelem(i,j))==0
if abs(lSetLocal(NBelem(i,j)))<=lmin
check=1;
tempindex=j;
end
end
ltot=ltot+abs(lSetLocal(NBelem(i,j)));
end
if check==1 & ltot/3.<=avlmin
eindex=i;
nindex=tempindex;
lmin=lSetLocal(NBelem(eindex,nindex));
avlmin=ltot/3.;
end
end
end
if maincheck==0
% Find New LS for point
xp=Node(NLocal(NBelem(eindex,nindex)),1);
yp=Node(NLocal(NBelem(eindex,nindex)),2);
count=0;
for i=1:3
if i~=nindex
count=count+1;
x(count)=Node(NLocal(NBelem(eindex,i)),1);
y(count)=Node(NLocal(NBelem(eindex,i)),2);
lloc(count)=newlSet(NBelem(eindex,i));
end
end
delxa=x(1)-xp;
delya=y(1)-yp;
delxb=x(2)-xp;
delyb=y(2)-yp;
N=[delxa delya; delxb delyb];
M=N^-1;
A=(M(1)*M(1)+M(2)*M(2));
B=(M(3)*M(3)+M(4)*M(4));
C=2.*(M(1)*M(3)+M(2)*M(4));
a=A+B+C;
b=-2.*lloc(1)*A-2.*lloc(2)*B-C*(lloc(1)+lloc(2));
c=lloc(1)*lloc(1)*A+lloc(2)*lloc(2)*B+lloc(1)*lloc(2)*C-1.;
templ1=(-b+sqrt(b*b-4.*a*c))/(2.*a);
templ2=(-b-sqrt(b*b-4.*a*c))/(2.*a);
if abs(templ1)>abs(templ2)
newlSet(NBelem(eindex,nindex))=templ1;
else
newlSet(NBelem(eindex,nindex))=templ2;
end
nstat(NBelem(eindex,nindex))=1;
end
end
% lSetLocal=newlSet;
% Update Global Level Set
for i=1:NBNodes
lSet(NLocal(i))=lSetLocal(i);
end
end
lSet'
[X Y]=meshgrid(0:0.25:8);
Z=zeros(33);
for i=1:1089
Z(i)=lSet(i);
end
surf(X,Y,Z)

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function [] = FESolveX()
% MATLAB based XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=10.;
% Define Section Properties
rho=1.;
% Generate Mesh
numElem=10;
charlen=len/numElem;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=2;
% initial interface position
dpos=5.;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
%storage
stored(1)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
if crdn1<=dpos
Tnew(2*elemNodes(e,1)-1)=1.;
end
end
% Define Time Step
dtime=0.05;
tsteps=20;
time=0.;
% penalty term
beta=40.;
% Loop through time steps
for ts=1:tsteps
% Get interface velocity
d(1)=dpos+charlen;
d(2)=dpos+3*charlen/4;
d(3)=dpos+charlen/4;
d(4)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
for j=1:4
if d(j)>=crdn1 & d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(elemNodes(e,1)*2-1);
tmp1b=Tnew(elemNodes(e,1)*2);
tmp2a=Tnew(elemNodes(e,2)*2-1);
tmp2b=Tnew(elemNodes(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
vel=(0.5/charlen)*(2*t(1)+t(2)-t(3)-2*t(4));
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(2*ndof);
Me=zeros(2*ndof);
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
enr=2;
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
if sign(theta(1))~=sign(theta(2))
% enriched element
enr=4;
% get interface position on element
point=(dpos-crdn1)/ajacob-1.;
% devide element for sub integration
len1=abs(-point-1.);
len2=abs(1.-point);
mid1=-1+len1/2.;
mid2=1-len2/2.;
gpx(1)=-(len1/2.)/sqrt(3.)+mid1;
gpx(2)=(len1/2.)/sqrt(3.)+mid1;
gpx(3)=-(len2/2.)/sqrt(3.)+mid2;
gpx(4)=(len2/2.)/sqrt(3.)+mid2;
w(1)=(len1/2.);
w(2)=(len1/2.);
w(3)=(len2/2.);
w(4)=(len2/2.);
fdofs(1)=2*elemNodes(e,1);
fdofs(2)=2*elemNodes(e,2);
else
% regular element - fix extra dofs
gpx(1)=-1/sqrt(3.);
gpx(2)=1/sqrt(3.);
w(1)=1.;
w(2)=1.;
end
% Loop Through Int Points
for i=1:enr;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(3)=(1.+c)/2.;
term=theta(1)*phi(1)+theta(2)*phi(3);
if term<0
cond=0.;
spec=0.01;
else
cond=1.;
spec=1.;
end
phi(2)=phi(1)*(abs(term)-abs(theta(1)));
phi(4)=phi(3)*(abs(term)-abs(theta(2)));
phic(1)=-0.5;
phic(3)=0.5;
dterm=sign(term)*(phic(1)*theta(1)+phic(3)*theta(2));
phic(2)=phic(1)*(abs(term)-abs(theta(1)))+phi(1)*dterm;
phic(4)=phic(3)*(abs(term)-abs(theta(2)))+phi(3)*dterm;
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
phix(3)=phic(3)/ajacob;
phix(4)=phic(4)/ajacob;
we=ajacob*w(i);
Ke=Ke+we*cond*phix'*phix;
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
if enr==4;
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
tpos=gm(1)*Tnew(1)+gm(2)*Tnew(2)+gm(3)*Tnew(3)+gm(4)*Tnew(4);
pen=beta*(gm'*gm);
pfL=beta*1*gm';
Ke=Ke+pen;
else
pen=zeros(4);
pfL=zeros(4,1);
end
% Assemble Global Matrices
gnum=2.*elemNodes(e,1)-1.;
for i=1:4;
for j=1:4;
K(gnum+j-1,gnum+i-1)=K(gnum+j-1,gnum+i-1)+Ke(j,i);
M(gnum+j-1,gnum+i-1)=M(gnum+j-1,gnum+i-1)+Me(j,i);
end
pforce(gnum+i-1)=pforce(gnum+i-1)+pfL(i);
end
end
%Remove inactive DOFs(Reduce Matrices)
T1=1;
RHS=M*Tnew;
iindex=0;
for i=1:ndof*numNodes;
check=1;
if i==fdofs(1)|i==fdofs(2)
check=0;
elseif mod(i,2)~=0 & i~=1
check=0;
end
if check==0
jindex=0;
iindex=iindex+1;
for j=1:ndof*numNodes;
check=1;
if j==fdofs(1)|j==fdofs(2)
check=0;
elseif mod(j,2)~=0 & j~=1
check=0;
end
if check==0
jindex=jindex+1;
Kred(iindex,jindex)=K(i,j);
Mred(iindex,jindex)=M(i,j);
end
end
Subr(iindex)=(K(i,1)+M(i,1))*T1;
RHSr(iindex)=RHS(i);
pforcer(iindex)=pforce(i);
end
end
%Solve
Mred+Kred;
StiffI=(Mred+Kred)^-1;
Tnewr=StiffI*(RHSr'-Subr'+pforcer');
iindex=0.;
for i=1:ndof*numNodes;
check=1;
if i==fdofs(1)|i==fdofs(2)
check=0;
elseif mod(i,2)~=0 & i~=1
check=0;
end
if check==0
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
else
Tnew(i)=0.;
end
end
Tnew(1)=1.;
Tnew
end
stored'

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function [] = GetF()
% set up grid
gd=0.;
numElem=4;
eLen=0.25;
for i=1:numElem+1
ndCrd(i)=gd;
gd=gd+eLen;
end
for i=1:numElem
elemNod(i,1)=i;
elemNod(i,2)=i+1;
end
% Initial level set
dpos=0.1;
for i=1:numElem+1
lSet(i)=sign(ndCrd(i)-dpos)*abs(dpos-ndCrd(i));
end
lSet'
for tstep=1:1
% Velocity BC
F=zeros(numElem+1,1);
for i=1:numElem
if sign(lSet(elemNod(i,1)))~=sign(lSet(elemNod(i,2)))
F(elemNod(i,1))= 0.0005;
F(elemNod(i,2))= 0.0005;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(numElem+1);
for i=1:numElem
pos(1)=-1/sqrt(3);
pos(2)=1/sqrt(3);
AfL=zeros(2);
AfLGLS=zeros(2);
for j=1:2
shp(1)=(1-pos(j))/2.;
shp(2)=(1+pos(j))/2.;
dshp(1)=-0.5;
dshp(2)=0.5;
rset=shp(1)*lSet(elemNod(i,1))+shp(2)*lSet(elemNod(i,2));
dls=dshp(1)*lSet(elemNod(i,1))+dshp(2)*lSet(elemNod(i,2));
AfL=AfL+shp'*sign(rset)*(dls*dshp);
AfLGLS=AfLGLS+(dshp'*dls)*(0.25/abs(dls))*(dls*dshp);
end
for k=1:2;
for j=1:2;
A(elemNod(i,j),elemNod(i,k))=A(elemNod(i,j),elemNod(i,k))+AfL(j,k)+AfLGLS(j,k)
end
end
end
% Apply BCs
RHS=zeros(numElem+1,1);
Sub=A*F;
iindex=0;
for i=1:numElem+1
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:numElem+1
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:numElem+1
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(numElem+1);
mMatGLS=zeros(numElem+1);
f1=zeros(numElem+1,1);
f2=zeros(numElem+1,1);
f3=zeros(numElem+1,1);
h=0.25;
visc=0.000;
for i=1:numElem
pos(1)=-1/sqrt(3);
pos(2)=1/sqrt(3);
mMatL=zeros(2);
mMatGLSL=zeros(2);
f1L=zeros(2,1);
f2L=zeros(2,1);
f3L=zeros(2,1);
for j=1:2
shp(1)=(1-pos(j))/2.;
shp(2)=(1+pos(j))/2.;
dshp(1)=-0.5;
dshp(2)=0.5;
Floc=shp(1)*F(elemNod(i,1))+shp(2)*F(elemNod(i,2));
rset=shp(1)*lSet(elemNod(i,1))+shp(2)*lSet(elemNod(i,2));
dls=dshp(1)*lSet(elemNod(i,1))+dshp(2)*lSet(elemNod(i,2));
mMatL=mMatL+shp'*shp;
mMatGLSL=mMatGLSL+((dshp'*(dls/abs(dls)))*Floc*(h/abs(Floc)))*shp;
f1L=f1L+shp'*Floc*abs(dls);
f2L=f2L+(dshp'*(dls/abs(dls))*Floc)*(h/abs(Floc))*Floc*abs(dls);
vs=h*((abs(visc+Floc*abs(dls)))/(abs(Floc*dls)+h));
f3L=f3L+vs*dshp'*dls;
end
for k=1:2;
for j=1:2;
mMat(elemNod(i,j),elemNod(i,k))=mMat(elemNod(i,j),elemNod(i,k))+mMatL(j,k);
mMatGLS(elemNod(i,j),elemNod(i,k))=mMatGLS(elemNod(i,j),elemNod(i,k))+mMatGLSL(j,k);
end
f1(elemNod(i,k))=f1(elemNod(i,k))+f1L(k);
f2(elemNod(i,k))=f2(elemNod(i,k))+f2L(k);
f3(elemNod(i,k))=f3(elemNod(i,k))+f3L(k);
end
end
dt=0.01;
lSet=lSet-((((mMat+mMatGLS)^-1)/dt)*(f1+f2+f3))';
lSet';
end

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function [] = GetF2D_()
clear all
% Define Main Solution Mesh
NumX=10;
NumY=10;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
centx=5.;
centy=5.;
rad=2.5;
for i=1:numNodes;
dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
lSet(i)=dist-rad;
end
% Plot initial level set
[X,Y]=meshgrid(0:1.:10);
Z=zeros(11);
for i=1:numNodes
Z(i)=lSet(i);
end
surf(X,Y,Z)
% LS Algorithm Parameters
bandwith=10.;
% Loop through timesteps
for tstep=1:1
% Identify Narrow Band Elements
NBindex=0;
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,2);
NBelem(NBindex,3)=Element(i,3);
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,3);
NBelem(NBindex,3)=Element(i,4);
end
end
% Velocity BC
F=zeros(numNodes,1);
for i=1:NBindex
if lSet(NBelem(i,1))~=lSet(NBelem(i,2)) || lSet(NBelem(i,1))~=lSet(NBelem(i,3))
F(NBelem(i,1))= 0.0005;
F(NBelem(i,2))= 0.0005;
F(NBelem(i,3))= 0.0005;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(numNodes);
for i=1:NBindex
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
AfL=zeros(4);
AfLGLS=zeros(4);
for j=1:4
g=gx(j);
h=hx(j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(2)=0.25*(1.+g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.+h);
phi(4)=0.25*(1.-g)*(1.+h);
phig(1)=0.25*-(1.-h);
phig(2)=0.25*(1.-h);
phig(3)=0.25*(1.+h);
phig(4)=0.25*-(1.+h);
rset=phi(1)*lSet(Element(i,1))+phi(2)*lSet(Element(i,2))+phi(3)*lSet(Element(i,3))+phi(4)*lSet(Element(i,4));
dls=phig(1)*lSet(Element(i,1))+phig(2)*lSet(Element(i,2))+phig(3)*lSet(Element(i,3))+phig(4)*lSet(Element(i,4));
AfL=AfL+(phi'*sign(rset))*(dls*phig);
AfLGLS=AfLGLS+(phig'*dls)*(1./abs(dls))*(dls*phig);
end
for k=1:4;
for j=1:4;
A(Element(i,j),Element(i,k))=A(Element(i,j),Element(i,k))+AfL(j,k)+AfLGLS(j,k);
end
end
end
% Apply BCs
RHS=zeros(numNodes,1);
Sub=A*F;
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:numNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(numNodes);
mMatGLS=zeros(numNodes);
f1=zeros(numNodes,1);
f2=zeros(numNodes,1);
f3=zeros(numNodes,1);
h=1.;
visc=0.000;
for i=1:numElem
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
mMatL=zeros(4);
mMatGLSL=zeros(4);
f1L=zeros(4,1);
f2L=zeros(4,1);
f3L=zeros(4,1);
for j=1:4
g=gx(j);
h=hx(j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(2)=0.25*(1.+g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.+h);
phi(4)=0.25*(1.-g)*(1.+h);
phig(1)=0.25*-(1.-h);
phig(2)=0.25*(1.-h);
phig(3)=0.25*(1.+h);
phig(4)=0.25*-(1.+h);
Floc=phi(1)*F(Element(i,1))+phi(2)*F(Element(i,2))+phi(3)*F(Element(i,3))+phi(4)*F(Element(i,4));
rset=phi(1)*lSet(Element(i,1))+phi(2)*lSet(Element(i,2))+phi(3)*lSet(Element(i,3))+phi(4)*lSet(Element(i,4));
dls=phig(1)*lSet(Element(i,1))+phig(2)*lSet(Element(i,2))+phig(3)*lSet(Element(i,3))+phig(4)*lSet(Element(i,4));
mMatL=mMatL+phi'*phi;
mMatGLSL=mMatGLSL+((phig'*(dls/abs(dls)))*Floc*(h/abs(Floc)))*phi;
f1L=f1L+phi'*Floc*abs(dls);
f2L=f2L+(phig'*(dls/abs(dls))*Floc)*(h/abs(Floc))*Floc*abs(dls);
vs=h*((abs(visc+Floc*abs(dls)))/(abs(Floc*dls)+h));
f3L=f3L+vs*phig'*dls;
end
for k=1:4;
for j=1:4;
mMat(Element(i,j),Element(i,k))=mMat(Element(i,j),Element(i,k))+mMatL(j,k);
mMatGLS(Element(i,j),Element(i,k))=mMatGLS(Element(i,j),Element(i,k))+mMatGLSL(j,k);
end
f1(Element(i,k))=f1(Element(i,k))+f1L(k);
f2(Element(i,k))=f2(Element(i,k))+f2L(k);
f3(Element(i,k))=f3(Element(i,k))+f3L(k);
end
end
dt=0.01;
lSet=lSet-((((mMat+mMatGLS)^-1)/dt)*(f1+f2+f3))';
end
%scatter3(Node(:,1), Node(:,2), lSet');

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function [] = GetF2D()
clear all
% Define Main Solution Mesh
NumX=10;
NumY=10;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
centx=5.;
centy=5.;
rad=2.5;
for i=1:numNodes;
dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
lSet(i)=dist-rad;
end
% Plot initial level set
[X,Y]=meshgrid(0:1.:10);
Z=zeros(11);
for i=1:numNodes
Z(i)=lSet(i);
end
surf(X,Y,Z)
% LS Algorithm Parameters
bandwith=10.;
% Loop through timesteps
for tstep=1:1
% Identify Narrow Band Elements
NBindex=0;
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,2);
NBelem(NBindex,3)=Element(i,3);
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,3);
NBelem(NBindex,3)=Element(i,4);
end
end
% Velocity BC
F=zeros(numNodes,1);
for i=1:NBindex
if lSet(NBelem(i,1))~=lSet(NBelem(i,2)) || lSet(NBelem(i,1))~=lSet(NBelem(i,3))
F(NBelem(i,1))= 0.0005;
F(NBelem(i,2))= 0.0005;
F(NBelem(i,3))= 0.0005;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(numNodes);
for i=1:NBindex
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
AfL=zeros(4);
AfLGLS=zeros(4);
for j=1:4
g=gx(j);
h=hx(j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(2)=0.25*(1.+g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.+h);
phi(4)=0.25*(1.-g)*(1.+h);
phig(1)=0.25*-(1.-h);
phig(2)=0.25*(1.-h);
phig(3)=0.25*(1.+h);
phig(4)=0.25*-(1.+h);
rset=phi(1)*lSet(Element(i,1))+phi(2)*lSet(Element(i,2))+phi(3)*lSet(Element(i,3))+phi(4)*lSet(Element(i,4));
dls=phig(1)*lSet(Element(i,1))+phig(2)*lSet(Element(i,2))+phig(3)*lSet(Element(i,3))+phig(4)*lSet(Element(i,4));
AfL=AfL+(phi'*sign(rset))*(dls*phig);
AfLGLS=AfLGLS+(phig'*dls)*(1./abs(dls))*(dls*phig);
end
for k=1:4;
for j=1:4;
A(Element(i,j),Element(i,k))=A(Element(i,j),Element(i,k))+AfL(j,k)+AfLGLS(j,k)
end
end
end
% Apply BCs
RHS=zeros(numNodes,1);
Sub=A*F;
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:numNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(numNodes);
mMatGLS=zeros(numNodes);
f1=zeros(numNodes,1);
f2=zeros(numNodes,1);
f3=zeros(numNodes,1);
h=1.;
visc=0.000;
for i=1:numElem
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
mMatL=zeros(4);
mMatGLSL=zeros(4);
f1L=zeros(4,1);
f2L=zeros(4,1);
f3L=zeros(4,1);
for j=1:4
g=gx(j);
h=hx(j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(2)=0.25*(1.+g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.+h);
phi(4)=0.25*(1.-g)*(1.+h);
phig(1)=0.25*-(1.-h);
phig(2)=0.25*(1.-h);
phig(3)=0.25*(1.+h);
phig(4)=0.25*-(1.+h);
Floc=phi(1)*F(Element(i,1))+phi(2)*F(Element(i,2))+phi(3)*F(Element(i,3))+phi(4)*F(Element(i,4));
rset=phi(1)*lSet(Element(i,1))+phi(2)*lSet(Element(i,2))+phi(3)*lSet(Element(i,3))+phi(4)*lSet(Element(i,4));
dls=phig(1)*lSet(Element(i,1))+phig(2)*lSet(Element(i,2))+phig(3)*lSet(Element(i,3))+phig(4)*lSet(Element(i,4));
mMatL=mMatL+phi'*phi;
mMatGLSL=mMatGLSL+((phig'*(dls/abs(dls)))*Floc*(h/abs(Floc)))*phi;
f1L=f1L+phi'*Floc*abs(dls);
f2L=f2L+(phig'*(dls/abs(dls))*Floc)*(h/abs(Floc))*Floc*abs(dls);
vs=h*((abs(visc+Floc*abs(dls)))/(abs(Floc*dls)+h));
f3L=f3L+vs*phig'*dls;
end
for k=1:4;
for j=1:4;
mMat(Element(i,j),Element(i,k))=mMat(Element(i,j),Element(i,k))+mMatL(j,k);
mMatGLS(Element(i,j),Element(i,k))=mMatGLS(Element(i,j),Element(i,k))+mMatGLSL(j,k);
end
f1(Element(i,k))=f1(Element(i,k))+f1L(k);
f2(Element(i,k))=f2(Element(i,k))+f2L(k);
f3(Element(i,k))=f3(Element(i,k))+f3L(k);
end
end
dt=0.01;
lSet=lSet-((((mMat+mMatGLS)^-1)/dt)*(f1+f2+f3))';
end
%scatter3(Node(:,1), Node(:,2), lSet');

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function [] = GetF2D_T()
clear all
% Define Main Solution Mesh
NumX=2;
NumY=1;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
%centx=5.;
%centy=5.;
%rad=2.5;
%for i=1:numNodes;
% dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
% lSet(i)=dist-rad;
%end
for i=1:numNodes;
dist=Node(i,1)-0.1;
lSet(i)=dist;
end
% Plot initial level set
%[X]=meshgrid(0:1.:10);
%Z=zeros(11);
%for i=1:numNodes
% Z(i)=lSet(i);
%end
%surf(X,Z)
% LS Algorithm Parameters
bandwidth=10.;
% Loop through timesteps
for tstep=1:1
% Identify Narrow Band Elements
NBindex=0;
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,2);
NBelem(NBindex,3)=Element(i,3);
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,3);
NBelem(NBindex,3)=Element(i,4);
end
end
% Velocity BC
F=zeros(numNodes,1);
for i=1:NBindex
if sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,2)))||sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,3)))
F(NBelem(i,1))= 0.05;
F(NBelem(i,2))= 0.05;
F(NBelem(i,3))= 0.05;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(numNodes);
for i=1:NBindex
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
% Apply BCs
RHS=zeros(numNodes,1);
Sub=A*F;
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:numNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(numNodes);
mMatGLS=zeros(numNodes);
f1=zeros(numNodes,1);
f2=zeros(numNodes,1);
f3=zeros(numNodes,1);
h=1.;
visc=0.001;
for i=1:NBindex
mMatL=zeros(3);
mMatGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
mMatL=mMatL+(phi'*phi)/3.;
mMatGLSL=mMatGLSL+((delphi'*(delset/norm(delset)))*Floc*(h/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(h/abs(Floc))*Floc*norm(delset)/3.;
vs=h*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+h));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
mMat(NBelem(i,j),NBelem(i,k))=mMat(NBelem(i,j),NBelem(i,k))+mMatL(j,k);
mMatGLS(NBelem(i,j),NBelem(i,k))=mMatGLS(NBelem(i,j),NBelem(i,k))+mMatGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
(mMat+mMatGLS)^-1
f1+f2+f3
dt=0.0001;
-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))'
lSet=lSet-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))';
end
lSet';
%scatter3(Node(:,1), Node(:,2), lSet');

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function [] = GetF2D_T()
clear all
% Define Main Solution Mesh
NumX=2;
NumY=1;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
%centx=5.;
%centy=5.;
%rad=2.5;
%for i=1:numNodes;
% dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
% lSet(i)=dist-rad;
%end
for i=1:numNodes;
dist=Node(i,1)-1.5;
lSet(i)=dist;
end
% Plot initial level set
%[X]=meshgrid(0:1.:10);
%Z=zeros(11);
%for i=1:numNodes
% Z(i)=lSet(i);
%end
%surf(X,Z)
% LS Algorithm Parameters
bandwidth=10.;
% Loop through timesteps
for tstep=1:1
% Identify Narrow Band Elements
NBindex=0;
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,2);
NBelem(NBindex,3)=Element(i,3);
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,3);
NBelem(NBindex,3)=Element(i,4);
end
end
% Velocity BC
F=zeros(numNodes,1);
for i=1:NBindex
if sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,2)))||sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,3)))
F(NBelem(i,1))= 0.05;
F(NBelem(i,2))= 0.05;
F(NBelem(i,3))= 0.05;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(numNodes);
for i=1:NBindex
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
% Apply BCs
RHS=zeros(numNodes,1);
Sub=A*F;
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:numNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(numNodes);
mMatGLS=zeros(numNodes);
f1=zeros(numNodes,1);
f2=zeros(numNodes,1);
f3=zeros(numNodes,1);
h=1.;
visc=0.0005;
for i=1:NBindex
mMatL=zeros(3);
mMatGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
mMatL=mMatL+(phi'*phi)/3.;
mMatGLSL=mMatGLSL+((delphi'*(delset/norm(delset)))*Floc*(h/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(h/abs(Floc))*Floc*norm(delset)/3.;
vs=h*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+h));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
mMat(NBelem(i,j),NBelem(i,k))=mMat(NBelem(i,j),NBelem(i,k))+mMatL(j,k);
mMatGLS(NBelem(i,j),NBelem(i,k))=mMatGLS(NBelem(i,j),NBelem(i,k))+mMatGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
mMat
mMatGLS
(mMat+mMatGLS)^-1
f1+f2+f3
dt=0.1;
-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))'
lSet=lSet-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))';
end
lSet';
%scatter3(Node(:,1), Node(:,2), lSet');

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function [] = GetF2D_Z()
clear all
% Define Main Solution Mesh
NumX=8;
NumY=8;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
centx=4.;
centy=4.;
rad=2.1;
for i=1:numNodes;
dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
lSet(i)=dist-rad;
end
%for i=1:numNodes;
% dist=Node(i,1)-0.1;
% lSet(i)=dist;
%end
% Plot initial level set
[X Y]=meshgrid(0:1.:8);
Z=zeros(9);
for i=1:81
Z(i)=lSet(i);
end
surf(X,Y,Z)
% LS Algorithm Parameters
lSet'
bandwidth=10.;
% Loop through timesteps
for tstep=1:100
% Identify Narrow Band Elements
NBindex=0;
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,2);
NBelem(NBindex,3)=Element(i,3);
NBindex=NBindex+1;
NBelem(NBindex,1)=Element(i,1);
NBelem(NBindex,2)=Element(i,3);
NBelem(NBindex,3)=Element(i,4);
end
end
% Velocity BC
F=zeros(numNodes,1);
for i=1:NBindex
if sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,2)))||sign(lSet(NBelem(i,1)))~=sign(lSet(NBelem(i,3)))
F(NBelem(i,1))= 1.;
F(NBelem(i,2))= 1.;
F(NBelem(i,3))= 1.;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(numNodes);
for i=1:NBindex
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
% Apply BCs
RHS=zeros(numNodes,1);
Sub=A*F;
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:numNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:numNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(numNodes);
mMatGLS=zeros(numNodes);
f1=zeros(numNodes,1);
f2=zeros(numNodes,1);
f3=zeros(numNodes,1);
h2=0.00001;
visc=0.0005;
for i=1:NBindex
mMatL=zeros(3);
mMatGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NBelem(i,1),1);
y1=Node(NBelem(i,1),2);
x2=Node(NBelem(i,2),1);
y2=Node(NBelem(i,2),2);
x3=Node(NBelem(i,3),1);
y3=Node(NBelem(i,3),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSet(NBelem(i,1));lSet(NBelem(i,2));lSet(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
mMatL=mMatL+(phi'*phi)/3.;
mMatGLSL=mMatGLSL+((delphi'*(delset/norm(delset)))*Floc*(h2/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(h2/abs(Floc))*Floc*norm(delset)/3.;
vs=h2*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+h2));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
mMat(NBelem(i,j),NBelem(i,k))=mMat(NBelem(i,j),NBelem(i,k))+mMatL(j,k);
mMatGLS(NBelem(i,j),NBelem(i,k))=mMatGLS(NBelem(i,j),NBelem(i,k))+mMatGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
dt=0.01;
lSet=lSet-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))';
end
lSet'
%[X Y]=meshgrid(0:1.:8);
%Z=zeros(9);
%for i=1:81
% Z(i)=lSet(i);
%end
%surf(X,Y,Z)

2
Unpublished/XFEM/1D_Solver/testGrid.asv Normal file
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@ -0,0 +1,2 @@
function [] = testGrid()

4
Unpublished/XFEM/1D_Solver/testGrid.m Normal file
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@ -0,0 +1,4 @@
function [] = testGrid()
[X,Y]=meshgrid(1:1:10);
Z=2*X+Y
surf(X,Y,Z)

202
Unpublished/XFEM/2D_Solver/2D_XCor.for Normal file
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c 2D XFEM Corrosion Element
subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props,
1 nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,kinc,
2 jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags,
3 mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension phig(8),phih(8),phi(8),phix(8),phiy(8)
dimension crdnx(4),crdny(4),w(8),dndg(4),dndh(4)
dimension theta(4),rjac(2,2),rjaci(2,2)
c
parameter(zero=0.d0,one=1.d0)
c material property definition
thick = 1.
rho = 1.
beta=40.
vel=0.0
dpos=0.25+vel*time(2)
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=zero
do k2=1,ndofel
amatrx(k2,k1)=zero
enddo
enddo
if (lflags(1).eq.33) then
do icrd=1,4
crdnx(icrd)=coords(1,icrd)
crdny(icrd)=coords(2,icrd)
theta(icrd)=abs(crdnx(icrd)-dpos)*
1 sign(1.,crdnx(icrd)-dpos)
enddo
if sign(1.,theta(1))/=sign(1.,theta(2))then
c Enriched
ienr=8
elen=abs(crdnx(2)-crdnx(1))
frac=abs(dpos-crdnx(1))/elen
rlen1=2.*frac
rlen2=2.*(1.-frac)
rmid1=-1+rlen1/2.
rmid2=1-rlen2/2.
gx(1)=rmid1-(rlen1/2.)/sqrt(3.)
gx(2)=rmid1+(rlen1/2.)/sqrt(3.)
gx(3)=rmid1+(rlen1/2.)/sqrt(3.)
gx(4)=rmid1-(rlen1/2.)/sqrt(3.)
gx(5)=rmid2-(rlen2/2.)/sqrt(3.)
gx(6)=rmid2+(rlen2/2.)/sqrt(3.)
gx(7)=rmid2+(rlen2/2.)/sqrt(3.)
gx(8)=rmid2-(rlen2/2.)/sqrt(3.)
gpos=1/sqrt(3.)
hx(1)=-gpos
hx(2)=-gpos
hx(3)=+gpos
hx(4)=+gpos
hx(5)=-gpos
hx(6)=-gpos
hx(7)=+gpos
hx(8)=+gpos
do iw=1,4
w(iw)=frac/2.;
w(iw+4)=(1.-frac)/2.;
enddo
else
c Normal Shp Funs
ienr=4
gpos=1./sqrt(3.)
gx(1)=-gpos
gx(2)=gpos
gx(3)=gpos
gx(4)=-gpos
hx(1)=-gpos
hx(2)=-gpos
hx(3)=gpos
hx(4)=gpos
do iw=1,4
w(iw)=1.
enddo
endif
c assemble amatrx and rhs
do k=1,ienr
c loop through gauss pts
g=gx(k)
h=hx(k)
phi(1)=0.25*(1.-g)*(1.-h)
phi(3)=0.25*(1.+g)*(1.-h)
phi(5)=0.25*(1.+g)*(1.+h)
phi(7)=0.25*(1.-g)*(1.+h)
riLS=theta(1)*phi(1)+theta(2)*phi(3)+
1 theta(3)*phi(5)+theta(4)*phi(7)
if (riLS<0.)then
cond=0.
spec=0.01
else
cond=1.
spec=1.
endif
do iter=1,4
phi(2*iter)=phi(2*iter-1)*
1 (abs(riLS)-abs(theta(iter)))
enddo
phig(1)=0.25*-(1.-h)
phig(3)=0.25*(1.-h)
phig(5)=0.25*(1.+h)
phig(7)=0.25*-(1.+h)
phih(1)=0.25*-(1.-g)
phih(3)=0.25*-(1.+g)
phih(5)=0.25*(1.+g)
phih(7)=0.25*(1.-g)
diLSg=sign(1.,iLS)*(phig(1)*theta(1)+phig(3)*
1 theta(2)+phig(5)*theta(3)+phig(7)*theta(4))
diLSh=sign(1.,iLS)*(phih(1)*theta(1)+phih(3)*
1 theta(2)+phih(5)*theta(3)+phih(7)*theta(4))
do iter=1,4
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-
1 abs(theta(iter)))+phi(2*iter-1)*diLSg
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-
1 abs(theta(iter)))+phi(2*iter-1)*diLSh
enddo
rjac=0.
do iter=1,4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter)
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter)
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter)
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter)
enddo
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1)
rjaci(1,1)= rjac(2,2)/djac
rjaci(2,2)= rjac(1,1)/djac
rjaci(1,2)=-rjac(1,2)/djac
rjaci(2,1)=-rjac(2,1)/djac
do iter=1,8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter)
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter)
enddo
dtdx=zero
dtdy=zero
t =zero
told=zero
do i=1,8
dtdx=u(i)*phix(i)+dtdx
dtdy=u(i)*phiy(i)+dtdy
t=u(i)*phi(i)+t
told=(u(i)-du(i,nrhs))*phi(i)+told
end do
dtdt=(t-told)/dtime
we=w(k)*djac
do ki=1,8
c loop over nodes
rhs(ki,nrhs) = rhs(ki,nrhs) -
1 we*(phi(ki)*rho*spec*dtdt +
2 cond*(phix(ki)*dtdx + phiy(ki)*dtdy))
do kj=1,8
amatrx(ki,kj)= amatrx(ki,kj) +
1 we*(phi(ki)*phi(kj)*rho*spec/dtime +
1 cond*(phix(ki)*phix(kj)+phiy(ki)*phiy(kj)))
end do
end do
enddo
c if interface is in the element a penalty term is needed
if(enr==4)then
xi=point
gm(1)=(1.-xi)/2.
gm(3)=(1.+xi)/2.
term=theta(1)*gm(1)+theta(2)*gm(3)
gm(2)=gm(1)*(abs(term)-abs(theta(1)))
gm(4)=gm(3)*(abs(term)-abs(theta(2)))
term2=gm(1)*u(1)+gm(2)*u(2)+gm(3)*u(3)+gm(4)*u(4)
diff=abs(term2-1.)
c add penalty flux/force: BGtc
targetT=1.
do i=1,4
rhs(i,nrhs)=rhs(i,nrhs)+beta*gm(i)*diff
enddo
c find GtG
gm2=0.
do i=1,4
do j=1,4
gm2(i,j)=gm(i)*gm(j)
enddo
enddo
c add penalty stiffness
do i=1,4
do j=1,4
amatrx(i,j)=amatrx(i,j)+beta*gm2(i,j)
enddo
enddo
endif
end if
return
end

78
Unpublished/XFEM/2D_Solver/AbInp.inp Normal file
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@ -0,0 +1,78 @@
*Heading
** Job name: Job-1 Model name: Model-1
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=Part-1
*Node
1, 0., 0., 0.
2, 1., 0., 0.
3, 1., 1., 0.
4, 0., 1., 0.
5, 1., 2., 0.
6, 0., 2., 0.
*USER ELEMENT,NODES=4,TYPE=U1,PROP=1,COORDINATES=2,VAR=2,unsymm
11,
*Element, type=U1,ELSET=UEL
1, 1, 2,3,4
2, 4, 3,5,6
*UEL Property, Elset=UEL
1.
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=Part-1-1, part=Part-1
*End Instance
**
*Nset, nset=_PickedSet16, internal, instance=Part-1-1
1,3,5
*Nset, nset=_PickedSet17, internal, instance=Part-1-1
2,4,6
*Nset, nset=Set-6, instance=Part-1-1
1,3,5
*End Assembly
**
** MATERIALS
**
*Material, name=Material-1
*Conductivity
1.,
*Density
1.,
*Specific Heat
1.,
** ----------------------------------------------------------------
**
** Name: Predefined Field-1 Type: Temperature
*Initial Conditions, type=TEMPERATURE
_PickedSet16, 1.,0.
** Name: Predefined Field-2 Type: Temperature
*Initial Conditions, type=TEMPERATURE
_PickedSet17, 0.,0.
** STEP: Step-1
**
*Step, name=Step-1
*Heat Transfer, end=PERIOD, deltmx=100.
0.1, 1., 1e-09, 0.1,
**
** BOUNDARY CONDITIONS
**
** Name: BC-1 Type: Temperature
*Boundary
Set-6, 11, 11, 1.
**
** OUTPUT REQUESTS
**
*Restart, write, frequency=0
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
*Output, history, frequency=0
*End Step

331
Unpublished/XFEM/2D_Solver/F2DLevelSetFMM.m Normal file
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function [] = F2DLevelSetFMM()
clear all
% Define Main Solution Mesh
NumX=32;
NumY=32;
delX=0.25;
delY=0.25;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Initial Level Set
centx=4.;
centy=4.;
rad=2.1;
for i=1:numNodes;
dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
lSet(i)=dist-rad;
end
%for i=1:numNodes;
% dist=Node(i,1)-0.1;
% lSet(i)=dist;
%end
% Plot initial level set
[X Y]=meshgrid(0:0.25:8);
Z=zeros(33);
for i=1:1089
Z(i)=lSet(i);
end
surf(X,Y,Z)
% LS Algorithm Parameters
lSet'
bandwidth=10;
% Loop through timesteps
for tstep=1:10
% Identify Narrow Band Elements
NBElems=0;
NBNodes=0;
NGlobal=zeros(numNodes);
for i=1:numElem
check=0;
for iNd=1:4
if abs(lSet(Element(i,iNd)))<=bandwidth*delX
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
for j=1:4
if NGlobal(Element(i,j))==0
NBNodes=NBNodes+1;
NGlobal(Element(i,j))=NBNodes;
NLocal(NBNodes)=Element(i,j);
end
end
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,2));
NBelem(NBElems,3)=NGlobal(Element(i,3));
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,3));
NBelem(NBElems,3)=NGlobal(Element(i,4));
end
end
% Get local Level Set
for i=1:NBNodes
lSetLocal(i)=lSet(NLocal(i));
end
% Velocity BC
F=zeros(NBNodes,1);
for i=1:NBElems
L1=sign(lSetLocal(NBelem(i,1)));
L2=sign(lSetLocal(NBelem(i,2)));
L3=sign(lSetLocal(NBelem(i,3)));
if L1 ~= L2 || L1 ~= L3
F(NBelem(i,1))= 1.;
F(NBelem(i,2))= 1.;
F(NBelem(i,3))= 1.;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(NBNodes);
for i=1:NBElems
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSetLocal(NBelem(i,1));lSetLocal(NBelem(i,2));lSetLocal(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
% Apply BCs
RHS=zeros(NBNodes,1);
Sub=A*F;
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:NBNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(NBNodes);
mMatGLS=zeros(NBNodes);
f1=zeros(NBNodes,1);
f2=zeros(NBNodes,1);
f3=zeros(NBNodes,1);
h2=0.00001;
visc=0.0005;
for i=1:NBElems
mMatL=zeros(3);
mMatGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[lSetLocal(NBelem(i,1));lSetLocal(NBelem(i,2));lSetLocal(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
mMatL=mMatL+(phi'*phi)/3.;
mMatGLSL=mMatGLSL+((delphi'*(delset/norm(delset)))*Floc*(h2/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(h2/abs(Floc))*Floc*norm(delset)/3.;
vs=h2*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+h2));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
mMat(NBelem(i,j),NBelem(i,k))=mMat(NBelem(i,j),NBelem(i,k))+mMatL(j,k);
mMatGLS(NBelem(i,j),NBelem(i,k))=mMatGLS(NBelem(i,j),NBelem(i,k))+mMatGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
dt=0.01;
lSetLocal=lSetLocal-((((mMat+mMatGLS)^-1)*dt)*(f1+f2+f3))';
newlSet=lSetLocal;
% Reinitialize LS
nstat=zeros(NBNodes,1);
for i=1:NBElems
L1=sign(lSetLocal(NBelem(i,1)));
L2=sign(lSetLocal(NBelem(i,2)));
L3=sign(lSetLocal(NBelem(i,3)));
if L1 ~= L2 || L1 ~= L3
for j=1:3
nstat(NBelem(i,j))=1;
end
end
end
maincheck=0;
while(maincheck==0)
lmin=1000.;
avlmin=1000.;
eindex=0;
nindex=0;
maincheck=1;
for i=1:NBElems
if nstat(NBelem(i,1))+nstat(NBelem(i,2))+nstat(NBelem(i,3))==2
maincheck=0;
check=0;
ltot=0.;
for j=1:3
if nstat(NBelem(i,j))==0
if abs(lSetLocal(NBelem(i,j)))<=lmin
check=1;
tempindex=j;
end
end
ltot=ltot+abs(lSetLocal(NBelem(i,j)));
end
if check==1 & ltot/3.<=avlmin
eindex=i;
nindex=tempindex;
lmin=lSetLocal(NBelem(eindex,nindex));
avlmin=ltot/3.;
end
end
end
if maincheck==0
% Find New LS for point
xp=Node(NLocal(NBelem(eindex,nindex)),1);
yp=Node(NLocal(NBelem(eindex,nindex)),2);
count=0;
for i=1:3
if i~=nindex
count=count+1;
x(count)=Node(NLocal(NBelem(eindex,i)),1);
y(count)=Node(NLocal(NBelem(eindex,i)),2);
lloc(count)=newlSet(NBelem(eindex,i));
end
end
delxa=x(1)-xp;
delya=y(1)-yp;
delxb=x(2)-xp;
delyb=y(2)-yp;
N=[delxa delya; delxb delyb];
M=N^-1;
A=(M(1)*M(1)+M(2)*M(2));
B=(M(3)*M(3)+M(4)*M(4));
C=2.*(M(1)*M(3)+M(2)*M(4));
a=A+B+C;
b=-2.*lloc(1)*A-2.*lloc(2)*B-C*(lloc(1)+lloc(2));
c=lloc(1)*lloc(1)*A+lloc(2)*lloc(2)*B+lloc(1)*lloc(2)*C-1.;
templ1=(-b+sqrt(b*b-4.*a*c))/(2.*a);
templ2=(-b-sqrt(b*b-4.*a*c))/(2.*a);
if abs(templ1)>abs(templ2)
newlSet(NBelem(eindex,nindex))=templ1;
else
newlSet(NBelem(eindex,nindex))=templ2;
end
nstat(NBelem(eindex,nindex))=1;
end
end
% lSetLocal=newlSet;
% Update Global Level Set
for i=1:NBNodes
lSet(NLocal(i))=lSetLocal(i);
end
end
lSet'
[X Y]=meshgrid(0:0.25:8);
Z=zeros(33);
for i=1:1089
Z(i)=lSet(i);
end
surf(X,Y,Z)

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function [] = FESolve2DS()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=4;
NumY=1;
delX=0.25;
delY=0.25;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=1;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
stored(1)=dpos;
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(2*Element(e,n)-1)=1.;
end
if crdn<0.01
Bound(2*Element(e,n)-1)=1.;
end
end
end
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% Loop through time steps
for ts=1:tsteps
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4*ndof);
Me=zeros(4*ndof);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
end
% regular element - fix extra dofs
enr=4;
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
for iw=1:4
w(iw)=1.;
end
% Loop Through Int Points
for i=1:enr;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(2)=0.25*(1.+g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.+h);
phi(4)=0.25*(1.-g)*(1.+h);
cond=1.;
spec=1.;
phig(1)=0.25*-(1.-h);
phig(2)=0.25*(1.-h);
phig(3)=0.25*(1.+h);
phig(4)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(2)=0.25*-(1.+g);
phih(3)=0.25*(1.+g);
phih(4)=0.25*(1.-g);
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(iter)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(iter)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(iter)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(iter)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:4
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=w(i)*djac;
Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(j,i);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove inactive DOFs(Reduce Matrices)
A=K+M;
Sub=A*Bound;
RHS=M*Tnew-Sub+pforce;
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSR(iindex)=RHS(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
AR(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tnewr=(AR^-1)*RHSR';
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end
stored';

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function [] = FESolve2DS()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=4;
NumY=1;
delX=0.25;
delY=0.25;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Section Properties
rho=1.;
% Initial temperatures
Tnew=zeros(numNodes,1);
Bound=zeros(numNodes,1);
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=0.6
Tnew(Element(e,n))=1.;
end
if crdn<0.01
Bound(Element(e,n))=1.;
end
end
end
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% Loop through time steps
for ts=1:tsteps
K=zeros(numNodes,numNodes);
M=zeros(numNodes,numNodes);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4);
Me=zeros(4);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
end
% regular element - fix extra dofs
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
for iw=1:4
w(iw)=1.;
end
% Loop Through Int Points
for i=1:4;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(2)=0.25*(1.+g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.+h);
phi(4)=0.25*(1.-g)*(1.+h);
cond=1.;
spec=1.;
phig(1)=0.25*-(1.-h);
phig(2)=0.25*(1.-h);
phig(3)=0.25*(1.+h);
phig(4)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(2)=0.25*-(1.+g);
phih(3)=0.25*(1.+g);
phih(4)=0.25*(1.-g);
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(iter)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(iter)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(iter)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(iter)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:4
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=w(i)*djac;
Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Assemble Global Matrices
gnum(1)=Element(e,1);
gnum(2)=Element(e,2);
gnum(3)=Element(e,3);
gnum(4)=Element(e,4);
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(j,i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(j,i);
end
end
end
%Remove inactive DOFs(Reduce Matrices)
A=K+M;
Sub=A*Bound;
RHS=M*Tnew-Sub;
iindex=0;
for i=1:numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSR(iindex)=RHS(i);
jindex=0;
for j=1:numNodes;
if Bound(j)==0.;
jindex=jindex+1;
AR(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tnewr=(AR^-1)*RHSR';
iindex=0;
for i=1:numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end

