function [] = FESolveX2D()
% MATLAB based XFEM Solver
% J. Grogan (2012)
clear all
% Define Geometry
Lx=1;
Ly=1;
% Generate Mesh
numElemX=1;
numElemY=1;
numElem=numElemX*numElemY;
gapX=Lx/numElemX;
gapY=Ly/numElemY;
tmpX=0.
tmpY=0.
for i=1:numElemX+1
    for j=1:numElemY+1
        ndCrdX=



[ndCrds,elemNodes]=rectangularMesh(Lx,Ly,numElemX,numElemY);
xx=ndCrds(:,1);
yy=ndCrds(:,2);
drawingMesh(ndCrds,elemNodes,'Q4','k-');
numNodes=size(xx,1);
ndCrds
elemNodes
indx=1:numElem;
elemNodes(:,1)=indx;
elemNodes(:,2)=indx+1;
% dofs per node
ndof=2;
% Define Section Properties
rho=1.;
% 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'