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