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function [] = FESolveX2D()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=4;
NumY=1;
delX=0.25;
delY=0.25;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=2;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
stored(1)=dpos;
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(2*Element(e,n)-1)=1.;
end
if crdn<0.01
Bound(2*Element(e,n)-1)=1.;
end
end
end
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% penalty term
Penalty=00.;
% Loop through time steps
for ts=1:tsteps
eNodes=zeros(2*numNodes,1);
% Get interface velocity
d(1)=dpos+delX;
d(2)=dpos+3*delX/4;
d(3)=dpos+delX/4;
d(4)=dpos;
for e=1:numElem
crdn1=Node(Element(e,1),1);
crdn2=Node(Element(e,2),1);
for j=1:4
if d(j)>=crdn1 && d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(Element(e,1)*2-1);
tmp1b=Tnew(Element(e,1)*2);
tmp2a=Tnew(Element(e,2)*2-1);
tmp2b=Tnew(Element(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
% vel=-0.1*(0.5/delX)*(2*t(1)+t(2)-t(3)-2*t(4))
vel=0.0;
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4*ndof);
Me=zeros(4*ndof);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
theta(icrd)=abs(crdnx(icrd)-dpos)*sign(crdnx(icrd)-dpos);
end
if sign(theta(1))~=sign(theta(2))
% possible enriched element
npart=10;
enr=npart*npart;
for sdx=1:npart
for sdy=1:npart
midx=-1.-1./npart+(2./npart)*sdx;
midy=-1.-1./npart+(2./npart)*sdy;
subindex=npart*(sdy-1)+sdx;
gpos=1./(sqrt(3.)*npart);
gx(subindex,1)=midx-gpos;
gx(subindex,2)=midx+gpos;
gx(subindex,3)=midx+gpos;
gx(subindex,4)=midx-gpos;
hx(subindex,1)=midy-gpos;
hx(subindex,2)=midy-gpos;
hx(subindex,3)=midy+gpos;
hx(subindex,4)=midy+gpos;
end
end
% check if int points are on different sides of front
check=0;
for i=1:enr
for j=1:4
g=gx(i,j);
h=hx(i,j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
if i==1 && j==1
sgn=sign(iLS);
else
if sign(iLS)~=sgn
check=1;
end
end
end
end
if check==0
% regular element - fix extra dofs
enr=1;
gpos=1/sqrt(3.);
gx(1,1)=-gpos;
gx(1,2)=gpos;
gx(1,3)=gpos;
gx(1,4)=-gpos;
hx(1,1)=-gpos;
hx(1,2)=-gpos;
hx(1,3)=gpos;
hx(1,4)=gpos;
end
eNodes(2*Element(e,1))=1;
eNodes(2*Element(e,2))=1;
eNodes(2*Element(e,3))=1;
eNodes(2*Element(e,4))=1;
% enriched element
enr=8;
% get interface position on element
elen=abs(crdnx(2)-crdnx(1));
frac=abs(dpos-crdnx(1))/elen;
len1=2.*frac;
len2=2.*(1.-frac);
% devide element for sub integration
mid1=-1+len1/2.;
mid2=1-len2/2.;
gx(1)=mid1-(len1/2.)/sqrt(3.);
gx(2)=mid1+(len1/2.)/sqrt(3.);
gx(3)=mid1+(len1/2.)/sqrt(3.);
gx(4)=mid1-(len1/2.)/sqrt(3.);
gx(5)=mid2-(len2/2.)/sqrt(3.);
gx(6)=mid2+(len2/2.)/sqrt(3.);
gx(7)=mid2+(len2/2.)/sqrt(3.);
gx(8)=mid2-(len2/2.)/sqrt(3.);
gpos=1/sqrt(3.);
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=+gpos;
hx(4)=+gpos;
hx(5)=-gpos;
hx(6)=-gpos;
hx(7)=+gpos;
hx(8)=+gpos;
for iw=1:4
w(iw)=frac/2.;
w(iw+4)=(1.-frac)/2.;
end
else
% regular element - fix extra dofs
enr=4;
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
for iw=1:4
w(iw)=1.;
end
end
% Loop Through Int Points
for i=1:enr;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
cond=1.;
spec=1.;
for iter=1:4
if enr==8
phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(iter)));
else
phi(2*iter)=0.;
end
end
phig(1)=0.25*-(1.-h);
phig(3)=0.25*(1.-h);
phig(5)=0.25*(1.+h);
phig(7)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(3)=0.25*-(1.+g);
phih(5)=0.25*(1.+g);
phih(7)=0.25*(1.-g);
diLSg=sign(iLS)*(phig(1)*theta(1)+phig(3)*theta(2)+phig(5)*theta(3)+phig(7)*theta(4));
diLSh=sign(iLS)*(phih(1)*theta(1)+phih(3)*theta(2)+phih(5)*theta(3)+phih(7)*theta(4));
for iter=1:4
if enr==8
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSg;
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSh;
else
phig(2*iter)=0.;
phih(2*iter)=0.;
end
end
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=w(i)*djac;
Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
if enr==8;
count=0;
if sign(theta(1))~=sign(theta(2))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(2)));
xi(count)=f*(crdnx(2)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(2)-crdny(1))+crdny(1);
gi(count)=(2.*xi(count)-(crdnx(1)+crdnx(2)))/(-crdnx(1)+crdnx(2));
hi(count)=-1.;
end
if sign(theta(2))~=sign(theta(3))
count=count+1;
f=abs(theta(2))/(abs(theta(2))+abs(theta(3)));
xi(count)=f*(crdnx(3)-crdnx(2))+crdnx(2);
yi(count)=f*(crdny(3)-crdny(2))+crdny(2);
gi(count)=1.;
hi(count)=(2.*yi(count)-(crdny(2)+crdny(3)))/(-crdny(2)+crdny(3));
end
if sign(theta(3))~=sign(theta(4))
count=count+1;
f=abs(theta(3))/(abs(theta(3))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(3))+crdnx(3);
yi(count)=f*(crdny(4)-crdny(3))+crdny(3);
gi(count)=(2.*xi(count)-(crdnx(4)+crdnx(3)))/(-crdnx(4)+crdnx(3));
hi(count)=1.;
end
if sign(theta(1))~=sign(theta(4))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(4)-crdny(1))+crdny(1);
gi(count)=-1.;
hi(count)=(2.*yi(count)-(crdny(1)+crdny(4)))/(-crdny(4)+crdny(1));
end
c=zeros(2,1);
c=(c+1.);
for i=1:2;
G(i,1)=0.25*(1.-gi(i))*(1.-hi(i));
G(i,3)=0.25*(1.+gi(i))*(1.-hi(i));
G(i,5)=0.25*(1.+gi(i))*(1.+hi(i));
G(i,7)=0.25*(1.-gi(i))*(1.+hi(i));
G(i,2)=-G(i,1)*abs(theta(1));
G(i,4)=-G(i,3)*abs(theta(2));
G(i,6)=-G(i,5)*abs(theta(3));
G(i,8)=-G(i,7)*abs(theta(4));
end
pen=Penalty*(G'*G);
pfL=Penalty*G'*c;
Ke=Ke+pen;
else
pen=zeros(8);
pfL=zeros(8,1);
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(2*j-1,2*i-1);
K(gnum(j)+1,gnum(i))=K(gnum(j)+1,gnum(i))+Ke(2*j,2*i-1);
K(gnum(j),gnum(i)+1)=K(gnum(j),gnum(i)+1)+Ke(2*j-1,2*i);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i))=M(gnum(j)+1,gnum(i))+Me(2*j,2*i-1);
M(gnum(j),gnum(i)+1)=M(gnum(j),gnum(i)+1)+Me(2*j-1,2*i);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove NON-ENHANCED DOFs(Reduce Matrices)
iindex=0.;
for i=1:ndof*numNodes;
check=0;
if mod(i,2)==0 && eNodes(i)~=1
check=1;
end
if check==0
iindex=iindex+1;
TR1(iindex)=Tnew(i);
BR1(iindex)=Bound(i);
pforceR1(iindex)=pforce(i);
jindex=0;
for j=1:ndof*numNodes;
check=0;
if mod(j,2)==0 && eNodes(j)~=1
check=1;
end
if check==0
jindex=jindex+1;
MR1(iindex,jindex)=M(i,j);
KR1(iindex,jindex)=K(i,j);
end
end
end
end
AR1=KR1+MR1;
SubR1=AR1*BR1';
RHSR1=MR1*TR1'-SubR1+pforceR1';
% Apply Boundary Conditions
Biindex=0.;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
RHSR2(Biindex)=RHSR1(i);
jindex=0;
for j=1:iindex;
check=0;
if BR1(j)==0.;
jindex=jindex+1;
AR2(Biindex,jindex)=AR1(i,j);
end
end
end
end
%Solve
Tnewr=(AR2^-1)*RHSR2';
% Restore Matrices
Biindex=0;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
TR1(i)=Tnewr(Biindex);
end
end
iindex=0;
for i=1:ndof*numNodes;
check=0;
if mod(i,2)==0 && eNodes(i)~=1
check=1;
end
if check==0
iindex=iindex+1;
Tnew(i)=TR1(iindex);
end
end
Tnew
end
stored';

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function [] = FESolveX2D()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=4;
NumY=1;
delX=0.25;
delY=0.25;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=2;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
stored(1)=dpos;
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(2*Element(e,n)-1)=1.;
end
if crdn<0.01
Bound(2*Element(e,n)-1)=1.;
end
end
end
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% penalty term
Penalty=50.;
% Loop through time steps
for ts=1:tsteps
eNodes=zeros(2*numNodes,1);
% Get interface velocity
d(1)=dpos+delX;
d(2)=dpos+3*delX/4;
d(3)=dpos+delX/4;
d(4)=dpos;
for e=1:numElem
crdn1=Node(Element(e,1),1);
crdn2=Node(Element(e,2),1);
for j=1:4
if d(j)>=crdn1 && d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(Element(e,1)*2-1);
tmp1b=Tnew(Element(e,1)*2);
tmp2a=Tnew(Element(e,2)*2-1);
tmp2b=Tnew(Element(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
% vel=-0.1*(0.5/delX)*(2*t(1)+t(2)-t(3)-2*t(4))
vel=0.0;
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4*ndof);
Me=zeros(4*ndof);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
theta(icrd)=abs(crdnx(icrd)-dpos)*sign(crdnx(icrd)-dpos);
end
if sign(theta(1))~=sign(theta(2))
% possible enriched element
npart=10;
enr=npart*npart;
for sdx=1:npart
for sdy=1:npart
midx=-1.-1./npart+(2./npart)*sdx;
midy=-1.-1./npart+(2./npart)*sdy;
subindex=npart*(sdy-1)+sdx;
gpos=1./(sqrt(3.)*npart);
gx(subindex,1)=midx-gpos;
gx(subindex,2)=midx+gpos;
gx(subindex,3)=midx+gpos;
gx(subindex,4)=midx-gpos;
hx(subindex,1)=midy-gpos;
hx(subindex,2)=midy-gpos;
hx(subindex,3)=midy+gpos;
hx(subindex,4)=midy+gpos;
end
end
% check if int points are on different sides of front
check=0;
for i=1:enr
for j=1:4
g=gx(i,j);
h=hx(i,j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
if i==1 && j==1
sgn=sign(iLS);
else
if sign(iLS)~=sgn
check=1;
end
end
end
end
if check==0
% regular element - fix extra dofs
enr=1;
gpos=1/sqrt(3.);
gx(1,1)=-gpos;
gx(1,2)=gpos;
gx(1,3)=gpos;
gx(1,4)=-gpos;
hx(1,1)=-gpos;
hx(1,2)=-gpos;
hx(1,3)=gpos;
hx(1,4)=gpos;
end
else
% regular element - fix extra dofs
enr=1;
gpos=1/sqrt(3.);
gx(1,1)=-gpos;
gx(1,2)=gpos;
gx(1,3)=gpos;
gx(1,4)=-gpos;
hx(1,1)=-gpos;
hx(1,2)=-gpos;
hx(1,3)=gpos;
hx(1,4)=gpos;
end
% Loop Through Int Points
for i=1:enr
for j=1:4
g=gx(i,j);
h=hx(i,j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
cond=1.;
spec=1.;
for iter=1:4
phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(iter)));
end
phig(1)=0.25*-(1.-h);
phig(3)=0.25*(1.-h);
phig(5)=0.25*(1.+h);
phig(7)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(3)=0.25*-(1.+g);
phih(5)=0.25*(1.+g);
phih(7)=0.25*(1.-g);
diLSg=sign(iLS)*(phig(1)*theta(1)+phig(3)*theta(2)+phig(5)*theta(3)+phig(7)*theta(4));
diLSh=sign(iLS)*(phih(1)*theta(1)+phih(3)*theta(2)+phih(5)*theta(3)+phih(7)*theta(4));
for iter=1:4
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSg;
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSh;
end
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=djac;
Ke=Ke+(we*cond*(phix'*phix+phiy'*phiy))/double(enr);
Me=Me+((we*rho*spec*phi'*phi)/dtime)/double(enr);
end
end
% Add penalty term and get temp gradient on interface
if enr>1;
count=0;
if sign(theta(1))~=sign(theta(2))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(2)));
xi(count)=f*(crdnx(2)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(2)-crdny(1))+crdny(1);
gi(count)=(2.*xi(count)-(crdnx(1)+crdnx(2)))/(-crdnx(1)+crdnx(2));
hi(count)=-1.;
end
if sign(theta(2))~=sign(theta(3))
count=count+1;
f=abs(theta(2))/(abs(theta(2))+abs(theta(3)));
xi(count)=f*(crdnx(3)-crdnx(2))+crdnx(2);
yi(count)=f*(crdny(3)-crdny(2))+crdny(2);
gi(count)=1.;
hi(count)=(2.*yi(count)-(crdny(2)+crdny(3)))/(-crdny(2)+crdny(3));
end
if sign(theta(3))~=sign(theta(4))
count=count+1;
f=abs(theta(3))/(abs(theta(3))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(3))+crdnx(3);
yi(count)=f*(crdny(4)-crdny(3))+crdny(3);
gi(count)=(2.*xi(count)-(crdnx(4)+crdnx(3)))/(-crdnx(4)+crdnx(3));
hi(count)=1.;
end
if sign(theta(1))~=sign(theta(4))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(4)-crdny(1))+crdny(1);
gi(count)=-1.;
hi(count)=(2.*yi(count)-(crdny(1)+crdny(4)))/(-crdny(4)+crdny(1));
end
c=zeros(2,1);
c=(c+1.);
for i=1:2;
G(i,1)=0.25*(1.-gi(i))*(1.-hi(i));
G(i,3)=0.25*(1.+gi(i))*(1.-hi(i));
G(i,5)=0.25*(1.+gi(i))*(1.+hi(i));
G(i,7)=0.25*(1.-gi(i))*(1.+hi(i));
G(i,2)=-G(i,1)*abs(theta(1));
G(i,4)=-G(i,3)*abs(theta(2));
G(i,6)=-G(i,5)*abs(theta(3));
G(i,8)=-G(i,7)*abs(theta(4));
end
pen=Penalty*(G'*G);
pfL=Penalty*G'*c;
Ke=Ke+pen;
else
pen=zeros(8);
pfL=zeros(8,1);
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(2*j-1,2*i-1);
K(gnum(j)+1,gnum(i))=K(gnum(j)+1,gnum(i))+Ke(2*j,2*i-1);
K(gnum(j),gnum(i)+1)=K(gnum(j),gnum(i)+1)+Ke(2*j-1,2*i);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i))=M(gnum(j)+1,gnum(i))+Me(2*j,2*i-1);
M(gnum(j),gnum(i)+1)=M(gnum(j),gnum(i)+1)+Me(2*j-1,2*i);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove NON-ENHANCED DOFs(Reduce Matrices)
iindex=0.;
for i=1:ndof*numNodes;
check=0;
% if mod(i,2)==0 && eNodes(i)~=1
% check=1;
% end
if check==0
iindex=iindex+1;
TR1(iindex)=Tnew(i);
BR1(iindex)=Bound(i);
pforceR1(iindex)=pforce(i);
jindex=0;
for j=1:ndof*numNodes;
check=0;
% if mod(j,2)==0 && eNodes(j)~=1
% check=1;
% end
if check==0
jindex=jindex+1;
MR1(iindex,jindex)=M(i,j);
KR1(iindex,jindex)=K(i,j);
end
end
end
end
AR1=KR1+MR1;
SubR1=AR1*BR1';
RHSR1=MR1*TR1'-SubR1+pforceR1';
% Apply Boundary Conditions
Biindex=0.;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
RHSR2(Biindex)=RHSR1(i);
jindex=0;
for j=1:iindex;
check=0;
if BR1(j)==0.;
jindex=jindex+1;
AR2(Biindex,jindex)=AR1(i,j);
end
end
end
end
%Solve
Tnewr=(AR2^-1)*RHSR2';
% Restore Matrices
Biindex=0;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
TR1(i)=Tnewr(Biindex);
end
end
iindex=0;
for i=1:ndof*numNodes;
check=0;
% if mod(i,2)==0 && eNodes(i)~=1
% check=1;
% end
if check==0
iindex=iindex+1;
Tnew(i)=TR1(iindex);
end
end
Tnew
end
stored';

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function [] = FESolveX2DLS()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=3;
NumY=1;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=2;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.1;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
stored(1)=dpos;
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(2*Element(e,n)-1)=1.;
Bound(2*Element(e,n)-1)=1.;
end
end
end
% Define Time Step
dtime=0.1;
tsteps=20;
time=0.;
% penalty term
beta=80.;
% Loop through time steps
for ts=1:tsteps
% Get interface velocity
d(1)=dpos+delX;
d(2)=dpos+3*delX/4;
d(3)=dpos+delX/4;
d(4)=dpos;
for e=1:numElem
crdn1=Node(Element(e,1),1);
crdn2=Node(Element(e,2),1);
for j=1:4
if d(j)>=crdn1 & d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(Element(e,1)*2-1);
tmp1b=Tnew(Element(e,1)*2);
tmp2a=Tnew(Element(e,2)*2-1);
tmp2b=Tnew(Element(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
% vel=-0.1*(0.5/delX)*(2*t(1)+t(2)-t(3)-2*t(4))
vel=0.0;
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4*ndof);
Me=zeros(4*ndof);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
theta(icrd)=abs(crdnx(icrd)-dpos)*sign(crdnx(icrd)-dpos);
end
if sign(theta(1))~=sign(theta(2))
% enriched element
enr=8;
% get interface position on element
elen=abs(crdnx(2)-crdnx(1));
frac=abs(dpos-crdnx(1))/elen;
len1=2.*frac;
len2=2.*(1.-frac);
% devide element for sub integration
mid1=-1+len1/2.;
mid2=1-len2/2.;
gx(1)=mid1-(len1/2.)/sqrt(3.);
gx(2)=mid1+(len1/2.)/sqrt(3.);
gx(3)=mid1+(len1/2.)/sqrt(3.);
gx(4)=mid1-(len1/2.)/sqrt(3.);
gx(5)=mid2-(len2/2.)/sqrt(3.);
gx(6)=mid2+(len2/2.)/sqrt(3.);
gx(7)=mid2+(len2/2.)/sqrt(3.);
gx(8)=mid2-(len2/2.)/sqrt(3.);
gpos=1/sqrt(3.);
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=+gpos;
hx(4)=+gpos;
hx(5)=-gpos;
hx(6)=-gpos;
hx(7)=+gpos;
hx(8)=+gpos;
for iw=1:4
w(iw)=frac/2.;
w(iw+4)=(1.-frac)/2.;
end
else
% regular element - fix extra dofs
enr=4;
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
for iw=1:4
w(iw)=1.;
end
end
% Loop Through Int Points
for i=1:enr;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
if iLS<0.
cond=0.;
spec=0.01;
else
cond=1.;
spec=1.;
end
for iter=1:4
phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(iter)));
end
phig(1)=0.25*-(1.-h);
phig(3)=0.25*(1.-h);
phig(5)=0.25*(1.+h);
phig(7)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(3)=0.25*-(1.+g);
phih(5)=0.25*(1.+g);
phih(7)=0.25*(1.-g);
diLSg=sign(iLS)*(phig(1)*theta(1)+phig(3)*theta(2)+phig(5)*theta(3)+phig(7)*theta(4));
diLSh=sign(iLS)*(phih(1)*theta(1)+phih(3)*theta(2)+phih(5)*theta(3)+phih(7)*theta(4));
for iter=1:4
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSg;
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSh;
end
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=w(i)*djac;
Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
if enr==8;
xi=2.*frac-1;
yi=0.;
gm(1)=0.25*(1.-xi)*(1.-yi);
gm(3)=0.25*(1.+xi)*(1.-yi);
gm(5)=0.25*(1.+xi)*(1.+yi);
gm(7)=0.25*(1.-xi)*(1.+yi);
gm(2)=gm(1)*(-abs(theta(1)));
gm(4)=gm(3)*(-abs(theta(2)));
gm(6)=gm(5)*(-abs(theta(3)));
gm(8)=gm(7)*(-abs(theta(4)));
pen=beta*(gm'*gm);
pfL=beta*1.*gm';
Ke=Ke+pen;
else
pen=zeros(8);
pfL=zeros(8,1);
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(2*j-1,2*i-1);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove inactive DOFs(Reduce Matrices)
RHS=M*Tnew;
A=K+M;
Sub=A*Bound;
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i)+pforce(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
StiffI=Ared^-1;
Tnewr=(Ared^-1)*RHSred';
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end
stored'

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function [] = FESolveX2Db()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=2;
NumY=1;
delX=0.25;
delY=0.25;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=2;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.4;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
stored(1)=dpos;
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(2*Element(e,n)-1)=1.;
end
if crdn<0.01
Bound(2*Element(e,n)-1)=1.;
end
end
end
% Define Time Step
dtime=0.01;
tsteps=1;
time=0.;
% penalty term
Penalty=00.;
% Loop through time steps
for ts=1:tsteps
% Get interface velocity
d(1)=dpos+delX;
d(2)=dpos+3*delX/4;
d(3)=dpos+delX/4;
d(4)=dpos;
for e=1:numElem
crdn1=Node(Element(e,1),1);
crdn2=Node(Element(e,2),1);
for j=1:4
if d(j)>=crdn1 && d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(Element(e,1)*2-1);
tmp1b=Tnew(Element(e,1)*2);
tmp2a=Tnew(Element(e,2)*2-1);
tmp2b=Tnew(Element(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
% vel=-0.1*(0.5/delX)*(2*t(1)+t(2)-t(3)-2*t(4))
vel=0.0;
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4*ndof);
Me=zeros(4*ndof);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
theta(icrd)=abs(crdnx(icrd)-dpos)*sign(crdnx(icrd)-dpos);
end
if sign(theta(1))~=sign(theta(2))
% enriched element
enr=8;
% get interface position on element
elen=abs(crdnx(2)-crdnx(1));
frac=abs(dpos-crdnx(1))/elen;
len1=2.*frac;
len2=2.*(1.-frac);
% devide element for sub integration
mid1=-1+len1/2.;
mid2=1-len2/2.;
gx(1)=mid1-(len1/2.)/sqrt(3.);
gx(2)=mid1+(len1/2.)/sqrt(3.);
gx(3)=mid1+(len1/2.)/sqrt(3.);
gx(4)=mid1-(len1/2.)/sqrt(3.);
gx(5)=mid2-(len2/2.)/sqrt(3.);
gx(6)=mid2+(len2/2.)/sqrt(3.);
gx(7)=mid2+(len2/2.)/sqrt(3.);
gx(8)=mid2-(len2/2.)/sqrt(3.);
gpos=1/sqrt(3.);
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=+gpos;
hx(4)=+gpos;
hx(5)=-gpos;
hx(6)=-gpos;
hx(7)=+gpos;
hx(8)=+gpos;
for iw=1:4
w(iw)=frac/2.;
w(iw+4)=(1.-frac)/2.;
end
else
% regular element - fix extra dofs
enr=4;
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
for iw=1:4
w(iw)=1.;
end
end
% Loop Through Int Points
for i=1:enr;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
cond=1.;
spec=1.;
for iter=1:4
phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(iter)));
end
phig(1)=0.25*-(1.-h);
phig(3)=0.25*(1.-h);
phig(5)=0.25*(1.+h);
phig(7)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(3)=0.25*-(1.+g);
phih(5)=0.25*(1.+g);
phih(7)=0.25*(1.-g);
diLSg=sign(iLS)*(phig(1)*theta(1)+phig(3)*theta(2)+phig(5)*theta(3)+phig(7)*theta(4));
diLSh=sign(iLS)*(phih(1)*theta(1)+phih(3)*theta(2)+phih(5)*theta(3)+phih(7)*theta(4));
for iter=1:4
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSg;
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSh;
end
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=w(i)*djac;
Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
Me
% Add penalty term and get temp gradient on interface
if enr==8;
count=0;
if sign(theta(1))~=sign(theta(2))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(2)));
xi(count)=f*(crdnx(2)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(2)-crdny(1))+crdny(1);
gi(count)=(2.*xi(count)-(crdnx(1)+crdnx(2)))/(-crdnx(1)+crdnx(2));
hi(count)=-1.;
end
if sign(theta(2))~=sign(theta(3))
count=count+1;
f=abs(theta(2))/(abs(theta(2))+abs(theta(3)));
xi(count)=f*(crdnx(3)-crdnx(2))+crdnx(2);
yi(count)=f*(crdny(3)-crdny(2))+crdny(2);
gi(count)=1.;
hi(count)=(2.*yi(count)-(crdny(2)+crdny(3)))/(-crdny(2)+crdny(3));
end
if sign(theta(3))~=sign(theta(4))
count=count+1;
f=abs(theta(3))/(abs(theta(3))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(3))+crdnx(3);
yi(count)=f*(crdny(4)-crdny(3))+crdny(3);
gi(count)=(2.*xi(count)-(crdnx(4)+crdnx(3)))/(-crdnx(4)+crdnx(3));
hi(count)=1.;
end
if sign(theta(1))~=sign(theta(4))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(4)-crdny(1))+crdny(1);
gi(count)=-1.;
hi(count)=(2.*yi(count)-(crdny(1)+crdny(4)))/(-crdny(4)+crdny(1));
end
c=zeros(2,1);
c=(c+1.);
for i=1:2;
G(i,1)=0.25*(1.-gi(i))*(1.-hi(i));
G(i,3)=0.25*(1.+gi(i))*(1.-hi(i));
G(i,5)=0.25*(1.+gi(i))*(1.+hi(i));
G(i,7)=0.25*(1.-gi(i))*(1.+hi(i));
G(i,2)=-G(i,1)*abs(theta(1));
G(i,4)=-G(i,3)*abs(theta(2));
G(i,6)=-G(i,5)*abs(theta(3));
G(i,8)=-G(i,7)*abs(theta(4));
end
pen=Penalty*(G'*G);
pfL=Penalty*G'*c;
Ke=Ke+pen;
else
pen=zeros(8);
pfL=zeros(8,1);
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(2*j-1,2*i-1);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i))=M(gnum(j)+1,gnum(i))+Me(2*j,2*i-1);
M(gnum(j),gnum(i)+1)=M(gnum(j),gnum(i)+1)+Me(2*j-1,2*i);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove inactive DOFs(Reduce Matrices)
M
A=K+M;
Sub=A*Bound;
M*Tnew;
RHS=M*Tnew-Sub+pforce;
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSR(iindex)=RHS(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
AR(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tnewr=(AR^-1)*RHSR';
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end
stored';

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function [] = FESolveX2Db()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=4;
NumY=1;
delX=0.25;
delY=0.25;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=2;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(2*Element(e,n)-1)=1.;
end
if crdn<0.01
Bound(2*Element(e,n)-1)=1.;
end
end
end
% Define Time Step
dtime=0.01;
tsteps=1;
time=0.;
% penalty term
Penalty=00.;
% Loop through time steps
for ts=1:tsteps
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4*ndof);
Me=zeros(4*ndof);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
theta(icrd)=abs(crdnx(icrd)-dpos)*sign(crdnx(icrd)-dpos);
end
% if sign(theta(1))~=sign(theta(2))
if 1==2
% enriched element
enr=8;
% get interface position on element
elen=abs(crdnx(2)-crdnx(1));
frac=abs(dpos-crdnx(1))/elen;
len1=2.*frac;
len2=2.*(1.-frac);
% devide element for sub integration
mid1=-1+len1/2.;
mid2=1-len2/2.;
gx(1)=mid1-(len1/2.)/sqrt(3.);
gx(2)=mid1+(len1/2.)/sqrt(3.);
gx(3)=mid1+(len1/2.)/sqrt(3.);
gx(4)=mid1-(len1/2.)/sqrt(3.);
gx(5)=mid2-(len2/2.)/sqrt(3.);
gx(6)=mid2+(len2/2.)/sqrt(3.);
gx(7)=mid2+(len2/2.)/sqrt(3.);
gx(8)=mid2-(len2/2.)/sqrt(3.);
gpos=1/sqrt(3.);
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=+gpos;
hx(4)=+gpos;
hx(5)=-gpos;
hx(6)=-gpos;
hx(7)=+gpos;
hx(8)=+gpos;
for iw=1:4
w(iw)=frac/2.;
w(iw+4)=(1.-frac)/2.;
end
else
% regular element - fix extra dofs
enr=4;
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
for iw=1:4
w(iw)=1.;
end
end
% Loop Through Int Points
for i=1:enr;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
cond=1.;
spec=1.;
for iter=1:4
phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(iter)));
end
phig(1)=0.25*-(1.-h);
phig(3)=0.25*(1.-h);
phig(5)=0.25*(1.+h);
phig(7)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(3)=0.25*-(1.+g);
phih(5)=0.25*(1.+g);
phih(7)=0.25*(1.-g);
diLSg=sign(iLS)*(phig(1)*theta(1)+phig(3)*theta(2)+phig(5)*theta(3)+phig(7)*theta(4));
diLSh=sign(iLS)*(phih(1)*theta(1)+phih(3)*theta(2)+phih(5)*theta(3)+phih(7)*theta(4));
for iter=1:4
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSg;
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSh;
end
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=w(i)*djac;
B=[phix;phiy];
% Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Ke=Ke+we*cond*(B'*B);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
if enr==8;
count=0;
if sign(theta(1))~=sign(theta(2))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(2)));
xi(count)=f*(crdnx(2)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(2)-crdny(1))+crdny(1);
gi(count)=(2.*xi(count)-(crdnx(1)+crdnx(2)))/(-crdnx(1)+crdnx(2));
hi(count)=-1.;
end
if sign(theta(2))~=sign(theta(3))
count=count+1;
f=abs(theta(2))/(abs(theta(2))+abs(theta(3)));
xi(count)=f*(crdnx(3)-crdnx(2))+crdnx(2);
yi(count)=f*(crdny(3)-crdny(2))+crdny(2);
gi(count)=1.;
hi(count)=(2.*yi(count)-(crdny(2)+crdny(3)))/(-crdny(2)+crdny(3));
end
if sign(theta(3))~=sign(theta(4))
count=count+1;
f=abs(theta(3))/(abs(theta(3))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(3))+crdnx(3);
yi(count)=f*(crdny(4)-crdny(3))+crdny(3);
gi(count)=(2.*xi(count)-(crdnx(4)+crdnx(3)))/(-crdnx(4)+crdnx(3));
hi(count)=1.;
end
if sign(theta(1))~=sign(theta(4))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(4)-crdny(1))+crdny(1);
gi(count)=-1.;
hi(count)=(2.*yi(count)-(crdny(1)+crdny(4)))/(-crdny(4)+crdny(1));
end
c=zeros(2,1);
c=(c+1.);
for i=1:2;
G(i,1)=0.25*(1.-gi(i))*(1.-hi(i));
G(i,3)=0.25*(1.+gi(i))*(1.-hi(i));
G(i,5)=0.25*(1.+gi(i))*(1.+hi(i));
G(i,7)=0.25*(1.-gi(i))*(1.+hi(i));
G(i,2)=-G(i,1)*abs(theta(1));
G(i,4)=-G(i,3)*abs(theta(2));
G(i,6)=-G(i,5)*abs(theta(3));
G(i,8)=-G(i,7)*abs(theta(4));
end
pen=Penalty*(G'*G);
pfL=Penalty*G'*c;
Ke=Ke+pen;
else
pen=zeros(8);
pfL=zeros(8,1);
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(2*j-1,2*i-1);
K(gnum(j)+1,gnum(i))=K(gnum(j)+1,gnum(i))+Ke(2*j,2*i-1);
K(gnum(j),gnum(i)+1)=K(gnum(j),gnum(i)+1)+Ke(2*j-1,2*i);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i))=M(gnum(j)+1,gnum(i))+Me(2*j,2*i-1);
M(gnum(j),gnum(i)+1)=M(gnum(j),gnum(i)+1)+Me(2*j-1,2*i);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove inactive DOFs(Reduce Matrices)
A=K+M;
Sub=A*Bound;
RHS=M*Tnew-Sub+pforce;
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSR(iindex)=RHS(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
AR(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tnewr=(AR^-1)*RHSR';
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end

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! This Subroutine Implements the Level Set Method
! J. Grogan - 07/10/13
subroutine uexternaldb(lop,lrestart,time,dtime,kstep,kinc)
include 'aba_param.inc'
dimension time(2)
integer Element (100,4)
real Node(100,2),LSet(100)
!
if(lop==0)then
! Get Mesh Data
call getMesh(Element,Node,numElem,numNodes)
! Get Initial Level Set
call initialLSet(LSet,Node,numNodes)
elseif(lop==1)then
! Update Level Set
call updateLSet(Node,numNodes,Element,numElem,dtime,LSet)
endif
return
end
! This subroutine returns the finite element mesh connectivity data
subroutine getMesh(Element,Node,numElem,numNodes)
include 'aba_param.inc'
integer Element (100,4)
real Node(100,2)
character(256) outdir,jobname,input
call getoutdir(outdir,lenoutdir)
call getjobname(jobname,lenjobname)
filename=trim(outdir)//trim(jobname)//'.inp'
open(unit=107,file=filename,status='old')
read(107,*)input
do while (index(input,'*Node')==0)
read(107,*)input
enddo
ierr=0
numNodes=0
do while (ierr==0)
read(107,*)nodeNum,xcor,ycor,zcor
if(ierr==0)then
Node(nodeNum,1)=xcor
Node(nodeNum,2)=ycor
numNodes=numNodes+1
endif
enddo
do while (index(input,'*Element')==0)
read(107,*)input
enddo
numElem=0
do while (ierr==0)
read(107,*)elNum,n1,n2,n3,n4
if(ierr==0)then
Element(elNum,1)=n1
Element(elNum,2)=n2
Element(elNum,3)=n3
Element(elNum,4)=n4
numElem=numElem+1
endif
enddo
close(107)
end subroutine
! This subroutine calculates the initial Level Set
subroutine initialLSet(LSet,Node,numNodes)
include 'aba_param.inc'
real Node(100,2),LSet(100)
!centx=4.
!centy=4.
!rad=2.1
!do i=1,numNodes
! dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy))
! LSet(i)=dist-rad
!enddo
do i=1,numNodes
dist=Node(i,1)-0.1
LSet(i)=dist
enddo
end subroutine
! This subroutine updates the Level Set
subroutine updateLSet(Node,numNodes,Element,numElem,dtime,LSet)
include 'aba_param.inc'
integer Element(100,4),NBelem(100,4)
real Node(100,2),NGlobal(100,2),NLocal(100,2)
real LSet(100),LSetLocal(100),F(100),Fred(100)
real A(100,100),Ared(100,100),RHS(100),RHSred(100)
real M(100,100),MGL(100,100),f1(100),f2(100),f3(100)
! parameters
bandWidth=10.
av_Length=1.
h2=0.00001*av_Length
visc=0.0005
dt=0.01
! explicit update of Level Set
do istep=1:floor(dtime/dt)
! Identify Narrow Band Elements and Get Local Level Set
call getNarrowBand(NBelem,NBElems,NGlobal,NLocal,NBNodes,LSetLocal, &
& bandWidth,LSet,Element,numElem,numNodes)
! Identify Scalar Velocity on Nodes Crossed By Interface - F
call getF(F,LSetLocal,NBElems,NBNodes,NBelem)
! Get 'Stiffness' Matrix - A
call getA(A,Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal)
! Apply BCs
RHS=-matmul(A,F)
iindex=0
do i=1,NBNodes
if (F(i)==0.)then
iindex=iindex+1
RHSred(iindex)=RHS(i)
jindex=0
do j=1,NBNodes
if (F(j)==0.)then
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j)
endif
enddo
endif
enddo
! Solve for Fred
Fred=(Ared^-1)*RHSred'
! Get F
iindex=0
do i=1,NBNodes
if (F(i)==0.)then
iindex=iindex+1
F(i)=Fred(iindex)
endif
enddo
! Get Level Set Equation Terms
call getTerms(M,MGLS,f1,f2,f3,Node,NLocal,NBelem,NBNodes,NBElems,&
& LSetLocal,visc,h2,F)
LSetLocal=LSetLocal-((((M+MGLS)^-1)*dt)*(f1+f2+f3))'
! Reinitialize LS
!call fastMarch(LSetLocal,NBelem,Node,NLocal,NBNodes)
do i=1,NBNodes
LSet(NLocal(i))=LSetLocal(i)
end
enddo
end subroutine
! This subroutine identifies elements in the narrow band
subroutine getNarrowBand(NBelem,NBElems,NGlobal,NLocal,NBNodes,LSetLocal,&
& bandWidth,LSet,Element,numElem,numNodes)
include 'aba_param.inc'
integer Element(100,4),NBelem(100,4)
real Node(100,2),NGlobal(100,2),NLocal(100,2)
real LSet(100),LSetLocal(100)
! Identify Narrow Band Elements
NBElems=0
NBNodes=0
NGlobal=0.
do i=1,numElem
check=0
do iNd=1,4
if (abs(LSet(Element(i,iNd)))<=bandWidth*(delX+delY)/2.)then
check=1
endif
enddo
! If an element is in the narrow band split it into triangles
if (check==1)then
for j=1,4
if (NGlobal(Element(i,j))==0)then
NBNodes=NBNodes+1
NGlobal(Element(i,j))=NBNodes
NLocal(NBNodes)=Element(i,j)
endif
endddo
NBElems=NBElems+1
NBelem(NBElems,1)=NGlobal(Element(i,1))
NBelem(NBElems,2)=NGlobal(Element(i,2))
NBelem(NBElems,3)=NGlobal(Element(i,3))
NBElems=NBElems+1
NBelem(NBElems,1)=NGlobal(Element(i,1))
NBelem(NBElems,2)=NGlobal(Element(i,3))
NBelem(NBElems,3)=NGlobal(Element(i,4))
endif
enddo
! Get local Level Set
do i=1,NBNodes
LSetLocal(i)=LSet(NLocal(i))
enddo
end subroutine
! This subroutine extends the interface velocity throughout the computational domain
subroutine getF(F,LSetLocal,NBElems,NBNodes,NBelem)
include 'aba_param.inc'
real LSetLocal(100),F(100),L(3)
F=0.
do i=1,NBElems
do j=1,3
L(j)=sign(1.,LSetLocal(NBelem(i,j)))
enddo
if (L(1) /= L(2) .or. L(1) /= L(3))then
do j=1,3
F(NBelem(i,j))= 1.
enddo
endif
enddo
end subroutine
! This subroutine gets the 'stiffness' matrix - A
subroutine getA(A,Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal)
include 'aba_param.inc'
integer NBelem(100,4)
real Node(100,2),NLocal(100,2)
real LSetLocal(100),A(100,100),AfL(3,3),AfLGLS(3,3)
real gx(3),hx(3),phi(3),phig(3),phih(3),phix(3),phiy(3)
A=0.
do i=1,NBElems
gx(1)=2./3.
gx(2)=1./6.
gx(3)=1./6.
hx(1)=1./6.
hx(2)=1./6.
hx(3)=2./3.
AfL=0.
AfLGLS=0.
x1=Node(NLocal(NBelem(i,1)),1)
y1=Node(NLocal(NBelem(i,1)),2)
x2=Node(NLocal(NBelem(i,2)),1)
y2=Node(NLocal(NBelem(i,2)),2)
x3=Node(NLocal(NBelem(i,3)),1)
y3=Node(NLocal(NBelem(i,3)),2)
do j=1,3
g=gx(j)
h=hx(j)
phi(1)=1.-g-h
phi(2)=g
phi(3)=h
phig(1)=-1.
phig(2)=1.
phig(3)=0.
phih(1)=-1.
phih(2)=0.
phih(3)=1.
djac=2.*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2))
do k=1,3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k))
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k))
enddo
delphi=[phix;phiy]
nodalLset=[LSetLocal(NBelem(i,1));LSetLocal(NBelem(i,2));LSetLocal(NBelem(i,3))]
set=phi*nodalLset
delset=delphi*nodalLset
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.
enddo
do k=1,3
do j=1,3
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+AfL(j,k)+AfLGLS(j,k)
enddo
enddo
enddo
end subroutine
! This subroutine gets the neccessary terms for the level set equation
subroutine getTerms(M,MGLS,f1,f2,f3,Node,NLocal,NBelem,NBNodes,NBElems,&
& LSetLocal,visc,h2,F)
include 'aba_param.inc'
integer NBelem(100,4)
real Node(100,2),NLocal(100,2),LSetLocal(100)
real M(100,100),MGLS(100,100),f1(100),f2(100),f3(100)
real ML(3,3),MGLSL(3,3),f1L(3),f2L(3),f3L(3)
real gx(3),hx(3),phi(3),phig(3),phih(3),phix(3),phiy(3)
M=0.
MGLS=0.
f1=0.
f2=0.
f3=0.
do i=1,NBElems
ML=0.
MGLSL=0.
f1L=0.
f2L=0.
f3L=0.
gx(1)=2./3.
gx(2)=1./6.
gx(3)=1./6.
hx(1)=1./6.
hx(2)=1./6.
hx(3)=2./3.
x1=Node(NLocal(NBelem(i,1)),1)
y1=Node(NLocal(NBelem(i,1)),2)
x2=Node(NLocal(NBelem(i,2)),1)
y2=Node(NLocal(NBelem(i,2)),2)
x3=Node(NLocal(NBelem(i,3)),1)
y3=Node(NLocal(NBelem(i,3)),2)
do j=1,3
g=gx(j)
h=hx(j)
phi(1)=1.-g-h
phi(2)=g
phi(3)=h
phig(1)=-1.
phig(2)=1.
phig(3)=0.
phih(1)=-1.
phih(2)=0.
phih(3)=1.
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2))
do k=1,3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k))
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k))
end
delphi=[phix;phiy]
nodalLset=[LSetLocal(NBelem(i,1));LSetLocal(NBelem(i,2));LSetLocal(NBelem(i,3))]
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))]
delset=delphi*nodalLset;
Floc=phi*nodalF
ML=ML+(phi'*phi)/3.
MGLSL=MGLSL+((delphi'*(delset/norm(delset)))*Floc*(h2/abs(Floc)))*phi/3.
f1L=f1L+phi'*Floc*norm(delset)/3.
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(h2/abs(Floc))*Floc*norm(delset)/3.
vs=h2*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+h2))
f3L=f3L+vs*delphi'*delset/3.
enddo
do k=1,3
do j=1,3
M(NBelem(i,j),NBelem(i,k))=M(NBelem(i,j),NBelem(i,k))+ML(j,k)
MGLS(NBelem(i,j),NBelem(i,k))=MGLS(NBelem(i,j),NBelem(i,k))+MGLSL(j,k)
enddo
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k)
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k)
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k)
enddo
enddo
end subroutine
! This subroutine uses the fast marching method to re-initialize the level set
call fastMarch(LSetLocal,NBelem,Node,NLocal,NBNodes)
include 'aba_param.inc'
integer NBelem(100,4),nstat(100)
real Node(100,2),NLocal(100,2),LSetLocal(100),newlSet(100),L(3)
newlSet=LSetLocal
! Reinitialize LS
nstat=0
do i=1,NBElems
do j=1,3
L(j)=sign(1.,lSetLocal(NBelem(i,j)))
enddo
if (L(1) /= L(2) .or. L(1) /= L(3))then
do j=1,3
nstat(NBelem(i,j))=1
enddo
endif
enddo
maincheck=0
do while(maincheck==0)
lmin=1000.
avlmin=1000.
eindex=0
nindex=0
maincheck=1
do i=1,NBElems
if (nstat(NBelem(i,1))+nstat(NBelem(i,2))+nstat(NBelem(i,3))==2)then
maincheck=0
check=0
ltot=0.
do j=1,3
if (nstat(NBelem(i,j))==0)then
if (abs(lSetLocal(NBelem(i,j)))<=lmin)then
check=1
tempindex=j
endif
endif
ltot=ltot+abs(lSetLocal(NBelem(i,j)))
enddo
if (check==1 .and. ltot/3.<=avlmin)then
eindex=i
nindex=tempindex
lmin=lSetLocal(NBelem(eindex,nindex))
avlmin=ltot/3.
endif
endif
enddo
if (maincheck==0)then
! Find New LS for point
xp=Node(NLocal(NBelem(eindex,nindex)),1)
yp=Node(NLocal(NBelem(eindex,nindex)),2)
count=0
do i=1,3
if (i/=nindex)then
icount=icount+1
x(icount)=Node(NLocal(NBelem(eindex,i)),1)
y(icount)=Node(NLocal(NBelem(eindex,i)),2)
lloc(icount)=newlSet(NBelem(eindex,i))
endif
enddo
delxa=x(1)-xp
delya=y(1)-yp
delxb=x(2)-xp
delyb=y(2)-yp
N=[delxa delya; delxb delyb]
M=N^-1
A=(M(1)*M(1)+M(2)*M(2))
B=(M(3)*M(3)+M(4)*M(4))
C=2.*(M(1)*M(3)+M(2)*M(4))
a=A+B+C
b=-2.*lloc(1)*A-2.*lloc(2)*B-C*(lloc(1)+lloc(2))
c=lloc(1)*lloc(1)*A+lloc(2)*lloc(2)*B+lloc(1)*lloc(2)*C-1.
templ1=(-b+sqrt(b*b-4.*a*c))/(2.*a)
templ2=(-b-sqrt(b*b-4.*a*c))/(2.*a)
if (abs(templ1)>abs(templ2))then
newlSet(NBelem(eindex,nindex))=templ1
else
newlSet(NBelem(eindex,nindex))=templ2
endif
nstat(NBelem(eindex,nindex))=1
endif
enddo
LSetLocal=newlSet
end subroutine

