PFV4 convection matrix
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Revision as of 19:12, 15 March 2013 by Jonas Holdeman (Talk | contribs)
function [Cm,RowNdx,ColNdx,Rcm,RcNdx]=CMatW(eu,Xe,Elcon,nn2nft,Vdof) %CMatW - Returns the element convection matrix for the Hermite basis % functions with 3, 4, or 6 degrees-of-freedom and defined on a % 3-node (triangle) or 4-node (quadrilateral) element by the class % instance es using Gauss quadrature on the reference element. % The columns of the array Vdof must contain the 3, 4, or 6 nodal % degree-of-freedom vectors in the proper nodal order. % The degrees of freedom in Vdof are the stream function, the two components % u and v of the solenoidal velocity vector, and possibly the second % derivatives Pxx, Pxy, Pyy of the stream function. % % Usage: % [CM,Rndx,Cndx] = CMatW(es,Xe,Elcon,nn2nft,Vdof) % [CM,Rndx,Cndx,Rcm,RcNdx] = CMatW(es,Xe,Elcon,nn2nft,Vdof) % eu - handle for basis function definitions % Xe(1,:) - x-coordinates of corner nodes of element. % Xe(2,:) - y-coordinates of corner nodes of element. % Elcon - this element connectivity matrix % nn2nft - global number and type of DOF at each node % Vdof - (ndfxnnd) array of stream function, velocity components, and % perhaps second stream function derivatives to specify the velocity % over the element. % % Indirectly may use (handle passed by eu): % GQuad2 - function providing 2D rectangle quadrature rules. % TQuad2 - function providing 2D triangle quadrature rules. % % Jonas Holdeman, August 2007, revised March 2013 % ----------------- Constants and fixed data ------------------------- nnodes = eu.nnodes; % number of nodes per element (4); nndofs = eu.nndofs; % nndofs = number of dofs per node, (3|6); nedofs = nnodes*nndofs; nn = eu.nn; % defines local nodal order, [-1 -1; 1 -1; 1 1; -1 1] % ------------------------------------------------------------------------ persistent QQCM4; if isempty(QQCM4) QRorder = 3*eu.mxpowr-1; % =9 [QQCM4.xa, QQCM4.ya, QQCM4.wt, QQCM4.nq] = eu.hQuad(QRorder); end % if isempty... xa = QQCM4.xa; ya = QQCM4.ya; wt = QQCM4.wt; Nq = QQCM4.nq; % ------------------------------------------------------------------------ persistent ZZ_Sc; persistent ZZ_SXc; persistent ZZ_SYc; if (isempty(ZZ_Sc)||size(ZZ_Sc,2)~=Nq) % Evaluate and save/cache the set of shape functions at quadrature pts. ZZ_Sc=cell(nnodes,Nq); ZZ_SXc=cell(nnodes,Nq); ZZ_SYc=cell(nnodes,Nq); for k=1:Nq for m=1:nnodes ZZ_Sc{m,k}= eu.S(nn(m,:),xa(k),ya(k)); [ZZ_SXc{m,k},ZZ_SYc{m,k}]=eu.DS(nn(m,:),xa(k),ya(k)); end end end % if isempty... % ----------------------- End fixed data --------------------------------- if (nargout<4), ItrType=0; else ItrType=1; end % ItrType=1 for Newton iter affine = eu.isaffine(Xe); % affine? %affine = (sum(abs(Xe(:,1)-Xe(:,2)+Xe(:,3)-Xe(:,4)))<4*eps); % affine? Ti=cell(nnodes); % Jt=[x_q, x_r; y_q, y_r]; if affine % (J constant) Jt=Xe*eu.Gm(nn(:,:),eu.cntr(1),eu.cntr(2)); JtiD=[Jt(2,2),-Jt(1,2); -Jt(2,1),Jt(1,1)]; % det(J)*inv(J) if nndofs==3 TT=blkdiag(1,JtiD); elseif nndofs==4 TT=blkdiag(1,JtiD,Jt(1,1)*Jt(2,2)+Jt(1,2)*Jt(2,1)); else T2=[Jt(1,1)^2, 2*Jt(1,1)*Jt(1,2), Jt(1,2)^2; ... % alt Jt(1,1)*Jt(2,1), Jt(1,1)*Jt(2,2)+Jt(1,2)*Jt(2,1), Jt(1,2)*Jt(2,2); ... Jt(2,1)^2, 2*Jt(2,1)*Jt(2,2), Jt(2,2)^2]; TT=blkdiag(1,JtiD,T2); Bxy=Xe*eu.DGm(nn(:,:),0,0); % Second cross derivatives TT(5,2)= Bxy(2); TT(5,3)=-Bxy(1); end % nndofs... for m=1:nnodes, Ti{m}=TT; end Det=Jt(1,1)*Jt(2,2)-Jt(1,2)*Jt(2,1); % Determinant of Jt & J Jtd=Jt/Det; Ji=[Jt(2,2),-Jt(2,1); -Jt(1,2),Jt(1,1)]/Det; else for m=1:nnodes Jt=Xe*eu.Gm(nn(:,:),nn(m,1),nn(m,2)); JtiD=[Jt(2,2),-Jt(1,2); -Jt(2,1),Jt(1,1)]; % det(J)*inv(J) if nndofs==3 TT=blkdiag(1,JtiD); elseif nndofs==4 TT=blkdiag(1,JtiD,Jt(1,1)*Jt(2,2)+Jt(1,2)*Jt(2,1)); else T2=[Jt(1,1)^2, 2*Jt(1,1)*Jt(1,2), Jt(1,2)^2; ... % alt Jt(1,1)*Jt(2,1), Jt(1,1)*Jt(2,2)+Jt(1,2)*Jt(2,1),Jt(1,2)*Jt(2,2);... Jt(2,1)^2, 2*Jt(2,1)*Jt(2,2), Jt(2,2)^2]; TT=blkdiag(1,JtiD,T2); Bxy=Xe*eu.DGm(nn(:,:),nn(m,1),nn(m,2)); % Second cross derivatives TT(5,2)= Bxy(2); TT(5,3)=-Bxy(1); end % nndofs... Ti{m}=TT; end % for m=... end % affine % Allocate arrays Cm=zeros(nedofs,nedofs); Rcm=zeros(nedofs,1); S=zeros(2,nedofs); Sx=zeros(2,nedofs); Sy=zeros(2,nedofs); ND=1:nndofs; % Begin loop over Gauss-Legendre quadrature points. for k=1:Nq if ~affine Jt=Xe*eu.Gm(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 Jtd=Jt/Det; Ji=[Jt(2,2),-Jt(2,1); -Jt(1,2),Jt(1,1)]/Det; end % Initialize functions and derivatives at the quadrature point (xa,ya). for m=1:nnodes mm=nndofs*(m-1); S(:,mm+ND)=Jtd*ZZ_Sc{m,k}*Ti{m}; Sx(:,mm+ND)=Jtd*(Ji(1,1)*ZZ_SXc{m,k}+Ji(1,2)*ZZ_SYc{m,k})*Ti{m}; Sy(:,mm+ND)=Jtd*(Ji(2,1)*ZZ_SXc{m,k}+Ji(2,2)*ZZ_SYc{m,k})*Ti{m}; end % loop m % Compute the fluid velocity at the quadrature point. U = S*Vdof(:); % Submatrix ordered by psi,u,v Cm = Cm + S'*(U(1)*Sx+U(2)*Sy)*(wt(k)*Det); if (ItrType~=0) % iteration type is Newton Ux=Sx*Vdof(:); Uy=Sy*Vdof(:); Cm = Cm + S'*(Ux*S(1,:)+Uy*S(2,:))*(wt(k)*Det); Rcm=Rcm + S'*(U(1)*Ux+U(2)*Uy)*(wt(k)*Det); end % Cm & Rcm for Newton iteration end % end loop k over quadrature points gf=zeros(nedofs,1); m=0; for n=1:nnodes % Loop over element nodes gf(m+ND)=(nn2nft(Elcon(n),1)-1)+ND; % Get global freedoms m=m+nndofs; end RowNdx=repmat(gf,1,nedofs); % Row indices ColNdx=RowNdx'; % Col indices Cm = reshape(Cm,nedofs*nedofs,1); RowNdx=reshape(RowNdx,nedofs*nedofs,1); ColNdx=reshape(ColNdx,nedofs*nedofs,1); if(ItrType==0), Rcm=[]; RcNdx=[]; else RcNdx=gf; end return;