PFV convection matrix 2
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Matlab function CMat4424W.m for pressure-free velocity convection matrix.
function [Cm,RowNdx,ColNdx,Rcm,RcNdx]=CMat4424W(Xe,Elcon,nn2nft,Vdof,ItType) %CMAT4424G - Returns the element convection matrix for the quartic Hermite % basis functions on 4-node straight-sided quadrilateral elements with 6 DOF % per node using Gauss quadrature on the reference square. % The columns of the array Vdof must contain the six 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 the second derivatives % Pxx, Pxy, Pyy of the stream function. % The assumed nodal numbering starts with 1 at the lower left corner of the % element and proceeds counter-clockwise around the element. % % Usage: % [CM,Rndx,Cndx] = CMat4424W(Xe,Elcon,nn2nft,Vdof) % [CM,Rndx,Cndx,Rcm,RcNdx] = CMat4424W(Xe,Elcon,nn2nft,Vdof,ItType) % 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 - (6x4) array of stream function, velocity components, and second % stream function derivatives to specify the velocity over the element. % ItType - Iteration type, 0 for simple, 1 for Newton, default is simple % % Jonas Holdeman, August 2007, revised July 2011 % Constants and fixed data nd = 6; 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 8-point quadrature data once, on first call. % Gaussian weights and absissas to integrate 15th degree polynomials exactly. global GQ8; if (isempty(GQ8)) % Define 8-point quadrature data once, on first call. Aq=[-.960289856497536,-.796666477413627,-.525532409916329,-.183434642495650, ... .183434642495650, .525532409916329, .796666477413627, .960289856497536]; %Abs Hq=[ .101228536290376, .222381034453374, .313706645877887, .362683783378362, ... .362683783378362, .313706645877887, .222381034453374, .101228536290376]; %Wts GQ8.size=64; GQ8.xa=[Aq;Aq;Aq;Aq;Aq;Aq;Aq;Aq]; GQ8.ya=GQ8.xa'; Wt=[Hq;Hq;Hq;Hq;Hq;Hq;Hq;Hq]; GQ8.wt=Wt.*Wt'; end xa=GQ8.xa; ya=GQ8.ya; wt=GQ8.wt; Nq=GQ8.size; % Use GQ8 (8x8) for exact integration global Zs4424D2c; global ZS4424c; if (isempty(ZS4424c)|isempty(Zs4424D2c)|size(ZS4424c,2)~=Nq) % Evaluate and save/cache the set of shape functions at quadrature pts. Zs4424D2c=cell(4,Nq); ZS4424c=cell(4,Nq); for k=1:Nq for m=1:4 Zs4424D2c{m,k}=D3s(nn(m,:),xa(k),ya(k)); ZS4424c{m,k}= Sr(nn(m,:),xa(k),ya(k)); end end end % if(isempty(*)) % --------------- End fixed data ---------------- if (nargin<5 | isempty(ItType) | nargout<4) ItType=0; end % default iteration type is 'simple'. Ti=cell(4); % Jt=[x_q, x_r; y_q, y_r]; for m=1:4 Jt=Xe*GBL(nn(:,:),nn(m,1),nn(m,2)); JtiD=[Jt(2,2),-Jt(1,2); -Jt(2,1),Jt(1,1)]; % det(J)*inv(J) 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]; T6=blkdiag(1,JtiD,T2); Bxy=Xe*BLxy(nn(:,:),nn(m,1),nn(m,2)); % Second cross derivatives T6(5,2:3)=Bxy([2,1])'; Ti{m}=T6; end Cm=zeros(nd4,nd4); Rcm=zeros(nd4,1); S=zeros(2,nd4); Sx=zeros(2,nd4); Sy=zeros(2,nd4); % Pre-allocate arrays % Begin loop over Gauss-Legendre quadrature points. 