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function [] = XCOR1D()
% MATLAB based 1-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=1.;
% Define Section Properties
rho=1.;
% Generate Mesh
numElem=4;
charlen=len/numElem;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=2;
% initial interface position
dpos=0.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
%storage
stored(1)=dpos;
for e=1:numElem
for n=1:2
crdn1=ndCoords(elemNodes(e,n));
if crdn1<=dpos
Tnew(2*elemNodes(e,n)-1)=1.;
end
end
end
Bound(1)=1.;
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% penalty term
beta=100.;
% Loop through time steps
for ts=1:tsteps
eNodes=zeros(2*numNodes,1);
% Get interface velocity
d(1)=dpos+charlen;
d(2)=dpos+3*charlen/4;
d(3)=dpos+charlen/4;
d(4)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
for j=1:4
if d(j)>=crdn1 && d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(elemNodes(e,1)*2-1);
tmp1b=Tnew(elemNodes(e,1)*2);
tmp2a=Tnew(elemNodes(e,2)*2-1);
tmp2b=Tnew(elemNodes(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
vel=0.5*(0.5/charlen)*(2*t(1)+t(2)-t(3)-2*t(4));
vel=0.
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(2*ndof);
Me=zeros(2*ndof);
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
enr=2;
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
if sign(theta(1))~=sign(theta(2))
% enriched element
eNodes(2*elemNodes(e,1))=1;
eNodes(2*elemNodes(e,2))=1;
enr=4;
% get interface position on element
point=(dpos-crdn1)/ajacob-1.;
% devide element for sub integration
len1=abs(-point-1.);
len2=abs(1.-point);
mid1=-1+len1/2.;
mid2=1-len2/2.;
gpx(1)=-(len1/2.)/sqrt(3.)+mid1;
gpx(2)=(len1/2.)/sqrt(3.)+mid1;
gpx(3)=-(len2/2.)/sqrt(3.)+mid2;
gpx(4)=(len2/2.)/sqrt(3.)+mid2;
w(1)=(len1/2.);
w(2)=(len1/2.);
w(3)=(len2/2.);
w(4)=(len2/2.);
else
% regular element - fix extra dofs
gpx(1)=-1/sqrt(3.);
gpx(2)=1/sqrt(3.);
w(1)=1.;
w(2)=1.;
end
% Loop Through Int Points
for i=1:enr;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(3)=(1.+c)/2.;
term=theta(1)*phi(1)+theta(2)*phi(3);
% if term<0
% cond=0.00;
% spec=0.001;
% else
cond=1.;
spec=1.;
% end
if enr==4
phi(2)=phi(1)*(abs(term)-abs(theta(1)));
phi(4)=phi(3)*(abs(term)-abs(theta(2)));
else
phi(2)=0.0;
phi(4)=0.0;
end
phic(1)=-0.5;
phic(3)=0.5;
dterm=sign(term)*(phic(1)*theta(1)+phic(3)*theta(2));
if enr==4
phic(2)=phic(1)*(abs(term)-abs(theta(1)))+phi(1)*dterm;
phic(4)=phic(3)*(abs(term)-abs(theta(2)))+phi(3)*dterm;
else
phic(2)=0.0;
phic(4)=0.0;
end
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
phix(3)=phic(3)/ajacob;
phix(4)=phic(4)/ajacob;
we=ajacob*w(i);
Ke=Ke+we*cond*phix'*phix;
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
if enr==5;
Ke(1,2)=0.;
Me(1,2)=0.;
Ke(2,1)=0.;
Me(2,1)=0.;
Ke(1,4)=0.;
Me(1,4)=0.;
Ke(4,1)=0.;
Me(4,1)=0.;
Ke(3,2)=0.;
Me(3,2)=0.;
Ke(2,3)=0.;
Me(2,3)=0.;
Ke(4,3)=0.;
Me(4,3)=0.;
Ke(3,4)=0.;
Me(3,4)=0.;
end
% Add penalty term and get temp gradient on interface
if enr==4;
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
pen=beta*(gm'*gm);
pfL=beta*1*gm';
Ke=Ke+pen;
else
pen=zeros(4);
pfL=zeros(4,1);
end
% Assemble Global Matrices
gnum=2.*elemNodes(e,1)-1.;
for i=1:4;
for j=1:4;
K(gnum+j-1,gnum+i-1)=K(gnum+j-1,gnum+i-1)+Ke(j,i);
M(gnum+j-1,gnum+i-1)=M(gnum+j-1,gnum+i-1)+Me(j,i);
end
pforce(gnum+i-1)=pforce(gnum+i-1)+pfL(i);
end
end
%Remove NON-ENHANCED DOFs(Reduce Matrices)
iindex=0.;
for i=1:ndof*numNodes;
check=0;
if mod(i,2)==0 && eNodes(i)~=1
check=1;
end
if check==0
iindex=iindex+1;
TR1(iindex)=Tnew(i);
BR1(iindex)=Bound(i);
pforceR1(iindex)=pforce(i);
jindex=0;
for j=1:ndof*numNodes;
check=0;
if mod(j,2)==0 && eNodes(j)~=1
check=1;
end
if check==0
jindex=jindex+1;
MR1(iindex,jindex)=M(i,j);
KR1(iindex,jindex)=K(i,j);
end
end
end
end
AR1=KR1+MR1;
SubR1=AR1*BR1';
RHSR1=MR1*TR1'-SubR1+pforceR1';
% Apply Boundary Conditions
Biindex=0.;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
RHSR2(Biindex)=RHSR1(i);
jindex=0;
for j=1:iindex;
check=0;
if BR1(j)==0.;
jindex=jindex+1;
AR2(Biindex,jindex)=AR1(i,j);
end
end
end
end
%Solve
Tnewr=(AR2^-1)*RHSR2';
% Restore Matrices
Biindex=0;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
TR1(i)=Tnewr(Biindex);
end
end
iindex=0;
for i=1:ndof*numNodes;
check=0;
if mod(i,2)==0 && eNodes(i)~=1
check=1;
end
if check==0
iindex=iindex+1;
Tnew(i)=TR1(iindex);
end
end
Tnew
end
stored';

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function [] = XCOR1Db()
% MATLAB based 1-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=1.;
% Define Section Properties
rho=1.;
% Generate Mesh
numElem=4;
charlen=len/numElem;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=2;
% initial interface position
dpos=0.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
%storage
stored(1)=dpos;
for e=1:numElem
for n=1:2
crdn1=ndCoords(elemNodes(e,n));
if crdn1<=dpos
Tnew(2*elemNodes(e,n)-1)=1.;
end
end
end
Bound(1)=1.;
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% penalty term
beta=100.;
% Loop through time steps
for ts=1:tsteps
% Get interface velocity
d(1)=dpos+charlen;
d(2)=dpos+3*charlen/4;
d(3)=dpos+charlen/4;
d(4)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
for j=1:4
if d(j)>=crdn1 && d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(elemNodes(e,1)*2-1);
tmp1b=Tnew(elemNodes(e,1)*2);
tmp2a=Tnew(elemNodes(e,2)*2-1);
tmp2b=Tnew(elemNodes(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
vel=0.5*(0.5/charlen)*(2*t(1)+t(2)-t(3)-2*t(4));
vel=0.;
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(2*ndof);
Me=zeros(2*ndof);
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
enr=2;
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
if sign(theta(1))~=sign(theta(2))
% enriched element
enr=4;
% get interface position on element
point=(dpos-crdn1)/ajacob-1.;
% devide element for sub integration
len1=abs(-point-1.);
len2=abs(1.-point);
mid1=-1+len1/2.;
mid2=1-len2/2.;
gpx(1)=-(len1/2.)/sqrt(3.)+mid1;
gpx(2)=(len1/2.)/sqrt(3.)+mid1;
gpx(3)=-(len2/2.)/sqrt(3.)+mid2;
gpx(4)=(len2/2.)/sqrt(3.)+mid2;
w(1)=(len1/2.);
w(2)=(len1/2.);
w(3)=(len2/2.);
w(4)=(len2/2.);
else
% regular element - fix extra dofs
gpx(1)=-1/sqrt(3.);
gpx(2)=1/sqrt(3.);
w(1)=1.;
w(2)=1.;
end
% Loop Through Int Points
for i=1:enr;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(3)=(1.+c)/2.;
term=theta(1)*phi(1)+theta(2)*phi(3);
cond=1.;
spec=1.;
phi(2)=phi(1)*(abs(term)-abs(theta(1)));
phi(4)=phi(3)*(abs(term)-abs(theta(2)));
phic(1)=-0.5;
phic(3)=0.5;
dterm=sign(term)*(phic(1)*theta(1)+phic(3)*theta(2));
phic(2)=phic(1)*(abs(term)-abs(theta(1)))+phi(1)*dterm;
phic(4)=phic(3)*(abs(term)-abs(theta(2)))+phi(3)*dterm;
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
phix(3)=phic(3)/ajacob;
phix(4)=phic(4)/ajacob;
we=ajacob*w(i);
Ke=Ke+we*cond*phix'*phix
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
Me
% Add penalty term and get temp gradient on interface
if enr==4;
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
pen=beta*(gm'*gm);
pfL=beta*1*gm';
Ke=Ke+pen;
else
pen=zeros(4);
pfL=zeros(4,1);
end
% Assemble Global Matrices
gnum=2.*elemNodes(e,1)-1.;
for i=1:4;
for j=1:4;
K(gnum+j-1,gnum+i-1)=K(gnum+j-1,gnum+i-1)+Ke(j,i);
M(gnum+j-1,gnum+i-1)=M(gnum+j-1,gnum+i-1)+Me(j,i);
end
pforce(gnum+i-1)=pforce(gnum+i-1)+pfL(i);
end
end
A=K+M;
Sub=A*Bound;
M*Tnew;
RHS=M*Tnew-Sub+pforce;
% Apply Boundary Conditions
Biindex=0.;
for i=1:ndof*numNodes;
if Bound(i)==0.;
Biindex=Biindex+1;
RHSR(Biindex)=RHS(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
AR(Biindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tnewr=(AR^-1)*RHSR';
% Restore Matrices
Biindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
Biindex=Biindex+1;
Tnew(i)=Tnewr(Biindex);
end
end
Tnew
end
stored';

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function []=XCOR_2D()
clear all
% Define Mesh
NumX=2;
NumY=1;
delX=1.;
delY=1.;
numElem=NumX*NumY;
numNodes=(NumX+1)*(NumY+1);
Elength=(delX+delY)/2.;
[Node,Element]=buildMesh(NumX,NumY,delX,delY);
% Simulation Parameters
rho=1.;
Penalty=80.;
dtImp=0.1;
dtExp=0.01;
tsteps=4;
bandWidth=10.;
epsilon=0.00001;
visc=0.0005;
% Get Initial Level Set
LSetOld=initialLSet(Node,numNodes);
% plotLSet(NumX,NumY,delX,delY,LSet);
% Initial Conditions
Temp=zeros(numNodes*2,1);
for i=1:numNodes
if LSetOld(i)<=0
Temp(2*i-1)=1.;
end
end
% Boundary Conditions
Bound=zeros(numNodes*2,1);
for i=1:numNodes
if Node(i,1)<delX/10.
Bound(2*i-1)=1.;
end
end
% Loop through time steps
for ts=1:tsteps
% Update Level Set
LSetNew=updateLSet(Temp,Node,numNodes,Element,numElem,dtImp,dtExp,LSetOld,...
Elength,bandWidth,epsilon,visc);
% Solve for Temperature
Temp=getTemp(Node,Element,numNodes,numElem,LSetNew,Bound,Temp,Penalty,rho,dtImp,LSetOld);
LSetOld=LSetNew;
LSetOld'
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Create a linear quadrilateral FE mesh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [Node,Element]=buildMesh(NumX,NumY,delX,delY)
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This function updates the level set
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [LSet]=updateLSet(Temp,Node,numNodes,Element,numElem,dtImp,dtExp,LSet,...
Elength,bandWidth,epsilon,visc)
% parameters
charLen=epsilon*Elength;
for tstep=1:floor(dtImp/dtExp)
% Identify Narrow Band Elements and Get Local Level Set
[NBelem,NBElems,NGlobal,NLocal,NBNodes,LSetLocal]=getNarrowBand(bandWidth,...
Elength,LSet,Element,numElem,numNodes);
% Identify Scalar Velocity on Nodes Crossed By Interface - F
F=getF(Temp,LSetLocal,NBElems,NBNodes,NLocal,NBelem,Node,Elength);
% Get 'Stiffness' Matrix - A
A=getA(Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal);
% Apply BCs
RHS=-A*F;
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i);
jindex=0;
for j=1:NBNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
if iindex>0
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
end
% Get Level Set Equation Terms
[M,MGLS,f1,f2,f3]=getTerms(Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal,visc,charLen,F);
LSetLocal=LSetLocal-((((M+MGLS)^-1)*dtExp)*(f1+f2+f3))';
% Reinitialize LS
%LSetLocal=fastMarch(LSetLocal,NBelem,Node,NLocal,NBNodes,NBElems);
for i=1:NBNodes
LSet(NLocal(i))=LSetLocal(i);
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Find elements in narrow band and create map between
% global node labels and those in narrow band
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [NBelem,NBElems,NGlobal,NLocal,NBNodes,LSetLocal]=getNarrowBand(bandWidth,...
ELength,LSet,Element,numElem,numNodes)
% Identify Narrow Band Elements
NBElems=0;
NBNodes=0;
NGlobal=zeros(numNodes);
for i=1:numElem
check=0;
for iNd=1:4
if abs(LSet(Element(i,iNd)))<=bandWidth*ELength
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
for j=1:4
if NGlobal(Element(i,j))==0
NBNodes=NBNodes+1;
NGlobal(Element(i,j))=NBNodes;
NLocal(NBNodes)=Element(i,j);
end
end
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,2));
NBelem(NBElems,3)=NGlobal(Element(i,3));
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,3));
NBelem(NBElems,3)=NGlobal(Element(i,4));
end
end
% Get local Level Set
for i=1:NBNodes
LSetLocal(i)=LSet(NLocal(i));
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get Interface Normal Veloctiy 'F'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function F=getF(Temp,LSetLocal,NBElems,NBNodes,NLocal,NBelem,Node,ELength)
F=zeros(NBNodes,1);
eStat=zeros(NBElems,1);
nData=zeros(NBNodes,2);
for i=1:NBElems
for j=1:3
L(j)=LSetLocal(NBelem(i,j));
end
x11=Node(NLocal(NBelem(i,1)),1);
x12=Node(NLocal(NBelem(i,2)),1);
x13=Node(NLocal(NBelem(i,3)),1);
y11=Node(NLocal(NBelem(i,1)),2);
y12=Node(NLocal(NBelem(i,2)),2);
y13=Node(NLocal(NBelem(i,3)),2);
count=0.;
if sign(L(1)) ~= sign(L(2))
eStat(i)=1;
count=count+1;
f=abs(L(1))/(abs(L(1))+abs(L(2)));
xi(count)=f*(x12-x11)+x11;
yi(count)=f*(y12-y11)+y11;
end
if sign(L(1)) ~= sign(L(3))
eStat(i)=1;
count=count+1;
f=abs(L(1))/(abs(L(1))+abs(L(3)));
xi(count)=f*(x13-x11)+x11;
yi(count)=f*(y13-y11)+y11 ;
end
if sign(L(2)) ~= sign(L(3))
eStat(i)=1;
count=count+1;
f=abs(L(2))/(abs(L(2))+abs(L(3)));
xi(count)=f*(x13-x12)+x12;
yi(count)=f*(y13-y12)+y12 ;
end
if eStat(i)==1
n=[yi(2)-yi(1); xi(1)-xi(2)];
n=n/norm(n);
xd(1,1)=(xi(1)+xi(2))/2.;
xd(1,2)=(yi(1)+yi(2))/2.;
xd(2,1)=0.1*ELength*n(1)+xd(1,1);
xd(2,2)=0.1*ELength*n(2)+xd(1,2);
% Check if xd2 is in element
v0(1)=x11;
v0(2)=y11;
v1(1)=x12-x11;
v1(2)=y12-y11;
v2(1)=x13-x11;
v2(2)=y13-y11;
v(1)=xd(2,1);
v(2)=xd(2,2);
ra=((v(1)*v2(2)-v2(1)*v(2))-(v0(1)*v2(2)-v2(1)*v0(2)))/(v1(1)*v2(2)-v2(1)*v1(2));
rb=-((v(1)*v1(2)-v1(1)*v(2))-(v0(1)*v1(2)-v1(1)*v0(2)))/(v1(1)*v2(2)-v2(1)*v1(2));
check=0;
if ra>0. && rb>0. && ra+rb<1.
index=i;
x21=x11;
x22=x12;
x23=x13;
y21=y11;
y22=y12;
y23=y13;
else
for j=1:NBElems
tx1=Node(NLocal(NBelem(j,1)),1);
tx2=Node(NLocal(NBelem(j,2)),1);
tx3=Node(NLocal(NBelem(j,3)),1);
ty1=Node(NLocal(NBelem(j,1)),2);
ty2=Node(NLocal(NBelem(j,2)),2);
ty3=Node(NLocal(NBelem(j,3)),2);
v0(1)=tx1;
v0(2)=ty1;
v1(1)=tx2-tx1;
v1(2)=ty2-ty1;
v2(1)=tx3-tx1;
v2(2)=ty3-ty1;
v(1)=xd(2,1);
v(2)=xd(2,2);
ra=((v(1)*v2(2)-v2(1)*v(2))-(v0(1)*v2(2)-v2(1)*v0(2)))/(v1(1)*v2(2)-v2(1)*v1(2));
rb=-((v(1)*v1(2)-v1(1)*v(2))-(v0(1)*v1(2)-v1(1)*v0(2)))/(v1(1)*v2(2)-v2(1)*v1(2));
if ra>0. && rb>0. && ra+rb<1.
index=j;
x21=tx1;
x22=tx2;
x23=tx3;
y21=ty1;
y22=ty2;
y23=ty3;
end
end
end
Ae1=0.5*((x12*y13-x13*y12)+(y12-y13)*x11+(x13-x12)*y11);
Ae2=0.5*((x22*y23-x23*y22)+(y22-y23)*x21+(x23-x22)*y21);
N1=(1./(2.*Ae))*((y2-y3)*(xd(j,1)-x2)+(x3-x2)*(xd(j,2)-y2));
N2=(1./(2.*Ae))*((y3-y1)*(xd(j,1)-x3)+(x1-x3)*(xd(j,2)-y3));
N3=(1./(2.*Ae))*((y1-y2)*(xd(j,1)-x1)+(x2-x1)*(xd(j,2)-y1));
T1=Temp(2*NLocal(NBelem(i,1))-1);
T2=Temp(2*NLocal(NBelem(i,2))-1);
T3=Temp(2*NLocal(NBelem(i,3))-1);
a1=Temp(2*NLocal(NBelem(i,1)));
a2=Temp(2*NLocal(NBelem(i,2)));
a3=Temp(2*NLocal(NBelem(i,3)));
L1=LSetLocal(NBelem(i,1));
L2=LSetLocal(NBelem(i,2));
L3=LSetLocal(NBelem(i,3));
LS=abs(N1*L1+L2*N2+L3*N3);
p1=N1*(LS-abs(L1));
p2=N2*(LS-abs(L2));
p3=N3*(LS-abs(L3));
T(j)=N1*T1+N2*T2+N3*T3+p1*a1+p2*a2+p3*a3;
end
gradT=(T(2)-T(1))/(0.1*ELength);
for j=1:3
nData(NBelem(i,j),1)=nData(NBelem(i,j),1)+1.;
nData(NBelem(i,j),2)=nData(NBelem(i,j),2)+0.1*gradT;
end
end
end
for i=1:NBNodes
if nData(i,1)>0
F(i)=nData(i,2)/nData(i,1);
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get 'Stiffness' Matrix 'A'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [A]=getA(Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal)
A=zeros(NBNodes);
for i=1:NBElems
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[LSetLocal(NBelem(i,1));LSetLocal(NBelem(i,2));LSetLocal(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get terms for LS equation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [M,MGLS,f1,f2,f3]=getTerms(Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal,visc,charLen,F)
M=zeros(NBNodes);
MGLS=zeros(NBNodes);
f1=zeros(NBNodes,1);
f2=zeros(NBNodes,1);
f3=zeros(NBNodes,1);
for i=1:NBElems
ML=zeros(3);
MGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[LSetLocal(NBelem(i,1));LSetLocal(NBelem(i,2));LSetLocal(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
ML=ML+(phi'*phi)/3.;
MGLSL=MGLSL+((delphi'*(delset/norm(delset)))*Floc*(charLen/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(charLen/abs(Floc))*Floc*norm(delset)/3.;
vs=charLen*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+charLen));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
M(NBelem(i,j),NBelem(i,k))=M(NBelem(i,j),NBelem(i,k))+ML(j,k);
MGLS(NBelem(i,j),NBelem(i,k))=MGLS(NBelem(i,j),NBelem(i,k))+MGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Use Fast March Method to Reinitialize LS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function LSetLocal=fastMarch(LSetLocal,NBelem,Node,NLocal,NBNodes,NBElems)
newlSet=LSetLocal;
% Reinitialize LS
nstat=zeros(NBNodes,1);
for i=1:NBElems
for j=1:3
L(j)=sign(LSetLocal(NBelem(i,j)));
end
if L(1) ~= L(2) || L(1) ~= L(3)
for j=1:3
nstat(NBelem(i,j))=1;
end
end
end
maincheck=0;
while(maincheck==0)
lmin=1000.;
avlmin=1000.;
eindex=0;
nindex=0;
maincheck=1;
for i=1:NBElems
if nstat(NBelem(i,1))+nstat(NBelem(i,2))+nstat(NBelem(i,3))==2
maincheck=0;
check=0;
ltot=0.;
for j=1:3
if nstat(NBelem(i,j))==0
if abs(LSetLocal(NBelem(i,j)))<=lmin
check=1;
tempindex=j;
end
end
ltot=ltot+abs(LSetLocal(NBelem(i,j)));
end
if check==1 && ltot/3.<=avlmin
eindex=i;
nindex=tempindex;
lmin=LSetLocal(NBelem(eindex,nindex));
avlmin=ltot/3.;
end
end
end
if maincheck==0
% Find New LS for point
xp=Node(NLocal(NBelem(eindex,nindex)),1);
yp=Node(NLocal(NBelem(eindex,nindex)),2);
count=0;
for i=1:3
if i~=nindex
count=count+1;
x(count)=Node(NLocal(NBelem(eindex,i)),1);
y(count)=Node(NLocal(NBelem(eindex,i)),2);
lloc(count)=newlSet(NBelem(eindex,i));
end
end
delxa=x(1)-xp;
delya=y(1)-yp;
delxb=x(2)-xp;
delyb=y(2)-yp;
N=[delxa delya; delxb delyb];
M=N^-1;
A=(M(1)*M(1)+M(2)*M(2));
B=(M(3)*M(3)+M(4)*M(4));
C=2.*(M(1)*M(3)+M(2)*M(4));
a=A+B+C;
b=-2.*lloc(1)*A-2.*lloc(2)*B-C*(lloc(1)+lloc(2));
c=lloc(1)*lloc(1)*A+lloc(2)*lloc(2)*B+lloc(1)*lloc(2)*C-1.;
templ1=(-b+sqrt(b*b-4.*a*c))/(2.*a);
templ2=(-b-sqrt(b*b-4.*a*c))/(2.*a);
if abs(templ1)>abs(templ2)
newlSet(NBelem(eindex,nindex))=templ1;
else
newlSet(NBelem(eindex,nindex))=templ2;
end
nstat(NBelem(eindex,nindex))=1;
end
end
LSetLocal=newlSet;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Solve Implicit Porblem to Get Temperature
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function Temp=getTemp(Node,Element,numNodes,numElem,LSet,Bound,Temp,Penalty,rho,dtImp,LSetOld)
K=zeros(numNodes*2,numNodes*2);
M=zeros(numNodes*2,numNodes*2);
MStar=zeros(numNodes*2,numNodes*2);
pforce=zeros(numNodes*2,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(8);
Me=zeros(8);
MeStar=zeros(8);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
theta(icrd)=LSet(Element(e,icrd));
thetaO(icrd)=LSetOld(Element(e,icrd));
end
check=0;
for i=1:3
for j=i+1:4
if sign(theta(1))~=sign(theta(j))
check=1;
end
end
end
if check==1
% possible enriched element
npart=10;
enr=npart*npart;
for sdx=1:npart
for sdy=1:npart
midx=-1.-1./npart+(2./npart)*sdx;
midy=-1.-1./npart+(2./npart)*sdy;
subindex=npart*(sdy-1)+sdx;
gpos=1./(sqrt(3.)*npart);
gx(subindex,1)=midx-gpos;
gx(subindex,2)=midx+gpos;
gx(subindex,3)=midx+gpos;
gx(subindex,4)=midx-gpos;
hx(subindex,1)=midy-gpos;
hx(subindex,2)=midy-gpos;
hx(subindex,3)=midy+gpos;
hx(subindex,4)=midy+gpos;
end
end
% check if int points are on different sides of front
check=0;
for i=1:enr
for j=1:4
g=gx(i,j);
h=hx(i,j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
phiO=phi;
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
if i==1 && j==1
sgn=sign(iLS);
else
if sign(iLS)~=sgn
check=1;
end
end
end
end
if check==0
% regular element - fix extra dofs
enr=1;
gpos=1/sqrt(3.);
gx(1,1)=-gpos;
gx(1,2)=gpos;
gx(1,3)=gpos;
gx(1,4)=-gpos;
hx(1,1)=-gpos;
hx(1,2)=-gpos;
hx(1,3)=gpos;
hx(1,4)=gpos;
end
else
% regular element - fix extra dofs
enr=1;
gpos=1/sqrt(3.);
gx(1,1)=-gpos;
gx(1,2)=gpos;
gx(1,3)=gpos;
gx(1,4)=-gpos;
hx(1,1)=-gpos;
hx(1,2)=-gpos;
hx(1,3)=gpos;
hx(1,4)=gpos;
end
for i=1:enr
for j=1:4
g=gx(i,j);
h=hx(i,j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
phiO=phi;
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
iLSO=thetaO(1)*phi(1)+thetaO(2)*phi(3)+thetaO(3)*phi(5)+thetaO(4)*phi(7);
if iLS<0.
cond=0.;
spec=0.01;
else
cond=1.;
spec=1.;
end
if iLSO<0.
specO=0.01;
else
specO=1.;
end
for iter=1:4
phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(iter)));
phiO(2*iter)=phiO(2*iter-1)*(abs(iLSO)-abs(thetaO(iter)));
end
phig(1)=0.25*-(1.-h);
phig(3)=0.25*(1.-h);
phig(5)=0.25*(1.+h);
phig(7)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(3)=0.25*-(1.+g);
phih(5)=0.25*(1.+g);
phih(7)=0.25*(1.-g);
diLSg=sign(iLS)*(phig(1)*theta(1)+phig(3)*theta(2)+phig(5)*theta(3)+phig(7)*theta(4));
diLSh=sign(iLS)*(phih(1)*theta(1)+phih(3)*theta(2)+phih(5)*theta(3)+phih(7)*theta(4));
for iter=1:4
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSg;
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSh;
end
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=djac;
Ke=Ke+(we*cond*(phix'*phix+phiy'*phiy))/double(enr);
Me=Me+((we*rho*spec*phi'*phi)/dtImp)/double(enr);
MeStar=MeStar+((we*rho*specO*phi'*phiO)/dtImp)/double(enr);
end
end
% Add penalty term and get temp gradient on interface
if enr>1;
count=0;
if sign(theta(1))~=sign(theta(2))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(2)));
xi(count)=f*(crdnx(2)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(2)-crdny(1))+crdny(1);
gi(count)=(2.*xi(count)-(crdnx(1)+crdnx(2)))/(-crdnx(1)+crdnx(2));
hi(count)=-1.;
end
if sign(theta(2))~=sign(theta(3))
count=count+1;
f=abs(theta(2))/(abs(theta(2))+abs(theta(3)));
xi(count)=f*(crdnx(3)-crdnx(2))+crdnx(2);
yi(count)=f*(crdny(3)-crdny(2))+crdny(2);
gi(count)=1.;
hi(count)=(2.*yi(count)-(crdny(2)+crdny(3)))/(-crdny(2)+crdny(3));
end
if sign(theta(3))~=sign(theta(4))
count=count+1;
f=abs(theta(3))/(abs(theta(3))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(3))+crdnx(3);
yi(count)=f*(crdny(4)-crdny(3))+crdny(3);
gi(count)=(2.*xi(count)-(crdnx(4)+crdnx(3)))/(-crdnx(4)+crdnx(3));
hi(count)=1.;
end
if sign(theta(1))~=sign(theta(4))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(4)-crdny(1))+crdny(1);
gi(count)=-1.;
hi(count)=(2.*yi(count)-(crdny(1)+crdny(4)))/(-crdny(4)+crdny(1));
end
c=zeros(2,1);
c=(c+1.);
for i=1:2;
G(i,1)=0.25*(1.-gi(i))*(1.-hi(i));
G(i,3)=0.25*(1.+gi(i))*(1.-hi(i));
G(i,5)=0.25*(1.+gi(i))*(1.+hi(i));
G(i,7)=0.25*(1.-gi(i))*(1.+hi(i));
G(i,2)=-G(i,1)*abs(theta(1));
G(i,4)=-G(i,3)*abs(theta(2));
G(i,6)=-G(i,5)*abs(theta(3));
G(i,8)=-G(i,7)*abs(theta(4));
end
pen=Penalty*(G'*G);
pfL=Penalty*G'*c;
% pen=zeros(8);
% pfL=zeros(8,1);
Ke=Ke+pen;
else
pen=zeros(8);
pfL=zeros(8,1);
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(2*j-1,2*i-1);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
MStar(gnum(j),gnum(i))=MStar(gnum(j),gnum(i))+MeStar(2*j-1,2*i-1);
MStar(gnum(j)+1,gnum(i)+1)=MStar(gnum(j)+1,gnum(i)+1)+MeStar(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove inactive DOFs(Reduce Matrices)
RHS=MStar*Temp;
A=K+M;
Sub=A*Bound;
iindex=0;
for i=1:2*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i)+pforce(i);
jindex=0;
for j=1:2*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tempr=(Ared^-1)*RHSred';
iindex=0;
for i=1:2*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Temp(i)=Tempr(iindex);
end
end
Temp
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Generates the initial level set
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [LSet]=initialLSet(Node,numNodes)
%centx=4.;
%centy=4.;
%rad=2.1;
%for i=1:numNodes;
% dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
% LSet(i)=dist-rad;
%end
for i=1:numNodes;
dist=Node(i,1)-1.1;
LSet(i)=dist;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Plot the level set
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function []=plotLSet(NumX,NumY,delX,delY,LSet)
[X Y]=meshgrid(0:delX:delX*NumX,0:delY:delY*NumY);
Z=zeros(NumX+1,NumY+1);
for i=1:(NumX+1)*(NumY+1)
Z(i)=LSet(i);
end
surf(X,Y,Z)
end

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function []=XCOR_2D()
clear all
% Define Mesh
NumX=10;
NumY=1;
delX=1.;
delY=1.;
numElem=NumX*NumY;
numNodes=(NumX+1)*(NumY+1);
Elength=(delX+delY)/2.;
[Node,Element]=buildMesh(NumX,NumY,delX,delY);
% Simulation Parameters
rho=1.;
Penalty=200.;
dtImp=0.01;
dtExp=0.001;
tsteps=10;
bandWidth=10.;
epsilon=0.00001;
visc=0.0005;
% Get Initial Level Set
LSetOld=initialLSet(Node,numNodes);
% plotLSet(NumX,NumY,delX,delY,LSet);
% Initial Conditions
Temp=zeros(numNodes*2,1);
for i=1:numNodes
if LSetOld(i)<=0
Temp(2*i-1)=1.;
end
end
% Boundary Conditions
Bound=zeros(numNodes*2,1);
for i=1:numNodes
if Node(i,1)<delX/10.
Bound(2*i-1)=1.;
end
end
% Loop through time steps
for ts=1:tsteps
% Update Level Set
LSetNew=updateLSet(Temp,Node,numNodes,Element,numElem,dtImp,dtExp,LSetOld,...
Elength,bandWidth,epsilon,visc);
% Solve for Temperature
Temp=getTemp(Node,Element,numNodes,numElem,LSetNew,Bound,Temp,Penalty,rho,dtImp,LSetOld);
LSetOld=LSetNew;
LSetOld'
POUT(ts)=LSetOld(1);
end
POUT';
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Create a linear quadrilateral FE mesh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [Node,Element]=buildMesh(NumX,NumY,delX,delY)
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This function updates the level set
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [LSet]=updateLSet(Temp,Node,numNodes,Element,numElem,dtImp,dtExp,LSet,...
Elength,bandWidth,epsilon,visc)
% parameters
charLen=epsilon*Elength;
for tstep=1:floor(dtImp/dtExp)
% Identify Narrow Band Elements and Get Local Level Set
[NBelem,NBElems,NGlobal,NLocal,NBNodes,LSetLocal]=getNarrowBand(bandWidth,...
Elength,LSet,Element,numElem,numNodes);
% Identify Scalar Velocity on Nodes Crossed By Interface - F
F=getF(Temp,LSetLocal,NBElems,NBNodes,NLocal,NBelem,Node,Elength);
% Get 'Stiffness' Matrix - A
A=getA(Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal);
% Apply BCs
RHS=-A*F;
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i);
jindex=0;
for j=1:NBNodes
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
if iindex>0
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:NBNodes
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
end
% Get Level Set Equation Terms
[M,MGLS,f1,f2,f3]=getTerms(Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal,visc,charLen,F);
LSetLocal=LSetLocal-((((M+MGLS)^-1)*dtExp)*(f1+f2+f3))';
% Reinitialize LS
%LSetLocal=fastMarch(LSetLocal,NBelem,Node,NLocal,NBNodes,NBElems);
for i=1:NBNodes
LSet(NLocal(i))=LSetLocal(i);
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Find elements in narrow band and create map between
% global node labels and those in narrow band
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [NBelem,NBElems,NGlobal,NLocal,NBNodes,LSetLocal]=getNarrowBand(bandWidth,...
ELength,LSet,Element,numElem,numNodes)
% Identify Narrow Band Elements
NBElems=0;
NBNodes=0;
NGlobal=zeros(numNodes);
for i=1:numElem
check=0;
for iNd=1:4
if abs(LSet(Element(i,iNd)))<=bandWidth*ELength
check=1;
end
end
% If an element is in the narrow band split it into triangles
if check==1
for j=1:4
if NGlobal(Element(i,j))==0
NBNodes=NBNodes+1;
NGlobal(Element(i,j))=NBNodes;
NLocal(NBNodes)=Element(i,j);
end
end
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,2));
NBelem(NBElems,3)=NGlobal(Element(i,3));
NBElems=NBElems+1;
NBelem(NBElems,1)=NGlobal(Element(i,1));
NBelem(NBElems,2)=NGlobal(Element(i,3));
NBelem(NBElems,3)=NGlobal(Element(i,4));
end
end
% Get local Level Set
for i=1:NBNodes
LSetLocal(i)=LSet(NLocal(i));
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get Interface Normal Veloctiy 'F'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function F=getF(Temp,LSetLocal,NBElems,NBNodes,NLocal,NBelem,Node,ELength)
F=zeros(NBNodes,1);
eStat=zeros(NBElems,1);
nData=zeros(NBNodes,2);
for i=1:NBElems
for j=1:3
L(j)=LSetLocal(NBelem(i,j));
end
x11=Node(NLocal(NBelem(i,1)),1);
x12=Node(NLocal(NBelem(i,2)),1);
x13=Node(NLocal(NBelem(i,3)),1);
y11=Node(NLocal(NBelem(i,1)),2);
y12=Node(NLocal(NBelem(i,2)),2);
y13=Node(NLocal(NBelem(i,3)),2);
count=0.;
if sign(L(1)) ~= sign(L(2))
eStat(i)=1;
count=count+1;
f=abs(L(1))/(abs(L(1))+abs(L(2)));
xi(count)=f*(x12-x11)+x11;
yi(count)=f*(y12-y11)+y11;
end
if sign(L(1)) ~= sign(L(3))
eStat(i)=1;
count=count+1;
f=abs(L(1))/(abs(L(1))+abs(L(3)));
xi(count)=f*(x13-x11)+x11;
yi(count)=f*(y13-y11)+y11 ;
end
if sign(L(2)) ~= sign(L(3))
eStat(i)=1;
count=count+1;
f=abs(L(2))/(abs(L(2))+abs(L(3)));
xi(count)=f*(x13-x12)+x12;
yi(count)=f*(y13-y12)+y12 ;
end
if eStat(i)==1
n=[yi(2)-yi(1); xi(1)-xi(2)];
n=n/norm(n);
xd(1,1)=(xi(1)+xi(2))/2.;
xd(1,2)=(yi(1)+yi(2))/2.;
xd(2,1)=0.15*ELength*n(1)+xd(1,1);
xd(2,2)=0.15*ELength*n(2)+xd(1,2);
% Check if xd2 is in element
v0(1)=x11;
v0(2)=y11;
v1(1)=x12-x11;
v1(2)=y12-y11;
v2(1)=x13-x11;
v2(2)=y13-y11;
v(1)=xd(2,1);
v(2)=xd(2,2);
ra=((v(1)*v2(2)-v2(1)*v(2))-(v0(1)*v2(2)-v2(1)*v0(2)))/(v1(1)*v2(2)-v2(1)*v1(2));
rb=-((v(1)*v1(2)-v1(1)*v(2))-(v0(1)*v1(2)-v1(1)*v0(2)))/(v1(1)*v2(2)-v2(1)*v1(2));
check=0;
if ra>0. && rb>0. && ra+rb<1.
index=i;
x21=x11;
x22=x12;
x23=x13;
y21=y11;
y22=y12;
y23=y13;
else
for j=1:NBElems
tx1=Node(NLocal(NBelem(j,1)),1);
tx2=Node(NLocal(NBelem(j,2)),1);
tx3=Node(NLocal(NBelem(j,3)),1);
ty1=Node(NLocal(NBelem(j,1)),2);
ty2=Node(NLocal(NBelem(j,2)),2);
ty3=Node(NLocal(NBelem(j,3)),2);
v0(1)=tx1;
v0(2)=ty1;
v1(1)=tx2-tx1;
v1(2)=ty2-ty1;
v2(1)=tx3-tx1;
v2(2)=ty3-ty1;
v(1)=xd(2,1);
v(2)=xd(2,2);
ra=((v(1)*v2(2)-v2(1)*v(2))-(v0(1)*v2(2)-v2(1)*v0(2)))/(v1(1)*v2(2)-v2(1)*v1(2));
rb=-((v(1)*v1(2)-v1(1)*v(2))-(v0(1)*v1(2)-v1(1)*v0(2)))/(v1(1)*v2(2)-v2(1)*v1(2));
if ra>0. && rb>0. && ra+rb<1.
index=j;
x21=tx1;
x22=tx2;
x23=tx3;
y21=ty1;
y22=ty2;
y23=ty3;
end
end
end
Ae=0.5*((x12*y13-x13*y12)+(y12-y13)*x11+(x13-x12)*y11);
N1=(1./(2.*Ae))*((y12-y13)*(xd(1,1)-x12)+(x13-x12)*(xd(1,2)-y12));
N2=(1./(2.*Ae))*((y13-y11)*(xd(1,1)-x13)+(x11-x13)*(xd(1,2)-y13));
N3=(1./(2.*Ae))*((y11-y12)*(xd(1,1)-x11)+(x12-x11)*(xd(1,2)-y11));
T1=Temp(2*NLocal(NBelem(i,1))-1);
T2=Temp(2*NLocal(NBelem(i,2))-1);
T3=Temp(2*NLocal(NBelem(i,3))-1);
a1=Temp(2*NLocal(NBelem(i,1)));
a2=Temp(2*NLocal(NBelem(i,2)));
a3=Temp(2*NLocal(NBelem(i,3)));
L1=LSetLocal(NBelem(i,1));
L2=LSetLocal(NBelem(i,2));
L3=LSetLocal(NBelem(i,3));
LS=abs(N1*L1+L2*N2+L3*N3);
p1=N1*(LS-abs(L1));
p2=N2*(LS-abs(L2));
p3=N3*(LS-abs(L3));
T(1)=N1*T1+N2*T2+N3*T3+p1*a1+p2*a2+p3*a3;
Ae=0.5*((x22*y23-x23*y22)+(y22-y23)*x21+(x23-x22)*y21);
N1=(1./(2.*Ae))*((y22-y23)*(xd(2,1)-x22)+(x23-x22)*(xd(2,2)-y22));
N2=(1./(2.*Ae))*((y23-y21)*(xd(2,1)-x23)+(x21-x23)*(xd(2,2)-y23));
N3=(1./(2.*Ae))*((y21-y22)*(xd(2,1)-x21)+(x22-x21)*(xd(2,2)-y21));
T1=Temp(2*NLocal(NBelem(index,1))-1);
T2=Temp(2*NLocal(NBelem(index,2))-1);
T3=Temp(2*NLocal(NBelem(index,3))-1);
a1=Temp(2*NLocal(NBelem(index,1)));
a2=Temp(2*NLocal(NBelem(index,2)));
a3=Temp(2*NLocal(NBelem(index,3)));
L1=LSetLocal(NBelem(index,1));
L2=LSetLocal(NBelem(index,2));
L3=LSetLocal(NBelem(index,3));
LS=abs(N1*L1+L2*N2+L3*N3);
p1=N1*(LS-abs(L1));
p2=N2*(LS-abs(L2));
p3=N3*(LS-abs(L3));
T(2)=N1*T1+N2*T2+N3*T3+p1*a1+p2*a2+p3*a3
gradT=(T(2)-T(1))/(0.15*ELength);
for j=1:3
nData(NBelem(i,j),1)=nData(NBelem(i,j),1)+1.;
nData(NBelem(i,j),2)=nData(NBelem(i,j),2)+0.1*gradT;
end
end
end
for i=1:NBNodes
if nData(i,1)>0
F(i)=nData(i,2)/nData(i,1);
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get 'Stiffness' Matrix 'A'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [A]=getA(Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal)
A=zeros(NBNodes);
for i=1:NBElems
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
AfL=zeros(3);
AfLGLS=zeros(3);
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=2*abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[LSetLocal(NBelem(i,1));LSetLocal(NBelem(i,2));LSetLocal(NBelem(i,3))];
set=phi*nodalLset;
delset=delphi*nodalLset;
AfL=AfL+(phi'*sign(set))*(delset'*delphi)/3.;
AfLGLS=AfLGLS+(delphi'*delset)*(1./norm(delset))*(delset'*delphi)/3.;
end
sum=AfL+AfLGLS;
for k=1:3;
for j=1:3;
A(NBelem(i,j),NBelem(i,k))=A(NBelem(i,j),NBelem(i,k))+sum(j,k);
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get terms for LS equation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [M,MGLS,f1,f2,f3]=getTerms(Node,NLocal,NBelem,NBNodes,NBElems,LSetLocal,visc,charLen,F)
M=zeros(NBNodes);
MGLS=zeros(NBNodes);
f1=zeros(NBNodes,1);
f2=zeros(NBNodes,1);
f3=zeros(NBNodes,1);
for i=1:NBElems
ML=zeros(3);
MGLSL=zeros(3);
f1L=zeros(3,1);
f2L=zeros(3,1);
f3L=zeros(3,1);
gx(1)=2./3.;
gx(2)=1./6.;
gx(3)=1./6.;
hx(1)=1./6.;
hx(2)=1./6.;
hx(3)=2./3.;
x1=Node(NLocal(NBelem(i,1)),1);
y1=Node(NLocal(NBelem(i,1)),2);
x2=Node(NLocal(NBelem(i,2)),1);
y2=Node(NLocal(NBelem(i,2)),2);
x3=Node(NLocal(NBelem(i,3)),1);
y3=Node(NLocal(NBelem(i,3)),2);
for j=1:3
g=gx(j);
h=hx(j);
phi(1)=1.-g-h;
phi(2)=g;
phi(3)=h;
phig(1)=-1.;
phig(2)=1.;
phig(3)=0.;
phih(1)=-1.;
phih(2)=0.;
phih(3)=1.;
djac=abs(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2));
for k=1:3
phix(k)=(1./djac)*((-y1+y3)*phig(k)+(y1-y2)*phih(k));
phiy(k)=(1./djac)*((x1-x3)*phig(k)+(-x1+x2)*phih(k));
end
delphi=[phix;phiy];
nodalLset=[LSetLocal(NBelem(i,1));LSetLocal(NBelem(i,2));LSetLocal(NBelem(i,3))];
nodalF=[F(NBelem(i,1));F(NBelem(i,2));F(NBelem(i,3))];
delset=delphi*nodalLset;
Floc=phi*nodalF;
ML=ML+(phi'*phi)/3.;
MGLSL=MGLSL+((delphi'*(delset/norm(delset)))*Floc*(charLen/abs(Floc)))*phi/3.;
f1L=f1L+phi'*Floc*norm(delset)/3.;
f2L=f2L+(delphi'*(delset/norm(delset))*Floc)*(charLen/abs(Floc))*Floc*norm(delset)/3.;
vs=charLen*((abs(visc+Floc*norm(delset)))/(norm(Floc*delset)+charLen));
f3L=f3L+vs*delphi'*delset/3.;
end
for k=1:3;
for j=1:3;
M(NBelem(i,j),NBelem(i,k))=M(NBelem(i,j),NBelem(i,k))+ML(j,k);
MGLS(NBelem(i,j),NBelem(i,k))=MGLS(NBelem(i,j),NBelem(i,k))+MGLSL(j,k);
end
f1(NBelem(i,k))=f1(NBelem(i,k))+f1L(k);
f2(NBelem(i,k))=f2(NBelem(i,k))+f2L(k);
f3(NBelem(i,k))=f3(NBelem(i,k))+f3L(k);
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Use Fast March Method to Reinitialize LS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function LSetLocal=fastMarch(LSetLocal,NBelem,Node,NLocal,NBNodes,NBElems)
newlSet=LSetLocal;
% Reinitialize LS
nstat=zeros(NBNodes,1);
for i=1:NBElems
for j=1:3
L(j)=sign(LSetLocal(NBelem(i,j)));
end
if L(1) ~= L(2) || L(1) ~= L(3)
for j=1:3
nstat(NBelem(i,j))=1;
end
end
end
maincheck=0;
while(maincheck==0)
lmin=1000.;
avlmin=1000.;
eindex=0;
nindex=0;
maincheck=1;
for i=1:NBElems
if nstat(NBelem(i,1))+nstat(NBelem(i,2))+nstat(NBelem(i,3))==2
maincheck=0;
check=0;
ltot=0.;
for j=1:3
if nstat(NBelem(i,j))==0
if abs(LSetLocal(NBelem(i,j)))<=lmin
check=1;
tempindex=j;
end
end
ltot=ltot+abs(LSetLocal(NBelem(i,j)));
end
if check==1 && ltot/3.<=avlmin
eindex=i;
nindex=tempindex;
lmin=LSetLocal(NBelem(eindex,nindex));
avlmin=ltot/3.;
end
end
end
if maincheck==0
% Find New LS for point
xp=Node(NLocal(NBelem(eindex,nindex)),1);
yp=Node(NLocal(NBelem(eindex,nindex)),2);
count=0;
for i=1:3
if i~=nindex
count=count+1;
x(count)=Node(NLocal(NBelem(eindex,i)),1);
y(count)=Node(NLocal(NBelem(eindex,i)),2);
lloc(count)=newlSet(NBelem(eindex,i));
end
end
delxa=x(1)-xp;
delya=y(1)-yp;
delxb=x(2)-xp;
delyb=y(2)-yp;
N=[delxa delya; delxb delyb];
M=N^-1;
A=(M(1)*M(1)+M(2)*M(2));
B=(M(3)*M(3)+M(4)*M(4));
C=2.*(M(1)*M(3)+M(2)*M(4));
a=A+B+C;
b=-2.*lloc(1)*A-2.*lloc(2)*B-C*(lloc(1)+lloc(2));
c=lloc(1)*lloc(1)*A+lloc(2)*lloc(2)*B+lloc(1)*lloc(2)*C-1.;
templ1=(-b+sqrt(b*b-4.*a*c))/(2.*a);
templ2=(-b-sqrt(b*b-4.*a*c))/(2.*a);
if abs(templ1)>abs(templ2)
newlSet(NBelem(eindex,nindex))=templ1;
else
newlSet(NBelem(eindex,nindex))=templ2;
end
nstat(NBelem(eindex,nindex))=1;
end
end
LSetLocal=newlSet;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Solve Implicit Porblem to Get Temperature
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function Temp=getTemp(Node,Element,numNodes,numElem,LSet,Bound,Temp,Penalty,rho,dtImp,LSetOld)
K=zeros(numNodes*2,numNodes*2);
M=zeros(numNodes*2,numNodes*2);
MStar=zeros(numNodes*2,numNodes*2);
pforce=zeros(numNodes*2,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(8);
Me=zeros(8);
MeStar=zeros(8);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
theta(icrd)=LSet(Element(e,icrd));
thetaO(icrd)=LSetOld(Element(e,icrd));
end
check=0;
for i=1:3
for j=i+1:4
if sign(theta(1))~=sign(theta(j))
check=1;
end
end
end
if check==1
% possible enriched element
npart=10;
enr=npart*npart;
for sdx=1:npart
for sdy=1:npart
midx=-1.-1./npart+(2./npart)*sdx;
midy=-1.-1./npart+(2./npart)*sdy;
subindex=npart*(sdy-1)+sdx;
gpos=1./(sqrt(3.)*npart);
gx(subindex,1)=midx-gpos;
gx(subindex,2)=midx+gpos;
gx(subindex,3)=midx+gpos;
gx(subindex,4)=midx-gpos;
hx(subindex,1)=midy-gpos;
hx(subindex,2)=midy-gpos;
hx(subindex,3)=midy+gpos;
hx(subindex,4)=midy+gpos;
end
end
% check if int points are on different sides of front
check=0;
for i=1:enr
for j=1:4
g=gx(i,j);
h=hx(i,j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
phiO=phi;
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
if i==1 && j==1
sgn=sign(iLS);
else
if sign(iLS)~=sgn
check=1;
end
end
end
end
if check==0
% regular element - fix extra dofs
enr=1;
gpos=1/sqrt(3.);
gx(1,1)=-gpos;
gx(1,2)=gpos;
gx(1,3)=gpos;
gx(1,4)=-gpos;
hx(1,1)=-gpos;
hx(1,2)=-gpos;
hx(1,3)=gpos;
hx(1,4)=gpos;
end
else
% regular element - fix extra dofs
enr=1;
gpos=1/sqrt(3.);
gx(1,1)=-gpos;
gx(1,2)=gpos;
gx(1,3)=gpos;
gx(1,4)=-gpos;
hx(1,1)=-gpos;
hx(1,2)=-gpos;
hx(1,3)=gpos;
hx(1,4)=gpos;
end
for i=1:enr
for j=1:4
g=gx(i,j);
h=hx(i,j);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
phiO=phi;
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
iLSO=thetaO(1)*phi(1)+thetaO(2)*phi(3)+thetaO(3)*phi(5)+thetaO(4)*phi(7);
if iLS<0.
cond=0.;
spec=0.01;
else
cond=1.;
spec=1.;
end
if iLSO<0.
specO=0.01;
else
specO=1.;
end
for iter=1:4
phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(iter)));
phiO(2*iter)=phiO(2*iter-1)*(abs(iLSO)-abs(thetaO(iter)));
end
phig(1)=0.25*-(1.-h);
phig(3)=0.25*(1.-h);
phig(5)=0.25*(1.+h);
phig(7)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(3)=0.25*-(1.+g);
phih(5)=0.25*(1.+g);
phih(7)=0.25*(1.-g);
diLSg=sign(iLS)*(phig(1)*theta(1)+phig(3)*theta(2)+phig(5)*theta(3)+phig(7)*theta(4));
diLSh=sign(iLS)*(phih(1)*theta(1)+phih(3)*theta(2)+phih(5)*theta(3)+phih(7)*theta(4));
for iter=1:4
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSg;
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSh;
end
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=djac;
Ke=Ke+(we*cond*(phix'*phix+phiy'*phiy))/double(enr);
Me=Me+((we*rho*spec*phi'*phi)/dtImp)/double(enr);
MeStar=MeStar+((we*rho*specO*phi'*phiO)/dtImp)/double(enr);
end
end
% Add penalty term and get temp gradient on interface
if enr>1;
count=0;
if sign(theta(1))~=sign(theta(2))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(2)));
xi(count)=f*(crdnx(2)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(2)-crdny(1))+crdny(1);
gi(count)=(2.*xi(count)-(crdnx(1)+crdnx(2)))/(-crdnx(1)+crdnx(2));
hi(count)=-1.;
end
if sign(theta(2))~=sign(theta(3))
count=count+1;
f=abs(theta(2))/(abs(theta(2))+abs(theta(3)));
xi(count)=f*(crdnx(3)-crdnx(2))+crdnx(2);
yi(count)=f*(crdny(3)-crdny(2))+crdny(2);
gi(count)=1.;
hi(count)=(2.*yi(count)-(crdny(2)+crdny(3)))/(-crdny(2)+crdny(3));
end
if sign(theta(3))~=sign(theta(4))
count=count+1;
f=abs(theta(3))/(abs(theta(3))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(3))+crdnx(3);
yi(count)=f*(crdny(4)-crdny(3))+crdny(3);
gi(count)=(2.*xi(count)-(crdnx(4)+crdnx(3)))/(-crdnx(4)+crdnx(3));
hi(count)=1.;
end
if sign(theta(1))~=sign(theta(4))
count=count+1;
f=abs(theta(1))/(abs(theta(1))+abs(theta(4)));
xi(count)=f*(crdnx(4)-crdnx(1))+crdnx(1);
yi(count)=f*(crdny(4)-crdny(1))+crdny(1);
gi(count)=-1.;
hi(count)=(2.*yi(count)-(crdny(1)+crdny(4)))/(-crdny(4)+crdny(1));
end
c=zeros(2,1);
c=(c+1.);
for i=1:2;
G(i,1)=0.25*(1.-gi(i))*(1.-hi(i));
G(i,3)=0.25*(1.+gi(i))*(1.-hi(i));
G(i,5)=0.25*(1.+gi(i))*(1.+hi(i));
G(i,7)=0.25*(1.-gi(i))*(1.+hi(i));
G(i,2)=-G(i,1)*abs(theta(1));
G(i,4)=-G(i,3)*abs(theta(2));
G(i,6)=-G(i,5)*abs(theta(3));
G(i,8)=-G(i,7)*abs(theta(4));
end
pen=Penalty*(G'*G);
pfL=Penalty*G'*c;
% pen=zeros(8);
% pfL=zeros(8,1);
Ke=Ke+pen;
else
pen=zeros(8);
pfL=zeros(8,1);
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(2*j-1,2*i-1);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
MStar(gnum(j),gnum(i))=MStar(gnum(j),gnum(i))+MeStar(2*j-1,2*i-1);
MStar(gnum(j)+1,gnum(i)+1)=MStar(gnum(j)+1,gnum(i)+1)+MeStar(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove inactive DOFs(Reduce Matrices)
RHS=MStar*Temp;
A=K+M;
Sub=A*Bound;
iindex=0;
for i=1:2*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i)+pforce(i);
jindex=0;
for j=1:2*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tempr=(Ared^-1)*RHSred';
iindex=0;
for i=1:2*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Temp(i)=Tempr(iindex);
end
end
Temp
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Generates the initial level set
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [LSet]=initialLSet(Node,numNodes)
%centx=4.;
%centy=4.;
%rad=2.1;
%for i=1:numNodes;
% dist=sqrt((Node(i,1)-centx)*(Node(i,1)-centx)+(Node(i,2)-centy)*(Node(i,2)-centy));
% LSet(i)=dist-rad;
%end
for i=1:numNodes;
dist=Node(i,1)-4.6;
LSet(i)=dist;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Plot the level set
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function []=plotLSet(NumX,NumY,delX,delY,LSet)
[X Y]=meshgrid(0:delX:delX*NumX,0:delY:delY*NumY);
Z=zeros(NumX+1,NumY+1);
for i=1:(NumX+1)*(NumY+1)
Z(i)=LSet(i);
end
surf(X,Y,Z)
end

