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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Unpublished/XFEM2/1D_Solver/2DLevelSet.m Normal file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/XFEM2/1D_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/XFEM2/1D_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)

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Unpublished/XFEM2/1D_Solver/testGrid.asv Normal file
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function [] = testGrid()

4
Unpublished/XFEM2/1D_Solver/testGrid.m Normal file
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function [] = testGrid()
[X,Y]=meshgrid(1:1:10);
Z=2*X+Y
surf(X,Y,Z)