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phd-scripts/Unpublished/XFEM2/Full2D/XCOR_2D.m

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Add scripts and inp files.
2024-05-13 19:50:21 +00:00
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
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