62
Unpublished/XFEM/2D_Solver/uelmatinp.inp Normal file
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@ -0,0 +1,62 @@
*HEADING
Test for passing abaqus material to UELMAT: transient heat transfer
*RESTART,WRITE,NUMBER INTERVAL=10
*PREPRINT,MODEL=YES
*PART,NAME=part1
*NODE,NSET=NALL
1,0,0,0
2,1,0,0
3,0,1,0
4,1,1,0
5,0,2,0
6,1,2,0
*NSET,NSET=Left
1,3,5
*NSET,NSET=Right
2,4,6
*USER ELEMENT, TYPE=U1, NODES=4, COORDINATES=2,
INTEGRATION=4,TENSOR=TWOD
11,
*ELEMENT,TYPE=U1,ELSET=SOLID
1, 1,2,4,3
2, 3,4,6,5
*END PART
*ASSEMBLY,NAME=A1
*INSTANCE,NAME=I1,PART=PART1
*END INSTANCE
*Nset, nset=Set-6, instance=I1
1,3,5
*Nset, nset=Set-7, instance=I1
2,4,6
*END ASSEMBLY
*UEL PROPERTY, ELSET=I1.SOLID, MATERIAL=MAT_THERM
**************************************
***************************************
*MATERIAL,NAME=MAT_THERM
*CONDUCTIVITY
1.0,
*SPECIFIC HEAT
1.,
*DENSITY
1.,
*Initial Conditions, type=TEMPERATURE
Set-6, 1.,0.
*Initial Conditions, type=TEMPERATURE
Set-7, 0.,0.
*STEP
*HEAT TRANSFER, DELTMX=1.
0.1,1.0,,0.1
**
*BOUNDARY
Set-6,11,11,1.
*OUTPUT,FIELD,freq=1
*ELEMENT OUTPUT,ELSET=I1.SOLID
HFL,
*NODE OUTPUT,NSET=I1.NALL
NT,
*OUTPUT,HISTORY
*ELEMENT OUTPUT,ELSET=I1.SOLID
HFL,
*NODE OUTPUT,NSET=I1.NALL
NT11,
*END STEP

546
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@ -0,0 +1,546 @@
C
C User element accessing Abaqus materials
C Heat Transfer -- conduction
C
c*****************************************************************
subroutine uelmat(rhs, amatrx, svars, energy, ndofel, nrhs,
1 nsvars, props, nprops, coords, mcrd, nnode, u, du, v, a, jtype,
2 time, dtime, kstep, kinc, jelem, params, ndload, jdltyp, adlmag,
3 predef, npredf, lflags, mlvarx, ddlmag, mdload, pnewdt, jprops,
4 njpro, period, materiallib)
c
include 'aba_param.inc'
C
dimension rhs(mlvarx,*), amatrx(ndofel, ndofel), props(*),
1 svars(*), energy(*), coords(mcrd, nnode), u(ndofel),
2 du(mlvarx,*), v(ndofel), a(ndofel), time(2), params(*),
3 jdltyp(mdload,*), adlmag(mdload,*), ddlmag(mdload,*),
4 predef(2, npredf, nnode), lflags(*), jprops(*)
c
c local arrays
c
parameter (zero=0.d0, one=1.d0)
parameter (ndim=2, ndof=1, ninpt=4, nnodemax=4)
c
c ndim ... number of spatial dimensions
c ndof ... number of degrees of freedom per node
c ninpt ... number of integration points
c
dimension stiff(ndof*nnodemax,ndof*nnodemax),
1 force(ndof*nnodemax), shape(nnodemax), dshape(ndim,nnodemax),
2 xjaci(ndim,ndim), bmat(nnodemax*ndim), wght(ninpt)
dimension coords_ip(3),dfgrd0(3,3),dfgrd1(3,3),
1 drot(3,3)
dimension coords_new(mcrd,nnodemax)
c
dimension predef_loc(npredf),dpredef_loc(npredf),xx1(3,3),
1 xx1Old(3,3)
dimension xjaci_new(ndim,ndim),bmat_new(nnodemax*ndim)
dimension dtemdx(ndim),rhoUdotdg(3),flux(ndim),dfdt(ndim),
1 dfdg(ndim,ndim)
c
data wght /one, one, one, one/
c
c********************************************************************
c
c U1 = first-order, plane strain, full integration
c
c********************************************************************
if (lflags(3).eq.4) goto 999
c
c Preliminaries
c
pnewdtLocal = pnewdt
if(jtype .ne. 1) then
write(7,*)'Incorrect element type'
call xit
endif
c
c initialize rhs and lhs
c
do k1=1, ndof*nnode
rhs(k1, 1)= zero
do k2=1, ndof*nnode
amatrx(k1, k2)= zero
end do
end do
c
do k1=1,nnode
do k2=1,mcrd
kk = (k1-1)*mcrd + k2
coords_new(k2,k1) = coords(k2,k1) + u(kk)
end do
end do
c
c loop over integration points
c
do kintk = 1, ninpt
c
c initialization
c
rho = zero
rhoUdot = zero
rhoUdotdt = zero
rhoUdotdg = zero
do i=1, 3
rhoUdotdg(i) = zero
end do
do i=1, ndim
flux(i) = zero
dfdt(i) = zero
end do
do i=1, ndim
do j=1, ndim
dfdg(i,j) = zero
end do
end do
c
c evaluate shape functions and derivatives
c
call shapefcn(kintk,ninpt,nnode,ndim,shape,dshape)
c
c compute coordinates at the integration point
c
do k1=1, 3
coords_ip(k1) = zero
end do
do k1=1,nnode
do k2=1,mcrd
coords_ip(k2)=coords_ip(k2)+shape(k1)*coords(k2,k1)
end do
end do
c
if(npredf.gt.0) then
call tempfv(kintk,ninpt,nnode,ndim,shape,predef,
* npredf,predef_loc,dpredef_loc)
end if
c
c form B-matrix
c
djac = one
djac_new = one
call jacobian(jelem,mcrd,ndim,nnode,coords,dshape,
1 djac,xjaci,pnewdt,coords_new,xjaci_new,djac_new)
c
if (pnewdt .lt. pnewdtLocal) pnewdtLocal = pnewdt
c
call bmatrix(xjaci,dshape,nnode,ndim,bmat,xjaci_new,
1 bmat_new)
c
c compute temp. and temp. gradient
c
temp = zero
dtemp = zero
call settemp(ndofel,ndof,ndim,nnode,mlvarx,bmat,du,
* dstran,u,xx1,xx1Old,temp,dtemp,dtemdx,shape)
c
c get Abaqus material
c
rpl = zero
drpldt = zero
celent = one
call material_lib_ht(materiallib,rhoUdot,rhoUdotdt,rhoUdotdg,
* flux,dfdt,dfdg,rpl,drpldt,kintk,djac,predef_loc,
* dpredef_loc,npredf,temp,dtemp,dtemdx,celent,coords_ip)
c
c
c form stiffness matrix and internal force vector
c
call rhsjacobian(nnode,ndim,ndof,
1 wght(kintk),djac,rhoUdot,rhoUdotdt,rhoUdotdg,flux,
2 dfdt,dfdg,shape,bmat,stiff,force,dtime,lflags)
c
c assemble rhs and lhs
c
do k1=1, ndof*nnode
rhs(k1, 1) = rhs(k1, 1) - force(k1)
do k2=1, ndof*nnode
amatrx(k1, k2) = amatrx(k1, k2) + stiff(k1,k2)
end do
end do
end do ! end loop on material integration points
pnewdt = pnewdtLocal
c
999 continue
c
return
end
c*****************************************************************
c
c Compute shape fuctions
c
subroutine shapefcn(kintk,ninpt,nnode,ndim,dN,dNdz)
c
include 'aba_param.inc'
c
parameter (dmone=-1.0d0,one=1.0d0,four=4.0d0,eight=8.0d0,
1 gaussCoord=0.577350269d0)
parameter (maxElemNode=8,maxDof=3,i2d4node=24,i3d8node=38)
dimension dN(*),dNdz(ndim,*),coord24(2,4),coord38(3,8)
c
data coord24 /dmone, dmone,
2 one, dmone,
3 one, one,
4 dmone, one/
c
data coord38 /dmone, dmone, dmone,
2 one, dmone, dmone,
3 one, one, dmone,
4 dmone, one, dmone,
5 dmone, dmone, one,
6 one, dmone, one,
7 one, one, one,
8 dmone, one, one/
C
iCode = 0
if (ninpt.eq.4.and.nnode.eq.4.and.ndim.eq.2) then
iCode = 24
else if (ninpt.eq.8.and.nnode.eq.8.and.ndim.eq.3) then
iCode = 38
else
write (6,*) '***ERROR: The shape fuctions cannot be found'
end if
C
C 3D 8-nodes
C
if (iCode.eq.i3d8node) then
c
c determine (g,h,r)
c
g = coord38(1,kintk)*gaussCoord
h = coord38(2,kintk)*gaussCoord
r = coord38(3,kintk)*gaussCoord
c
c shape functions
dN(1) = (one - g)*(one - h)*(one - r)/eight
dN(2) = (one + g)*(one - h)*(one - r)/eight
dN(3) = (one + g)*(one + h)*(one - r)/eight
dN(4) = (one - g)*(one + h)*(one - r)/eight
dN(5) = (one - g)*(one - h)*(one + r)/eight
dN(6) = (one + g)*(one - h)*(one + r)/eight
dN(7) = (one + g)*(one + h)*(one + r)/eight
dN(8) = (one - g)*(one + h)*(one + r)/eight
c
c derivative d(Ni)/d(g)
dNdz(1,1) = -(one - h)*(one - r)/eight
dNdz(1,2) = (one - h)*(one - r)/eight
dNdz(1,3) = (one + h)*(one - r)/eight
dNdz(1,4) = -(one + h)*(one - r)/eight
dNdz(1,5) = -(one - h)*(one + r)/eight
dNdz(1,6) = (one - h)*(one + r)/eight
dNdz(1,7) = (one + h)*(one + r)/eight
dNdz(1,8) = -(one + h)*(one + r)/eight
c
c derivative d(Ni)/d(h)
dNdz(2,1) = -(one - g)*(one - r)/eight
dNdz(2,2) = -(one + g)*(one - r)/eight
dNdz(2,3) = (one + g)*(one - r)/eight
dNdz(2,4) = (one - g)*(one - r)/eight
dNdz(2,5) = -(one - g)*(one + r)/eight
dNdz(2,6) = -(one + g)*(one + r)/eight
dNdz(2,7) = (one + g)*(one + r)/eight
dNdz(2,8) = (one - g)*(one + r)/eight
c
c derivative d(Ni)/d(r)
dNdz(3,1) = -(one - g)*(one - h)/eight
dNdz(3,2) = -(one + g)*(one - h)/eight
dNdz(3,3) = -(one + g)*(one + h)/eight
dNdz(3,4) = -(one - g)*(one + h)/eight
dNdz(3,5) = (one - g)*(one - h)/eight
dNdz(3,6) = (one + g)*(one - h)/eight
dNdz(3,7) = (one + g)*(one + h)/eight
dNdz(3,8) = (one - g)*(one + h)/eight
C
C 2D 4-nodes
C
else if (iCode.eq.i2d4node) then
c
c determine (g,h)
c
g = coord24(1,kintk)*gaussCoord
h = coord24(2,kintk)*gaussCoord
c
c shape functions
dN(1) = (one - g)*(one - h)/four;
dN(2) = (one + g)*(one - h)/four;
dN(3) = (one + g)*(one + h)/four;
dN(4) = (one - g)*(one + h)/four;
c
c derivative d(Ni)/d(g)
dNdz(1,1) = -(one - h)/four;
dNdz(1,2) = (one - h)/four;
dNdz(1,3) = (one + h)/four;
dNdz(1,4) = -(one + h)/four;
c
c derivative d(Ni)/d(h)
dNdz(2,1) = -(one - g)/four;
dNdz(2,2) = -(one + g)/four;
dNdz(2,3) = (one + g)/four;
dNdz(2,4) = (one - g)/four;
end if
c
return
end
c*****************************************************************
c Get local predefined fileds
c
subroutine tempfv(kintk,ninpt,nnode,ndim,shape,predef,
* npredf,predef_loc,dpredef_loc)
c
include 'aba_param.inc'
c
dimension shape(nnode),predef(2,npredf,nnode)
dimension predef_loc(npredf),dpredef_loc(npredf)
parameter (zero=0.d0)
c
do k1=1,npredf
predef_loc(k1) = zero
dpredef_loc(k1) = zero
do k2=1,nnode
predef_loc(k1) = predef_loc(k1)+
& (predef(1,k1,k2)-predef(2,k1,k2))*shape(k2)
dpredef_loc(k1) = dpredef_loc(k1)+predef(2,k1,k2)*shape(k2)
end do
end do
c
return
end
c*****************************************************************
c Compute jacobian matrix
c
subroutine jacobian(jelem,mcrd,ndim,nnode,
1 coords,dshape,djac,xjaci,pnewdt,coords_new,xjaci_new,
2 djac_new)
c
c Notation: ndim ....... element dimension
c nnode ..... number of nodes
c coords ..... coordinates of nodes
c dshape ..... derivs of shape fcn
c djac ....... determinant of Jacobian
c xjaci ...... inverse of Jacobian matrix
c
c
include 'aba_param.inc'
parameter(zero=0.d0, fourth=0.25d0, maxDof=3)
dimension xjac(maxDof,maxDof), xjaci(ndim,*), coords(mcrd,*)
dimension dshape(ndim,*),coords_new(mcrd,*)
dimension xjac_new(maxDof,maxDof), xjaci_new(ndim,*)
c
do i = 1, ndim
do j = 1, ndim
xjac(i,j) = zero
xjaci(i,j) = zero
xjac_new(i,j) = zero
xjaci_new(i,j) = zero
end do
end do
c
do inod= 1, nnode
do idim = 1, ndim
do jdim = 1, ndim
xjac(idim,jdim) = xjac(idim,jdim) +
1 dshape(jdim,inod)*coords(idim,inod)
end do
end do
end do
C
C ndim == 3
C
if (ndim.eq.3) then
djac = xjac(1,1)*xjac(2,2)*xjac(3,3) +
& xjac(2,1)*xjac(3,2)*xjac(1,3) +
& xjac(3,1)*xjac(2,3)*xjac(1,2) -
& xjac(3,1)*xjac(2,2)*xjac(1,3) -
& xjac(2,1)*xjac(1,2)*xjac(3,3) -
& xjac(1,1)*xjac(2,3)*xjac(3,2)
if (djac .gt. zero) then
! jacobian is positive - o.k.
xjaci(1,1) = (xjac(2,2)*xjac(3,3)-xjac(2,3)*xjac(3,2))/djac
xjaci(1,2) = (xjac(1,3)*xjac(3,2)-xjac(1,2)*xjac(3,3))/djac
xjaci(1,3) = (xjac(1,2)*xjac(2,3)-xjac(1,3)*xjac(2,2))/djac
!
xjaci(2,1) = (xjac(2,3)*xjac(3,1)-xjac(2,1)*xjac(3,3))/djac
xjaci(2,2) = (xjac(1,1)*xjac(3,3)-xjac(1,3)*xjac(3,1))/djac
xjaci(2,3) = (xjac(1,3)*xjac(2,1)-xjac(1,1)*xjac(2,3))/djac
!
xjaci(3,1) = (xjac(2,1)*xjac(3,2)-xjac(2,2)*xjac(3,1))/djac
xjaci(3,2) = (xjac(1,2)*xjac(3,1)-xjac(1,1)*xjac(3,2))/djac
xjaci(3,3) = (xjac(1,1)*xjac(2,2)-xjac(1,2)*xjac(2,1))/djac
else
! negative or zero jacobian
write(7,*)'WARNING: element',jelem,'has neg.
1 Jacobian'
pnewdt = fourth
endif
C
C ndim == 2
C
else if (ndim.eq.2) then
djac = xjac(1,1)*xjac(2,2) - xjac(1,2)*xjac(2,1)
djac_new = xjac_new(1,1)*xjac_new(2,2)
* - xjac_new(1,2)*xjac_new(2,1)
if (djac .gt. zero) then
! jacobian is positive - o.k.
xjaci(1,1) = xjac(2,2)/djac
xjaci(2,2) = xjac(1,1)/djac
xjaci(1,2) = -xjac(1,2)/djac
xjaci(2,1) = -xjac(2,1)/djac
else
! negative or zero jacobian
write(7,*)'WARNING: element',jelem,'has neg.
1 Jacobian'
pnewdt = fourth
endif
end if
return
end
c*****************************************************************
c
c Compute the B matrix
c
subroutine bmatrix(xjaci,dshape,nnode,ndim,bmat,
* xjaci_new,bmat_new)
c
c Notation:
c bmat(i) .....dN1/dx, dN1/dy, dN2/dx, dN2/dy..
c xjaci ...... inverse Jabobian matrix
c dshape ......derivative of shape functions
c
include 'aba_param.inc'
c
parameter (zero=0.d0)
dimension bmat(*), dshape(ndim,*)
dimension xjaci(ndim,*)
dimension xjaci_new(ndim,*),bmat_new(*)
do i = 1, nnode*ndim
bmat(i) = zero
bmat_new(i) = zero
end do
do inod = 1, nnode
do ider = 1, ndim
do idim = 1, ndim
irow = idim + (inod - 1)*ndim
bmat(irow) = bmat(irow) +
1 xjaci(idim,ider)*dshape(ider,inod)
end do
end do
end do
return
end
c*****************************************************************
c
c Set temperatures
c
subroutine settemp(ndofel,ndof,ndim,nnode,
1 mlvarx,bmat,du,dstran,u,xx1,xx1Old,temp,dtemp,dtemdx,dN)
c
c
c
include 'aba_param.inc'
parameter(zero=0.d0, one=1.d0)
dimension dstran(*), bmat(ndim,*),
1 du(mlvarx, *), xdu(3), xx1(3,*),
2 u(ndofel), utmp(3),
3 utmpOld(3),xx1Old(3,*),eps(3,3),dInvFold(3,3)
dimension dtemdx(*),dN(*)
C
c
c****************************************************************
c Compute temp, dtemp, and temp gradient at the material point
c****************************************************************
c
temp = zero
dtemp = zero
do iNode=1, nnode
temp = temp + dN(iNode)*u(iNode)
dtemp = dtemp + dN(iNode)*du(iNode,1)
end do
do iDof = 1, ndim
dtemdx(iDof) = zero
do iNode=1, nnode
dtemdx(iDof) = dtemdx(iDof) + bmat(idof,iNode)*u(iNode)
end do
end do
c
return
end
c*****************************************************************
c
c Compute element jacobian and nodal forces
c
subroutine rhsjacobian(nnode,ndim,ndof,
1 weight,djac,rhoUdot,rhoUdotdt,rhoUdotdg,flux,dfdt,
2 dfdg,dN,bmat,stiff,force,dtime,lflags)
c
c Stiffness matrix and internal force contributions at
c material integration point
c
include 'aba_param.inc'
parameter(zero=0.d0,maxDof=3)
dimension stiff(ndof*nnode,*)
dimension force(*)
dimension flux(*),dfdt(*),dfdg(ndim,*),rhoUdotdg(*)
dimension dN(*),bmat(ndim,*),lflags(*)
do i = 1, ndof*nnode
force(i) = zero
do j = 1, ndof*nnode
stiff(j,i) = zero
end do
end do
dvol=weight*djac
do nodj=1, nnode
if (lflags(1).eq.32.or.lflags(1).eq.33)
& force(nodj) = dN(nodj)*rhoUdot*dvol
ccc force(nodj) = dN(nodj)*(rhoUdot-rpl)*dvol
do jDof=1, ndim
force(nodj) = force(nodj)+bmat(jDof,nodj)*flux(jDof)*dvol
do nodi=1, nnode
ccc stiff(nodj,nodi) = stiff(nodj,nodi) +
ccc * bmat(jDof,nodj)*dN(nodi)*dfdt(jDof)*dvol
do iDof=1, ndim
stiff(nodj,nodi) = stiff(nodj,nodi) +
* bmat(jDof,nodj)*bmat(iDof,nodi)*dfdg(jDof,iDof)*dvol
end do
end do
end do
end do
do nodj=1, nnode
do nodi=1, nnode
do iDof=1, ndim
ccc stiff(nodj,nodi) = stiff(nodj,nodi) +
ccc * dN(nodj)*bmat(iDof,nodi)*rhoUdotdg(iDof)*dvol
end do
end do
end do
c
if (lflags(1).eq.32.or.lflags(1).eq.33) then
do nodj=1, nnode
do nodi=1, nnode
stiff(nodj,nodi) = stiff(nodj,nodi) +
* rhoUdotdt*dN(nodj)*dN(nodi)*dvol/dtime
end do
end do
end if
c
return
end

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subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props,
1 nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,kinc,
2 jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags,
3 mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),dndg(4),dndh(4),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension gx(4),hx(4),gwei(4),dN(4),phix(8),phiy(8)
c
parameter(zero=0.d0,one=1.d0)
C MATERIAL PROPERTY DEFINITION
thick = 1.
rho = 1.
spec = 1.
conduc = 1.
C INITIALIZATION (NRHS=1)
do k1=1,ndofel
rhs(k1,nrhs)=zero
do k2=1,ndofel
amatrx(k2,k1)=zero
enddo
enddo
if (lflags(1).eq.33) then
gpos=1./sqrt(3.)
gx(1)=-gpos
gx(2)=gpos
gx(3)=gpos
gx(4)=-gpos
hx(1)=-gpos
hx(2)=-gpos
hx(3)=gpos
hx(4)=gpos
do i=1,4
gwei(i)=1.
enddo
c assemble amatrx and rhs
do k=1,4
c loop through gauss pts
g=gx(k)
h=hx(k)
c shape functions
dN(1) = (one - g)*(one - h)/4.
dN(2) = (one + g)*(one - h)/4.
dN(3) = (one + g)*(one + h)/4.
dN(4) = (one - g)*(one + h)/4.
c derivative d(Ni)/d(g)
dNdg(1) = -(one - h)/4.
dNdg(2) = (one - h)/4.
dNdg(3) = (one + h)/4.
dNdg(4) = -(one + h)/4.
c derivative d(Ni)/d(h)
dNdh(1) = -(one - g)/4.
dNdh(2) = -(one + g)/4.
dNdh(3) = (one + g)/4.
dNdh(4) = (one - g)/4.
c derivative dx/dg,dx/dh,dy/dg,dy/dh
dxdg=zero
dxdh=zero
dydg=zero
dydh=zero
do i=1,4
dxdg=dxdg+coords(1,i)*dNdg(i)
dxdh=dxdh+coords(1,i)*dNdh(i)
dydg=dydg+coords(2,i)*dNdg(i)
dydh=dydh+coords(2,i)*dNdh(i)
enddo
c calculation of jacobian
ajacob=(dxdg*dydh-dxdh*dydg)
c derivative dn/dx,dn/dy
do i=1,4
phix(i)=(dNdg(i)*dydh-dNdh(i)*dydg)/ajacob
phiy(i)=(dNdh(i)*dxdg-dNdg(i)*dxdh)/ajacob
enddo
dtdx=zero
dtdy=zero
t =zero
told=zero
do i=1,4
dtdx=u(i)*phix(i)+dtdx
dtdy=u(i)*phiy(i)+dtdy
t=u(i)*dn(i)+t
told=(u(i)-du(i,nrhs))*dn(i)+told
end do
cond=1.
dcdt=zero
dtdt=(t-told)/dtime
we=gwei(k)*ajacob
do ki=1,4
c loop over nodes
rhs(ki,nrhs) = rhs(ki,nrhs) -
1 we*(dN(ki)*rho*spec*dtdt +
2 cond*(phix(ki)*dtdx + phiy(ki)*dtdy))
do kj=1,4
amatrx(ki,kj)= amatrx(ki,kj) +
1 we*(dn(ki)*dn(kj)*rho*spec/dtime +
1 cond*(phix(ki)*phix(kj) + phiy(ki)*phiy(kj)))
end do
end do
enddo
end if
return
end

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c 2D XFEM Corrosion Element
subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props,
1 nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,kinc,
2 jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags,
3 mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension phig(8),phih(8),phi(8),phix(8),phiy(8)
dimension crdnx(4),crdny(4),dndg(4),dndh(4)
dimension theta(4),rjac(2,2),rjaci(2,2)
dimension gx(100,4),hx(100,4),xi(2),yi(2),gi(2),hi(2)
dimension c(2),gp(2,8),gm2(8,8)
c
parameter(zero=0.d0,one=1.d0)
c material property definition
thick = 1.
rho = 1.
beta=0.
dpos=0.6
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=zero
do k2=1,ndofel
amatrx(k2,k1)=zero
enddo
enddo
if (lflags(1).eq.33) then
do icrd=1,4
crdnx(icrd)=coords(1,icrd)
crdny(icrd)=coords(2,icrd)
theta(icrd)=abs(crdnx(icrd)-dpos)*
1 sign(1.,crdnx(icrd)-dpos)
enddo
c if (sign(1.,theta(1))/=sign(1.,theta(2)))then
if (2==1)then
c possible enriched element
npart=10
rpart=float(npart)
ienr=npart*npart
do isdx=1,npart
do isdy=1,npart
rmidx=-1.-1./rpart+(2./rpart)*float(isdx)
rmidy=-1.-1./rpart+(2./rpart)*float(isdy)
isubindex=npart*(isdy-1)+isdx
gpos=1./(sqrt(3.)*rpart)
gx(isubindex,1)=rmidx-gpos
gx(isubindex,2)=rmidx+gpos
gx(isubindex,3)=rmidx+gpos
gx(isubindex,4)=rmidx-gpos
hx(isubindex,1)=rmidy-gpos
hx(isubindex,2)=rmidy-gpos
hx(isubindex,3)=rmidy+gpos
hx(isubindex,4)=rmidy+gpos
enddo
enddo
c check if int points are on different sides of front
icheck=0
do i=1,ienr
do j=1,4
g=gx(i,j)
h=hx(i,j)
phi(1)=0.25*(1.-g)*(1.-h)
phi(3)=0.25*(1.+g)*(1.-h)
phi(5)=0.25*(1.+g)*(1.+h)
phi(7)=0.25*(1.-g)*(1.+h)
rLS=theta(1)*phi(1)+theta(2)*phi(3)
1 +theta(3)*phi(5)+theta(4)*phi(7)
if (i==1 .and. j==1)then
sgn=sign(1.,rLS)
else
if (sign(1.,rLS)/=sgn)then
icheck=1
endif
endif
enddo
enddo
if (icheck==0)then
c regular element - fix extra dofs
ienr=1
gpos=1/sqrt(3.)
gx(1,1)=-gpos
gx(1,2)=gpos
gx(1,3)=gpos
gx(1,4)=-gpos
hx(1,1)=-gpos
hx(1,2)=-gpos
hx(1,3)=gpos
hx(1,4)=gpos
endif
else
c Normal Shp Funs
ienr=1
gpos=1/sqrt(3.)
gx(1,1)=-gpos
gx(1,2)=gpos
gx(1,3)=gpos
gx(1,4)=-gpos
hx(1,1)=-gpos
hx(1,2)=-gpos
hx(1,3)=gpos
hx(1,4)=gpos
endif
c assemble amatrx and rhs
do i=1,ienr
do j=1,4
g=gx(i,j)
h=hx(i,j)
phi(1)=0.25*(1.-g)*(1.-h)
phi(3)=0.25*(1.+g)*(1.-h)
phi(5)=0.25*(1.+g)*(1.+h)
phi(7)=0.25*(1.-g)*(1.+h)
rLS=theta(1)*phi(1)+theta(2)*phi(3)
1 +theta(3)*phi(5)+theta(4)*phi(7)
cond=1.
spec=1.
do iter=1,4
phi(2*iter)=phi(2*iter-1)*
1 (abs(rLS)-abs(theta(iter)))
enddo
phig(1)=0.25*-(1.-h)
phig(3)=0.25*(1.-h)
phig(5)=0.25*(1.+h)
phig(7)=0.25*-(1.+h)
phih(1)=0.25*-(1.-g)
phih(3)=0.25*-(1.+g)
phih(5)=0.25*(1.+g)
phih(7)=0.25*(1.-g)
diLSg=sign(1.,rLS)*(phig(1)*theta(1)+phig(3)*
1 theta(2)+phig(5)*theta(3)+phig(7)*theta(4))
diLSh=sign(1.,rLS)*(phih(1)*theta(1)+phih(3)*
1 theta(2)+phih(5)*theta(3)+phih(7)*theta(4))
do iter=1,4
phig(2*iter)=phig(2*iter-1)*(abs(rLS)-
1 abs(theta(iter)))+phi(2*iter-1)*diLSg
phih(2*iter)=phih(2*iter-1)*(abs(rLS)-
1 abs(theta(iter)))+phi(2*iter-1)*diLSh
enddo
rjac=zero
do iter=1,4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter)
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter)
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter)
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter)
enddo
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1)
rjaci(1,1)= rjac(2,2)/djac
rjaci(2,2)= rjac(1,1)/djac
rjaci(1,2)=-rjac(1,2)/djac
rjaci(2,1)=-rjac(2,1)/djac
do iter=1,8
phix(iter)=rjaci(1,1)*phig(iter)+
1 rjaci(1,2)*phih(iter)
phiy(iter)=rjaci(2,1)*phig(iter)+
1 rjaci(2,2)*phih(iter)
enddo
dtdx=zero
dtdy=zero
t =zero
told=zero
do iter=1,8
dtdx=u(iter)*phix(iter)+dtdx
dtdy=u(iter)*phiy(iter)+dtdy
t=u(iter)*phi(iter)+t
told=(u(iter)-du(iter,nrhs))*phi(iter)+told
end do
dtdt=(t-told)/dtime
we=djac
do ki=1,8
c loop over nodes
rhs(ki,nrhs) = rhs(ki,nrhs) -
1 (we/float(ienr))*(phi(ki)*rho*spec*dtdt+
2 cond*(phix(ki)*dtdx + phiy(ki)*dtdy))
do kj=1,8
amatrx(ki,kj)=amatrx(ki,kj)+(we/float(ienr))
1 *(phi(ki)*phi(kj)*rho*spec/dtime+cond
2 *(phix(ki)*phix(kj)+phiy(ki)*phiy(kj)))
end do
end do
enddo
enddo
c if interface is in the element a penalty term is needed
if(ienr>1)then
icount=0
if (sign(1.,theta(1))/=sign(1.,theta(2)))then
icount=icount+1
f=abs(theta(1))/(abs(theta(1))+abs(theta(2)))
xi(icount)=f*(crdnx(2)-crdnx(1))+crdnx(1)
yi(icount)=f*(crdny(2)-crdny(1))+crdny(1)
gi(icount)=(2.*xi(icount)-(crdnx(1)+crdnx(2)))
1 /(-crdnx(1)+crdnx(2))
hi(icount)=-1.
endif
if (sign(1.,theta(2))/=sign(1.,theta(3)))then
icount=icount+1
f=abs(theta(2))/(abs(theta(2))+abs(theta(3)))
xi(icount)=f*(crdnx(3)-crdnx(2))+crdnx(2)
yi(icount)=f*(crdny(3)-crdny(2))+crdny(2)
gi(icount)=1.
hi(icount)=(2.*yi(icount)-(crdny(2)+crdny(3)))
1 /(-crdny(2)+crdny(3))
endif
if (sign(1.,theta(3))/=sign(1.,theta(4)))then
icount=icount+1
f=abs(theta(3))/(abs(theta(3))+abs(theta(4)))
xi(icount)=f*(crdnx(4)-crdnx(3))+crdnx(3)
yi(icount)=f*(crdny(4)-crdny(3))+crdny(3)
gi(icount)=(2.*xi(icount)-(crdnx(4)+crdnx(3)))
1 /(-crdnx(4)+crdnx(3))
hi(icount)=1.
endif
if (sign(1.,theta(1))/=sign(1.,theta(4)))then
icount=icount+1
f=abs(theta(1))/(abs(theta(1))+abs(theta(4)))
xi(icount)=f*(crdnx(4)-crdnx(1))+crdnx(1)
yi(icount)=f*(crdny(4)-crdny(1))+crdny(1)
gi(icount)=-1.
hi(icount)=(2.*yi(icount)-(crdny(1)+crdny(4)))
1 /(-crdny(4)+crdny(1))
endif
c(1)=1.
c(2)=1.
do iter=1,2
Gp(iter,1)=0.25*(1.-gi(iter))*(1.-hi(iter))
Gp(iter,3)=0.25*(1.+gi(iter))*(1.-hi(iter))
Gp(iter,5)=0.25*(1.+gi(iter))*(1.+hi(iter))
Gp(iter,7)=0.25*(1.-gi(iter))*(1.+hi(iter))
Gp(iter,2)=-Gp(iter,1)*abs(theta(1))
Gp(iter,4)=-Gp(iter,3)*abs(theta(2))
Gp(iter,6)=-Gp(iter,5)*abs(theta(3))
Gp(iter,8)=-Gp(iter,7)*abs(theta(4))
enddo
do i=1,8
rhs(i,nrhs)=rhs(i,nrhs)+
1 beta*(Gp(1,i)*c(1)+Gp(2,i)*c(2))
enddo
c find GtG
gm2=0.
do i=1,8
do j=1,8
gm2(i,j)=gp(1,i)*gp(1,j)+gp(2,i)*gp(2,j)
enddo
enddo
c add penalty stiffness
do i=1,8
do j=1,8
amatrx(i,j)=amatrx(i,j)+beta*gm2(i,j)
enddo
enddo
endif
end if
return
end

