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