PFV diffusion matrix
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(Created page with "Function '''DMatW.m''' for pressure-free velocity diffusion matrix <pre> function [Dm,RowNdx,ColNdx]=DMatW(Xe,Elcon,nn2nft) %DMATW - Returns the affine-mapped element diffusion...")
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(Created page with "Function '''DMatW.m''' for pressure-free velocity diffusion matrix <pre> function [Dm,RowNdx,ColNdx]=DMatW(Xe,Elcon,nn2nft) %DMATW - Returns the affine-mapped element diffusion...")
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Revision as of 03:44, 29 June 2011
Function DMatW.m for pressure-free velocity diffusion matrix
function [Dm,RowNdx,ColNdx]=DMatW(Xe,Elcon,nn2nft) %DMATW - Returns the affine-mapped element diffusion matrix for the simple cubic Hermite % basis functions on 4-node straight-sided quadrilateral elements with 3 DOF % per node using Gauss quadrature on the reference square and row/col indices. % % Cubic-complete, fully-conforming, divergence-free, Hermite basis % functions on 4-node rectangular elements with 3 DOF per node using % Gauss quadrature on the 2x2 reference square. % The assumed nodal numbering starts with 1 at the lower left corner % of the element and proceeds counter-clockwise around the element. % Uses second derivatives of stream function. % % Usage: % [Dm,Rndx,Cndx] = DMatW(Xe,Elcon,nn2nft) % Xe(1,:) - x-coordinates of corner nodes of element. % Xe(2,:) - y-coordinates of corner nodes of element. % and shape of the element. It is constant for affine elements. % Elcon - connectivity matrix for this element. % nn2nft - global number and type of DOF at each node % % Jonas Holdeman, August 2007, revised June 2011 % Constants and fixed data nd = 3; nd4=4*nd; ND=1:nd; % nd = number of dofs per node, nn=[-1 -1; 1 -1; 1 1; -1 1]; % defines local nodal order % Define 4-point quadrature data once, on first call. % Gaussian weights and absissas to integrate 7th degree polynomials exactly. global GQ4; if (isempty(GQ4)) % Define 4-point quadrature data once, on first call. Aq=[-.861136311594053,-.339981043584856,.339981043584856, .861136311594053]; %Abs Hq=[ .347854845137454, .652145154862546,.652145154862546, .347854845137454]; %Wts GQ4.size=16; GQ4.xa=[Aq;Aq;Aq;Aq]; GQ4.ya=GQ4.xa'; wt=[Hq;Hq;Hq;Hq]; GQ4.wt=wt.*wt'; end xa=GQ4.xa; ya=GQ4.ya; wt=GQ4.wt; Nq=GQ4.size; % ----------------------------------------------- global Zs3412d2; if (isempty(Zs3412d2)|size(Zs3412d2,2)~=Nq) % Evaluate and save/cache the set of shape functions at quadrature pts. Zs3412d2=cell(4,Nq); for k=1:Nq for m=1:4 Zs3412d2{m,k}=D3s(nn(m,:),xa(k),ya(k)); end end end % if(isempty(*)) % --------------- End fixed data ---------------- Ti=cell(4); for m=1:4 Jt=Xe*GBL(nn(:,:),nn(m,1),nn(m,2)); % transpose of Jacobian at node m JtiD=[Jt(2,2),-Jt(1,2); -Jt(2,1),Jt(1,1)]; % det(J)*inv(Jt) Ti{m}=blkdiag(1,JtiD); end % Move Jacobian evaluation inside k-loop for general convex quadrilateral. % Jt=[x_q, x_r; y_q, y_r]; Dm=zeros(nd4,nd4); Sx=zeros(2,nd4); Sy=zeros(2,nd4); % Pre-allocate arrays for k=1:Nq Jt=Xe*GBL(nn(:,:),xa(k),ya(k)); % transpose of Jacobian at (xa,ya) Det=Jt(1,1)*Jt(2,2)-Jt(1,2)*Jt(2,1); % Determinant of Jt & J TL=[Jt(2,2)^2, -2*Jt(2,1)*Jt(2,2), Jt(2,1)^2; ... -Jt(1,2)*Jt(2,2), Jt(1,1)*Jt(2,2)+Jt(2,1)*Jt(1,2), -Jt(1,1)*Jt(2,1); ... Jt(1,2)^2, -2*Jt(1,1)*Jt(1,2), Jt(1,1)^2]/Det^2; % Initialize functions and derivatives at the quadrature point (xa,ya). for m=1:4 mm=nd*(m-1); Ds = TL*Zs3412d2{m,k}*Ti{m}; Sx(:,mm+ND) = [Ds(2,:); -Ds(1,:)]; % [Pyx, -Pxx] Sy(:,mm+ND) = [Ds(3,:); -Ds(2,:)]; % [Pyy, -Pxy] end % loop m Dm = Dm+(Sx'*Sx+Sy'*Sy)*(wt(k)*Det); end % end loop k over quadrature points gf=zeros(nd4,1); m=0; for n=1:4 % Loop over element nodes gf(m+ND)=(nn2nft(1,Elcon(n))-1)+ND; % Get global freedoms m=m+nd; end RowNdx=repmat(gf,1,nd4); % Row indices ColNdx=RowNdx'; % Col indices Dm = reshape(Dm,nd4*nd4,1); RowNdx=reshape(RowNdx,nd4*nd4,1); ColNdx=reshape(ColNdx,nd4*nd4,1); return; % ------------------------------------------------------------------- function P2=D3s(ni,q,r) % Second derivatives [Pxx; Pxy; Pyy] of simple cubic stream function. qi=ni(1); q0=q*ni(1); q1=1+q0; ri=ni(2); r0=r*ni(2); r1=1+r0; P2=[-.75*qi^2*(r0+1)*q0, 0, -.25*qi*(r0+1)*(3*q0+1); ... .125*qi*ri*(4-3*(q^2+r^2)), .125*qi*(r0+1)*(3*r0-1), ... -.125*ri*(q0+1)*(3*q0-1); -.75*ri^2*(q0+1)*r0, .25*ri*(q0+1)*(3*r0+1), 0] ; return; function G=GBL(ni,q,r) % Transposed gradient (derivatives) of scalar bilinear mapping function. % The parameter ni can be a vector of coordinate pairs. G=[.25*ni(:,1).*(1+ni(:,2).*r), .25*ni(:,2).*(1+ni(:,1).*q)]; return;