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c 2D XFEM Corrosion Element
subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props,
1 nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,kinc,
2 jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags,
3 mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension phig(8),phih(8),phi(8),phix(8),phiy(8)
dimension crdnx(4),crdny(4),dndg(4),dndh(4)
dimension theta(4),rjac(2,2),rjaci(2,2)
dimension gx(100,4),hx(100,4),xi(2),yi(2),gi(2),hi(2)
dimension c(2),gp(2,8),gm2(8,8)
c
parameter(zero=0.d0,one=1.d0)
c material property definition
thick = 1.
rho = 1.
beta=0.
dpos=0.6
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=zero
do k2=1,ndofel
amatrx(k2,k1)=zero
enddo
enddo
if (lflags(1).eq.33) then
do icrd=1,4
crdnx(icrd)=coords(1,icrd)
crdny(icrd)=coords(2,icrd)
theta(icrd)=abs(crdnx(icrd)-dpos)*
1 sign(1.,crdnx(icrd)-dpos)
enddo
c if (sign(1.,theta(1))/=sign(1.,theta(2)))then
if (2==1)then
c possible enriched element
npart=10
rpart=float(npart)
ienr=npart*npart
do isdx=1,npart
do isdy=1,npart
rmidx=-1.-1./rpart+(2./rpart)*float(isdx)
rmidy=-1.-1./rpart+(2./rpart)*float(isdy)
isubindex=npart*(isdy-1)+isdx
gpos=1./(sqrt(3.)*rpart)
gx(isubindex,1)=rmidx-gpos
gx(isubindex,2)=rmidx+gpos
gx(isubindex,3)=rmidx+gpos
gx(isubindex,4)=rmidx-gpos
hx(isubindex,1)=rmidy-gpos
hx(isubindex,2)=rmidy-gpos
hx(isubindex,3)=rmidy+gpos
hx(isubindex,4)=rmidy+gpos
enddo
enddo
c check if int points are on different sides of front
icheck=0
do i=1,ienr
do j=1,4
g=gx(i,j)
h=hx(i,j)
phi(1)=0.25*(1.-g)*(1.-h)
phi(3)=0.25*(1.+g)*(1.-h)
phi(5)=0.25*(1.+g)*(1.+h)
phi(7)=0.25*(1.-g)*(1.+h)
rLS=theta(1)*phi(1)+theta(2)*phi(3)
1 +theta(3)*phi(5)+theta(4)*phi(7)
if (i==1 .and. j==1)then
sgn=sign(1.,rLS)
else
if (sign(1.,rLS)/=sgn)then
icheck=1
endif
endif
enddo
enddo
if (check==0)then
c regular element - fix extra dofs
ienr=1
gpos=1/sqrt(3.)
gx(1,1)=-gpos
gx(1,2)=gpos
gx(1,3)=gpos
gx(1,4)=-gpos
hx(1,1)=-gpos
hx(1,2)=-gpos
hx(1,3)=gpos
hx(1,4)=gpos
endif
else
c Normal Shp Funs
ienr=1
gpos=1/sqrt(3.)
gx(1,1)=-gpos
gx(1,2)=gpos
gx(1,3)=gpos
gx(1,4)=-gpos
hx(1,1)=-gpos
hx(1,2)=-gpos
hx(1,3)=gpos
hx(1,4)=gpos
endif
c assemble amatrx and rhs
do i=1,ienr
do j=1,4
g=gx(i,j)
h=hx(i,j)
phi(1)=0.25*(1.-g)*(1.-h)
phi(3)=0.25*(1.+g)*(1.-h)
phi(5)=0.25*(1.+g)*(1.+h)
phi(7)=0.25*(1.-g)*(1.+h)
rLS=theta(1)*phi(1)+theta(2)*phi(3)
1 +theta(3)*phi(5)+theta(4)*phi(7)
cond=1.
spec=1.
do iter=1,4
phi(2*iter)=phi(2*iter-1)*
1 (abs(rLS)-abs(theta(iter)))
enddo
phig(1)=0.25*-(1.-h)
phig(3)=0.25*(1.-h)
phig(5)=0.25*(1.+h)
phig(7)=0.25*-(1.+h)
phih(1)=0.25*-(1.-g)
phih(3)=0.25*-(1.+g)
phih(5)=0.25*(1.+g)
phih(7)=0.25*(1.-g)
diLSg=sign(1.,rLS)*(phig(1)*theta(1)+phig(3)*
1 theta(2)+phig(5)*theta(3)+phig(7)*theta(4))
diLSh=sign(1.,rLS)*(phih(1)*theta(1)+phih(3)*
1 theta(2)+phih(5)*theta(3)+phih(7)*theta(4))
do iter=1,4
phig(2*iter)=phig(2*iter-1)*(abs(rLS)-
1 abs(theta(iter)))+phi(2*iter-1)*diLSg
phih(2*iter)=phih(2*iter-1)*(abs(rLS)-
1 abs(theta(iter)))+phi(2*iter-1)*diLSh
enddo
rjac=zero
do iter=1,4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter)
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter)
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter)
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter)
enddo
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1)
rjaci(1,1)= rjac(2,2)/djac
rjaci(2,2)= rjac(1,1)/djac
rjaci(1,2)=-rjac(1,2)/djac
rjaci(2,1)=-rjac(2,1)/djac
do iter=1,8
phix(iter)=rjaci(1,1)*phig(iter)+
1 rjaci(1,2)*phih(iter)
phiy(iter)=rjaci(2,1)*phig(iter)+
1 rjaci(2,2)*phih(iter)
enddo
dtdx=zero
dtdy=zero
t =zero
told=zero
do iter=1,8
dtdx=u(iter)*phix(iter)+dtdx
dtdy=u(iter)*phiy(iter)+dtdy
t=u(iter)*phi(iter)+t
told=(u(iter)-du(iter,nrhs))*phi(iter)+told
end do
dtdt=(t-told)/dtime
we=djac
do ki=1,8
c loop over nodes
rhs(ki,nrhs) = rhs(ki,nrhs) -
1 (we/float(ienr))*(phi(ki)*rho*spec*dtdt+
2 cond*(phix(ki)*dtdx + phiy(ki)*dtdy))
do kj=1,8
amatrx(ki,kj)=amatrx(ki,kj)+(we/float(ienr))
1 *(phi(ki)*phi(kj)*rho*spec/dtime+cond
2 *(phix(ki)*phix(kj)+phiy(ki)*phiy(kj)))
end do
end do
enddo
enddo
c if interface is in the element a penalty term is needed
if(ienr>1)then
icount=0
if (sign(1.,theta(1))/=sign(1.,theta(2)))then
icount=icount+1
f=abs(theta(1))/(abs(theta(1))+abs(theta(2)))
xi(icount)=f*(crdnx(2)-crdnx(1))+crdnx(1)
yi(icount)=f*(crdny(2)-crdny(1))+crdny(1)
gi(icount)=(2.*xi(icount)-(crdnx(1)+crdnx(2)))
1 /(-crdnx(1)+crdnx(2))
hi(icount)=-1.
endif
if (sign(1.,theta(2))/=sign(1.,theta(3)))then
icount=icount+1
f=abs(theta(2))/(abs(theta(2))+abs(theta(3)))
xi(icount)=f*(crdnx(3)-crdnx(2))+crdnx(2)
yi(icount)=f*(crdny(3)-crdny(2))+crdny(2)
gi(icount)=1.
hi(icount)=(2.*yi(icount)-(crdny(2)+crdny(3)))
1 /(-crdny(2)+crdny(3))
endif
if (sign(1.,theta(3))/=sign(1.,theta(4)))then
icount=icount+1
f=abs(theta(3))/(abs(theta(3))+abs(theta(4)))
xi(icount)=f*(crdnx(4)-crdnx(3))+crdnx(3)
yi(icount)=f*(crdny(4)-crdny(3))+crdny(3)
gi(icount)=(2.*xi(icount)-(crdnx(4)+crdnx(3)))
1 /(-crdnx(4)+crdnx(3))
hi(icount)=1.
endif
if (sign(1.,theta(1))/=sign(1.,theta(4)))then
icount=icount+1
f=abs(theta(1))/(abs(theta(1))+abs(theta(4)))
xi(icount)=f*(crdnx(4)-crdnx(1))+crdnx(1)
yi(icount)=f*(crdny(4)-crdny(1))+crdny(1)
gi(icount)=-1.
hi(icount)=(2.*yi(icount)-(crdny(1)+crdny(4)))
1 /(-crdny(4)+crdny(1))
endif
c(1)=1.
c(2)=1.
do iter=1,2
Gp(iter,1)=0.25*(1.-gi(iter))*(1.-hi(iter))
Gp(iter,3)=0.25*(1.+gi(iter))*(1.-hi(iter))
Gp(iter,5)=0.25*(1.+gi(iter))*(1.+hi(iter))
Gp(iter,7)=0.25*(1.-gi(iter))*(1.+hi(iter))
Gp(iter,2)=-Gp(iter,1)*abs(theta(1))
Gp(iter,4)=-Gp(iter,3)*abs(theta(2))
Gp(iter,6)=-Gp(iter,5)*abs(theta(3))
Gp(iter,8)=-Gp(iter,7)*abs(theta(4))
enddo
do i=1,8
rhs(i,nrhs)=rhs(i,nrhs)+
1 beta*(Gp(1,i)*c(1)+Gp(2,i)*c(2))
enddo
c find GtG
gm2=0.
do i=1,8
do j=1,8
gm2(i,j)=gp(1,i)*gp(1,j)+gp(2,i)*gp(2,j)
enddo
enddo
c add penalty stiffness
do i=1,8
do j=1,8
amatrx(i,j)=amatrx(i,j)+beta*gm2(i,j)
enddo
enddo
endif
end if
return
end

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c 2D XFEM Corrosion Element
subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props,
1 nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,kinc,
2 jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags,
3 mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension phig(8),phih(8),phi(8),phix(8),phiy(8)
dimension crdnx(4),crdny(4),dndg(4),dndh(4)
dimension theta(4),rjac(2,2),rjaci(2,2)
dimension gx(100,4),hx(100,4),xi(2),yi(2),gi(2),hi(2)
dimension c(2),gp(2,8),gm2(8,8)
c
parameter(zero=0.d0,one=1.d0)
c material property definition
thick = 1.
rho = 1.
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=zero
do k2=1,ndofel
amatrx(k2,k1)=zero
enddo
enddo
if (lflags(1).eq.33) then
do icrd=1,4
crdnx(icrd)=coords(1,icrd)
crdny(icrd)=coords(2,icrd)
enddo
c Normal Shp Funs
ienr=1
gpos=1/sqrt(3.)
gx(1,1)=-gpos
gx(1,2)=gpos
gx(1,3)=gpos
gx(1,4)=-gpos
hx(1,1)=-gpos
hx(1,2)=-gpos
hx(1,3)=gpos
hx(1,4)=gpos
c assemble amatrx and rhs
do i=1,ienr
do j=1,4
g=gx(i,j)
h=hx(i,j)
phi(1)=0.25*(1.-g)*(1.-h)
phi(2)=0.25*(1.+g)*(1.-h)
phi(3)=0.25*(1.+g)*(1.+h)
phi(4)=0.25*(1.-g)*(1.+h)
cond=1.
spec=1.
phig(1)=0.25*-(1.-h)
phig(2)=0.25*(1.-h)
phig(3)=0.25*(1.+h)
phig(4)=0.25*-(1.+h)
phih(1)=0.25*-(1.-g)
phih(2)=0.25*-(1.+g)
phih(3)=0.25*(1.+g)
phih(4)=0.25*(1.-g)
rjac=zero
do iter=1,4
rjac(1,1)=rjac(1,1)+phig(iter)*crdnx(iter)
rjac(1,2)=rjac(1,2)+phig(iter)*crdny(iter)
rjac(2,1)=rjac(2,1)+phih(iter)*crdnx(iter)
rjac(2,2)=rjac(2,2)+phih(iter)*crdny(iter)
enddo
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1)
rjaci(1,1)= rjac(2,2)/djac
rjaci(2,2)= rjac(1,1)/djac
rjaci(1,2)=-rjac(1,2)/djac
rjaci(2,1)=-rjac(2,1)/djac
do iter=1,4
phix(iter)=rjaci(1,1)*phig(iter)+
1 rjaci(1,2)*phih(iter)
phiy(iter)=rjaci(2,1)*phig(iter)+
1 rjaci(2,2)*phih(iter)
enddo
dtdx=zero
dtdy=zero
t =zero
told=zero
do iter=1,4
dtdx=u(iter)*phix(iter)+dtdx
dtdy=u(iter)*phiy(iter)+dtdy
t=u(iter)*phi(iter)+t
told=(u(iter)-du(iter,nrhs))*phi(iter)+told
end do
dtdt=(t-told)/dtime
we=djac
do ki=1,4
c loop over nodes
rhs(ki,nrhs) = rhs(ki,nrhs) -
1 (we/float(ienr))*(phi(ki)*rho*spec*dtdt+
2 cond*(phix(ki)*dtdx + phiy(ki)*dtdy))
do kj=1,4
amatrx(ki,kj)=amatrx(ki,kj)+(we/float(ienr))
1 *(phi(ki)*phi(kj)*rho*spec/dtime+cond
2 *(phix(ki)*phix(kj)+phiy(ki)*phiy(kj)))
end do
end do
enddo
enddo
end if
return
end

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*Heading
** Job name: Job-1 Model name: Model-1
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=Part-1
*Node
1, 0., 0., 0.
2, 0.25, 0., 0.
3, 0.5, 0., 0.
4, 0.75, 0., 0.
5, 1., 0., 0.
6, 0., 0.25, 0.
7, 0.25, 0.25, 0.
8, 0.5, 0.25, 0.
9, 0.75, 0.25, 0.
10, 1., 0.25, 0.
*USER ELEMENT,NODES=4,TYPE=U1,PROP=1,COORDINATES=2,VAR=2,unsymm
11,
*Element, type=U1,ELSET=UEL
1, 1, 2,7,6
2, 2, 3,8,7
3, 3, 4,9,8
4, 4, 5,10,9
*UEL Property, Elset=UEL
1.
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=Part-1-1, part=Part-1
*End Instance
**
*Nset, nset=_PickedSet16, internal, instance=Part-1-1
1,2,3,6,7,8
*Nset, nset=_PickedSet17, internal, instance=Part-1-1
4,5,9,10
*Nset, nset=Set-6, instance=Part-1-1
1,6
*End Assembly
**
** MATERIALS
**
*Material, name=Material-1
*Conductivity
1.,
*Density
1.,
*Specific Heat
1.,
** ----------------------------------------------------------------
**
** Name: Predefined Field-1 Type: Temperature
*Initial Conditions, type=TEMPERATURE
_PickedSet16, 1.,0.
** Name: Predefined Field-2 Type: Temperature
*Initial Conditions, type=TEMPERATURE
_PickedSet17, 0.,0.
** STEP: Step-1
**
*Step, name=Step-1
*Heat Transfer, end=PERIOD, deltmx=100.
0.01, 0.1, 1e-09, 0.01,
**
** BOUNDARY CONDITIONS
**
** Name: BC-1 Type: Temperature
*Boundary
Set-6, 11, 11, 1.
**
** OUTPUT REQUESTS
**
*Restart, write, frequency=0
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
*Output, history, frequency=0
*End Step

79
Unpublished/XFEM/AbaqusUEL/AbInp1d.inp Normal file
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@ -0,0 +1,79 @@
*Heading
** Job name: Job-1 Model name: Model-1
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=Part-1
*Node
1, 0., 0., 0.
2, 0.25, 0., 0.
3, 0.5, 0., 0.
4, 0.75, 0., 0.
5, 1., 0., 0.
*USER ELEMENT,NODES=2,TYPE=U1,PROP=1,COORDINATES=1,VAR=2,unsymm
11,12
*Element, type=U1,ELSET=UEL
1, 1, 2,
2, 2, 3,
3, 3,4
4, 4,5
*UEL Property, Elset=UEL
1.
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=Part-1-1, part=Part-1
*End Instance
**
*Nset, nset=_PickedSet16, internal, instance=Part-1-1
1,2,3
*Nset, nset=_PickedSet17, internal, instance=Part-1-1
4,5,6
*Nset, nset=Set-6, instance=Part-1-1
1,
*End Assembly
**
** MATERIALS
**
*Material, name=Material-1
*Conductivity
1.,
*Density
1.,
*Specific Heat
1.,
** ----------------------------------------------------------------
**
** Name: Predefined Field-1 Type: Temperature
*Initial Conditions, type=TEMPERATURE
_PickedSet16, 1.,0.
** Name: Predefined Field-2 Type: Temperature
*Initial Conditions, type=TEMPERATURE
_PickedSet17, 0.,0.
** STEP: Step-1
**
*Step, name=Step-1
*Heat Transfer, end=PERIOD, deltmx=100.
0.01, 0.1, 1e-09, 0.01,
**
** BOUNDARY CONDITIONS
**
** Name: BC-1 Type: Temperature
*Boundary
Set-6, 11, 11, 1.
**
** OUTPUT REQUESTS
**
*Restart, write, frequency=0
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
*Output, history, frequency=0
*End Step

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c***********************************************************
subroutine uelmat(rhs,amatrx,svars,energy,ndofel,nrhs,
1 nsvars,props,nprops,coords,mcrd,nnode,u,du,
2 v,a,jtype,time,dtime,kstep,kinc,jelem,params,
3 ndload,jdltyp,adlmag,predef,npredf,lflags,mlvarx,
4 ddlmag,mdload,pnewdt,jprops,njpro,period,
5 materiallib)
c
include 'aba_param.inc'
C
dimension rhs(mlvarx,*), amatrx(ndofel, ndofel), props(*),
1 svars(*), energy(*), coords(mcrd, nnode), u(ndofel),
2 du(mlvarx,*), v(ndofel), a(ndofel), time(2), params(*),
3 jdltyp(mdload,*), adlmag(mdload,*), ddlmag(mdload,*),
4 predef(2, npredf, nnode), lflags(*), jprops(*)
parameter (zero=0.d0, dmone=-1.0d0, one=1.d0, four=4.0d0,
1 fourth=0.25d0,gaussCoord=0.577350269d0)
parameter (ndim=2, ndof=2, nshr=1,nnodemax=4,
1 ntens=4, ninpt=4, nsvint=4)
c
dimension stiff(ndof*nnodemax,ndof*nnodemax),
1 force(ndof*nnodemax), shape(nnodemax), dshape(ndim,nnodemax),
2 xjac(ndim,ndim),xjaci(ndim,ndim), bmat(nnodemax*ndim),
3 statevLocal(nsvint),stress(ntens), ddsdde(ntens, ntens),
4 stran(ntens), dstran(ntens), wght(ninpt)
c
dimension predef_loc(npredf),dpredef_loc(npredf),
1 defGrad(3,3),utmp(3),xdu(3),stiff_p(3,3),force_p(3)
dimension coord24(2,4),coords_ip(3)
data coord24 /dmone, dmone,
2 one, dmone,
3 one, one,
4 dmone, one/
c
data wght /one, one, one, one/
c*************************************************************
c U1 = first-order, plane strain, full integration
c*************************************************************
c define mass matrix as identity at start of analysis
if (lflags(3).eq.4) then
amatrx = zero
do i=1, ndofel
amatrx(i,i) = one
end do
goto 999
end if
c properties
thickness = 0.1d0
c initialize rhs and k
rhs = zero
amatrx = zero
c loop over integration points
do kintk = 1, ninpt
c determine gauss point in local sys (g,h)
c takes form: [4 3]
c [1 2]
g = coord24(1,kintk)*gaussCoord
h = coord24(2,kintk)*gaussCoord
c shape functions
shape(1) = (one - g)*(one - h)/four
shape(2) = (one + g)*(one - h)/four
shape(3) = (one + g)*(one + h)/four
shape(4) = (one - g)*(one + h)/four
c derivative d(Ni)/d(g)
dshape(1,1) = -(one - h)/four
dshape(1,2) = (one - h)/four
dshape(1,3) = (one + h)/four
dshape(1,4) = -(one + h)/four
c derivative d(Ni)/d(h)
dshape(2,1) = -(one - g)/four
dshape(2,2) = -(one + g)/four
dshape(2,3) = (one + g)/four
dshape(2,4) = (one - g)/four
c compute global coordinates of the ip
coords_ip = zero
do k1=1,nnode
do k2=1,mcrd
coords_ip(k2)=coords_ip(k2)+shape(k1)*coords(k2,k1)
end do
end do
c form b-matrix
djac = one
xjac = zero
xjaci = zero
c Get Jacobian
do inod= 1, nnode
do idim = 1, ndim
do jdim = 1, ndim
xjac(jdim,idim) = xjac(jdim,idim) +
1 dshape(jdim,inod)*coords(idim,inod)
end do
end do
end do
c Get Det of Jacobian
djac = xjac(1,1)*xjac(2,2) - xjac(1,2)*xjac(2,1)
if (djac .gt. zero) then
! jacobian is positive - invert it
xjaci(1,1) = xjac(2,2)/djac
xjaci(2,2) = xjac(1,1)/djac
xjaci(1,2) = -xjac(1,2)/djac
xjaci(2,1) = -xjac(2,1)/djac
else
! negative or zero jacobian - reduce time inc.
write(7,*)'WARNING: element',jelem,'has neg. Jacobian'
pnewdt = fourth
endif
if (pnewdt .lt. pnewdtLocal) pnewdtLocal = pnewdt
c Build B matrix
bmat = zero
do inod = 1, nnode
do ider = 1, ndim
do idim = 1, ndim
irow = idim + (inod - 1)*ndim
bmat(irow) = bmat(irow) +
1 xjaci(idim,ider)*dshape(ider,inod)
end do
end do
end do
c get strain inc
dstran(i) = zero
c set deformation gradient to Identity matrix
defGrad = zero
do k1=1,3
defGrad(k1,k1) = one
end do
c compute incremental strains
do nodi = 1, nnode
incr_row = (nodi - 1)*ndof
do i = 1, ndof
xdu(i)= du(i + incr_row,1)
utmp(i) = u(i + incr_row)
end do
dNidx = bmat(1+(nodi-1)*ndim)
dNidy = bmat(2+(nodi-1)*ndim)
dstran(1) = dstran(1)+dNidx*xdu(1)
dstran(2) = dstran(2)+dNidy*xdu(2)
dstran(4) = dstran(4)+dNidy*xdu(1)+dNidx*xdu(2)
c deformation gradient (prob not required for umat)
defGrad(1,1) = defGrad(1,1) + dNidx*utmp(1)
defGrad(1,2) = defGrad(1,2) + dNidy*utmp(1)
defGrad(2,1) = defGrad(2,1) + dNidx*utmp(2)
defGrad(2,2) = defGrad(2,2) + dNidy*utmp(2)
end do
c call constitutive routine
isvinc= (kintk-1)*nsvint
c prepare arrays for entry into material routines
do i = 1, nsvint
statevLocal(i)=svars(i+isvinc)
end do
c state variables
do k1=1,ntens
stran(k1) = statevLocal(k1)
stress(k1) = zero
end do
do i=1, ntens
do j=1, ntens
ddsdde(i,j) = zero
end do
ddsdde(i,j) = one
enddo
c compute characteristic element length
celent = sqrt(djac*dble(ninpt))
dvmat = djac*thickness
c
dvdv0 = one
call material_lib_mech(materiallib,stress,ddsdde,
1 stran,dstran,kintk,dvdv0,dvmat,defGrad,
2 predef_loc,dpredef_loc,npredf,celent,coords_ip)
c
do k1=1,ntens
statevLocal(k1) = stran(k1) + dstran(k1)
end do
isvinc= (kintk-1)*nsvint
c update element state variables
do i = 1, nsvint
svars(i+isvinc)=statevLocal(i)
end do
c form stiffness matrix and internal force vector
dNjdx = zero
dNjdy = zero
force = zero
stiff = zero
dvol= wght(kintk)*djac
do nodj = 1, nnode
incr_col = (nodj - 1)*ndof
dNjdx = bmat(1+(nodj-1)*ndim)
dNjdy = bmat(2+(nodj-1)*ndim)
force_p(1) = dNjdx*stress(1) + dNjdy*stress(4)
force_p(2) = dNjdy*stress(2) + dNjdx*stress(4)
do jdof = 1, ndof
jcol = jdof + incr_col
force(jcol) = force(jcol)+force_p(jdof)*dvol
end do
do nodi = 1, nnode
incr_row = (nodi -1)*ndof
dNidx = bmat(1+(nodi-1)*ndim)
dNidy = bmat(2+(nodi-1)*ndim)
stiff_p(1,1) = dNidx*ddsdde(1,1)*dNjdx
& + dNidy*ddsdde(4,4)*dNjdy
& + dNidx*ddsdde(1,4)*dNjdy
& + dNidy*ddsdde(4,1)*dNjdx
stiff_p(1,2) = dNidx*ddsdde(1,2)*dNjdy
& + dNidy*ddsdde(4,4)*dNjdx
& + dNidx*ddsdde(1,4)*dNjdx
& + dNidy*ddsdde(4,2)*dNjdy
stiff_p(2,1) = dNidy*ddsdde(2,1)*dNjdx
& + dNidx*ddsdde(4,4)*dNjdy
& + dNidy*ddsdde(2,4)*dNjdy
& + dNidx*ddsdde(4,1)*dNjdx
stiff_p(2,2) = dNidy*ddsdde(2,2)*dNjdy
& + dNidx*ddsdde(4,4)*dNjdx
& + dNidy*ddsdde(2,4)*dNjdx
& + dNidx*ddsdde(4,2)*dNjdy
do jdof = 1, ndof
icol = jdof + incr_col
do idof = 1, ndof
irow = idof + incr_row
stiff(irow,icol) = stiff(irow,icol) +
& stiff_p(idof,jdof)*dvol
end do
end do
end do
end do
c assemble rhs and lhs
do k1=1, ndof*nnode
rhs(k1, 1) = rhs(k1, 1) - force(k1)
do k2=1, ndof*nnode
amatrx(k1, k2) = amatrx(k1, k2) + stiff(k1,k2)
end do
end do
end do ! end loop on material integration points
pnewdt = pnewdtLocal
c
999 continue
c
return
end

293
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c User subroutine UEL XFEM
subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,
1 props,nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,
1 kstep,kinc,jelem,params,ndload,jdltyp,adlmag,predef,npredf,
1 lflags,mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c ABAQUS defined variables:
dimension rhs(mlvarx,*), amatrx(ndofel,ndofel), props(*),
1 svars(nsvars), energy(8), coords(mcrd,nnode), u(ndofel),
1 du(mlvarx,*), v(ndofel), a(ndofel), time(2), params(*),
1 jdltyp(mdload,*), adlmag(mdload,*), ddlmag(mdload,*),
1 predef(2,npredf,nnode), lflags(*), jprops(*)
c
character*256 outdir
integer lenoutdir
integer i,j,k,pss,orderq(3),gint,flag,dimens
integer ncracks,maxncp,nelmx,nnodx,typexe(nnode),ix(nnode)
integer,parameter :: mpg=1650
integer,allocatable:: typex(:,:),ncp(:)
real*8 e, nu
real*8 f(ndofel)
real*8 sg(3,mpg),xypg(2,mpg),xe(8),ye(8),xyc0(2),xycprev(2)
real*8, allocatable:: xyc(:,:,:),dist(:,:),elemgg(:,:)
real*8, allocatable:: batg(:,:),dbatg(:,:),jatg(:)
c
c Read real and integer properties set at the ABAQUS input file
e = props(1)
nu = props(2)
pss = jprops(1)
orderq(1) = jprops(2)
orderq(2) = jprops(3)
orderq(3) = jprops(4)
dimens = jprops(5)
c Read the working directory
call getoutdir(outdir,lenoutdir)
c read number of cracks, max number of crack path points,
c number of enriched elements and enriched nodes.
open(68,file=outdir(1:lenoutdir)//’\files\gginfox’)
read(68,*) ncracks,maxncp,nelmx,nnodx
close(68)
c Allocate dimensions
allocate (typex(nnodx,2), ncp(ncracks))
allocate (xyc(ncracks,maxncp,2), dist(nnodx,3), elemgg(nelmx,10))
c read coordinates of path points for each crack
open(68,file=outdir(1:lenoutdir)//’\files\ggxyc’)
do i=1,ncracks
read(68,*) ncp(i)
do j=1,ncp(i)
read(68,*) (xyc(i,j,k),k=1,2)
end do
end do
close(68)
c Read list of enriched nodes, type of enrichment and distances
open(68,file=outdir(1:lenoutdir)//’\files\ggnodex’)
do i=1,nnodx
read(68,*) (typex(i,j),j=1,2),(dist(i,j),j=2,3)
dist(i,1)=typex(i,1)
end do
close(68)
c read list of enriched elements, type of enrichment and intersection points
open(68,file=outdir(1:lenoutdir)//’\files\ggelemx’)
do i=1,nelmx
read(68,*) (elemgg(i,j),j=1,10)
end do
close(68)
c call initializing routines for matrix and vectors
call initializem(rhs,ndofel,nrhs)
call initializem(amatrx,ndofel,ndofel)
call initializev(energy,8)
call initializev(svars,nsvars)
c verification of element type (type=12 for enriched element)
if (jtype.eq.12) then
c **************************************
c * 4 node enriched element with *
c * up to 12 dof/node for x-fem *
c **************************************
if (lflags(1).eq.71) then
c coupled thermal-stress, steady state analysis
if (lflags(3).eq.1) then
c Routine that defines the location of integration points according to
c the appropriate subdivision. This enables to know the total number of
c integration points for the current element, stored in gint, and whether
c the element is subdivided for integration (flag=1) or not.
CALL int2d_X(JELEM,NelmX,ElemGG,MCRD,NNODE,COORDS,orderQ,
1 NCracks,maxNCP,NCP,XYC,gint,sg,Xe,Ye,flag,mpg,xypg,
1 XYC0,XYCPrev)
c Allocate dimensions once the total number of integration points gint is known
allocate(batg(3*gint,ndofel),dbatg(3*gint,ndofel),jatg(gint))
call initializem(batg,3*gint,ndofel)
call initializem(dbatg,3*gint,ndofel)
call initializev(jatg,gint)
c Search of the enrichment type for the nodes of the current element.
c The keys to the enrichment types are stored in the element vector TypeXe
call typexelement(outdir,lenoutdir,jelem,nnode,nelmx,ix,typexe)
c element stiffness matrix computation, stored in amatrx
call k_u12(e,nu,amatrx,ndofel,nnode,dimens,mcrd,
coords,pss,nnodx,ix,typexe,dist,xyc0,xycprev,
gint,sg,xe,ye,flag,batg,dbatg,jatg)
c Routine that multiplies AMATRX times U to obtain the force vector F
c at the end of the current increment
call mult_v(amatrx,ndofel,ndofel,u,f,ndofel)
c compute the residual force vector
do i=1,ndofel
rhs(i,1) = rhs(i,1) - f(i)
end do
c Compute stresses at Gauss points for post-processing purposes
c Store them as SVARS for output to the results file (.fil)
call svars_u12(jtype,jelem,svars,nsvars,u,ndofel,batg,
1 dbatg,jatg,gint,mpg,xypg)
end if
end if
end if
return
end
C Element stiffness matrix. Subroutine: K U12
subroutine k_u12(e,nu,amatrx,ndofel,nnode,dimens,mcrd,
1 COORDS,PSS,NnodX,ix,TypeXe,Dist,XYC0,XYCPrev,
1 gint,sg,Xe,Ye,flag,BatG,DBatG,JatG)
implicit none
integer ndofel,nnode,dimens,mcrd,pss,nnodx,gint,flag,pos
integer l,i,j,kk,typexe(nnode),ix(nnode)
real*8 e,nu,dist(nnodx,3),sg(3,*)
real*8 amatrx(ndofel,ndofel),xyc0(2),xycprev(2)
real*8 xe(2*nnode),ye(2*nnode),coords(mcrd,nnode),xl(dimens,nnode)
real*8 xsj(gint),shp(3,4)
real*8 dnf(nnode,2,4),fnode(nnode,4),h,hnode(nnode)
real*8 b(3,ndofel), db(3,ndofel), bt(ndofel,3), d(3,3)
real*8 batg(3*gint,ndofel),dbatg(3*gint,ndofel),jatg(gint)
logical nodetype1,nodetype2
c NOTES:
c Routine shapef2D is called to compute standard shape functions,
c derivatives and jacobian at integration points. This routine outputs:
c shp(3,*) - Shape functions and derivatives at point
c shp(1,i) = dN_i/dx = dN_i/dx1
c shp(2,i) = dN_i/dy = dN_i/dx2
c shp(3,i) = N_i
c xsj - Jacobian determinant at point
c Local coordinates of integration points are passed in sg(1,*), sg(2,*)
c Integration weights are passed in sg(3,*)
c Initialize AMATRX and logical variables
call initializem(amatrx,ndofel,ndofel)
NodeType1=.false.
NodeType2=.false.
c Reduce info passed thru COORDS (3D) to xl (2D)
do i=1,dimens
do j=1,nnode
xl(i,j)=coords(i,j)
end do
end do
c Define constitutive stress-strain elastic matrix
call calc_d(pss,d,e,nu)
c Specify the type of nodal enrichment
do i=1,nnode
if (typexe(i).eq.1) then
nodetype1=.true.
elseif (typexe(i).eq.2) then
nodetype2=.true.
end if
end do
c Numerical integration loop over gint integration points
DO l = 1,gint
c Compute shape functions, derivatives and jacobian at integration point
call shapef2d(sg(1,l),xl,shp,xsj(l),dimens,nnode,ix,.false.)
if (flag.eq.1) then !element is subdivided for integration
xsj(l) = sg(3,l) !the integration weight includes the jacobian
else !element is not subdivided. standard integration
xsj(l) = xsj(l)*sg(3,l)
endif
c Value of the Heaviside function at integration point
c (This call is also used to store the values of H
c at nodes of the element for modified enrichment)
if (nodetype1) then
call heaviside(nnodx,dist,nnode,ix,shp,h,hnode)
endif
c Derivatives of shape functions Ni times enrichment functions Fj at integration point
c (This call is also used to compute the derivatives of shape functions Ni times
c enrichment functions Fj at nodes of the element for modified enrichment)
if (nodetype2) then
call fcracktip(xyc0,xycprev,shp,xe,ye,dnf,fnode)
endif
c STIFFNESS MATRIX COMPUTATION:
c Assembly of element matrix B (denoted as B) at integration point
call initializem(b,3,ndofel)
pos=1
c loop over nodes
do i= 1,nnode
c Contribution to B of derivatives of standard shape functions
B(1,Pos) = shp(1,i)
B(2,Pos+1)= shp(2,i)
B(3,Pos) = shp(2,i)
B(3,Pos+1)= shp(1,i)
c Contribution to B of derivatives of shape functions times Heaviside function
if (typexe(i).eq.1) then
b(1,2+pos) = shp(1,i)*(h-hnode(i))
b(2,3+pos) = shp(2,i)*(h-hnode(i))
b(3,2+pos) = shp(2,i)*(h-hnode(i))
b(3,3+pos) = shp(1,i)*(h-hnode(i))
c Contribution to B of derivatives of shape functions times crack tip functions
elseif(typexe(i).eq.2) then
do kk= 1,4
b(1,2*kk+2+pos)= dnf(i,1,kk)-shp(1,i)*fnode(i,kk)
b(2,2*kk+3+pos)= dnf(i,2,kk)-shp(2,i)*fnode(i,kk)
b(3,2*kk+2+pos)= dnf(i,2,kk)-shp(2,i)*fnode(i,kk)
b(3,2*kk+3+pos)= dnf(i,1,kk)-shp(1,i)*fnode(i,kk)
end do
end if
Pos=Pos+12 !Each node has 12 dof
end do ! i = end loop over element nodes
db=matmul(d,b) ! matrix d*b
bt=transpose(b) ! b transpose
c Integration of BT*D*B
amatrx= amatrx + matmul(bt,db)*xsj(l)
c store information at each integration point for further post-processing
do i=1,3
do j=1,ndofel
batg(3*(l-1)+i,j)=b(i,j)
dbatg(3*(l-1)+i,j)=db(i,j)
end do
end do
jatg(l)=xsj(l)
end do ! l = end loop for each integration point
return
end
c
SUBROUTINE SVARS_U12(JTYPE,JELEM,SVARS,NSVARS,U,Dof,BatG,DBatG,
* JatG,gint,mpg,xypg)
c Calculates and/or stores the following magnitudes at the element integration points,
c storing them in SVARS: strains, stresses, strain energy density, dv/dx, du/dy, jacobian,
c dNi/dx, dNi/dy, global coordinates of integration points.
IMPLICIT NONE
INTEGER i,j,k,NSVARS, Dof, gint, JTYPE,JELEM,mpg
REAL*8 SVARS(NSVARS), U(Dof),BatG(3*gint,Dof),DBatG(3*gint,Dof)
REAL*8 JatG(gint),B(3,Dof),DB(3,Dof),Bdvdx(3,Dof),Bdudy(3,Dof)
REAL*8 EPS(3),SIG(3),W,dvdx(3),dudy(3),JAC,xypg(2,mpg)
c &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
39
c &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
c First value stored in SVARS is the total number of integration points
c of the enriched element
SVARS(1)=gint
DO i=1,gint
JAC=JatG(i)
DO k=1,3
DO j=1,Dof
B(k,j)=BatG(3*(i-1)+k,j)
Bdvdx(k,j)=B(k,j) ! For computation of dv/dx
Bdudy(k,j)=B(k,j) ! For computation of du/dy
DB(k,j)=DBatG(3*(i-1)+k,j)
END DO
END DO
CALL MULT_V(B,3,Dof,U,EPS,3) ! Compute strains EPS
CALL MULT_V(DB,3,Dof,U,SIG,3) ! Compute stresses SIG
W=0.5d0*(EPS(1)*SIG(1)+EPS(2)*SIG(2)+EPS(3)*SIG(3))
c Computation of dv/dx & du/dy
c Set to zero positions in the 3rd row of B associated with dN/dy
DO j=1,Dof,2
Bdvdx(3,j)=0.0d0
END DO
CALL MULT_V(Bdvdx,3,Dof,U,dvdx,3) !compute dv/dx, stored in dvdx(3)
c Set to zero positions in the 3rd row of B associated with dN/dx
DO j=2,Dof,2
Bdudy(3,j)=0.0d0
END DO
CALL MULT_V(Bdudy,3,Dof,U,dudy,3) !compute du/dy, stored in dudy(3)
c Store in SVARS the following information at integration points
SVARS(1+20*(i-1)+1)=EPS(1)
SVARS(1+20*(i-1)+2)=EPS(2)
SVARS(1+20*(i-1)+3)=EPS(3)
SVARS(1+20*(i-1)+4)=SIG(1)
SVARS(1+20*(i-1)+5)=SIG(2)
SVARS(1+20*(i-1)+6)=SIG(3)
SVARS(1+20*(i-1)+7)=W
SVARS(1+20*(i-1)+8)=dvdx(3)
SVARS(1+20*(i-1)+9)=dudy(3)
SVARS(1+20*(i-1)+10)=JAC ! Jacobian includes integration weight
c Store in SVARS the shape functions derivatives dNi/dx, dNi/dy for external computation
c of dq/dx, dq/dy (used in domain interaction integrals).
c (we take them from the positions associated with the standard dofs)
SVARS(1+20*(i-1)+11)=B(1,1)
SVARS(1+20*(i-1)+12)=B(1,13)
SVARS(1+20*(i-1)+13)=B(1,25)
SVARS(1+20*(i-1)+14)=B(1,37)
SVARS(1+20*(i-1)+15)=B(2,2)
SVARS(1+20*(i-1)+16)=B(2,14)
SVARS(1+20*(i-1)+17)=B(2,26)
SVARS(1+20*(i-1)+18)=B(2,38)
Store in SVARS the global coordinates of integration points
SVARS(1+20*(i-1)+19)=xypg(1,i)
SVARS(1+20*(i-1)+20)=xypg(2,i)
END DO !i loop over all integration points of the element
RETURN
END

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subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,
1 props,nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,
2 dtime,kstep,kinc,jelem,params,ndload,jdltyp,adlmag,
3 predef,npredf,lflags,mlvarx,ddlmag,mdload,pnewdt,
4 jprops,njprop,period)
c
include 'aba_param.inc'
parameter ( zero = 0.d0, half = 0.5d0, one = 1.d0 )
c
c This is a linear truss element for Abaqus/Standard
c general static analysis in 1D space (aligned to x-axis) only.
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),
1 svars(nsvars),energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),
5 jprops(*)
c
c assign section properties
area = props(1)
e = props(2)
anu = props(3)
rho = props(4)
c calculate stiffness and mass
alen = abs(coords(1,2)-coords(1,1))
ak = area*e/alen
am = half*area*rho*alen
c Initialize Arrays
rhs = zero
amatrx = zero
if (lflags(3).eq.1) then
C Stiffness and Force
C Get Stiffness Matrix
amatrx(1,1) = ak
amatrx(4,4) = ak
amatrx(1,4) = -ak
amatrx(4,1) = -ak
c Get Internal Contrib to Residual Force
rhs(1,1) = -ak*(u(1)-u(4))
rhs(4,1) = -ak*(u(4)-u(1))
c Get External Contrib to Residual Force
do kdload = 1, ndload
if (jdltyp(kdload,1).eq.1001) then
rhs(4,1) = rhs(4,1)+adlmag(kdload,1)
end if
end do
else if (lflags(3).eq.2) then
c Stiffness
amatrx(1,1) = ak
amatrx(4,4) = ak
amatrx(1,4) = -ak
amatrx(4,1) = -ak
else if (lflags(3).eq.4) then
c Mass
do k1 = 1, ndofel
amatrx(k1,k1) = am
end do
else if (lflags(3).eq.5) then
print *,'oops'
else if (lflags(3).eq.6) then
C Mass and Force
do k1 = 1, ndofel
amatrx(k1,k1) = am
end do
rhs(1,1) = -ak*(u(1)-u(4))
rhs(4,1) = -ak*(u(4)-u(1))
END IF
c
return
end

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c 1-D Moving Interface User Element - JGrogan 2012.
c Subroutine UEXTERNALDB
c Calculates interface velocity and position at the start of each increment
c and passes it to UEL via common block. It requires a list of current nodal
c coordinates, and 'T' and 'a' degrees of freedom.
subroutine uexternaldb(lop,lrestart,time,dtime,kstep,kinc)
c
include 'aba_param.inc'
c
real dpos, npos(6), ndof(6)
common dpos,npos,ndof
dimension time(2)
c
if (lop==0)then
c initialise common blocks
dpos=0.
npos=0.
tn=0.
an=0.
print *,'npos',npos,'extrn1',time
print *,'ndof',ndof,'extrn1',time
else
print *,'npos',npos,'extrn',time
print *,'ndof',ndof,'extrn',time
endif
return
end
c
c Subroutine UEL
c Calculates element mass and stiffness matrices and residual flux
c vector for Abaqus NR Solver.
subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props
1 ,nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,
2 kinc,jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags
3 ,mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension gpx(4),gwei(4),phi(4),phix(4),phic(4),gm(4),gm2(4,4)
dimension theta(2)
real dpos, npos(6), ndof(6)
common dpos,npos,ndof
c
c level set calculation
c store nodal positions and temperatures
npos(jelem)=coords(1,1)
npos(jelem+1)=coords(1,2)
ndof(2*jelem-1)=u(1)
ndof(2*jelem)=u(2)
dpos1=0.2+0.4*time(1)
c print *,'npos',npos,'uel',time
c print *,'ndof',ndof,'uel',time
c material property definition
rho = 1.
spec = 1.
c penalty term
beta=40.
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=0.
do k2=1,ndofel
amatrx(k2,k1)=0.
enddo
enddo
if (lflags(3).eq.4) return
c transient analysis
if (lflags(1).eq.33) then
c determine node level set params
crdn1=coords(1,1)
crdn2=coords(1,2)
theta(1)=abs(crdn1-dpos1)*sign(1.,crdn1-dpos1)
theta(2)=abs(crdn2-dpos1)*sign(1.,crdn2-dpos1)
enr=2
elen=abs(crdn2-crdn1)
ajacob=elen/2.
if (sign(1.,theta(1))/=sign(1.,theta(2)))then
c enriched element
enr=4
point=(dpos1-crdn1)/ajacob-1.
rlen1=abs(-point-1.)
rlen2=abs(1.-point)
rmid1=-1.+rlen1/2.
rmid2=1.-rlen2/2.
c Get int point locations and weights
gpx(1)=-(rlen1/2.)/sqrt(3.)+rmid1
gpx(2)=(rlen1/2.)/sqrt(3.)+rmid1
gpx(3)=-(rlen2/2.)/sqrt(3.)+rmid2
gpx(4)=(rlen2/2.)/sqrt(3.)+rmid2
gwei(1)=(rlen1/2.)
gwei(2)=(rlen1/2.)
gwei(3)=(rlen2/2.)
gwei(4)=(rlen2/2.)
else
c regular element
gpx(1)=-1./sqrt(3.)
gpx(2)=1./sqrt(3.)
gwei(1)=1.
gwei(2)=1.
endif
c assemble amatrx and rhs
do k=1,enr
c loop through gauss pts: i
c=gpx(k)
c get ip level set value: Oi
c get shape functions and derivatives
c Ni
phi(1)=(1.-c)/2.
phi(3)=(1.+c)/2.
term=theta(1)*phi(1)+theta(2)*phi(3)
if (term<0.)then
cond=0.
spec=0.1
else
cond=1.
spec=1.
endif
if(enr==4)then
phi(2)=phi(1)*(abs(term)-abs(theta(1)))
phi(4)=phi(3)*(abs(term)-abs(theta(2)))
else
phi(2)=0.
phi(4)=0.
endif
c dNdci
phic(1)=-0.5
phic(3)=0.5
dterm=sign(1.,term)*(phic(1)*theta(1)+phic(3)*theta(2))
if(enr==4)then
phic(2)=phic(1)*(abs(term)-abs(theta(1)))
1 +phi(1)*dterm
phic(4)=phic(3)*(abs(term)-abs(theta(2)))
1 +phi(3)*dterm
else
phic(2)=0.
phic(4)=0.
endif
c dNdxi
phix(1)=phic(1)*(1./ajacob)
phix(2)=phic(2)*(1./ajacob)
phix(3)=phic(3)*(1./ajacob)
phix(4)=phic(4)*(1./ajacob)
c interpolate temperatures Tbar to int point: i
dtdx=u(1)*phix(1)+u(2)*phix(2)
1 +u(3)*phix(3)+u(4)*phix(4)
t=u(1)*phi(1)+u(2)*phi(2)
1 +u(3)*phi(3)+u(4)*phi(4)
told=(u(1)-du(1,nrhs))*phi(1)+(u(2)-du(2,nrhs))*phi(2)+
1 (u(3)-du(3,nrhs))*phi(3)+(u(4)-du(4,nrhs))*phi(4)
c other housekeeping
dtdt=(t-told)/dtime
we=gwei(k)*ajacob
c Assemble Element Stiffness Matrix and Add to Global
do ki=1,4
c loop over nodes
rhs(ki,nrhs)=rhs(ki,nrhs)-we*(phi(ki)*rho*spec*dtdt
1 + cond*(phix(ki)*dtdx))
do kj=1,4
amatrx(ki,kj)=amatrx(ki,kj)+we*(phi(ki)*phi(kj)
1 *rho*spec/dtime+cond*(phix(ki)*phix(kj)))
end do
end do
enddo
end if
c if interface is in the element an penalty term is needed
if(enr==4)then
xi=point
gm(1)=(1.-xi)/2.
gm(3)=(1.+xi)/2.
term=theta(1)*gm(1)+theta(2)*gm(3)
gm(2)=gm(1)*(abs(term)-abs(theta(1)))
gm(4)=gm(3)*(abs(term)-abs(theta(2)))
term2=gm(1)*u(1)+gm(2)*u(2)+gm(3)*u(3)+gm(4)*u(4)
diff=abs(term2-1.)
c add penalty flux/force: BGtc
targetT=1.
do i=1,4
rhs(i,nrhs)=rhs(i,nrhs)+beta*gm(i)*diff
enddo
c find GtG
gm2=0.
do i=1,4
do j=1,4
gm2(i,j)=gm(i)*gm(j)
enddo
enddo
c add penalty stiffness
do i=1,4
do j=1,4
amatrx(i,j)=amatrx(i,j)+beta*gm2(i,j)
enddo
enddo
endif
return
end

