% SET CVFEM COEFICIENTS %zero out coefficient for i=1:n %loop on nodes %zero out coefficient ap(i)=0.0; %node point coefficient V(i)=0.0; %volume of CV for j=1:n_s(i)+1 a(i,j)=0.0; %neighbor coefficient end % Set loopcount % At internal nodes--recognized by S(i,n_s(i)+1)<>0 % the nodes in the support match the elements in the support % Hence loop over every node in the node i support: loopcount=n_s(i) % At boundary nodes--recognized by S(i,n_s(i)+1)=0 % the nodes in the support are one larger than the elements in the support % Hence only loop over the first n_s-1 nodes: loopcount=n_s(i) loopcount=n_s(i); if S(i,n_s(i)+1)==0 loopcount=n_s(i)-1; end

% Pick out each element in turn from i_th support for j=1:loopcount k1=S(i,j); %The numbers of the nodes in the j_th elemnt are k2=S(i,j+1); % i, k1=S(i,J), and k2=S(i,j+1) xL(1) = x(i); %xL=local x-coordinate of a node within a element xL(2) = x(k1); xL(3) = x(k2); yL(1) = y(i); %yL=local y-coordinate of a node within a element yL(2) = y(k1); yL(3) = y(k2); % Vele= element-volume Vele=(xL(2)*yL(3)-xL(3)*yL(2)-xL(1)*yL(3)+xL(1)*yL(2)... +yL(1)*xL(3)-yL(1)*xL(2))/2; Nx(1)= (yL(2)-yL(3))/(2*Vele); %derivative of shape function wrt x Nx(2)= (yL(3)-yL(1))/(2*Vele); Nx(3)= (yL(1)-yL(2))/(2*Vele); Ny(1)=-(xL(2)-xL(3))/(2*Vele); %derivative of shape function wrt y Ny(2)=-(xL(3)-xL(1))/(2*Vele); Ny(3)=-(xL(1)-xL(2))/(2*Vele);

% Face 1--face area normal components delx= [(xL(1)+xL(2)+xL(3))/3]-[(xL(1)+xL(2))/2]; dely= [(yL(1)+yL(2)+yL(3))/3]-[(yL(1)+yL(2))/2]; % Face 1--Diffusion Coefficients ap(i) = ap(i) + (- Nx(1) * dely + Ny(1) * delx); a(i,j) = a(i,j) + ( Nx(2) * dely - Ny(2) * delx); a(i,j+1) = a(i,j+1) + ( Nx(3) * dely - Ny(3) * delx); % Face 2--face area normal components delx= [(xL(1)+xL(3))/2]-[(xL(1)+xL(2)+xL(3))/3]; dely= [(yL(1)+yL(3))/2]-[(yL(1)+yL(2)+yL(3))/3]; % Face 2--Diffusion Coefficients ap(i) = ap(i) + (- Nx(1) * dely + Ny(1) * delx); a(i,j) = a(i,j) + ( Nx(2) * dely - Ny(2) * delx); a(i,j+1) = a(i,j+1) + ( Nx(3) * dely - Ny(3) * delx); end %end loop on nodes/elements in support % For internal nodes LAST node in region of support is also the FIRST % Hence coefficient for last node needs to be added to first node % and then removed a(i,1)=a(i,1)+a(i,n_s(i)+1); a(i,n_s(i)+1)=0; end %end loop on nodes in domain

The above will generate coefficients in the equation—assuming that there is no flow across segment of Control Vol along boundary Need This term In a simple treatment we write boundary contribution at node k in form i If boundary is insulated that we set If boundary has fixed value T =T fix Then we set And equation takes the form Where A large value—forcing solution of (1) at node i to return value (1)

%SET BOUNDARY for i=1:n %Default values BC(i)=0; %coefficient part BB(i)=0; %constant part end % Set value = 0 on r = r_outer boundary segment 2 for k=1:n_b(2) BC(B(2,k))=1e18; end %set value =1 on r = r_inner boundary segment 4 for k=1:n_b(4) BC(B(4,k))=1e18; BB(B(4,k))=1e18; end %

Test problem steady sate heat conduction in annulus BUT Solved in Cartesian Form in With boundary conditions Using Grid generation program as given in notes –develop a CVFEM Matlab Code (internal coefficients, boundary coefficients, solver) By Tuesday 17 hand complete matlab program and plot that comparing numerical sol With analytical