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 Jtd=Jt/Det; 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); S(:,mm+ND)=Jtd*ZS4424c{m,k}*Ti{m}; Ds = TL*Zs4424D2c{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 % 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 (ItType~=0) % iteration type is Newton SOMETHING WRONG HERE !! 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(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 Cm = reshape(Cm,nd4*nd4,1); RowNdx=reshape(RowNdx,nd4*nd4,1); ColNdx=reshape(ColNdx,nd4*nd4,1); if(ItType==0) Rcm=[]; RcNdx=[]; else RcNdx=gf; end return; % ---------------------------------------------------------------------------- function p2=D3s(ni,q,r) % Second derivatives [Pxx; Pxy; Pyy] of quartic stream function. qi=ni(1); q0=q*ni(1); q1=1+q0; ri=ni(2); r0=r*ni(2); r1=1+r0; p2=[-3/32*qi^2*q0*(1+r0)^2*(r0*(10*q^2+3*r^2-7)+20-20*q^2-6*r^2), 3/32*qi^2/ri*q0*(1+r0)^3*(1-r0)*(5-3*r0), ... -3/16*qi*(1-q^2)*(1+r0)^2*(2-r0)*(1+5*q0), -1/16*(1+q0)*(1+r0)^2*(2-r0)*(1+2*q0-5*q^2), ... -1/8*qi/ri*(1+r0)^2*(1-r0)*(1+3*q0), -3/32*qi^2/ri^2*q0*(1+r0)^3*(1-r0)^2; ... 9/64*qi*ri*(1-q^2)*(1-r^2)*(6-5*q^2-5*r^2), -3/64*qi*(1-q^2)*(1+r0)^2*(1-3*r0)*(7-5*r0), ... 3/64*ri*(1+q0)^2*(1-r^2)*(1-3*q0)*(7-5*q0), 3/64*ri/qi*(1+q0)^2*(1-r^2)*(1-q0)*(1-5*q0), ... 1/16*(1+q0)*(1+r0)*(1-3*q0)*(1-3*r0), 3/64*qi/ri*(1-q^2)*(1+r0)^2*(1-r0)*(1-5*r0); ... -3/32*ri^2*r0*(1+q0)^2*(q0*(10*r^2+3*q^2-7)+20-20*r^2-6*q^2), 3/16*ri*(1+q0)^2*(1-r^2)*(2-q0)*(1+5*r0), ... -3/32*ri^2*r0/qi*(1+q0)^2*(1-q^2)*(5-3*q0), -3/32*ri^2/qi^2*r0*(1+q0)*(1-q^2)^2, ... -1/8*ri/qi*(1+q0)*(1-q^2)*(1+3*r0), -1/16*(1+q0)^2*(1+r0)*(2-q0)*(1+2*r0-5*r^2)] ; return; function S=Sr(ni,q,r) %SR Array of solenoidal basis functions. qi=ni(1); q0=q*ni(1); q1=1+q0; ri=ni(2); r0=r*ni(2); r1=1+r0; % array of solenoidal vectors S=[3/64*ri*(1+q0)^2*(1-r^2)*((1+q0)*(8-9*q0+3*q^2)+(2-q0)*(1-5*r^2)), ... -1/64*(1+q0)^2*(1+r0)^2*(2-q0)*(7-26*r0+15*r^2), ... 3/64*ri/qi*(1+q0)^3*(1-r^2)*(1-q0)*(5-3*q0), ... 3/64*ri/qi^2*(1+q0)^3*(1-r^2)*(1-q0)^2, ... 1/16/qi*(1+q0)^2*(1+r0)*(1-q0)*(1-3*r0), ... 1/64/ri*(1+q0)^2*(1+r0)^2*(1-r0)*(1-5*r0)*(2-q0); ... -3/64*qi*(1+r0)^2*(1-q^2)*((1+r0)*(8-9*r0+3*r^2)+(2-r0)*(1-5*q^2)), ... 3/64*qi/ri*(1+r0)^3*(1-q^2)*(1-r0)*(5-3*r0), ... -1/64*(1+r0)^2*(1+q0)^2*(2-r0)*(7-26*q0+15*q^2), ... -1/64/qi*(1+r0)^2*(1+q0)^2*(1-q0)*(1-5*q0)*(2-r0), ... -1/16/ri*(1+r0)^2*(1+q0)*(1-r0)*(1-3*q0), ... -3/64*qi/ri^2*(1+r0)^3*(1-q^2)*(1-r0)^2]; return; function G=GBL(ni,q,r) % Transposed gradient (derivatives) of scalar bilinear mapping function for straight-sised elements. % 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; function D=BLxy(ni,q,r) % Transposed second cross-derivative of scalar bilinear mapping function. % The parameter ni can be a vector of coordinate pairs. D=[.25*ni(:,1).*ni(:,2)]; return;