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c Subroutine UEL
c Calculates element mass and stiffness matrices and residual flux
c vector for Abaqus NR Solver.
subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props
1 ,nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,
2 kinc,jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags
3 ,mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension gpx(4),gwei(4),phi(4),phix(4),phic(4),gm(4),gm2(4,4)
dimension theta(2)
real dpos, npos(6), ndof(6)
common dpos,npos,ndof
c
c level set calculation
c store nodal positions and temperatures
dpos1=0.6
c material property definition
rho = 1.
spec = 1.
c penalty term
beta=100.
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=0.
do k2=1,ndofel
amatrx(k2,k1)=0.
enddo
enddo
if (lflags(3).eq.4) return
c transient analysis
if (lflags(1).eq.33) then
c determine node level set params
crdn1=coords(1,1)
crdn2=coords(1,2)
theta(1)=abs(crdn1-dpos1)*sign(1.,crdn1-dpos1)
theta(2)=abs(crdn2-dpos1)*sign(1.,crdn2-dpos1)
enr=2
elen=abs(crdn2-crdn1)
ajacob=elen/2.
if (sign(1.,theta(1))/=sign(1.,theta(2)))then
c enriched element
enr=4
point=(dpos1-crdn1)/ajacob-1.
rlen1=abs(-point-1.)
rlen2=abs(1.-point)
rmid1=-1.+rlen1/2.
rmid2=1.-rlen2/2.
c Get int point locations and weights
gpx(1)=-(rlen1/2.)/sqrt(3.)+rmid1
gpx(2)=(rlen1/2.)/sqrt(3.)+rmid1
gpx(3)=-(rlen2/2.)/sqrt(3.)+rmid2
gpx(4)=(rlen2/2.)/sqrt(3.)+rmid2
gwei(1)=(rlen1/2.)
gwei(2)=(rlen1/2.)
gwei(3)=(rlen2/2.)
gwei(4)=(rlen2/2.)
else
c regular element
gpx(1)=-1./sqrt(3.)
gpx(2)=1./sqrt(3.)
gwei(1)=1.
gwei(2)=1.
endif
c assemble amatrx and rhs
do k=1,enr
c loop through gauss pts: i
c=gpx(k)
c get ip level set value: Oi
c get shape functions and derivatives
c Ni
phi(1)=(1.-c)/2.
phi(3)=(1.+c)/2.
term=theta(1)*phi(1)+theta(2)*phi(3)
c if (term<0.)then
c cond=0.
c spec=0.1
c else
cond=1.
spec=1.
c endif
c if(enr==4)then
phi(2)=phi(1)*(abs(term)-abs(theta(1)))
phi(4)=phi(3)*(abs(term)-abs(theta(2)))
c else
c phi(2)=0.
c phi(4)=0.
c endif
c dNdci
phic(1)=-0.5
phic(3)=0.5
dterm=sign(1.,term)*(phic(1)*theta(1)+phic(3)*theta(2))
c if(enr==4)then
phic(2)=phic(1)*(abs(term)-abs(theta(1)))
1 +phi(1)*dterm
phic(4)=phic(3)*(abs(term)-abs(theta(2)))
1 +phi(3)*dterm
c else
c phic(2)=0.
c phic(4)=0.
c endif
c dNdxi
phix(1)=phic(1)*(1./ajacob)
phix(2)=phic(2)*(1./ajacob)
phix(3)=phic(3)*(1./ajacob)
phix(4)=phic(4)*(1./ajacob)
c interpolate temperatures Tbar to int point: i
dtdx=u(1)*phix(1)+u(2)*phix(2)
1 +u(3)*phix(3)+u(4)*phix(4)
t=u(1)*phi(1)+u(2)*phi(2)
1 +u(3)*phi(3)+u(4)*phi(4)
told=(u(1)-du(1,nrhs))*phi(1)+(u(2)-du(2,nrhs))*phi(2)+
1 (u(3)-du(3,nrhs))*phi(3)+(u(4)-du(4,nrhs))*phi(4)
c other housekeeping
dtdt=(t-told)/dtime
we=gwei(k)*ajacob
c Assemble Element Stiffness Matrix and Add to Global
do ki=1,4
c loop over nodes
rhs(ki,nrhs)=rhs(ki,nrhs)-we*(phi(ki)*rho*spec*dtdt
1 + cond*(phix(ki)*dtdx))
do kj=1,4
amatrx(ki,kj)=amatrx(ki,kj)+we*(phi(ki)*phi(kj)
1 *rho*spec/dtime+cond*(phix(ki)*phix(kj)))
end do
end do
enddo
end if
c if interface is in the element a penalty term is needed
if(enr==4)then
xi=point
gm(1)=(1.-xi)/2.
gm(3)=(1.+xi)/2.
term=theta(1)*gm(1)+theta(2)*gm(3)
gm(2)=gm(1)*(abs(term)-abs(theta(1)))
gm(4)=gm(3)*(abs(term)-abs(theta(2)))
term2=gm(1)*u(1)+gm(2)*u(2)+gm(3)*u(3)+gm(4)*u(4)
diff=abs(term2-1.)
c add penalty flux/force: BGtc
do i=1,4
rhs(i,nrhs)=rhs(i,nrhs)+beta*gm(i)*diff
enddo
c find GtG
gm2=0.
do i=1,4
do j=1,4
gm2(i,j)=gm(i)*gm(j)
enddo
enddo
c add penalty stiffness
do i=1,4
do j=1,4
amatrx(i,j)=amatrx(i,j)+beta*gm2(i,j)
enddo
enddo
endif
return
end

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subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props
1 ,nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,
2 kinc,jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags
3 ,mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension gpx(9),gwei(9),phi(8),phix(8),phic(8),stiffk(2,2)
dimension stiffm(2,2)
c
c print *,u(1),u(2),du(1,nhrs),du(2,nhrs),time(1),lflags(3)
c material property definition
rho = 1.
spec = 1.
conduc = 1.
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=0.
do k2=1,ndofel
amatrx(k2,k1)=0.
stiffk(k2,k1)=0.
stiffm(k2,k1)=0.
enddo
enddo
if (lflags(3).eq.4) return
c transient analysis
if (lflags(1).eq.33) then
c determine gauss point locations
gpx(1)=-1./sqrt(3.)
gpx(2)=1./sqrt(3.)
gwei(1)=1.
gwei(2)=1.
c assemble amatrx and rhs
do k=1,2
c loop through gauss pts
c=gpx(k)
c get shape functions and derivatives
phi(1)=(1.-c)/2.
phi(2)=(1.+c)/2.
phic(1)=-0.5
phic(2)=0.5
dxdc=abs(coords(1,2)-coords(1,1))/2.
ajacob=dxdc
phix(1)=phic(1)*(1./ajacob)
phix(2)=phic(2)*(1./ajacob)
c interpolate temperatures to int points
dtdx=u(1)*phix(1)+u(2)*phix(2)
t=u(1)*phi(1)+u(2)*phi(2)
told=(u(1)-du(1,nrhs))*phi(1)+(u(2)-du(2,nrhs))*phi(2)
c other housekeeping
cond=conduc
dtdt=(t-told)/dtime
we=gwei(k)*ajacob
c Assemble Element Stiffness Matrix and Add to Global
do ki=1,2
c loop over nodes
rhs(ki,nrhs)=rhs(ki,nrhs)-we*(phi(ki)*rho*spec*dtdt
1 + cond*(phix(ki)*dtdx))
do kj=1,2
stiffk(ki,kj)=stiffk(ki,kj)+
1 we*cond*(phix(ki)*phix(kj))
stiffm(ki,kj)=stiffm(ki,kj)+
1 we*(phi(ki)*phi(kj)*rho*spec)/dtime
end do
end do
do i=1,2
do j=1,2
amatrx(i,j)=stiffk(i,j)+stiffm(i,j)
enddo
enddo
enddo
end if
return
end

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subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props
1 ,nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,
2 kinc,jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags
3 ,mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension gpx(9),gwei(9),phi(8),phix(8),phic(8)
c
c print *,u(1),u(2),du(1,nhrs),du(2,nhrs),time(1),lflags(3)
c level set calculation
dpos=0.01*time
c
c material property definition
rho = 1.
spec = 1.
conduc = 1.
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=0.
do k2=1,ndofel
amatrx(k2,k1)=0.
enddo
enddo
if (lflags(3).eq.4) return
c transient analysis
if (lflags(1).eq.33) then
c determine gauss point locations
gpx(1)=-1./sqrt(3.)
gpx(2)=1./sqrt(3.)
gwei(1)=1.
gwei(2)=1.
c determine node level sets
rnl1=abs(coords(1,1)-dpos)
rnl2=abs(coords(1,2)-dpos)
c get jacobian
dxdc=abs(coords(1,2)-coords(1,1))/2.
ajacob=dxdc
c assemble amatrx and rhs
do k=1,4
c loop through gauss pts
c=gpx(k)
c get shape functions and derivatives
phi(1)=(1.-c)/2.
phi(2)=(1.+c)/2.
c get ip position
pos=coord(1,1)+ajacob(1.+c)
phi(3)=abs(c)
phic(1)=-0.5
phic(2)=0.5
phix(1)=phic(1)*(1./ajacob)
phix(2)=phic(2)*(1./ajacob)
c interpolate temperatures to int points
dtdx=u(1)*phix(1)+u(2)*phix(2)
t=u(1)*phi(1)+u(2)*phi(2)
told=(u(1)-du(1,nrhs))*phi(1)+(u(2)-du(2,nrhs))*phi(2)
c other housekeeping
cond=conduc
dtdt=(t-told)/dtime
we=gwei(k)*ajacob
c Assemble Element Stiffness Matrix and Add to Global
do ki=1,2
c loop over nodes
rhs(ki,nrhs)=rhs(ki,nrhs)-we*(phi(ki)*rho*spec*dtdt
1 + cond*(phix(ki)*dtdx))
do kj=1,2
amatrx(ki,kj)=amatrx(ki,kj)+we*(phi(ki)*phi(kj)
1 *rho*spec/dtime+cond*(phix(ki)*phix(kj)))
end do
end do
enddo
end if
return
end

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c 1-D Moving Interface User Element - JGrogan 2012.
c Subroutine UEXTERNALDB
c Calculates interface velocity and position at the start of each increment
c and passes it to UEL via common block. It requires a list of current nodal
c coordinates, and 'T' and 'a' degrees of freedom.
subroutine uexternaldb(lop,lrestart,time,dtime,kstep,kinc)
c
include 'aba_param.inc'
c
real dpos, npos(6), ndof(6)
common dpos,npos,ndof
dimension time(2)
c
if (lop==0)then
c initialise common blocks
dpos=0.
npos=0.
tn=0.
an=0.
print *,'npos',npos,'extrn1',time
print *,'ndof',ndof,'extrn1',time
else
print *,'npos',npos,'extrn',time
print *,'ndof',ndof,'extrn',time
endif
return
end
c
c Subroutine UEL
c Calculates element mass and stiffness matrices and residual flux
c vector for Abaqus NR Solver.
subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props
1 ,nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,
2 kinc,jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags
3 ,mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension gpx(4),gwei(4),phi(4),phix(4),phic(4),gm(4),gm2(4,4)
dimension theta(2)
real dpos, npos(6), ndof(6)
common dpos,npos,ndof
c
c level set calculation
c store nodal positions and temperatures
npos(jelem)=coords(1,1)
npos(jelem+1)=coords(1,2)
ndof(2*jelem-1)=u(1)
ndof(2*jelem)=u(2)
dpos1=0.2+0.4*time(1)
c print *,'npos',npos,'uel',time
c print *,'ndof',ndof,'uel',time
c material property definition
rho = 1.
spec = 1.
c penalty term
beta=40.
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=0.
do k2=1,ndofel
amatrx(k2,k1)=0.
enddo
enddo
if (lflags(3).eq.4) return
c transient analysis
if (lflags(1).eq.33) then
c determine node level set params
crdn1=coords(1,1)
crdn2=coords(1,2)
theta(1)=abs(crdn1-dpos1)*sign(1.,crdn1-dpos1)
theta(2)=abs(crdn2-dpos1)*sign(1.,crdn2-dpos1)
enr=2
elen=abs(crdn2-crdn1)
ajacob=elen/2.
if (sign(1.,theta(1))/=sign(1.,theta(2)))then
c enriched element
enr=4
point=(dpos1-crdn1)/ajacob-1.
rlen1=abs(-point-1.)
rlen2=abs(1.-point)
rmid1=-1.+rlen1/2.
rmid2=1.-rlen2/2.
c Get int point locations and weights
gpx(1)=-(rlen1/2.)/sqrt(3.)+rmid1
gpx(2)=(rlen1/2.)/sqrt(3.)+rmid1
gpx(3)=-(rlen2/2.)/sqrt(3.)+rmid2
gpx(4)=(rlen2/2.)/sqrt(3.)+rmid2
gwei(1)=(rlen1/2.)
gwei(2)=(rlen1/2.)
gwei(3)=(rlen2/2.)
gwei(4)=(rlen2/2.)
else
c regular element
gpx(1)=-1./sqrt(3.)
gpx(2)=1./sqrt(3.)
gwei(1)=1.
gwei(2)=1.
endif
c assemble amatrx and rhs
do k=1,enr
c loop through gauss pts: i
c=gpx(k)
c get ip level set value: Oi
c get shape functions and derivatives
c Ni
phi(1)=(1.-c)/2.
phi(3)=(1.+c)/2.
term=theta(1)*phi(1)+theta(2)*phi(3)
if (term<0.)then
cond=0.
spec=0.1
else
cond=1.
spec=1.
endif
if(enr==4)then
phi(2)=phi(1)*(abs(term)-abs(theta(1)))
phi(4)=phi(3)*(abs(term)-abs(theta(2)))
else
phi(2)=0.
phi(4)=0.
endif
c dNdci
phic(1)=-0.5
phic(3)=0.5
dterm=sign(1.,term)*(phic(1)*theta(1)+phic(3)*theta(2))
if(enr==4)then
phic(2)=phic(1)*(abs(term)-abs(theta(1)))
1 +phi(1)*dterm
phic(4)=phic(3)*(abs(term)-abs(theta(2)))
1 +phi(3)*dterm
else
phic(2)=0.
phic(4)=0.
endif
c dNdxi
phix(1)=phic(1)*(1./ajacob)
phix(2)=phic(2)*(1./ajacob)
phix(3)=phic(3)*(1./ajacob)
phix(4)=phic(4)*(1./ajacob)
c interpolate temperatures Tbar to int point: i
dtdx=u(1)*phix(1)+u(2)*phix(2)
1 +u(3)*phix(3)+u(4)*phix(4)
t=u(1)*phi(1)+u(2)*phi(2)
1 +u(3)*phi(3)+u(4)*phi(4)
told=(u(1)-du(1,nrhs))*phi(1)+(u(2)-du(2,nrhs))*phi(2)+
1 (u(3)-du(3,nrhs))*phi(3)+(u(4)-du(4,nrhs))*phi(4)
c other housekeeping
dtdt=(t-told)/dtime
we=gwei(k)*ajacob
c Assemble Element Stiffness Matrix and Add to Global
do ki=1,4
c loop over nodes
rhs(ki,nrhs)=rhs(ki,nrhs)-we*(phi(ki)*rho*spec*dtdt
1 + cond*(phix(ki)*dtdx))
do kj=1,4
amatrx(ki,kj)=amatrx(ki,kj)+we*(phi(ki)*phi(kj)
1 *rho*spec/dtime+cond*(phix(ki)*phix(kj)))
end do
end do
enddo
end if
c if interface is in the element an penalty term is needed
if(enr==4)then
xi=point
gm(1)=(1.-xi)/2.
gm(3)=(1.+xi)/2.
term=theta(1)*gm(1)+theta(2)*gm(3)
gm(2)=gm(1)*(abs(term)-abs(theta(1)))
gm(4)=gm(3)*(abs(term)-abs(theta(2)))
term2=gm(1)*u(1)+gm(2)*u(2)+gm(3)*u(3)+gm(4)*u(4)
diff=abs(term2-1.)
c add penalty flux/force: BGtc
targetT=1.
do i=1,4
rhs(i,nrhs)=rhs(i,nrhs)+beta*gm(i)*diff
enddo
c find GtG
gm2=0.
do i=1,4
do j=1,4
gm2(i,j)=gm(i)*gm(j)
enddo
enddo
c add penalty stiffness
do i=1,4
do j=1,4
amatrx(i,j)=amatrx(i,j)+beta*gm2(i,j)
enddo
enddo
endif
return
end

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subroutine uel(rhs,amatrx,svars,energy,ndofel,nrhs,nsvars,props
1 ,nprops,coords,mcrd,nnode,u,du,v,a,jtype,time,dtime,kstep,
2 kinc,jelem,params,ndload,jdltyp,adlmag,predef,npredf,lflags
3 ,mlvarx,ddlmag,mdload,pnewdt,jprops,njprop,period)
c
include 'aba_param.inc'
c
dimension rhs(mlvarx,*),amatrx(ndofel,ndofel),svars(nsvars),
1 energy(8),props(*),coords(mcrd,nnode),
2 u(ndofel),du(mlvarx,*),v(ndofel),a(ndofel),time(2),
3 params(3),jdltyp(mdload,*),adlmag(mdload,*),
4 ddlmag(mdload,*),predef(2,npredf,nnode),lflags(*),jprops(*)
c
dimension gx(4),hx(4),phi(4),phix(4),phiy(4),phig(4),phih(4)
dimension rjac(2,2),rjaci(2,2)
c
c material property definition
rho = 1.
spec = 1.
cond = 1.
c initialization (nrhs=1)
do k1=1,ndofel
rhs(k1,nrhs)=0.
do k2=1,ndofel
amatrx(k2,k1)=0.
enddo
enddo
if (lflags(3).eq.4) return
c transient analysis
if (lflags(1).eq.33) then
c determine gauss point locations
gpos=1./sqrt(3.)
gx(1)=-gpos
gx(2)=gpos
gx(3)=gpos
gx(4)=-gpos
hx(1)=-gpos
hx(2)=-gpos
hx(3)=gpos
hx(4)=gpos
c assemble amatrx and rhs
do k=1,4
c loop through gauss pts
g=gx(k)
h=hx(k)
c get shape functions and derivatives
phi(1)=0.25*(1.-g)*(1.-h)
phi(2)=0.25*(1.+g)*(1.-h)
phi(3)=0.25*(1.+g)*(1.+h)
phi(4)=0.25*(1.-g)*(1.+h)
phig(1)=0.25*-(1.-h)
phig(2)=0.25*(1.-h)
phig(3)=0.25*(1.+h)
phig(4)=0.25*-(1.+h)
phih(1)=0.25*-(1.-g)
phih(2)=0.25*-(1.+g)
phih(3)=0.25*(1.+g)
phih(4)=0.25*(1.-g)
c get ip coords
crdx=0.
crdy=0.
do k1=1,4
crdx=crdx+phi(k1)*coords(1,k1)
crdy=crdy+phi(k1)*coords(2,k1)
end do
c get jacobian
rjac=0.
do i=1,4
rjac(1,1)=rjac(1,1)+phig(i)*coords(1,i)
rjac(1,2)=rjac(1,2)+phig(i)*coords(2,i)
rjac(2,1)=rjac(2,1)+phih(i)*coords(1,i)
rjac(2,2)=rjac(2,2)+phih(i)*coords(2,i)
enddo
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1)
print *,djac
rjaci(1,1) = rjac(2,2)/djac
rjaci(2,2) = rjac(1,1)/djac
rjaci(1,2) = -rjac(1,2)/djac
rjaci(2,1) = -rjac(2,1)/djac
c get b matrix
phix(1)=rjaci(1,1)*phig(1)+rjaci(1,2)*phih(1)
phiy(1)=rjaci(2,1)*phig(1)+rjaci(2,2)*phih(1)
phix(2)=rjaci(1,1)*phig(2)+rjaci(1,2)*phih(2)
phiy(2)=rjaci(2,1)*phig(2)+rjaci(2,2)*phih(2)
phix(3)=rjaci(1,1)*phig(3)+rjaci(1,2)*phih(3)
phiy(3)=rjaci(2,1)*phig(3)+rjaci(2,2)*phih(3)
phix(4)=rjaci(1,1)*phig(4)+rjaci(1,2)*phih(4)
phiy(4)=rjaci(2,1)*phig(4)+rjaci(2,2)*phih(4)
c interpolate temperatures to int points
dtdx=u(1)*phix(1)+u(2)*phix(2)
1 +u(3)*phix(3)+u(4)*phix(4)
dtdy=u(1)*phiy(1)+u(2)*phiy(2)
1 +u(3)*phiy(3)+u(4)*phiy(4)
t=u(1)*phi(1)+u(2)*phi(2)+u(3)*phi(3)+u(4)*phi(4)
told=(u(1)-du(1,nrhs))*phi(1)+(u(2)-du(2,nrhs))*phi(2)+
1 (u(3)-du(3,nrhs))*phi(3)+(u(4)-du(4,nrhs))*phi(4)
c other housekeeping
dtdt=(t-told)/dtime
we=djac
c Assemble Element Stiffness Matrix and Add to Global
do ki=1,4
c loop over nodes
rhs(ki,nrhs)=rhs(ki,nrhs)-we*(phi(ki)*rho*spec*dtdt
1 + cond*(phix(ki)*dtdx+phiy(ki)*dtdy))
do kj=1,4
amatrx(ki,kj)=amatrx(ki,kj)+we*(phi(ki)*phi(kj)
1 *rho*spec/dtime+cond*(phix(ki)*phix(kj)+
1 phiy(ki)*phiy(kj)))
end do
end do
enddo
end if
return
end

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SUBROUTINE UEL(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS,PROPS,
1 NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,DTIME,KSTEP,KINC,
2 JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,PREDEF,NPREDF,LFLAGS,
3 MLVARX,DDLMAG,MDLOAD,PNEWDT,JPROPS,NJPROP,PERIOD)
C
INCLUDE 'aba_param.inc'
C
DIMENSION RHS(MLVARX,*),AMATRX(NDOFEL,NDOFEL),SVARS(NSVARS),
1 ENERGY(8),PROPS(*),COORDS(MCRD,NNODE),
2 U(NDOFEL),DU(MLVARX,*),V(NDOFEL),A(NDOFEL),TIME(2),
3 PARAMS(3),JDLTYP(MDLOAD,*),ADLMAG(MDLOAD,*),
4 DDLMAG(MDLOAD,*),PREDEF(2,NPREDF,NNODE),LFLAGS(*),JPROPS(*)
C
DIMENSION GPX(9),GPY(9),GWEI(9),PHI(8),PHIX(8),PHIY(8),PHIC(8),
1 PHIE(8),IFACE(9),GWE(3),AR(3)
C
PARAMETER(ZERO=0.D0,TWOHUN=200.D0,FIVHUN=500.D0,CONDUC=204.D0)
DATA IFACE/1,5,2,6,3,7,4,8,1/
C
C
C MATERIAL PROPERTY DEFINITION
C
THICK = PROPS(1)
RHO = PROPS(2)
SPEC = PROPS(3)
C
C INITIALIZATION (NRHS=1)
C
DO 6 K1=1,NDOFEL
RHS(K1,NRHS)=ZERO
DO 4 K2=1,NDOFEL
AMATRX(K2,K1)=ZERO
4 CONTINUE
6 CONTINUE
C
IF (LFLAGS(3).EQ.4) RETURN
C
C TRANSIENT ANALYSIS
C
IF (LFLAGS(1).EQ.33) THEN
C
C DETERMINE GAUSS POINT LOCATIONS
C
SUBROUTINE GSPT(GPX,GPY)
INCLUDE 'aba_param.inc'
DIMENSION AR(3),GPX(9),GPY(9)
C
PARAMETER(ZERO=0.D0,ONENEG=-1.D0,ONE=1.D0,SIX=6.D0,TEN=10.D0)
C
C GPX: X COORDINATE OF GAUSS PT
C GPY: Y COORDINATE OF GAUSS PT
C
R=SQRT(SIX/TEN)
AR(1)=-1.
AR(2)=0.
AR(3)=1.
DO 10 I=1,3
DO 10 J=1,3
NUMGP=(I-1)*3+J
GPX(NUMGP)=AR(I)*R
GPY(NUMGP)=AR(J)*R
10 CONTINUE
RETURN
END
CALL GSPT(GPX,GPY)
C
C DETERMINE GAUSS WEIGHTS
C
CALL GSWT(GWEI,GWE)
C
C ASSEMBLE AMATRX AND RHS
C
DO 300 K=1,9
C LOOP THROUGH GAUSS PTS
C=GPX(K)
E=GPY(K)
CALL DER(C,E,GPX,GPY,GWEI,PHI,PHIX,PHIY,PHIC,PHIE
1 ,DXDC,DXDE,DYDC,DYDE,AJACOB,COORDS,MCRD,NNODE)
DTDX=ZERO
DTDY=ZERO
T =ZERO
TOLD=ZERO
DO I=1,8
DTDX=U(I)*PHIX(I)+DTDX
DTDY=U(I)*PHIY(I)+DTDY
T =U(I)*PHI(I)+T
TOLD=(U(I)-DU(I,NRHS))*PHI(I)+TOLD
END DO
C CHECK ON TEMPERATURE DEPENDENCE OF THERMAL CONDUCTIVITY
COND=CONDUC
DCDT=ZERO
DTDT=(T-TOLD)/DTIME
WE=GWEI(K)*AJACOB
DO KI=1,8
C LOOP OVER NODES
RHS(KI,NRHS) = RHS(KI,NRHS) -
1 WE*(PHI(KI)*RHO*SPEC*DTDT +
2 COND*(PHIX(KI)*DTDX + PHIY(KI)*DTDY))
DO KJ=1,8
AMATRX(KI,KJ)= AMATRX(KI,KJ) + WE*(PHI(KI)*PHI(KJ)*RHO*
1 SPEC/DTIME + COND*(PHIX(KI)*PHIX(KJ) + PHIY(KI)*
2 PHIY(KJ)) + DCDT*PHI(KJ)*(PHIX(KI)*DTDX +
3 PHIY(KI)*DTDY))
END DO
END DO
300 CONTINUE
C
RETURN
END
C
C
SUBROUTINE GSWT(GWEI,GWE)
INCLUDE 'aba_param.inc'
DIMENSION GWEI(9),GWE(3)
C
PARAMETER(FIVE=5.D0,EIGHT=8.D0,NINE=9.D0)
C
C GWEI : GAUSS WEIGHT
C
GWE(1)=FIVE/NINE
GWE(2)=EIGHT/NINE
GWE(3)=FIVE/NINE
DO 10 I=1,3
DO 10 J=1,3
NUMGP=(I-1)*3+J
GWEI(NUMGP)=GWE(I)*GWE(J)
10 CONTINUE
RETURN
END
C
SUBROUTINE DER(C,E,GPX,GPY,GWEI,PHI,PHIX,PHIY,PHIC,PHIE,
1 DXDC,DXDE,DYDC,DYDE,AJACOB,COORDS,MCRD,NNODE)
INCLUDE 'aba_param.inc'
DIMENSION PHI(8),PHIX(8),PHIY(8),PHIC(8),PHIE(8),
1 COORDS(MCRD,NNODE)
C
PARAMETER(ZERO=0.D0,FOURTH=0.25D0,HALF=0.5D0,ONE=1.D0,TWO=2.D0)
C
C INTERPOLATION FUNCTIONS
C
PHI(1) = FOURTH*(ONE-C)*(ONE-E)*(-C-E-ONE)
PHI(2) = FOURTH*(ONE+C)*(ONE-E)*(C-E-ONE)
PHI(3) = FOURTH*(ONE+C)*(ONE+E)*(C+E-ONE)
PHI(4) = FOURTH*(ONE-C)*(ONE+E)*(-C+E-ONE)
PHI(5) = HALF*(ONE-C*C)*(ONE-E)
PHI(6) = HALF*(ONE+C)*(ONE-E*E)
PHI(7) = HALF*(ONE-C*C)*(ONE+E)
PHI(8) = HALF*(ONE-C)*(ONE-E*E)
C
C DERIVATIVES WRT TO C
C
PHIC(1) = FOURTH*(ONE-E)*(TWO*C+E)
PHIC(2) = FOURTH*(ONE-E)*(TWO*C-E)
PHIC(3) = FOURTH*(ONE+E)*(TWO*C+E)
PHIC(4) = FOURTH*(ONE+E)*(TWO*C-E)
PHIC(5) = -C*(ONE-E)
PHIC(6) = HALF*(ONE-E*E)
PHIC(7) = -C*(ONE+E)
PHIC(8) = -HALF*(ONE-E*E)
C
C DERIVATIVES WRT TO E
C
PHIE(1) = FOURTH*(ONE-C)*(TWO*E+C)
PHIE(2) = FOURTH*(ONE+C)*(TWO*E-C)
PHIE(3) = FOURTH*(ONE+C)*(TWO*E+C)
PHIE(4) = FOURTH*(ONE-C)*(TWO*E-C)
PHIE(5) = -HALF*(ONE-C*C)
PHIE(6) = -E*(ONE+C)
PHIE(7) = HALF*(ONE-C*C)
PHIE(8) = -E*(ONE-C)
DXDC=ZERO
DXDE=ZERO
DYDC=ZERO
DYDE=ZERO
DO 3 I=1,8
DXDC=DXDC+COORDS(1,I)*PHIC(I)
DXDE=DXDE+COORDS(1,I)*PHIE(I)
DYDC=DYDC+COORDS(2,I)*PHIC(I)
DYDE=DYDE+COORDS(2,I)*PHIE(I)
3 CONTINUE
C
C CALCULATION OF JACOBIAN
C
AJACOB=(DXDC*DYDE-DXDE*DYDC)
C
C DERIVATIVES WRT TO X AND Y
C
DO 5 I=1,8
PHIX(I)=(PHIC(I)*DYDE-PHIE(I)*DYDC)/AJACOB
PHIY(I)=(PHIE(I)*DXDC-PHIC(I)*DXDE)/AJACOB
5 CONTINUE
RETURN
END

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SUBROUTINE UEL(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS,
1 PROPS,NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,
2 DTIME,KSTEP,KINC,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,
3 PREDEF,NPREDF,LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,
4 JPROPS,NJPROP,PERIOD)
C
INCLUDE 'ABA_PARAM.INC'
PARAMETER ( ZERO = 0.D0, HALF = 0.5D0, ONE = 1.D0 )
C
DIMENSION RHS(MLVARX,*),AMATRX(NDOFEL,NDOFEL),
1 SVARS(NSVARS),ENERGY(8),PROPS(*),COORDS(MCRD,NNODE),
2 U(NDOFEL),DU(MLVARX,*),V(NDOFEL),A(NDOFEL),TIME(2),
3 PARAMS(3),JDLTYP(MDLOAD,*),ADLMAG(MDLOAD,*),
4 DDLMAG(MDLOAD,*),PREDEF(2,NPREDF,NNODE),LFLAGS(*),
5 JPROPS(*)
DIMENSION SRESID(6)
C
C UEL SUBROUTINE FOR A HORIZONTAL TRUSS ELEMENT
C
C SRESID - stores the static residual at time t+dt
C SVARS - In 1-6, contains the static residual at time t
C upon entering the routine. SRESID is copied to
C SVARS(1-6) after the dynamic residual has been
C calculated.
C - For half-increment residual calculations: In 7-12,
C contains the static residual at the beginning
C of the previous increment. SVARS(1-6) are copied
C into SVARS(7-12) after the dynamic residual has
C been calculated.
C
AREA = PROPS(1)
E = PROPS(2)
ANU = PROPS(3)
RHO = PROPS(4)
C
ALEN = ABS(COORDS(1,2)-COORDS(1,1))
AK = AREA*E/ALEN
AM = HALF*AREA*RHO*ALEN
C
DO K1 = 1, NDOFEL
SRESID(K1) = ZERO
DO KRHS = 1, NRHS
RHS(K1,KRHS) = ZERO
END DO
DO K2 = 1, NDOFEL
AMATRX(K2,K1) = ZERO
END DO
END DO
C
IF (LFLAGS(3).EQ.1) THEN
C Normal incrementation
IF (LFLAGS(1).EQ.1 .OR. LFLAGS(1).EQ.2) THEN
C *STATIC
AMATRX(1,1) = AK
AMATRX(4,4) = AK
AMATRX(1,4) = -AK
AMATRX(4,1) = -AK
IF (LFLAGS(4).NE.0) THEN
FORCE = AK*(U(4)-U(1))
DFORCE = AK*(DU(4,1)-DU(1,1))
SRESID(1) = -DFORCE
SRESID(4) = DFORCE
RHS(1,1) = RHS(1,1)-SRESID(1)
RHS(4,1) = RHS(4,1)-SRESID(4)
ENERGY(2) = HALF*FORCE*(DU(4,1)-DU(1,1))
* + HALF*DFORCE*(U(4)-U(1))
* + HALF*DFORCE*(DU(4,1)-DU(1,1))
ELSE
FORCE = AK*(U(4)-U(1))
SRESID(1) = -FORCE
SRESID(4) = FORCE
RHS(1,1) = RHS(1,1)-SRESID(1)
RHS(4,1) = RHS(4,1)-SRESID(4)
DO KDLOAD = 1, NDLOAD
IF (JDLTYP(KDLOAD,1).EQ.1001) THEN
RHS(4,1) = RHS(4,1)+ADLMAG(KDLOAD,1)
ENERGY(8) = ENERGY(8)+(ADLMAG(KDLOAD,1)
* - HALF*DDLMAG(KDLOAD,1))*DU(4,1)
IF (NRHS.EQ.2) THEN
C Riks
RHS(4,2) = RHS(4,2)+DDLMAG(KDLOAD,1)
END IF
END IF
END DO
ENERGY(2) = HALF*FORCE*(U(4)-U(1))
END IF
ELSE IF (LFLAGS(1).EQ.11 .OR. LFLAGS(1).EQ.12) THEN
C *DYNAMIC
ALPHA = PARAMS(1)
BETA = PARAMS(2)
GAMMA = PARAMS(3)
C
DADU = ONE/(BETA*DTIME**2)
DVDU = GAMMA/(BETA*DTIME)
C
DO K1 = 1, NDOFEL
AMATRX(K1,K1) = AM*DADU
RHS(K1,1) = RHS(K1,1)-AM*A(K1)
END DO
AMATRX(1,1) = AMATRX(1,1)+(ONE+ALPHA)*AK
AMATRX(4,4) = AMATRX(4,4)+(ONE+ALPHA)*AK
AMATRX(1,4) = AMATRX(1,4)-(ONE+ALPHA)*AK
AMATRX(4,1) = AMATRX(4,1)-(ONE+ALPHA)*AK
FORCE = AK*(U(4)-U(1))
SRESID(1) = -FORCE
SRESID(4) = FORCE
RHS(1,1) = RHS(1,1) -
* ((ONE+ALPHA)*SRESID(1)-ALPHA*SVARS(1))
RHS(4,1) = RHS(4,1) -
* ((ONE+ALPHA)*SRESID(4)-ALPHA*SVARS(4))
ENERGY(1) = ZERO
DO K1 = 1, NDOFEL
SVARS(K1+6) = SVARS(k1)
SVARS(K1) = SRESID(K1)
ENERGY(1) = ENERGY(1)+HALF*V(K1)*AM*V(K1)
END DO
ENERGY(2) = HALF*FORCE*(U(4)-U(1))
END IF
ELSE IF (LFLAGS(3).EQ.2) THEN
C Stiffness matrix
AMATRX(1,1) = AK
AMATRX(4,4) = AK
AMATRX(1,4) = -AK
AMATRX(4,1) = -AK
ELSE IF (LFLAGS(3).EQ.4) THEN
C Mass matrix
DO K1 = 1, NDOFEL
AMATRX(K1,K1) = AM
END DO
ELSE IF (LFLAGS(3).EQ.5) THEN
C Half-increment residual calculation
ALPHA = PARAMS(1)
FORCE = AK*(U(4)-U(1))
SRESID(1) = -FORCE
SRESID(4) = FORCE
RHS(1,1) = RHS(1,1)-AM*A(1)-(ONE+ALPHA)*SRESID(1)
* + HALF*ALPHA*( SVARS(1)+SVARS(7) )
RHS(4,1) = RHS(4,1)-AM*A(4)-(ONE+ALPHA)*SRESID(4)
* + HALF*ALPHA*( SVARS(4)+SVARS(10) )
ELSE IF (LFLAGS(3).EQ.6) THEN
C Initial acceleration calculation
DO K1 = 1, NDOFEL
AMATRX(K1,K1) = AM
END DO
FORCE = AK*(U(4)-U(1))
SRESID(1) = -FORCE
SRESID(4) = FORCE
RHS(1,1) = RHS(1,1)-SRESID(1)
RHS(4,1) = RHS(4,1)-SRESID(4)
ENERGY(1) = ZERO
DO K1 = 1, NDOFEL
SVARS(K1) = SRESID(K1)
ENERGY(1) = ENERGY(1)+HALF*V(K1)*AM*V(K1)
END DO
ENERGY(2) = HALF*FORCE*(U(4)-U(1))
ELSE IF (LFLAGS(3).EQ.100) THEN
C Output for perturbations
IF (LFLAGS(1).EQ.1 .OR. LFLAGS(1).EQ.2) THEN
C *STATIC
FORCE = AK*(U(4)-U(1))
DFORCE = AK*(DU(4,1)-DU(1,1))
SRESID(1) = -DFORCE
SRESID(4) = DFORCE
RHS(1,1) = RHS(1,1)-SRESID(1)
RHS(4,1) = RHS(4,1)-SRESID(4)
ENERGY(2) = HALF*FORCE*(DU(4,1)-DU(1,1))
* + HALF*DFORCE*(U(4)-U(1))
* + HALF*DFORCE*(DU(4,1)-DU(1,1))
DO KVAR = 1, NSVARS
SVARS(KVAR) = ZERO
END DO
SVARS(1) = RHS(1,1)
SVARS(4) = RHS(4,1)
ELSE IF (LFLAGS(1).EQ.41) THEN
C *FREQUENCY
DO KRHS = 1, NRHS
DFORCE = AK*(DU(4,KRHS)-DU(1,KRHS))
SRESID(1) = -DFORCE
SRESID(4) = DFORCE
RHS(1,KRHS) = RHS(1,KRHS)-SRESID(1)
RHS(4,KRHS) = RHS(4,KRHS)-SRESID(4)
END DO
DO KVAR = 1, NSVARS
SVARS(KVAR) = ZERO
END DO
SVARS(1) = RHS(1,1)
SVARS(4) = RHS(4,1)
END IF
END IF
C
RETURN
END

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c***********************************************************
subroutine uelmat(rhs,amatrx,svars,energy,ndofel,nrhs,
1 nsvars,props,nprops,coords,mcrd,nnode,u,du,
2 v,a,jtype,time,dtime,kstep,kinc,jelem,params,
3 ndload,jdltyp,adlmag,predef,npredf,lflags,mlvarx,
4 ddlmag,mdload,pnewdt,jprops,njpro,period,
5 materiallib)
c
include 'aba_param.inc'
C
dimension rhs(mlvarx,*), amatrx(ndofel, ndofel), props(*),
1 svars(*), energy(*), coords(mcrd, nnode), u(ndofel),
2 du(mlvarx,*), v(ndofel), a(ndofel), time(2), params(*),
3 jdltyp(mdload,*), adlmag(mdload,*), ddlmag(mdload,*),
4 predef(2, npredf, nnode), lflags(*), jprops(*)
parameter (zero=0.d0, dmone=-1.0d0, one=1.d0, four=4.0d0,
1 fourth=0.25d0,gaussCoord=0.577350269d0)
parameter (ndim=2, ndof=2, nshr=1,nnodemax=4,
1 ntens=4, ninpt=4, nsvint=4)
c
c ndim ... number of spatial dimensions
c ndof ... number of degrees of freedom per node
c nshr ... number of shear stress component
c ntens ... total number of stress tensor components
c (=ndi+nshr)
c ninpt ... number of integration points
c nsvint... number of state variables per integration pt
c (strain)
c
dimension stiff(ndof*nnodemax,ndof*nnodemax),
1 force(ndof*nnodemax), shape(nnodemax), dshape(ndim,nnodemax),
2 xjac(ndim,ndim),xjaci(ndim,ndim), bmat(nnodemax*ndim),
3 statevLocal(nsvint),stress(ntens), ddsdde(ntens, ntens),
4 stran(ntens), dstran(ntens), wght(ninpt)
c
dimension predef_loc(npredf),dpredef_loc(npredf),
1 defGrad(3,3),utmp(3),xdu(3),stiff_p(3,3),force_p(3)
dimension coord24(2,4),coords_ip(3)
data coord24 /dmone, dmone,
2 one, dmone,
3 one, one,
4 dmone, one/
c
data wght /one, one, one, one/
c
c*************************************************************
c
c U1 = first-order, plane strain, full integration
c
c State variables: each integration point has nsvint SDVs
c
c isvinc=(npt-1)*nsvint ... integration point counter
c statev(1+isvinc ) ... strain
c
c*************************************************************
if (lflags(3).eq.4) then
do i=1, ndofel
do j=1, ndofel
amatrx(i,j) = zero
end do
amatrx(i,i) = one
end do
goto 999
end if
c
c PRELIMINARIES
c
pnewdtLocal = pnewdt
if(jtype .ne. 1) then
write(7,*)'Incorrect element type'
call xit
endif
if(nsvars .lt. ninpt*nsvint) then
write(7,*)'Increase the number of SDVs to', ninpt*nsvint
call xit
endif
thickness = 0.1d0
c
c INITIALIZE RHS AND LHS
c
do k1=1, ndof*nnode
rhs(k1, 1)= zero
do k2=1, ndof*nnode
amatrx(k1, k2)= zero
end do
end do
c
c LOOP OVER INTEGRATION POINTS
c
do kintk = 1, ninpt
c
c EVALUATE SHAPE FUNCTIONS AND THEIR DERIVATIVES
c
c determine (g,h)
c
g = coord24(1,kintk)*gaussCoord
h = coord24(2,kintk)*gaussCoord
c
c shape functions
shape(1) = (one - g)*(one - h)/four;
shape(2) = (one + g)*(one - h)/four;
shape(3) = (one + g)*(one + h)/four;
shape(4) = (one - g)*(one + h)/four;
c
c derivative d(Ni)/d(g)
dshape(1,1) = -(one - h)/four;
dshape(1,2) = (one - h)/four;
dshape(1,3) = (one + h)/four;
dshape(1,4) = -(one + h)/four;
c
c derivative d(Ni)/d(h)
dshape(2,1) = -(one - g)/four;
dshape(2,2) = -(one + g)/four;
dshape(2,3) = (one + g)/four;
dshape(2,4) = (one - g)/four;
c
c compute coordinates at the integration point
c
do k1=1, 3
coords_ip(k1) = zero
end do
do k1=1,nnode
do k2=1,mcrd
coords_ip(k2)=coords_ip(k2)+shape(k1)*coords(k2,k1)
end do
end do
c
c INTERPOLATE FIELD VARIABLES
c
if(npredf.gt.0) then
do k1=1,npredf
predef_loc(k1) = zero
dpredef_loc(k1) = zero
do k2=1,nnode
predef_loc(k1) =
& predef_loc(k1)+
& (predef(1,k1,k2)-predef(2,k1,k2))*shape(k2)
dpredef_loc(k1) =
& dpredef_loc(k1)+predef(2,k1,k2)*shape(k2)
end do
end do
end if
c
c FORM B-MATRIX
c
djac = one
c
do i = 1, ndim
do j = 1, ndim
xjac(i,j) = zero
xjaci(i,j) = zero
end do
end do
c
do inod= 1, nnode
do idim = 1, ndim
do jdim = 1, ndim
xjac(jdim,idim) = xjac(jdim,idim) +
1 dshape(jdim,inod)*coords(idim,inod)
end do
end do
end do
djac = xjac(1,1)*xjac(2,2) - xjac(1,2)*xjac(2,1)
if (djac .gt. zero) then
! jacobian is positive - o.k.
xjaci(1,1) = xjac(2,2)/djac
xjaci(2,2) = xjac(1,1)/djac
xjaci(1,2) = -xjac(1,2)/djac
xjaci(2,1) = -xjac(2,1)/djac
else
! negative or zero jacobian
write(7,*)'WARNING: element',jelem,'has neg.
1 Jacobian'
pnewdt = fourth
endif
if (pnewdt .lt. pnewdtLocal) pnewdtLocal = pnewdt
c
do i = 1, nnode*ndim
bmat(i) = zero
end do
do inod = 1, nnode
do ider = 1, ndim
do idim = 1, ndim
irow = idim + (inod - 1)*ndim
bmat(irow) = bmat(irow) +
1 xjaci(idim,ider)*dshape(ider,inod)
end do
end do
end do
c
c CALCULATE INCREMENTAL STRAINS
c
do i = 1, ntens
dstran(i) = zero
end do
!
! set deformation gradient to Identity matrix
do k1=1,3
do k2=1,3
defGrad(k1,k2) = zero
end do
defGrad(k1,k1) = one
end do
c
c COMPUTE INCREMENTAL STRAINS
c
do nodi = 1, nnode
incr_row = (nodi - 1)*ndof
do i = 1, ndof
xdu(i)= du(i + incr_row,1)
utmp(i) = u(i + incr_row)
end do
dNidx = bmat(1 + (nodi-1)*ndim)
dNidy = bmat(2 + (nodi-1)*ndim)
dstran(1) = dstran(1) + dNidx*xdu(1)
dstran(2) = dstran(2) + dNidy*xdu(2)
dstran(4) = dstran(4) +
1 dNidy*xdu(1) +
2 dNidx*xdu(2)
c deformation gradient
defGrad(1,1) = defGrad(1,1) + dNidx*utmp(1)
defGrad(1,2) = defGrad(1,2) + dNidy*utmp(1)
defGrad(2,1) = defGrad(2,1) + dNidx*utmp(2)
defGrad(2,2) = defGrad(2,2) + dNidy*utmp(2)
end do
c
c CALL CONSTITUTIVE ROUTINE
c
isvinc= (kintk-1)*nsvint ! integration point increment
c
c prepare arrays for entry into material routines
c
do i = 1, nsvint
statevLocal(i)=svars(i+isvinc)
end do
c
c state variables
c
do k1=1,ntens
stran(k1) = statevLocal(k1)
stress(k1) = zero
end do
c
do i=1, ntens
do j=1, ntens
ddsdde(i,j) = zero
end do
ddsdde(i,j) = one
enddo
c
c compute characteristic element length
c
celent = sqrt(djac*dble(ninpt))
dvmat = djac*thickness
c
dvdv0 = one
call material_lib_mech(materiallib,stress,ddsdde,
1 stran,dstran,kintk,dvdv0,dvmat,defGrad,
2 predef_loc,dpredef_loc,npredf,celent,coords_ip)
c
do k1=1,ntens
statevLocal(k1) = stran(k1) + dstran(k1)
end do
c
isvinc= (kintk-1)*nsvint ! integration point increment
c
c update element state variables
c
do i = 1, nsvint
svars(i+isvinc)=statevLocal(i)
end do
c
c form stiffness matrix and internal force vector
c
dNjdx = zero
dNjdy = zero
do i = 1, ndof*nnode
force(i) = zero
do j = 1, ndof*nnode
stiff(j,i) = zero
end do
end do
dvol= wght(kintk)*djac
do nodj = 1, nnode
incr_col = (nodj - 1)*ndof
dNjdx = bmat(1+(nodj-1)*ndim)
dNjdy = bmat(2+(nodj-1)*ndim)
force_p(1) = dNjdx*stress(1) + dNjdy*stress(4)
force_p(2) = dNjdy*stress(2) + dNjdx*stress(4)
do jdof = 1, ndof
jcol = jdof + incr_col
force(jcol) = force(jcol) +
& force_p(jdof)*dvol
end do
do nodi = 1, nnode
incr_row = (nodi -1)*ndof
dNidx = bmat(1+(nodi-1)*ndim)
dNidy = bmat(2+(nodi-1)*ndim)
stiff_p(1,1) = dNidx*ddsdde(1,1)*dNjdx
& + dNidy*ddsdde(4,4)*dNjdy
& + dNidx*ddsdde(1,4)*dNjdy
& + dNidy*ddsdde(4,1)*dNjdx
stiff_p(1,2) = dNidx*ddsdde(1,2)*dNjdy
& + dNidy*ddsdde(4,4)*dNjdx
& + dNidx*ddsdde(1,4)*dNjdx
& + dNidy*ddsdde(4,2)*dNjdy
stiff_p(2,1) = dNidy*ddsdde(2,1)*dNjdx
& + dNidx*ddsdde(4,4)*dNjdy
& + dNidy*ddsdde(2,4)*dNjdy
& + dNidx*ddsdde(4,1)*dNjdx
stiff_p(2,2) = dNidy*ddsdde(2,2)*dNjdy
& + dNidx*ddsdde(4,4)*dNjdx
& + dNidy*ddsdde(2,4)*dNjdx
& + dNidx*ddsdde(4,2)*dNjdy
do jdof = 1, ndof
icol = jdof + incr_col
do idof = 1, ndof
irow = idof + incr_row
stiff(irow,icol) = stiff(irow,icol) +
& stiff_p(idof,jdof)*dvol
end do
end do
end do
end do
c
c assemble rhs and lhs
c
do k1=1, ndof*nnode
rhs(k1, 1) = rhs(k1, 1) - force(k1)
do k2=1, ndof*nnode
amatrx(k1, k2) = amatrx(k1, k2) + stiff(k1,k2)
end do
end do
end do ! end loop on material integration points
pnewdt = pnewdtLocal
c
999 continue
c
return
end

