function [] = FESolveX2D()
% MATLAB based 2-D XFEM Solver
% J. Grogan (2012)
clear all
% Define Mesh
NumX=4;
NumY=1;
delX=0.25;
delY=0.25;
for j=1:NumY+1
   for i=1:NumX+1
       index=i+(NumX+1)*(j-1);
       Node(index,1)=single((i-1.))*delX;
       Node(index,2)=single((j-1.))*delY;
   end
end
numNodes=(NumX+1)*(NumY+1);
for j=1:NumY
    for i=1:NumX
        index=i+NumX*(j-1);
        Element(index,1)=i+(NumX+1)*(j-1);
        Element(index,2)=i+(NumX+1)*(j-1)+1;
        Element(index,3)=i+(NumX+1)*(j)+1;
        Element(index,4)=i+(NumX+1)*(j);
    end
end
numElem=(NumX)*(NumY);
% dofs per node
ndof=2;
% Define Section Properties
rho=1.;
% initial interface position
dpos=0.6;
% Initial temperatures
Tnew=zeros(numNodes*2,1);
Bound=zeros(numNodes*2,1);
stored(1)=dpos;
for e=1:numElem
    for n=1:4
        crdn=Node(Element(e,n),1);
        if crdn<=dpos
            Tnew(2*Element(e,n)-1)=1.;
        end
        if crdn<0.01
            Bound(2*Element(e,n)-1)=1.;
        end
    end
end
% Define Time Step
dtime=0.01;
tsteps=10;
time=0.;
% penalty term
Penalty=50.;
% Loop through time steps
for ts=1:tsteps
    eNodes=zeros(2*numNodes,1);
    % Get interface velocity
     d(1)=dpos+delX;
     d(2)=dpos+3*delX/4;
     d(3)=dpos+delX/4;
     d(4)=dpos;
     for e=1:numElem
         crdn1=Node(Element(e,1),1);
         crdn2=Node(Element(e,2),1);
        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(Element(e,1)*2-1);
                   tmp1b=Tnew(Element(e,1)*2);
                   tmp2a=Tnew(Element(e,2)*2-1);
                   tmp2b=Tnew(Element(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=-0.1*(0.5/delX)*(2*t(1)+t(2)-t(3)-2*t(4))
    vel=0.0;
    % 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(4*ndof);
        Me=zeros(4*ndof);
        for icrd=1:4;
            crdnx(icrd)=Node(Element(e,icrd),1);
            crdny(icrd)=Node(Element(e,icrd),2);
            theta(icrd)=abs(crdnx(icrd)-dpos)*sign(crdnx(icrd)-dpos);
        end   
        if sign(theta(1))~=sign(theta(2))
        % 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);
                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          
        % Loop Through Int Points
        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);
                iLS=theta(1)*phi(1)+theta(2)*phi(3)+theta(3)*phi(5)+theta(4)*phi(7);
                cond=1.;
                spec=1.;                                                  
                for iter=1:4
                    phi(2*iter)=phi(2*iter-1)*(abs(iLS)-abs(theta(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)/dtime)/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;  
            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))=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';