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c***********************************************************
subroutine uelmat(rhs,amatrx,svars,energy,ndofel,nrhs,
1 nsvars,props,nprops,coords,mcrd,nnode,u,du,
2 v,a,jtype,time,dtime,kstep,kinc,jelem,params,
3 ndload,jdltyp,adlmag,predef,npredf,lflags,mlvarx,
4 ddlmag,mdload,pnewdt,jprops,njpro,period,
5 materiallib)
c
include 'aba_param.inc'
C
dimension rhs(mlvarx,*), amatrx(ndofel, ndofel), props(*),
1 svars(*), energy(*), coords(mcrd, nnode), u(ndofel),
2 du(mlvarx,*), v(ndofel), a(ndofel), time(2), params(*),
3 jdltyp(mdload,*), adlmag(mdload,*), ddlmag(mdload,*),
4 predef(2, npredf, nnode), lflags(*), jprops(*)
parameter (zero=0.d0, dmone=-1.0d0, one=1.d0, four=4.0d0,
1 fourth=0.25d0,gaussCoord=0.577350269d0)
parameter (ndim=2, ndof=2, nshr=1,nnodemax=4,
1 ntens=4, ninpt=4, nsvint=4)
c
c ndim ... number of spatial dimensions
c ndof ... number of degrees of freedom per node
c nshr ... number of shear stress component
c ntens ... total number of stress tensor components
c (=ndi+nshr)
c ninpt ... number of integration points
c nsvint... number of state variables per integration pt
c (strain)
c
dimension stiff(ndof*nnodemax,ndof*nnodemax),
1 force(ndof*nnodemax), shape(nnodemax), dshape(ndim,nnodemax),
2 xjac(ndim,ndim),xjaci(ndim,ndim), bmat(nnodemax*ndim),
3 statevLocal(nsvint),stress(ntens), ddsdde(ntens, ntens),
4 stran(ntens), dstran(ntens), wght(ninpt)
c
dimension predef_loc(npredf),dpredef_loc(npredf),
1 defGrad(3,3),utmp(3),xdu(3),stiff_p(3,3),force_p(3)
dimension coord24(2,4),coords_ip(3)
data coord24 /dmone, dmone,
2 one, dmone,
3 one, one,
4 dmone, one/
c
data wght /one, one, one, one/
c
c*************************************************************
c
c U1 = first-order, plane strain, full integration
c
c State variables: each integration point has nsvint SDVs
c
c isvinc=(npt-1)*nsvint ... integration point counter
c statev(1+isvinc ) ... strain
c
c*************************************************************
if (lflags(3).eq.4) then
do i=1, ndofel
do j=1, ndofel
amatrx(i,j) = zero
end do
amatrx(i,i) = one
end do
goto 999
end if
c
c PRELIMINARIES
c
pnewdtLocal = pnewdt
if(jtype .ne. 1) then
write(7,*)'Incorrect element type'
call xit
endif
if(nsvars .lt. ninpt*nsvint) then
write(7,*)'Increase the number of SDVs to', ninpt*nsvint
call xit
endif
thickness = 0.1d0
c
c INITIALIZE RHS AND LHS
c
do k1=1, ndof*nnode
rhs(k1, 1)= zero
do k2=1, ndof*nnode
amatrx(k1, k2)= zero
end do
end do
c
c LOOP OVER INTEGRATION POINTS
c
do kintk = 1, ninpt
c
c EVALUATE SHAPE FUNCTIONS AND THEIR DERIVATIVES
c
c determine (g,h)
c
g = coord24(1,kintk)*gaussCoord
h = coord24(2,kintk)*gaussCoord
c
c shape functions
shape(1) = (one - g)*(one - h)/four;
shape(2) = (one + g)*(one - h)/four;
shape(3) = (one + g)*(one + h)/four;
shape(4) = (one - g)*(one + h)/four;
c
c derivative d(Ni)/d(g)
dshape(1,1) = -(one - h)/four;
dshape(1,2) = (one - h)/four;
dshape(1,3) = (one + h)/four;
dshape(1,4) = -(one + h)/four;
c
c derivative d(Ni)/d(h)
dshape(2,1) = -(one - g)/four;
dshape(2,2) = -(one + g)/four;
dshape(2,3) = (one + g)/four;
dshape(2,4) = (one - g)/four;
c
c compute coordinates at the integration point
c
do k1=1, 3
coords_ip(k1) = zero
end do
do k1=1,nnode
do k2=1,mcrd
coords_ip(k2)=coords_ip(k2)+shape(k1)*coords(k2,k1)
end do
end do
c
c INTERPOLATE FIELD VARIABLES
c
if(npredf.gt.0) then
do k1=1,npredf
predef_loc(k1) = zero
dpredef_loc(k1) = zero
do k2=1,nnode
predef_loc(k1) =
& predef_loc(k1)+
& (predef(1,k1,k2)-predef(2,k1,k2))*shape(k2)
dpredef_loc(k1) =
& dpredef_loc(k1)+predef(2,k1,k2)*shape(k2)
end do
end do
end if
c
c FORM B-MATRIX
c
djac = one
c
do i = 1, ndim
do j = 1, ndim
xjac(i,j) = zero
xjaci(i,j) = zero
end do
end do
c
do inod= 1, nnode
do idim = 1, ndim
do jdim = 1, ndim
xjac(jdim,idim) = xjac(jdim,idim) +
1 dshape(jdim,inod)*coords(idim,inod)
end do
end do
end do
djac = xjac(1,1)*xjac(2,2) - xjac(1,2)*xjac(2,1)
if (djac .gt. zero) then
! jacobian is positive - o.k.
xjaci(1,1) = xjac(2,2)/djac
xjaci(2,2) = xjac(1,1)/djac
xjaci(1,2) = -xjac(1,2)/djac
xjaci(2,1) = -xjac(2,1)/djac
else
! negative or zero jacobian
write(7,*)'WARNING: element',jelem,'has neg.
1 Jacobian'
pnewdt = fourth
endif
if (pnewdt .lt. pnewdtLocal) pnewdtLocal = pnewdt
c
do i = 1, nnode*ndim
bmat(i) = zero
end do
do inod = 1, nnode
do ider = 1, ndim
do idim = 1, ndim
irow = idim + (inod - 1)*ndim
bmat(irow) = bmat(irow) +
1 xjaci(idim,ider)*dshape(ider,inod)
end do
end do
end do
c
c CALCULATE INCREMENTAL STRAINS
c
do i = 1, ntens
dstran(i) = zero
end do
!
! set deformation gradient to Identity matrix
do k1=1,3
do k2=1,3
defGrad(k1,k2) = zero
end do
defGrad(k1,k1) = one
end do
c
c COMPUTE INCREMENTAL STRAINS
c
do nodi = 1, nnode
incr_row = (nodi - 1)*ndof
do i = 1, ndof
xdu(i)= du(i + incr_row,1)
utmp(i) = u(i + incr_row)
end do
dNidx = bmat(1 + (nodi-1)*ndim)
dNidy = bmat(2 + (nodi-1)*ndim)
dstran(1) = dstran(1) + dNidx*xdu(1)
dstran(2) = dstran(2) + dNidy*xdu(2)
dstran(4) = dstran(4) +
1 dNidy*xdu(1) +
2 dNidx*xdu(2)
c deformation gradient
defGrad(1,1) = defGrad(1,1) + dNidx*utmp(1)
defGrad(1,2) = defGrad(1,2) + dNidy*utmp(1)
defGrad(2,1) = defGrad(2,1) + dNidx*utmp(2)
defGrad(2,2) = defGrad(2,2) + dNidy*utmp(2)
end do
c
c CALL CONSTITUTIVE ROUTINE
c
isvinc= (kintk-1)*nsvint ! integration point increment
c
c prepare arrays for entry into material routines
c
do i = 1, nsvint
statevLocal(i)=svars(i+isvinc)
end do
c
c state variables
c
do k1=1,ntens
stran(k1) = statevLocal(k1)
stress(k1) = zero
end do
c
do i=1, ntens
do j=1, ntens
ddsdde(i,j) = zero
end do
ddsdde(i,j) = one
enddo
c
c compute characteristic element length
c
celent = sqrt(djac*dble(ninpt))
dvmat = djac*thickness
c
dvdv0 = one
call material_lib_mech(materiallib,stress,ddsdde,
1 stran,dstran,kintk,dvdv0,dvmat,defGrad,
2 predef_loc,dpredef_loc,npredf,celent,coords_ip)
c
do k1=1,ntens
statevLocal(k1) = stran(k1) + dstran(k1)
end do
c
isvinc= (kintk-1)*nsvint ! integration point increment
c
c update element state variables
c
do i = 1, nsvint
svars(i+isvinc)=statevLocal(i)
end do
c
c form stiffness matrix and internal force vector
c
dNjdx = zero
dNjdy = zero
do i = 1, ndof*nnode
force(i) = zero
do j = 1, ndof*nnode
stiff(j,i) = zero
end do
end do
dvol= wght(kintk)*djac
do nodj = 1, nnode
incr_col = (nodj - 1)*ndof
dNjdx = bmat(1+(nodj-1)*ndim)
dNjdy = bmat(2+(nodj-1)*ndim)
force_p(1) = dNjdx*stress(1) + dNjdy*stress(4)
force_p(2) = dNjdy*stress(2) + dNjdx*stress(4)
do jdof = 1, ndof
jcol = jdof + incr_col
force(jcol) = force(jcol) +
& force_p(jdof)*dvol
end do
do nodi = 1, nnode
incr_row = (nodi -1)*ndof
dNidx = bmat(1+(nodi-1)*ndim)
dNidy = bmat(2+(nodi-1)*ndim)
stiff_p(1,1) = dNidx*ddsdde(1,1)*dNjdx
& + dNidy*ddsdde(4,4)*dNjdy
& + dNidx*ddsdde(1,4)*dNjdy
& + dNidy*ddsdde(4,1)*dNjdx
stiff_p(1,2) = dNidx*ddsdde(1,2)*dNjdy
& + dNidy*ddsdde(4,4)*dNjdx
& + dNidx*ddsdde(1,4)*dNjdx
& + dNidy*ddsdde(4,2)*dNjdy
stiff_p(2,1) = dNidy*ddsdde(2,1)*dNjdx
& + dNidx*ddsdde(4,4)*dNjdy
& + dNidy*ddsdde(2,4)*dNjdy
& + dNidx*ddsdde(4,1)*dNjdx
stiff_p(2,2) = dNidy*ddsdde(2,2)*dNjdy
& + dNidx*ddsdde(4,4)*dNjdx
& + dNidy*ddsdde(2,4)*dNjdx
& + dNidx*ddsdde(4,2)*dNjdy
do jdof = 1, ndof
icol = jdof + incr_col
do idof = 1, ndof
irow = idof + incr_row
stiff(irow,icol) = stiff(irow,icol) +
& stiff_p(idof,jdof)*dvol
end do
end do
end do
end do
c
c assemble rhs and lhs
c
do k1=1, ndof*nnode
rhs(k1, 1) = rhs(k1, 1) - force(k1)
do k2=1, ndof*nnode
amatrx(k1, k2) = amatrx(k1, k2) + stiff(k1,k2)
end do
end do
end do ! end loop on material integration points
pnewdt = pnewdtLocal
c
999 continue
c
return
end

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c***********************************************************
subroutine uelmat(rhs,amatrx,svars,energy,ndofel,nrhs,
1 nsvars,props,nprops,coords,mcrd,nnode,u,du,
2 v,a,jtype,time,dtime,kstep,kinc,jelem,params,
3 ndload,jdltyp,adlmag,predef,npredf,lflags,mlvarx,
4 ddlmag,mdload,pnewdt,jprops,njpro,period,
5 materiallib)
c
include 'aba_param.inc'
C
dimension rhs(mlvarx,*), amatrx(ndofel, ndofel), props(*),
1 svars(*), energy(*), coords(mcrd, nnode), u(ndofel),
2 du(mlvarx,*), v(ndofel), a(ndofel), time(2), params(*),
3 jdltyp(mdload,*), adlmag(mdload,*), ddlmag(mdload,*),
4 predef(2, npredf, nnode), lflags(*), jprops(*)
parameter (zero=0.d0, dmone=-1.0d0, one=1.d0, four=4.0d0,
1 fourth=0.25d0,gaussCoord=0.577350269d0)
parameter (ndim=2, ndof=2, nshr=1,nnodemax=4,
1 ntens=4, ninpt=4, nsvint=4)
c
c ndim ... number of spatial dimensions
c ndof ... number of degrees of freedom per node
c nshr ... number of shear stress component
c ntens ... total number of stress tensor components
c (=ndi+nshr)
c ninpt ... number of integration points
c nsvint... number of state variables per integration pt
c (strain)
c
dimension stiff(ndof*nnodemax,ndof*nnodemax),
1 force(ndof*nnodemax), shape(nnodemax), dshape(ndim,nnodemax),
2 xjac(ndim,ndim),xjaci(ndim,ndim), bmat(nnodemax*ndim),
3 statevLocal(nsvint),stress(ntens), ddsdde(ntens, ntens),
4 stran(ntens), dstran(ntens), wght(ninpt)
c
dimension predef_loc(npredf),dpredef_loc(npredf),
1 defGrad(3,3),utmp(3),xdu(3),stiff_p(3,3),force_p(3)
dimension coord24(2,4),coords_ip(3)
data coord24 /dmone, dmone,
2 one, dmone,
3 one, one,
4 dmone, one/
c
data wght /one, one, one, one/
c
c*************************************************************
c
c U1 = first-order, plane strain, full integration
c
c State variables: each integration point has nsvint SDVs
c
c isvinc=(npt-1)*nsvint ... integration point counter
c statev(1+isvinc ) ... strain
c
c*************************************************************
if (lflags(3).eq.4) then
do i=1, ndofel
do j=1, ndofel
amatrx(i,j) = zero
end do
amatrx(i,i) = one
end do
goto 999
end if
c
c PRELIMINARIES
c
pnewdtLocal = pnewdt
if(jtype .ne. 1) then
write(7,*)'Incorrect element type'
call xit
endif
if(nsvars .lt. ninpt*nsvint) then
write(7,*)'Increase the number of SDVs to', ninpt*nsvint
call xit
endif
thickness = 0.1d0
c
c INITIALIZE RHS AND LHS
c
do k1=1, ndof*nnode
rhs(k1, 1)= zero
do k2=1, ndof*nnode
amatrx(k1, k2)= zero
end do
end do
c
c LOOP OVER INTEGRATION POINTS
c
do kintk = 1, ninpt
c
c EVALUATE SHAPE FUNCTIONS AND THEIR DERIVATIVES
c
c determine (g,h)
c
g = coord24(1,kintk)*gaussCoord
h = coord24(2,kintk)*gaussCoord
c
c shape functions
shape(1) = (one - g)*(one - h)/four;
shape(2) = (one + g)*(one - h)/four;
shape(3) = (one + g)*(one + h)/four;
shape(4) = (one - g)*(one + h)/four;
c
c derivative d(Ni)/d(g)
dshape(1,1) = -(one - h)/four;
dshape(1,2) = (one - h)/four;
dshape(1,3) = (one + h)/four;
dshape(1,4) = -(one + h)/four;
c
c derivative d(Ni)/d(h)
dshape(2,1) = -(one - g)/four;
dshape(2,2) = -(one + g)/four;
dshape(2,3) = (one + g)/four;
dshape(2,4) = (one - g)/four;
c
c compute coordinates at the integration point
c
do k1=1, 3
coords_ip(k1) = zero
end do
do k1=1,nnode
do k2=1,mcrd
coords_ip(k2)=coords_ip(k2)+shape(k1)*coords(k2,k1)
end do
end do
c
c INTERPOLATE FIELD VARIABLES
c
if(npredf.gt.0) then
do k1=1,npredf
predef_loc(k1) = zero
dpredef_loc(k1) = zero
do k2=1,nnode
predef_loc(k1) =
& predef_loc(k1)+
& (predef(1,k1,k2)-predef(2,k1,k2))*shape(k2)
dpredef_loc(k1) =
& dpredef_loc(k1)+predef(2,k1,k2)*shape(k2)
end do
end do
end if
c
c FORM B-MATRIX
c
djac = one
c
do i = 1, ndim
do j = 1, ndim
xjac(i,j) = zero
xjaci(i,j) = zero
end do
end do
c
do inod= 1, nnode
do idim = 1, ndim
do jdim = 1, ndim
xjac(jdim,idim) = xjac(jdim,idim) +
1 dshape(jdim,inod)*coords(idim,inod)
end do
end do
end do
djac = xjac(1,1)*xjac(2,2) - xjac(1,2)*xjac(2,1)
if (djac .gt. zero) then
! jacobian is positive - o.k.
xjaci(1,1) = xjac(2,2)/djac
xjaci(2,2) = xjac(1,1)/djac
xjaci(1,2) = -xjac(1,2)/djac
xjaci(2,1) = -xjac(2,1)/djac
else
! negative or zero jacobian
write(7,*)'WARNING: element',jelem,'has neg.
1 Jacobian'
pnewdt = fourth
endif
if (pnewdt .lt. pnewdtLocal) pnewdtLocal = pnewdt
c
do i = 1, nnode*ndim
bmat(i) = zero
end do
do inod = 1, nnode
do ider = 1, ndim
do idim = 1, ndim
irow = idim + (inod - 1)*ndim
bmat(irow) = bmat(irow) +
1 xjaci(idim,ider)*dshape(ider,inod)
end do
end do
end do
c
c CALCULATE INCREMENTAL STRAINS
c
do i = 1, ntens
dstran(i) = zero
end do
!
! set deformation gradient to Identity matrix
do k1=1,3
do k2=1,3
defGrad(k1,k2) = zero
end do
defGrad(k1,k1) = one
end do
c
c COMPUTE INCREMENTAL STRAINS
c
do nodi = 1, nnode
incr_row = (nodi - 1)*ndof
do i = 1, ndof
xdu(i)= du(i + incr_row,1)
utmp(i) = u(i + incr_row)
end do
dNidx = bmat(1 + (nodi-1)*ndim)
dNidy = bmat(2 + (nodi-1)*ndim)
dstran(1) = dstran(1) + dNidx*xdu(1)
dstran(2) = dstran(2) + dNidy*xdu(2)
dstran(4) = dstran(4) +
1 dNidy*xdu(1) +
2 dNidx*xdu(2)
c deformation gradient
defGrad(1,1) = defGrad(1,1) + dNidx*utmp(1)
defGrad(1,2) = defGrad(1,2) + dNidy*utmp(1)
defGrad(2,1) = defGrad(2,1) + dNidx*utmp(2)
defGrad(2,2) = defGrad(2,2) + dNidy*utmp(2)
end do
c
c CALL CONSTITUTIVE ROUTINE
c
isvinc= (kintk-1)*nsvint ! integration point increment
c
c prepare arrays for entry into material routines
c
do i = 1, nsvint
statevLocal(i)=svars(i+isvinc)
end do
c
c state variables
c
do k1=1,ntens
stran(k1) = statevLocal(k1)
stress(k1) = zero
end do
c
do i=1, ntens
do j=1, ntens
ddsdde(i,j) = zero
end do
ddsdde(i,j) = one
enddo
c
c compute characteristic element length
c
celent = sqrt(djac*dble(ninpt))
dvmat = djac*thickness
c
dvdv0 = one
call material_lib_mech(materiallib,stress,ddsdde,
1 stran,dstran,kintk,dvdv0,dvmat,defGrad,
2 predef_loc,dpredef_loc,npredf,celent,coords_ip)
c
do k1=1,ntens
statevLocal(k1) = stran(k1) + dstran(k1)
end do
c
isvinc= (kintk-1)*nsvint ! integration point increment
c
c update element state variables
c
do i = 1, nsvint
svars(i+isvinc)=statevLocal(i)
end do
c
c form stiffness matrix and internal force vector
c
dNjdx = zero
dNjdy = zero
do i = 1, ndof*nnode
force(i) = zero
do j = 1, ndof*nnode
stiff(j,i) = zero
end do
end do
dvol= wght(kintk)*djac
do nodj = 1, nnode
incr_col = (nodj - 1)*ndof
dNjdx = bmat(1+(nodj-1)*ndim)
dNjdy = bmat(2+(nodj-1)*ndim)
force_p(1) = dNjdx*stress(1) + dNjdy*stress(4)
force_p(2) = dNjdy*stress(2) + dNjdx*stress(4)
do jdof = 1, ndof
jcol = jdof + incr_col
force(jcol) = force(jcol) +
& force_p(jdof)*dvol
end do
do nodi = 1, nnode
incr_row = (nodi -1)*ndof
dNidx = bmat(1+(nodi-1)*ndim)
dNidy = bmat(2+(nodi-1)*ndim)
stiff_p(1,1) = dNidx*ddsdde(1,1)*dNjdx
& + dNidy*ddsdde(4,4)*dNjdy
& + dNidx*ddsdde(1,4)*dNjdy
& + dNidy*ddsdde(4,1)*dNjdx
stiff_p(1,2) = dNidx*ddsdde(1,2)*dNjdy
& + dNidy*ddsdde(4,4)*dNjdx
& + dNidx*ddsdde(1,4)*dNjdx
& + dNidy*ddsdde(4,2)*dNjdy
stiff_p(2,1) = dNidy*ddsdde(2,1)*dNjdx
& + dNidx*ddsdde(4,4)*dNjdy
& + dNidy*ddsdde(2,4)*dNjdy
& + dNidx*ddsdde(4,1)*dNjdx
stiff_p(2,2) = dNidy*ddsdde(2,2)*dNjdy
& + dNidx*ddsdde(4,4)*dNjdx
& + dNidy*ddsdde(2,4)*dNjdx
& + dNidx*ddsdde(4,2)*dNjdy
do jdof = 1, ndof
icol = jdof + incr_col
do idof = 1, ndof
irow = idof + incr_row
stiff(irow,icol) = stiff(irow,icol) +
& stiff_p(idof,jdof)*dvol
end do
end do
end do
end do
c
c assemble rhs and lhs
c
do k1=1, ndof*nnode
rhs(k1, 1) = rhs(k1, 1) - force(k1)
do k2=1, ndof*nnode
amatrx(k1, k2) = amatrx(k1, k2) + stiff(k1,k2)
end do
end do
end do ! end loop on material integration points
pnewdt = pnewdtLocal
c
999 continue
c
return
end

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*HEADING
Test for passing abaqus material to UELMAT: transient heat transfer
*RESTART,WRITE,NUMBER INTERVAL=10
*PREPRINT,MODEL=YES
*PART,NAME=part1
*NODE,NSET=NALL
1,0,0,0
2,1,0,0
3,0,1,0
4,1,1,0
5,0,2,0
6,1,2,0
*NSET,NSET=Left
1,3,5
*NSET,NSET=Right
2,4,6
*USER ELEMENT, TYPE=U1, NODES=4, COORDINATES=2,
INTEGRATION=4,TENSOR=TWOD
11,
*ELEMENT,TYPE=U1,ELSET=SOLID
1, 1,2,4,3
2, 3,4,6,5
*END PART
*ASSEMBLY,NAME=A1
*INSTANCE,NAME=I1,PART=PART1
*END INSTANCE
*Nset, nset=Set-6, instance=I1
1,3,5
*Nset, nset=Set-7, instance=I1
2,4,6
*END ASSEMBLY
*UEL PROPERTY, ELSET=I1.SOLID, MATERIAL=MAT_THERM
**************************************
***************************************
*MATERIAL,NAME=MAT_THERM
*CONDUCTIVITY
1.0,
*SPECIFIC HEAT
1.,
*DENSITY
1.,
*Initial Conditions, type=TEMPERATURE
Set-6, 1.,0.
*Initial Conditions, type=TEMPERATURE
Set-7, 0.,0.
*STEP
*HEAT TRANSFER, DELTMX=1.
0.1,1.0,,0.1
**
*BOUNDARY
Set-6,11,11,1.
*OUTPUT,FIELD,freq=1
*ELEMENT OUTPUT,ELSET=I1.SOLID
HFL,
*NODE OUTPUT,NSET=I1.NALL
NT,
*OUTPUT,HISTORY
*ELEMENT OUTPUT,ELSET=I1.SOLID
HFL,
*NODE OUTPUT,NSET=I1.NALL
NT11,
*END STEP

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C USER INPUT FOR ADAPTIVE MESH CONSTRAINT
C
SUBROUTINE UMESHMOTION(UREF,ULOCAL,NODE,NNDOF,
$ LNODETYPE,ALOCAL,NDIM,TIME,DTIME,PNEWDT,
$ KSTEP,KINC,KMESHSWEEP,JMATYP,JGVBLOCK,LSMOOTH)
C
include 'ABA_PARAM.INC'
C
C USER DEFINED DIMENSION STATEMENTS
C
CHARACTER*80 PARTNAME
DIMENSION ARRAY(1000),JPOS(15),HFARRAY(1000)
DIMENSION ULOCAL(*),UGLOBAL(NDIM),TLOCAL(NDIM)
DIMENSION JGVBLOCK(*),JMATYP(*)
DIMENSION NODELIST(100),JELEMLIST(10),JELEMTYPE(10)
DIMENSION ALOCAL(NDIM,*)
DIMENSION UTEMP(2)
C
C The dimensions of the variables ARRAY and JARRAY
C must be set equal to or greater than 15
C
CALL GETPARTINFO(NODE,0,PARTNAME,LOCNUM,JRCD)
CALL GETVRN(LOCNUM,'COORD',ARRAY,JRCD,JGVBLOCK,LTRN)
NELEMS=10
CALL GETNODETOELEMCONN(NODE, NELEMS, JELEMLIST, JELEMTYPE,
$ JRCD, JGVBLOCK)
CALL GETVRMAVGATNODE(NODE,1,'HFL',HFARRAY,JRCD,JELEMLIST,
$ NELEMS,JMATYP,JGVBLOCK)
C PRINT *,'****'
C PRINT *, HFARRAY(1),HFARRAY(2),HFARRAY(3),HFARRAY(4)
C PRINT *,NODE,TIME
FluxX=HFARRAY(2)
FluxY=HFARRAY(3)
FluxZ=HFARRAY(4)
if(abs(FluxX)<0.001)FluxX=0.
if(abs(FluxY)<0.001)FluxY=0.
if(abs(FluxZ)<0.001)FluxZ=0.
UGLOBAL(1) = -2.*FluxX
UGLOBAL(2) = -2.*FluxY
UGLOBAL(3) = 0.
DO I=1,NDIM
TLOCAL(I)=0.
DO J=1,NDIM
TLOCAL(I)=TLOCAL(I)+UGLOBAL(J)*ALOCAL(J,I)
ENDDO
ENDDO
DO I=1,NDIM
ULOCAL(I)=TLOCAL(I)
ENDDO
lsmooth=1
C
RETURN
END

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c J. Grogan, 2012
c -------------------------------------------------------------------
SUBROUTINE USDFLD(FIELD,STATEV,PNEWDT,DIRECT,T,CELENT,
1 TIME,DTIME,CMNAME,ORNAME,NFIELD,NSTATV,NOEL,NPT,LAYER,
2 KSPT,KSTEP,KINC,NDI,NSHR,COORD,JMAC,JMATYP,MATLAYO,
3 LACCFLA)
C
INCLUDE 'ABA_PARAM.INC'
C
CHARACTER*80 CMNAME,ORNAME
CHARACTER*3 FLGRAY(15)
DIMENSION FIELD(NFIELD),STATEV(NSTATV),DIRECT(3,3),
1 T(3,3),TIME(2)
DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),
1 COORD(*)
c -------------------------------------------------------------------
field(1)=0.
return
end subroutine

220
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c J. Grogan, 2012
c -------------------------------------------------------------------
SUBROUTINE USDFLD(FIELD,STATEV,PNEWDT,DIRECT,T,CELENT,
1 TIME,DTIME,CMNAME,ORNAME,NFIELD,NSTATV,NOEL,NPT,LAYER,
2 KSPT,KSTEP,KINC,NDI,NSHR,COORD,JMAC,JMATYP,MATLAYO,
3 LACCFLA)
C
INCLUDE 'ABA_PARAM.INC'
C
CHARACTER*80 CMNAME,ORNAME
CHARACTER*3 FLGRAY(15)
DIMENSION FIELD(NFIELD),STATEV(NSTATV),DIRECT(3,3),
1 T(3,3),TIME(2)
DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),
1 COORD(*)
c -------------------------------------------------------------------
field(1)=0.
print *, coord(1),time(1),dtime,T,'****'
return
end subroutine
SUBROUTINE UEL(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS,
1 PROPS,NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,
2 DTIME,KSTEP,KINC,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,
3 PREDEF,NPREDF,LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,
4 JPROPS,NJPROP,PERIOD)
C
INCLUDE 'ABA_PARAM.INC'
PARAMETER ( ZERO = 0.D0, HALF = 0.5D0, ONE = 1.D0 )
C
DIMENSION RHS(MLVARX,*),AMATRX(NDOFEL,NDOFEL),
1 SVARS(NSVARS),ENERGY(8),PROPS(*),COORDS(MCRD,NNODE),
2 U(NDOFEL),DU(MLVARX,*),V(NDOFEL),A(NDOFEL),TIME(2),
3 PARAMS(3),JDLTYP(MDLOAD,*),ADLMAG(MDLOAD,*),
4 DDLMAG(MDLOAD,*),PREDEF(2,NPREDF,NNODE),LFLAGS(*),
5 JPROPS(*)
DIMENSION SRESID(6)
C
C UEL SUBROUTINE FOR A HORIZONTAL TRUSS ELEMENT
C
C SRESID - stores the static residual at time t+dt
C SVARS - In 1-6, contains the static residual at time t
C upon entering the routine. SRESID is copied to
C SVARS(1-6) after the dynamic residual has been
C calculated.
C - For half-increment residual calculations: In 7-12,
C contains the static residual at the beginning
C of the previous increment. SVARS(1-6) are copied
C into SVARS(7-12) after the dynamic residual has
C been calculated.
C
AREA = PROPS(1)
E = PROPS(2)
ANU = PROPS(3)
RHO = PROPS(4)
C
ALEN = ABS(COORDS(1,2)-COORDS(1,1))
AK = AREA*E/ALEN
AM = HALF*AREA*RHO*ALEN
C
DO K1 = 1, NDOFEL
SRESID(K1) = ZERO
DO KRHS = 1, NRHS
RHS(K1,KRHS) = ZERO
END DO
DO K2 = 1, NDOFEL
AMATRX(K2,K1) = ZERO
END DO
END DO
C
IF (LFLAGS(3).EQ.1) THEN
C Normal incrementation
IF (LFLAGS(1).EQ.1 .OR. LFLAGS(1).EQ.2) THEN
C *STATIC
AMATRX(1,1) = AK
AMATRX(4,4) = AK
AMATRX(1,4) = -AK
AMATRX(4,1) = -AK
IF (LFLAGS(4).NE.0) THEN
FORCE = AK*(U(4)-U(1))
DFORCE = AK*(DU(4,1)-DU(1,1))
SRESID(1) = -DFORCE
SRESID(4) = DFORCE
RHS(1,1) = RHS(1,1)-SRESID(1)
RHS(4,1) = RHS(4,1)-SRESID(4)
ENERGY(2) = HALF*FORCE*(DU(4,1)-DU(1,1))
* + HALF*DFORCE*(U(4)-U(1))
* + HALF*DFORCE*(DU(4,1)-DU(1,1))
ELSE
FORCE = AK*(U(4)-U(1))
SRESID(1) = -FORCE
SRESID(4) = FORCE
RHS(1,1) = RHS(1,1)-SRESID(1)
RHS(4,1) = RHS(4,1)-SRESID(4)
DO KDLOAD = 1, NDLOAD
IF (JDLTYP(KDLOAD,1).EQ.1001) THEN
RHS(4,1) = RHS(4,1)+ADLMAG(KDLOAD,1)
ENERGY(8) = ENERGY(8)+(ADLMAG(KDLOAD,1)
* - HALF*DDLMAG(KDLOAD,1))*DU(4,1)
IF (NRHS.EQ.2) THEN
C Riks
RHS(4,2) = RHS(4,2)+DDLMAG(KDLOAD,1)
END IF
END IF
END DO
ENERGY(2) = HALF*FORCE*(U(4)-U(1))
END IF
ELSE IF (LFLAGS(1).EQ.11 .OR. LFLAGS(1).EQ.12) THEN
C *DYNAMIC
ALPHA = PARAMS(1)
BETA = PARAMS(2)
GAMMA = PARAMS(3)
C
DADU = ONE/(BETA*DTIME**2)
DVDU = GAMMA/(BETA*DTIME)
C
DO K1 = 1, NDOFEL
AMATRX(K1,K1) = AM*DADU
RHS(K1,1) = RHS(K1,1)-AM*A(K1)
END DO
AMATRX(1,1) = AMATRX(1,1)+(ONE+ALPHA)*AK
AMATRX(4,4) = AMATRX(4,4)+(ONE+ALPHA)*AK
AMATRX(1,4) = AMATRX(1,4)-(ONE+ALPHA)*AK
AMATRX(4,1) = AMATRX(4,1)-(ONE+ALPHA)*AK
FORCE = AK*(U(4)-U(1))
SRESID(1) = -FORCE
SRESID(4) = FORCE
RHS(1,1) = RHS(1,1) -
* ((ONE+ALPHA)*SRESID(1)-ALPHA*SVARS(1))
RHS(4,1) = RHS(4,1) -
* ((ONE+ALPHA)*SRESID(4)-ALPHA*SVARS(4))
ENERGY(1) = ZERO
DO K1 = 1, NDOFEL
SVARS(K1+6) = SVARS(k1)
SVARS(K1) = SRESID(K1)
ENERGY(1) = ENERGY(1)+HALF*V(K1)*AM*V(K1)
END DO
ENERGY(2) = HALF*FORCE*(U(4)-U(1))
END IF
ELSE IF (LFLAGS(3).EQ.2) THEN
C Stiffness matrix
AMATRX(1,1) = AK
AMATRX(4,4) = AK
AMATRX(1,4) = -AK
AMATRX(4,1) = -AK
ELSE IF (LFLAGS(3).EQ.4) THEN
C Mass matrix
DO K1 = 1, NDOFEL
AMATRX(K1,K1) = AM
END DO
ELSE IF (LFLAGS(3).EQ.5) THEN
C Half-increment residual calculation
ALPHA = PARAMS(1)
FORCE = AK*(U(4)-U(1))
SRESID(1) = -FORCE
SRESID(4) = FORCE
RHS(1,1) = RHS(1,1)-AM*A(1)-(ONE+ALPHA)*SRESID(1)
* + HALF*ALPHA*( SVARS(1)+SVARS(7) )
RHS(4,1) = RHS(4,1)-AM*A(4)-(ONE+ALPHA)*SRESID(4)
* + HALF*ALPHA*( SVARS(4)+SVARS(10) )
ELSE IF (LFLAGS(3).EQ.6) THEN
C Initial acceleration calculation
DO K1 = 1, NDOFEL
AMATRX(K1,K1) = AM
END DO
FORCE = AK*(U(4)-U(1))
SRESID(1) = -FORCE
SRESID(4) = FORCE
RHS(1,1) = RHS(1,1)-SRESID(1)
RHS(4,1) = RHS(4,1)-SRESID(4)
ENERGY(1) = ZERO
DO K1 = 1, NDOFEL
SVARS(K1) = SRESID(K1)
ENERGY(1) = ENERGY(1)+HALF*V(K1)*AM*V(K1)
END DO
ENERGY(2) = HALF*FORCE*(U(4)-U(1))
ELSE IF (LFLAGS(3).EQ.100) THEN
C Output for perturbations
IF (LFLAGS(1).EQ.1 .OR. LFLAGS(1).EQ.2) THEN
C *STATIC
FORCE = AK*(U(4)-U(1))
DFORCE = AK*(DU(4,1)-DU(1,1))
SRESID(1) = -DFORCE
SRESID(4) = DFORCE
RHS(1,1) = RHS(1,1)-SRESID(1)
RHS(4,1) = RHS(4,1)-SRESID(4)
ENERGY(2) = HALF*FORCE*(DU(4,1)-DU(1,1))
* + HALF*DFORCE*(U(4)-U(1))
* + HALF*DFORCE*(DU(4,1)-DU(1,1))
DO KVAR = 1, NSVARS
SVARS(KVAR) = ZERO
END DO
SVARS(1) = RHS(1,1)
SVARS(4) = RHS(4,1)
ELSE IF (LFLAGS(1).EQ.41) THEN
C *FREQUENCY
DO KRHS = 1, NRHS
DFORCE = AK*(DU(4,KRHS)-DU(1,KRHS))
SRESID(1) = -DFORCE
SRESID(4) = DFORCE
RHS(1,KRHS) = RHS(1,KRHS)-SRESID(1)
RHS(4,KRHS) = RHS(4,KRHS)-SRESID(4)
END DO
DO KVAR = 1, NSVARS
SVARS(KVAR) = ZERO
END DO
SVARS(1) = RHS(1,1)
SVARS(4) = RHS(4,1)
END IF
END IF
C
RETURN
END

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function [] = 2DSimp()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=3;
NumY=3;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j);
Element(index,4)=i+(NumX+1)*(j)+1;
end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=1;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.25;
% Initial temperatures
Tnew=zeros(numNodes,1);
Bound=zeros(numNodes,1);
stored(1)=dpos;
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(Element(e,n))=1.;
Bound(Element(e,n))=1.;
end
end
end
% Define Time Step
dtime=0.05;
tsteps=20;
time=0.;
% Loop through time steps
for ts=1:tsteps
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4*ndof);
Me=zeros(4*ndof);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
end
% regular element - fix extra dofs
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=-gpos;
hx(4)=gpos;
% Loop Through Int Points
for i=1:4;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(2)=0.25*(1.+g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.+h);
phi(4)=0.25*(1.-g)*(1.+h);
cond=1.;
spec=1.;
phig(1)=0.25*-(1.-h);
phig(2)=0.25*(1.-h);
phig(3)=0.25*(1.+h);
phig(4)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(2)=0.25*-(1.+g);
phih(3)=0.25*(1.+g);
phih(4)=0.25*(1.-g);
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(iter)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(iter)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(iter)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(iter)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:4
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=djac;
Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
% Assemble Global Matrices
gnum(1)=Element(e,1);
gnum(2)=Element(e,2);
gnum(3)=Element(e,3);
gnum(4)=Element(e,4);
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(j,i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(j,i);
end
end
end
%Remove inactive DOFs(Reduce Matrices)
RHS=M*Tnew;
A=K+M;
Sub=A*Bound;
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
StiffI=Ared^-1;
Tnewr=(Ared^-1)*RHSred';
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end
stored'

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*Heading
** Job name: Job-1 Model name: Model-1
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=Part-1
*Node
1, 0., 0., 0.
2, 0.2, 0., 0.
3, 0.4, 0., 0.
4, 0.6, 0., 0.
5, 0.8, 0., 0.
6, 1., 0., 0.
7, 1.2, 0., 0.
8, 1.4, 0., 0.
9, 1.6, 0., 0.
10, 1.8, 0., 0.
11, 2., 0., 0.
*USER ELEMENT,NODES=2,TYPE=U8008,PROP=1,COORDINATES=1,VAR=2,unsymm
11,12
*Element, type=U8008,ELSET=UEL
1, 1, 2
2, 2, 3
3, 3, 4
4, 4, 5
5, 5, 6
6, 6, 7
7, 7, 8
8, 8, 9
9, 9, 10
10, 10, 11
*UEL Property, Elset=UEL
1.
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=Part-1-1, part=Part-1
*End Instance
**
*Nset, nset=_PickedSet16, internal, instance=Part-1-1
1,
*Nset, nset=_PickedSet17, internal, instance=Part-1-1
2,
*Nset, nset=Set-6, instance=Part-1-1
1,
*End Assembly
**
** MATERIALS
**
*Material, name=Material-1
*Conductivity
1.,
*Density
1.,
*Specific Heat
1.,
** ----------------------------------------------------------------
**
** Name: Predefined Field-1 Type: Temperature
*Initial Conditions, type=TEMPERATURE
_PickedSet16, 1.,0.
** Name: Predefined Field-2 Type: Temperature
*Initial Conditions, type=TEMPERATURE
_PickedSet17, 0.,0.
** STEP: Step-1
**
*Step, name=Step-1
*Heat Transfer, end=PERIOD, deltmx=100.
0.1, 1.6, 1e-09, 0.1,
**
** BOUNDARY CONDITIONS
**
** Name: BC-1 Type: Temperature
*Boundary
Set-6, 11, 11, 1.
**
** OUTPUT REQUESTS
**
*Restart, write, frequency=0
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
*Output, history, frequency=0
*End Step

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function [] = FESolve()
% MATLAB based FE Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=1.;
% Define Section Properties
rho=1.;
spec=1.;
cond=1.;
% Generate Mesh
numElem=10;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% Initialize Conditions
Tnew=zeros(numNodes,1);
Tnew(1)=1.;
% Define Time Step
dtime=.1;
tsteps=10;
time=0.;
% Loop through time steps
for ts=1:tsteps;
time=time+dtime;
K=zeros(numNodes,numNodes);
M=zeros(numNodes,numNodes);
% Loop Through Elements
for e=1:numElem;
Ke=zeros(2);
Me=zeros(2);
gpx(1)=-1./sqrt(3.);
gpx(2)=1./sqrt(3.);
ajacob=abs(ndCoords(elemNodes(e,2))-ndCoords(elemNodes(e,1)))/2.;
% Loop Through Int Points
for i=1:2;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(2)=(1.+c)/2.;
phic(1)=-0.5;
phic(2)=0.5;
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
we=ajacob;
Ke=Ke+we*cond*phix'*phix;
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Assemble Global Matrices
K(elemNodes(e,1),elemNodes(e,1))=K(elemNodes(e,1),elemNodes(e,1))+Ke(1,1);
K(elemNodes(e,1),elemNodes(e,2))=K(elemNodes(e,1),elemNodes(e,2))+Ke(1,2);
K(elemNodes(e,2),elemNodes(e,1))=K(elemNodes(e,2),elemNodes(e,1))+Ke(2,1);
K(elemNodes(e,2),elemNodes(e,2))=K(elemNodes(e,2),elemNodes(e,2))+Ke(2,2);
M(elemNodes(e,1),elemNodes(e,1))=M(elemNodes(e,1),elemNodes(e,1))+Me(1,1);
M(elemNodes(e,1),elemNodes(e,2))=M(elemNodes(e,1),elemNodes(e,2))+Me(1,2);
M(elemNodes(e,2),elemNodes(e,1))=M(elemNodes(e,2),elemNodes(e,1))+Me(2,1);
M(elemNodes(e,2),elemNodes(e,2))=M(elemNodes(e,2),elemNodes(e,2))+Me(2,2);
end
%Apply Boundary Conditions (Reduce Matrices)
T1=1;
RHS=M*Tnew;
for i=1:numNodes-1;
for j=1:numNodes-1;
Kred(i,j)=K(i+1,j+1);
Mred(i,j)=M(i+1,j+1);
end
Subr(i)=(K(i+1,1)+M(i+1,1))*T1;
RHSr(i)=RHS(i+1);
end
%Solve
StiffI=(Mred+Kred)^-1;
Tnewr=StiffI*(RHSr'-Subr');
for i=1:numNodes-1;
Tnew(i+1)=Tnewr(i);
end
Tnew(1)=1.;
Tnew
end

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function [] = FESolveSimp()
% MATLAB based 1-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=10.;
% Define Section Properties
rho=1.;
% Generate Mesh
numElem=10;
charlen=len/numElem;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=1;
% Initial temperatures
Tnew=zeros(numNodes,1);
Bound=zeros(numNodes,1);
Tnew(1)=1.;
Bound(1)=1.;
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% Loop through time steps
for ts=1:tsteps
K=zeros(numNodes,numNodes);
M=zeros(numNodes,numNodes);
% Loop Through Elements
for e=1:numElem
Ke=zeros(2*ndof);
Me=zeros(2*ndof);
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
gpx(1)=-1/sqrt(3.);
gpx(2)=1/sqrt(3.);
w(1)=1.;
w(2)=1.;
% Loop Through Int Points
for i=1:2;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(2)=(1.+c)/2.;
cond=1.;
spec=1.;
phic(1)=-0.5;
phic(2)=0.5;
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
we=ajacob*w(i);
Ke=Ke+we*cond*phix'*phix;
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Assemble Global Matrices
gnum=elemNodes(e,1);
for i=1:2;
for j=1:2;
K(gnum+j-1,gnum+i-1)=K(gnum+j-1,gnum+i-1)+Ke(j,i);
M(gnum+j-1,gnum+i-1)=M(gnum+j-1,gnum+i-1)+Me(j,i);
end
end
end
%Remove NON-ENHANCED DOFs(Reduce Matrices)
iindex=0.;
A=K+M;
Sub=A*Bound;
RHS=M*Tnew-Sub;
% Apply Boundary Conditions
for i=1:numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSR(iindex)=RHS(i);
jindex=0;
for j=1:numNodes;
if Bound(j)==0.;
jindex=jindex+1;
AR(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tnewr=(AR^-1)*RHSR';
% Restore Matrices
iindex=0;
for i=1:numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end

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function [] = FESolveX()
% MATLAB based 1-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=10.;
% Define Section Properties
rho=1.;
% Generate Mesh
numElem=40;
charlen=len/numElem;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=2;
% initial interface position
dpos=4.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
%storage
stored(1)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
if crdn1<=dpos
Tnew(2*elemNodes(e,1)-1)=1.;
end
end
% Define Time Step
dtime=0.1;
tsteps=10;
time=0.;
% penalty term
beta=200.;
% Loop through time steps
for ts=1:tsteps
% Get interface velocity
d(1)=dpos+charlen;
d(2)=dpos+3*charlen/4;
d(3)=dpos+charlen/4;
d(4)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
for j=1:4
if d(j)>=crdn1 & d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(elemNodes(e,1)*2-1);
tmp1b=Tnew(elemNodes(e,1)*2);
tmp2a=Tnew(elemNodes(e,2)*2-1);
tmp2b=Tnew(elemNodes(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
vel=1.*(0.5/charlen)*(2*t(1)+t(2)-t(3)-2*t(4));
vel=0.;
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(2*ndof);
Me=zeros(2*ndof);
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
enr=2;
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
if sign(theta(1))~=sign(theta(2))
% enriched element
enr=4;
% get interface position on element
point=(dpos-crdn1)/ajacob-1.;
% devide element for sub integration
len1=abs(-point-1.);
len2=abs(1.-point);
mid1=-1+len1/2.;
mid2=1-len2/2.;
gpx(1)=-(len1/2.)/sqrt(3.)+mid1;
gpx(2)=(len1/2.)/sqrt(3.)+mid1;
gpx(3)=-(len2/2.)/sqrt(3.)+mid2;
gpx(4)=(len2/2.)/sqrt(3.)+mid2;
w(1)=(len1/2.);
w(2)=(len1/2.);
w(3)=(len2/2.);
w(4)=(len2/2.);
fdofs(1)=2*elemNodes(e,1);
fdofs(2)=2*elemNodes(e,2);
else
% regular element - fix extra dofs
gpx(1)=-1/sqrt(3.);
gpx(2)=1/sqrt(3.);
w(1)=1.;
w(2)=1.;
end
% Loop Through Int Points
for i=1:enr;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(3)=(1.+c)/2.;
term=theta(1)*phi(1)+theta(2)*phi(3);
if term<0
cond=0.;
spec=0.01;
else
cond=1.;
spec=1.;
end
phi(2)=phi(1)*(abs(term)-abs(theta(1)));
phi(4)=phi(3)*(abs(term)-abs(theta(2)));
phic(1)=-0.5;
phic(3)=0.5;
dterm=sign(term)*(phic(1)*theta(1)+phic(3)*theta(2));
phic(2)=phic(1)*(abs(term)-abs(theta(1)))+phi(1)*dterm;
phic(4)=phic(3)*(abs(term)-abs(theta(2)))+phi(3)*dterm;
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
phix(3)=phic(3)/ajacob;
phix(4)=phic(4)/ajacob;
we=ajacob*w(i);
Ke=Ke+we*cond*phix'*phix;
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
if enr==4;
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
tpos=gm(1)*Tnew(1)+gm(2)*Tnew(2)+gm(3)*Tnew(3)+gm(4)*Tnew(4);
pen=beta*(gm'*gm);
pfL=beta*1*gm';
Ke=Ke+pen;
else
pen=zeros(4);
pfL=zeros(4,1);
end
% Assemble Global Matrices
gnum=2.*elemNodes(e,1)-1.;
for i=1:4;
for j=1:4;
K(gnum+j-1,gnum+i-1)=K(gnum+j-1,gnum+i-1)+Ke(j,i);
M(gnum+j-1,gnum+i-1)=M(gnum+j-1,gnum+i-1)+Me(j,i);
end
pforce(gnum+i-1)=pforce(gnum+i-1)+pfL(i);
end
end
%Remove inactive DOFs(Reduce Matrices)
T1=1;
RHS=M*Tnew;
iindex=0;
for i=1:ndof*numNodes;
check=1;
if i==fdofs(1)|i==fdofs(2)
check=0;
elseif mod(i,2)~=0 & i~=1
check=0;
end
if check==0
jindex=0;
iindex=iindex+1;
for j=1:ndof*numNodes;
check=1;
if j==fdofs(1)|j==fdofs(2)
check=0;
elseif mod(j,2)~=0 & j~=1
check=0;
end
if check==0
jindex=jindex+1;
Kred(iindex,jindex)=K(i,j);
Mred(iindex,jindex)=M(i,j);
end
end
Subr(iindex)=(K(i,1)+M(i,1))*T1;
RHSr(iindex)=RHS(i);
pforcer(iindex)=pforce(i);
end
end
%Solve
StiffI=(Mred+Kred)^-1;
Tnewr=StiffI*(RHSr'-Subr'+pforcer');
iindex=0.;
for i=1:ndof*numNodes;
check=1;
if i==fdofs(1)|i==fdofs(2)
check=0;
elseif mod(i,2)~=0 & i~=1
check=0;
end
if check==0
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
else
Tnew(i)=0.;
end
end
Tnew(1)=1.;
Tnew
end
stored'

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function [] = FESolveX2D()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=4;
NumY=1;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=2;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.1;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
stored(1)=dpos;
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(2*Element(e,n)-1)=1.;
Bound(2*Element(e,n)-1)=1.;
end
end
end
% Define Time Step
dtime=0.1;
tsteps=10;
time=0.;
% penalty term
beta=80.;
% Loop through time steps
for ts=1:tsteps
% Get interface velocity
d(1)=dpos+delX;
d(2)=dpos+3*delX/4;
d(3)=dpos+delX/4;
d(4)=dpos;
for e=1:numElem
crdn1=Node(Element(e,1),1);
crdn2=Node(Element(e,2),1);
for j=1:4
if d(j)>=crdn1 & d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(Element(e,1)*2-1);
tmp1b=Tnew(Element(e,1)*2);
tmp2a=Tnew(Element(e,2)*2-1);
tmp2b=Tnew(Element(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
% vel=-0.1*(0.5/delX)*(2*t(1)+t(2)-t(3)-2*t(4))
vel=0.0;
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4*ndof);
Me=zeros(4*ndof);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
theta(icrd)=abs(crdnx(icrd)-dpos)*sign(crdnx(icrd)-dpos);
end
if sign(theta(1))~=sign(theta(2))
% enriched element
enr=8;
% get interface position on element
elen=abs(crdnx(2)-crdnx(1));
frac=abs(dpos-crdnx(1))/elen;
len1=2.*frac;
len2=2.*(1.-frac);
% devide element for sub integration
mid1=-1+len1/2.;
mid2=1-len2/2.;
gx(1)=mid1-(len1/2.)/sqrt(3.);
gx(2)=mid1+(len1/2.)/sqrt(3.);
gx(3)=mid1+(len1/2.)/sqrt(3.);
gx(4)=mid1-(len1/2.)/sqrt(3.);
gx(5)=mid2-(len2/2.)/sqrt(3.);
gx(6)=mid2+(len2/2.)/sqrt(3.);
gx(7)=mid2+(len2/2.)/sqrt(3.);
gx(8)=mid2-(len2/2.)/sqrt(3.);
gpos=1/sqrt(3.);
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=+gpos;
hx(4)=+gpos;
hx(5)=-gpos;
hx(6)=-gpos;
hx(7)=+gpos;
hx(8)=+gpos;
for iw=1:4
w(iw)=frac/2.;
w(iw+4)=(1.-frac)/2.;
end
else
% regular element - fix extra dofs
enr=4;
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
for iw=1:4
w(iw)=1.;
end
end
% Loop Through Int Points
for i=1:enr;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
if iLS<0.
cond=0.;
spec=0.01;
else
cond=1.;
spec=1.;
end
for iter=1:4
phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(iter)));
end
phig(1)=0.25*-(1.-h);
phig(3)=0.25*(1.-h);
phig(5)=0.25*(1.+h);
phig(7)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(3)=0.25*-(1.+g);
phih(5)=0.25*(1.+g);
phih(7)=0.25*(1.-g);
diLSg=sign(iLS)*(phig(1)*theta(1)+phig(3)*theta(2)+phig(5)*theta(3)+phig(7)*theta(4));
diLSh=sign(iLS)*(phih(1)*theta(1)+phih(3)*theta(2)+phih(5)*theta(3)+phih(7)*theta(4));
for iter=1:4
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSg;
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSh;
end
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=w(i)*djac;
Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
if enr==8;
xi=2.*frac-1;
yi=0.;
gm(1)=0.25*(1.-xi)*(1.-yi);
gm(3)=0.25*(1.+xi)*(1.-yi);
gm(5)=0.25*(1.+xi)*(1.+yi);
gm(7)=0.25*(1.-xi)*(1.+yi);
gm(2)=gm(1)*(-abs(theta(1)));
gm(4)=gm(3)*(-abs(theta(2)));
gm(6)=gm(5)*(-abs(theta(3)));
gm(8)=gm(7)*(-abs(theta(4)));
pen=beta*(gm'*gm);
pfL=beta*1.*gm';
Ke=Ke+pen;
else
pen=zeros(8);
pfL=zeros(8,1);
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(2*j-1,2*i-1);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove inactive DOFs(Reduce Matrices)
RHS=M*Tnew;
A=K+M;
Sub=A*Bound;
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i)+pforce(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
StiffI=Ared^-1;
Tnewr=(Ared^-1)*RHSred';
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end
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function [] = FESolveX2DLS()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=3;
NumY=1;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=2;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.1;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
stored(1)=dpos;
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(2*Element(e,n)-1)=1.;
Bound(2*Element(e,n)-1)=1.;
end
end
end
% Define Time Step
dtime=0.1;
tsteps=20;
time=0.;
% penalty term
beta=80.;
% Loop through time steps
for ts=1:tsteps
% Get interface velocity
d(1)=dpos+delX;
d(2)=dpos+3*delX/4;
d(3)=dpos+delX/4;
d(4)=dpos;
for e=1:numElem
crdn1=Node(Element(e,1),1);
crdn2=Node(Element(e,2),1);
for j=1:4
if d(j)>=crdn1 & d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(Element(e,1)*2-1);
tmp1b=Tnew(Element(e,1)*2);
tmp2a=Tnew(Element(e,2)*2-1);
tmp2b=Tnew(Element(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
% vel=-0.1*(0.5/delX)*(2*t(1)+t(2)-t(3)-2*t(4))
vel=0.0;
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4*ndof);
Me=zeros(4*ndof);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
theta(icrd)=abs(crdnx(icrd)-dpos)*sign(crdnx(icrd)-dpos);
end
if sign(theta(1))~=sign(theta(2))
% enriched element
enr=8;
% get interface position on element
elen=abs(crdnx(2)-crdnx(1));
frac=abs(dpos-crdnx(1))/elen;
len1=2.*frac;
len2=2.*(1.-frac);
% devide element for sub integration
mid1=-1+len1/2.;
mid2=1-len2/2.;
gx(1)=mid1-(len1/2.)/sqrt(3.);
gx(2)=mid1+(len1/2.)/sqrt(3.);
gx(3)=mid1+(len1/2.)/sqrt(3.);
gx(4)=mid1-(len1/2.)/sqrt(3.);
gx(5)=mid2-(len2/2.)/sqrt(3.);
gx(6)=mid2+(len2/2.)/sqrt(3.);
gx(7)=mid2+(len2/2.)/sqrt(3.);
gx(8)=mid2-(len2/2.)/sqrt(3.);
gpos=1/sqrt(3.);
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=+gpos;
hx(4)=+gpos;
hx(5)=-gpos;
hx(6)=-gpos;
hx(7)=+gpos;
hx(8)=+gpos;
for iw=1:4
w(iw)=frac/2.;
w(iw+4)=(1.-frac)/2.;
end
else
% regular element - fix extra dofs
enr=4;
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
for iw=1:4
w(iw)=1.;
end
end
% Loop Through Int Points
for i=1:enr;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.-h);
phi(5)=0.25*(1.+g)*(1.+h);
phi(7)=0.25*(1.-g)*(1.+h);
iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
if iLS<0.
cond=0.;
spec=0.01;
else
cond=1.;
spec=1.;
end
for iter=1:4
phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(iter)));
end
phig(1)=0.25*-(1.-h);
phig(3)=0.25*(1.-h);
phig(5)=0.25*(1.+h);
phig(7)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(3)=0.25*-(1.+g);
phih(5)=0.25*(1.+g);
phih(7)=0.25*(1.-g);
diLSg=sign(iLS)*(phig(1)*theta(1)+phig(3)*theta(2)+phig(5)*theta(3)+phig(7)*theta(4));
diLSh=sign(iLS)*(phih(1)*theta(1)+phih(3)*theta(2)+phih(5)*theta(3)+phih(7)*theta(4));
for iter=1:4
phig(2*iter)=phig(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSg;
phih(2*iter)=phih(2*iter-1)*(abs(iLS)-abs(theta(iter)))+phi(2*iter-1)*diLSh;
end
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(2*iter-1)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(2*iter-1)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(2*iter-1)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(2*iter-1)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:8
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=w(i)*djac;
Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
if enr==8;
xi=2.*frac-1;
yi=0.;
gm(1)=0.25*(1.-xi)*(1.-yi);
gm(3)=0.25*(1.+xi)*(1.-yi);
gm(5)=0.25*(1.+xi)*(1.+yi);
gm(7)=0.25*(1.-xi)*(1.+yi);
gm(2)=gm(1)*(-abs(theta(1)));
gm(4)=gm(3)*(-abs(theta(2)));
gm(6)=gm(5)*(-abs(theta(3)));
gm(8)=gm(7)*(-abs(theta(4)));
pen=beta*(gm'*gm);
pfL=beta*1.*gm';
Ke=Ke+pen;
else
pen=zeros(8);
pfL=zeros(8,1);
end
% Assemble Global Matrices
gnum(1)=2*Element(e,1)-1;
gnum(2)=2*Element(e,2)-1;
gnum(3)=2*Element(e,3)-1;
gnum(4)=2*Element(e,4)-1;
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(2*j-1,2*i-1);
K(gnum(j)+1,gnum(i)+1)=K(gnum(j)+1,gnum(i)+1)+Ke(2*j,2*i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(2*j-1,2*i-1);
M(gnum(j)+1,gnum(i)+1)=M(gnum(j)+1,gnum(i)+1)+Me(2*j,2*i);
end
end
for i=1:4;
pforce(gnum(i))=pforce(gnum(i))+pfL(2*i-1);
pforce(gnum(i)+1)=pforce(gnum(i)+1)+pfL(2*i);
end
end
%Remove inactive DOFs(Reduce Matrices)
RHS=M*Tnew;
A=K+M;
Sub=A*Bound;
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i)+pforce(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
StiffI=Ared^-1;
Tnewr=(Ared^-1)*RHSred';
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end
stored'

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function [] = FESolveXT()
% MATLAB based 1-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=10.;
% Define Section Properties
rho=1.;
% Generate Mesh
numElem=10;
charlen=len/numElem;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=2;
% initial interface position
dpos=4.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
%storage
stored(1)=dpos;
for e=1:numElem
for n=1:2
crdn1=ndCoords(elemNodes(e,n));
if crdn1<=dpos
Tnew(2*elemNodes(e,n)-1)=1.;
Bound(2*elemNodes(e,n)-1)=1.;
end
end
end
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% penalty term
beta=50.;
% Loop through time steps
for ts=1:tsteps
% Get interface velocity
d(1)=dpos+charlen;
d(2)=dpos+3*charlen/4;
d(3)=dpos+charlen/4;
d(4)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
for j=1:4
if d(j)>=crdn1 & d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(elemNodes(e,1)*2-1);
tmp1b=Tnew(elemNodes(e,1)*2);
tmp2a=Tnew(elemNodes(e,2)*2-1);
tmp2b=Tnew(elemNodes(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
vel=1.*(0.5/charlen)*(2*t(1)+t(2)-t(3)-2*t(4));
vel=0.
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(2*ndof);
Me=zeros(2*ndof);
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
enr=2;
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
if sign(theta(1))~=sign(theta(2))
% enriched element
enr=4;
% get interface position on element
point=(dpos-crdn1)/ajacob-1.;
% devide element for sub integration
len1=abs(-point-1.);
len2=abs(1.-point);
mid1=-1+len1/2.;
mid2=1-len2/2.;
gpx(1)=-(len1/2.)/sqrt(3.)+mid1;
gpx(2)=(len1/2.)/sqrt(3.)+mid1;
gpx(3)=-(len2/2.)/sqrt(3.)+mid2;
gpx(4)=(len2/2.)/sqrt(3.)+mid2;
w(1)=(len1/2.);
w(2)=(len1/2.);
w(3)=(len2/2.);
w(4)=(len2/2.);
else
% regular element - fix extra dofs
gpx(1)=-1/sqrt(3.);
gpx(2)=1/sqrt(3.);
w(1)=1.;
w(2)=1.;
end
% Loop Through Int Points
for i=1:enr;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(3)=(1.+c)/2.;
term=theta(1)*phi(1)+theta(2)*phi(3);
if term<0
cond=0.;
spec=0.01;
else
cond=1.;
spec=1.;
end
phi(2)=phi(1)*(abs(term)-abs(theta(1)));
phi(4)=phi(3)*(abs(term)-abs(theta(2)));
phic(1)=-0.5;
phic(3)=0.5;
dterm=sign(term)*(phic(1)*theta(1)+phic(3)*theta(2));
phic(2)=phic(1)*(abs(term)-abs(theta(1)))+phi(1)*dterm;
phic(4)=phic(3)*(abs(term)-abs(theta(2)))+phi(3)*dterm;
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
phix(3)=phic(3)/ajacob;
phix(4)=phic(4)/ajacob;
we=ajacob*w(i);
Ke=Ke+we*cond*phix'*phix;
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
if enr==4;
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
tpos=gm(1)*Tnew(1)+gm(2)*Tnew(2)+gm(3)*Tnew(3)+gm(4)*Tnew(4);
pen=beta*(gm'*gm);
pfL=beta*1*gm';
Ke=Ke+pen;
else
pen=zeros(4);
pfL=zeros(4,1);
end
% Assemble Global Matrices
gnum=2.*elemNodes(e,1)-1.;
for i=1:4;
for j=1:4;
K(gnum+j-1,gnum+i-1)=K(gnum+j-1,gnum+i-1)+Ke(j,i);
M(gnum+j-1,gnum+i-1)=M(gnum+j-1,gnum+i-1)+Me(j,i);
end
pforce(gnum+i-1)=pforce(gnum+i-1)+pfL(i);
end
end
%Remove inactive DOFs(Reduce Matrices)
RHS=M*Tnew;
A=K+M;
Sub=A*Bound;
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i)+pforce(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tnewr=(Ared^-1)*RHSred';
iindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end
stored'

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function [] = FESolveXT2()
% MATLAB based 1-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=1.;
% Define Section Properties
rho=1.;
% Generate Mesh
numElem=16;
charlen=len/numElem;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=2;
% initial interface position
dpos=0.4;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
%storage
stored(1)=dpos;
for e=1:numElem
for n=1:2
crdn1=ndCoords(elemNodes(e,n));
if crdn1<=dpos
Tnew(2*elemNodes(e,n)-1)=1.;
end
end
end
Bound(1)=1.;
% Define Time Step
dtime=0.001;
tsteps=100;
time=0.;
% penalty term
beta=100.;
% Loop through time steps
for ts=1:tsteps
eNodes=zeros(2*numNodes,1);
% Get interface velocity
d(1)=dpos+charlen;
d(2)=dpos+3*charlen/4;
d(3)=dpos+charlen/4;
d(4)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
for j=1:4
if d(j)>=crdn1 && d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(elemNodes(e,1)*2-1);
tmp1b=Tnew(elemNodes(e,1)*2);
tmp2a=Tnew(elemNodes(e,2)*2-1);
tmp2b=Tnew(elemNodes(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
vel=0.5*(0.5/charlen)*(2*t(1)+t(2)-t(3)-2*t(4));
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(2*ndof);
Me=zeros(2*ndof);
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
enr=2;
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
if sign(theta(1))~=sign(theta(2))
% enriched element
eNodes(2*elemNodes(e,1))=1;
eNodes(2*elemNodes(e,2))=1;
enr=4;
% get interface position on element
point=(dpos-crdn1)/ajacob-1.;
% devide element for sub integration
len1=abs(-point-1.);
len2=abs(1.-point);
mid1=-1+len1/2.;
mid2=1-len2/2.;
gpx(1)=-(len1/2.)/sqrt(3.)+mid1;
gpx(2)=(len1/2.)/sqrt(3.)+mid1;
gpx(3)=-(len2/2.)/sqrt(3.)+mid2;
gpx(4)=(len2/2.)/sqrt(3.)+mid2;
w(1)=(len1/2.);
w(2)=(len1/2.);
w(3)=(len2/2.);
w(4)=(len2/2.);
else
% regular element - fix extra dofs
gpx(1)=-1/sqrt(3.);
gpx(2)=1/sqrt(3.);
w(1)=1.;
w(2)=1.;
end
% Loop Through Int Points
for i=1:enr;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(3)=(1.+c)/2.;
term=theta(1)*phi(1)+theta(2)*phi(3);
% if term<0
% cond=0.00;
% spec=0.001;
% else
cond=1.;
spec=1.;
% end
if enr==4
phi(2)=phi(1)*(abs(term)-abs(theta(1)));
phi(4)=phi(3)*(abs(term)-abs(theta(2)));
else
phi(2)=0.0;
phi(4)=0.0;
end
phic(1)=-0.5;
phic(3)=0.5;
dterm=sign(term)*(phic(1)*theta(1)+phic(3)*theta(2));
if enr==4
phic(2)=phic(1)*(abs(term)-abs(theta(1)))+phi(1)*dterm;
phic(4)=phic(3)*(abs(term)-abs(theta(2)))+phi(3)*dterm;
else
phic(2)=0.0;
phic(4)=0.0;
end
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
phix(3)=phic(3)/ajacob;
phix(4)=phic(4)/ajacob;
we=ajacob*w(i);
Ke=Ke+we*cond*phix'*phix;
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
if enr==5;
Ke(1,2)=0.;
Me(1,2)=0.;
Ke(2,1)=0.;
Me(2,1)=0.;
Ke(1,4)=0.;
Me(1,4)=0.;
Ke(4,1)=0.;
Me(4,1)=0.;
Ke(3,2)=0.;
Me(3,2)=0.;
Ke(2,3)=0.;
Me(2,3)=0.;
Ke(4,3)=0.;
Me(4,3)=0.;
Ke(3,4)=0.;
Me(3,4)=0.;
end
% Add penalty term and get temp gradient on interface
if enr==4;
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
pen=beta*(gm'*gm);
pfL=beta*1*gm';
Ke=Ke+pen;
else
pen=zeros(4);
pfL=zeros(4,1);
end
Ke;
Me;
% Assemble Global Matrices
gnum=2.*elemNodes(e,1)-1.;
for i=1:4;
for j=1:4;
K(gnum+j-1,gnum+i-1)=K(gnum+j-1,gnum+i-1)+Ke(j,i);
M(gnum+j-1,gnum+i-1)=M(gnum+j-1,gnum+i-1)+Me(j,i);
end
pforce(gnum+i-1)=pforce(gnum+i-1)+pfL(i);
end
end
%Remove NON-ENHANCED DOFs(Reduce Matrices)
iindex=0.;
for i=1:ndof*numNodes;
check=0;
if mod(i,2)==0 && eNodes(i)~=1
check=1;
end
if check==0
iindex=iindex+1;
TR1(iindex)=Tnew(i);
BR1(iindex)=Bound(i);
pforceR1(iindex)=pforce(i);
jindex=0;
for j=1:ndof*numNodes;
check=0;
if mod(j,2)==0 && eNodes(j)~=1
check=1;
end
if check==0
jindex=jindex+1;
MR1(iindex,jindex)=M(i,j);
KR1(iindex,jindex)=K(i,j);
end
end
end
end
AR1=KR1+MR1;
SubR1=AR1*BR1';
RHSR1=MR1*TR1'-SubR1+pforceR1';
% Apply Boundary Conditions
Biindex=0.;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
RHSR2(Biindex)=RHSR1(i);
jindex=0;
for j=1:iindex;
check=0;
if BR1(j)==0.;
jindex=jindex+1;
AR2(Biindex,jindex)=AR1(i,j);
end
end
end
end
%Solve
Tnewr=(AR2^-1)*RHSR2';
% Restore Matrices
Biindex=0;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
TR1(i)=Tnewr(Biindex);
end
end
iindex=0;
for i=1:ndof*numNodes;
check=0;
if mod(i,2)==0 && eNodes(i)~=1
check=1;
end
if check==0
iindex=iindex+1;
Tnew(i)=TR1(iindex);
end
end
Tnew
end
stored'

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function [] = GetF()
% set up grid
gd=0.;
numElem=4;
eLen=0.25;
for i=1:numElem+1
ndCrd(i)=gd;
gd=gd+eLen;
end
for i=1:numElem
elemNod(i,1)=i;
elemNod(i,2)=i+1;
end
% Initial level set
dpos=0.1;
for i=1:numElem+1
lSet(i)=sign(ndCrd(i)-dpos)*abs(dpos-ndCrd(i));
end
lSet'
for tstep=1:120
% Velocity BC
F=zeros(numElem+1,1);
for i=1:numElem
if sign(lSet(elemNod(i,1)))~=sign(lSet(elemNod(i,2)))
F(elemNod(i,1))= 0.0005;
F(elemNod(i,2))= 0.0005;
end
end
% Assemble 'Stiffness' Matrices
A=zeros(numElem+1);
for i=1:numElem
pos(1)=-1/sqrt(3);
pos(2)=1/sqrt(3);
AfL=zeros(2);
AfLGLS=zeros(2);
for j=1:2
shp(1)=(1-pos(j))/2.;
shp(2)=(1+pos(j))/2.;
dshp(1)=-0.5;
dshp(2)=0.5;
rset=shp(1)*lSet(elemNod(i,1))+shp(2)*lSet(elemNod(i,2));
dls=dshp(1)*lSet(elemNod(i,1))+dshp(2)*lSet(elemNod(i,2));
AfL=AfL+shp'*sign(rset)*(dls*dshp);
AfLGLS=AfLGLS+(dshp'*dls)*(0.25/abs(dls))*(dls*dshp);
end
for k=1:2;
for j=1:2;
A(elemNod(i,j),elemNod(i,k))=A(elemNod(i,j),elemNod(i,k))+AfL(j,k)+AfLGLS(j,k);
end
end
end
% Apply BCs
RHS=zeros(numElem+1,1);
Sub=A*F;
iindex=0;
for i=1:numElem+1
if F(i)==0.
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
Fred=0.;
jindex=0;
for j=1:numElem+1
if F(j)==0.
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
% Solve for Fred
Fred=(Ared^-1)*RHSred';
% Get F
iindex=0;
for i=1:numElem+1
if F(i)==0.
iindex=iindex+1;
F(i)=Fred(iindex);
end
end
% Update level set
mMat=zeros(numElem+1);
mMatGLS=zeros(numElem+1);
f1=zeros(numElem+1,1);
f2=zeros(numElem+1,1);
f3=zeros(numElem+1,1);
h=0.25;
visc=0.000;
for i=1:numElem
pos(1)=-1/sqrt(3);
pos(2)=1/sqrt(3);
mMatL=zeros(2);
mMatGLSL=zeros(2);
f1L=zeros(2,1);
f2L=zeros(2,1);
f3L=zeros(2,1);
for j=1:2
shp(1)=(1-pos(j))/2.;
shp(2)=(1+pos(j))/2.;
dshp(1)=-0.5;
dshp(2)=0.5;
Floc=shp(1)*F(elemNod(i,1))+shp(2)*F(elemNod(i,2));
rset=shp(1)*lSet(elemNod(i,1))+shp(2)*lSet(elemNod(i,2));
dls=dshp(1)*lSet(elemNod(i,1))+dshp(2)*lSet(elemNod(i,2));
mMatL=mMatL+shp'*shp;
mMatGLSL=mMatGLSL+((dshp'*(dls/abs(dls)))*Floc*(h/abs(Floc)))*shp;
f1L=f1L+shp'*Floc*abs(dls);
f2L=f2L+(dshp'*(dls/abs(dls))*Floc)*(h/abs(Floc))*Floc*abs(dls);
vs=h*((abs(visc+Floc*abs(dls)))/(abs(Floc*dls)+h));
f3L=f3L+vs*dshp'*dls;
end
for k=1:2;
for j=1:2;
mMat(elemNod(i,j),elemNod(i,k))=mMat(elemNod(i,j),elemNod(i,k))+mMatL(j,k);
mMatGLS(elemNod(i,j),elemNod(i,k))=mMatGLS(elemNod(i,j),elemNod(i,k))+mMatGLSL(j,k);
end
f1(elemNod(i,k))=f1(elemNod(i,k))+f1L(k);
f2(elemNod(i,k))=f2(elemNod(i,k))+f2L(k);
f3(elemNod(i,k))=f3(elemNod(i,k))+f3L(k);
end
end
dt=0.01;
lSet=lSet-((((mMat+mMatGLS)^-1)/dt)*(f1+f2+f3))';
lSet'
end

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function [] = S2D()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=10;
NumY=1;
delX=1.;
delY=1.;
for j=1:NumY+1
for i=1:NumX+1
index=i+(NumX+1)*(j-1);
Node(index,1)=single((i-1.))*delX;
Node(index,2)=single((j-1.))*delY;
end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
for i=1:NumX
index=i+NumX*(j-1);
Element(index,1)=i+(NumX+1)*(j-1);
Element(index,2)=i+(NumX+1)*(j-1)+1;
Element(index,3)=i+(NumX+1)*(j)+1;
Element(index,4)=i+(NumX+1)*(j);
end
end
numElem=(NumX)*(NumY);
% Define Section Properties
rho=1.;
cond=1.;
spec=1.;
% initial interface position
dpos=0.25;
% Initial temperatures
Tnew=zeros(numNodes,1);
Bound=zeros(numNodes,1);
for e=1:numElem
for n=1:4
crdn=Node(Element(e,n),1);
if crdn<=dpos
Tnew(Element(e,n))=1.;
Bound(Element(e,n))=1.;
end
end
end
% Define Time Step
dtime=0.1;
tsteps=10;
% Loop through time steps
for ts=1:tsteps
K=zeros(numNodes,numNodes);
M=zeros(numNodes,numNodes);
% Loop Through Elements
for e=1:numElem
Ke=zeros(4);
Me=zeros(4);
for icrd=1:4;
crdnx(icrd)=Node(Element(e,icrd),1);
crdny(icrd)=Node(Element(e,icrd),2);
end
gpos=1/sqrt(3.);
gx(1)=-gpos;
gx(2)=gpos;
gx(3)=gpos;
gx(4)=-gpos;
hx(1)=-gpos;
hx(2)=-gpos;
hx(3)=gpos;
hx(4)=gpos;
% Loop Through Int Points
for i=1:4;
g=gx(i);
h=hx(i);
phi(1)=0.25*(1.-g)*(1.-h);
phi(2)=0.25*(1.+g)*(1.-h);
phi(3)=0.25*(1.+g)*(1.+h);
phi(4)=0.25*(1.-g)*(1.+h);
phig(1)=0.25*-(1.-h);
phig(2)=0.25*(1.-h);
phig(3)=0.25*(1.+h);
phig(4)=0.25*-(1.+h);
phih(1)=0.25*-(1.-g);
phih(2)=0.25*-(1.+g);
phih(3)=0.25*(1.+g);
phih(4)=0.25*(1.-g);
rjac=zeros(2,2);
for iter=1:4
rjac(1,1)=rjac(1,1)+phig(iter)*crdnx(iter);
rjac(1,2)=rjac(1,2)+phig(iter)*crdny(iter);
rjac(2,1)=rjac(2,1)+phih(iter)*crdnx(iter);
rjac(2,2)=rjac(2,2)+phih(iter)*crdny(iter);
end
djac=rjac(1,1)*rjac(2,2)-rjac(1,2)*rjac(2,1);
rjaci(1,1)= rjac(2,2)/djac;
rjaci(2,2)= rjac(1,1)/djac;
rjaci(1,2)=-rjac(1,2)/djac;
rjaci(2,1)=-rjac(2,1)/djac ;
for iter=1:4
phix(iter)=rjaci(1,1)*phig(iter)+rjaci(1,2)*phih(iter);
phiy(iter)=rjaci(2,1)*phig(iter)+rjaci(2,2)*phih(iter);
end
we=djac;
Ke=Ke+we*cond*(phix'*phix+phiy'*phiy);
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Assemble Global Matrices
gnum(1)=Element(e,1);
gnum(2)=Element(e,2);
gnum(3)=Element(e,3);
gnum(4)=Element(e,4);
for i=1:4;
for j=1:4;
K(gnum(j),gnum(i))=K(gnum(j),gnum(i))+Ke(j,i);
M(gnum(j),gnum(i))=M(gnum(j),gnum(i))+Me(j,i);
end
end
end
%Remove inactive DOFs(Reduce Matrices)
RHS=M*Tnew;
A=K+M;
Sub=A*Bound;
iindex=0;
for i=1:numNodes;
if Bound(i)==0.;
iindex=iindex+1;
RHSred(iindex)=RHS(i)-Sub(i);
jindex=0;
for j=1:numNodes;
if Bound(j)==0.;
jindex=jindex+1;
Ared(iindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tnewr=(Ared^-1)*RHSred';
iindex=0;
for i=1:numNodes;
if Bound(i)==0.;
iindex=iindex+1;
Tnew(i)=Tnewr(iindex);
end
end
Tnew
end

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function [] = XCOR1D()
% MATLAB based 1-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=1.;
% Define Section Properties
rho=1.;
% Generate Mesh
numElem=4;
charlen=len/numElem;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=2;
% initial interface position
dpos=0.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
%storage
stored(1)=dpos;
for e=1:numElem
for n=1:2
crdn1=ndCoords(elemNodes(e,n));
if crdn1<=dpos
Tnew(2*elemNodes(e,n)-1)=1.;
end
end
end
Bound(1)=1.;
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% penalty term
beta=100.;
% Loop through time steps
for ts=1:tsteps
eNodes=zeros(2*numNodes,1);
% Get interface velocity
d(1)=dpos+charlen;
d(2)=dpos+3*charlen/4;
d(3)=dpos+charlen/4;
d(4)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
for j=1:4
if d(j)>=crdn1 && d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(elemNodes(e,1)*2-1);
tmp1b=Tnew(elemNodes(e,1)*2);
tmp2a=Tnew(elemNodes(e,2)*2-1);
tmp2b=Tnew(elemNodes(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
vel=0.5*(0.5/charlen)*(2*t(1)+t(2)-t(3)-2*t(4));
vel=0.
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(2*ndof);
Me=zeros(2*ndof);
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
enr=2;
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
if sign(theta(1))~=sign(theta(2))
% enriched element
eNodes(2*elemNodes(e,1))=1;
eNodes(2*elemNodes(e,2))=1;
enr=4;
% get interface position on element
point=(dpos-crdn1)/ajacob-1.;
% devide element for sub integration
len1=abs(-point-1.);
len2=abs(1.-point);
mid1=-1+len1/2.;
mid2=1-len2/2.;
gpx(1)=-(len1/2.)/sqrt(3.)+mid1;
gpx(2)=(len1/2.)/sqrt(3.)+mid1;
gpx(3)=-(len2/2.)/sqrt(3.)+mid2;
gpx(4)=(len2/2.)/sqrt(3.)+mid2;
w(1)=(len1/2.);
w(2)=(len1/2.);
w(3)=(len2/2.);
w(4)=(len2/2.);
else
% regular element - fix extra dofs
gpx(1)=-1/sqrt(3.);
gpx(2)=1/sqrt(3.);
w(1)=1.;
w(2)=1.;
end
% Loop Through Int Points
for i=1:enr;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(3)=(1.+c)/2.;
term=theta(1)*phi(1)+theta(2)*phi(3);
% if term<0
% cond=0.00;
% spec=0.001;
% else
cond=1.;
spec=1.;
% end
if enr==4
phi(2)=phi(1)*(abs(term)-abs(theta(1)));
phi(4)=phi(3)*(abs(term)-abs(theta(2)));
else
phi(2)=0.0;
phi(4)=0.0;
end
phic(1)=-0.5;
phic(3)=0.5;
dterm=sign(term)*(phic(1)*theta(1)+phic(3)*theta(2));
if enr==4
phic(2)=phic(1)*(abs(term)-abs(theta(1)))+phi(1)*dterm;
phic(4)=phic(3)*(abs(term)-abs(theta(2)))+phi(3)*dterm;
else
phic(2)=0.0;
phic(4)=0.0;
end
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
phix(3)=phic(3)/ajacob;
phix(4)=phic(4)/ajacob;
we=ajacob*w(i);
Ke=Ke+we*cond*phix'*phix;
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
if enr==5;
Ke(1,2)=0.;
Me(1,2)=0.;
Ke(2,1)=0.;
Me(2,1)=0.;
Ke(1,4)=0.;
Me(1,4)=0.;
Ke(4,1)=0.;
Me(4,1)=0.;
Ke(3,2)=0.;
Me(3,2)=0.;
Ke(2,3)=0.;
Me(2,3)=0.;
Ke(4,3)=0.;
Me(4,3)=0.;
Ke(3,4)=0.;
Me(3,4)=0.;
end
% Add penalty term and get temp gradient on interface
if enr==4;
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
pen=beta*(gm'*gm);
pfL=beta*1*gm';
Ke=Ke+pen;
else
pen=zeros(4);
pfL=zeros(4,1);
end
Ke;
Me;
% Assemble Global Matrices
gnum=2.*elemNodes(e,1)-1.;
for i=1:4;
for j=1:4;
K(gnum+j-1,gnum+i-1)=K(gnum+j-1,gnum+i-1)+Ke(j,i);
M(gnum+j-1,gnum+i-1)=M(gnum+j-1,gnum+i-1)+Me(j,i);
end
pforce(gnum+i-1)=pforce(gnum+i-1)+pfL(i);
end
end
%Remove NON-ENHANCED DOFs(Reduce Matrices)
iindex=0.;
for i=1:ndof*numNodes;
check=0;
if mod(i,2)==0 && eNodes(i)~=1
check=1;
end
if check==0
iindex=iindex+1;
TR1(iindex)=Tnew(i);
BR1(iindex)=Bound(i);
pforceR1(iindex)=pforce(i);
jindex=0;
for j=1:ndof*numNodes;
check=0;
if mod(j,2)==0 && eNodes(j)~=1
check=1;
end
if check==0
jindex=jindex+1;
MR1(iindex,jindex)=M(i,j);
KR1(iindex,jindex)=K(i,j);
end
end
end
end
AR1=KR1+MR1;
SubR1=AR1*BR1';
RHSR1=MR1*TR1'-SubR1+pforceR1';
% Apply Boundary Conditions
Biindex=0.;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
RHSR2(Biindex)=RHSR1(i);
jindex=0;
for j=1:iindex;
check=0;
if BR1(j)==0.;
jindex=jindex+1;
AR2(Biindex,jindex)=AR1(i,j);
end
end
end
end
%Solve
Tnewr=(AR2^-1)*RHSR2';
% Restore Matrices
Biindex=0;
for i=1:iindex;
if BR1(i)==0.;
Biindex=Biindex+1;
TR1(i)=Tnewr(Biindex);
end
end
iindex=0;
for i=1:ndof*numNodes;
check=0;
if mod(i,2)==0 && eNodes(i)~=1
check=1;
end
if check==0
iindex=iindex+1;
Tnew(i)=TR1(iindex);
end
end
Tnew
end
stored';

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function [] = XCOR1Db()
% MATLAB based 1-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
len=1.;
% Define Section Properties
rho=1.;
% Generate Mesh
numElem=4;
charlen=len/numElem;
ndCoords=linspace(0,len,numElem+1);
numNodes=size(ndCoords,2);
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=2;
% initial interface position
dpos=0.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
%storage
stored(1)=dpos;
for e=1:numElem
for n=1:2
crdn1=ndCoords(elemNodes(e,n));
if crdn1<=dpos
Tnew(2*elemNodes(e,n)-1)=1.;
end
end
end
Bound(1)=1.;
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% penalty term
beta=100.;
% Loop through time steps
for ts=1:tsteps
eNodes=zeros(2*numNodes,1);
% Get interface velocity
d(1)=dpos+charlen;
d(2)=dpos+3*charlen/4;
d(3)=dpos+charlen/4;
d(4)=dpos;
for e=1:numElem
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
for j=1:4
if d(j)>=crdn1 && d(j)<crdn2
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
point=(d(j)-crdn1)/ajacob-1.;
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
tmp1a=Tnew(elemNodes(e,1)*2-1);
tmp1b=Tnew(elemNodes(e,1)*2);
tmp2a=Tnew(elemNodes(e,2)*2-1);
tmp2b=Tnew(elemNodes(e,2)*2);
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
t(j)=gm(1)*tmp1a+gm(2)*tmp1b+gm(3)*tmp2a+gm(4)*tmp2b;
end
end
end
vel=0.5*(0.5/charlen)*(2*t(1)+t(2)-t(3)-2*t(4));
vel=0.
% Update interface position
dpos=dpos+vel*dtime;
stored(ts+1)=dpos;
K=zeros(numNodes*ndof,numNodes*ndof);
M=zeros(numNodes*ndof,numNodes*ndof);
pforce=zeros(numNodes*ndof,1);
% Loop Through Elements
for e=1:numElem
Ke=zeros(2*ndof);
Me=zeros(2*ndof);
crdn1=ndCoords(elemNodes(e,1));
crdn2=ndCoords(elemNodes(e,2));
theta(1)=abs(crdn1-dpos)*sign(crdn1-dpos);
theta(2)=abs(crdn2-dpos)*sign(crdn2-dpos);
enr=2;
elen=abs(crdn2-crdn1);
ajacob=elen/2.;
if sign(theta(1))~=sign(theta(2))
% enriched element
eNodes(2*elemNodes(e,1))=1;
eNodes(2*elemNodes(e,2))=1;
enr=4;
% get interface position on element
point=(dpos-crdn1)/ajacob-1.;
% devide element for sub integration
len1=abs(-point-1.);
len2=abs(1.-point);
mid1=-1+len1/2.;
mid2=1-len2/2.;
gpx(1)=-(len1/2.)/sqrt(3.)+mid1;
gpx(2)=(len1/2.)/sqrt(3.)+mid1;
gpx(3)=-(len2/2.)/sqrt(3.)+mid2;
gpx(4)=(len2/2.)/sqrt(3.)+mid2;
w(1)=(len1/2.);
w(2)=(len1/2.);
w(3)=(len2/2.);
w(4)=(len2/2.);
else
% regular element - fix extra dofs
gpx(1)=-1/sqrt(3.);
gpx(2)=1/sqrt(3.);
w(1)=1.;
w(2)=1.;
end
% Loop Through Int Points
for i=1:enr;
c=gpx(i);
phi(1)=(1.-c)/2.;
phi(3)=(1.+c)/2.;
term=theta(1)*phi(1)+theta(2)*phi(3);
cond=1.;
spec=1.;
phi(2)=phi(1)*(abs(term)-abs(theta(1)));
phi(4)=phi(3)*(abs(term)-abs(theta(2)));
phic(1)=-0.5;
phic(3)=0.5;
dterm=sign(term)*(phic(1)*theta(1)+phic(3)*theta(2));
phic(2)=phic(1)*(abs(term)-abs(theta(1)))+phi(1)*dterm;
phic(4)=phic(3)*(abs(term)-abs(theta(2)))+phi(3)*dterm;
phix(1)=phic(1)/ajacob;
phix(2)=phic(2)/ajacob;
phix(3)=phic(3)/ajacob;
phix(4)=phic(4)/ajacob;
we=ajacob*w(i);
Ke=Ke+we*cond*phix'*phix;
Me=Me+(we*rho*spec*phi'*phi)/dtime;
end
% Add penalty term and get temp gradient on interface
if enr==4;
xi=point;
gm(1)=(1.-xi)/2.;
gm(3)=(1.+xi)/2.;
term=theta(1)*gm(1)+theta(2)*gm(3);
gm(2)=gm(1)*(abs(term)-abs(theta(1)));
gm(4)=gm(3)*(abs(term)-abs(theta(2)));
pen=beta*(gm'*gm);
pfL=beta*1*gm';
Ke=Ke+pen;
else
pen=zeros(4);
pfL=zeros(4,1);
end
% Assemble Global Matrices
gnum=2.*elemNodes(e,1)-1.;
for i=1:4;
for j=1:4;
K(gnum+j-1,gnum+i-1)=K(gnum+j-1,gnum+i-1)+Ke(j,i);
M(gnum+j-1,gnum+i-1)=M(gnum+j-1,gnum+i-1)+Me(j,i);
end
pforce(gnum+i-1)=pforce(gnum+i-1)+pfL(i);
end
end
A=K+M;
Sub=A*Bound;
RHS=M*Tnew-Sub+pforce;
% Apply Boundary Conditions
Biindex=0.;
for i=1:ndof*numNodes;
if Bound(i)==0.;
Biindex=Biindex+1;
RHSR(Biindex)=RHS(i);
jindex=0;
for j=1:ndof*numNodes;
if Bound(j)==0.;
jindex=jindex+1;
AR(Biindex,jindex)=A(i,j);
end
end
end
end
%Solve
Tnewr=(AR^-1)*RHSR';
% Restore Matrices
Biindex=0;
for i=1:ndof*numNodes;
if Bound(i)==0.;
Biindex=Biindex+1;
Tnew(i)=Tnewr(Biindex);
end
end
Tnew
end
stored';