PFV Buoyancy matrix 2
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Function BMat4424SW.m for quartic pressure-free buoyancy matrix
function [Bm,RowNdx,ColNdx]=BMat4424SW(Xe, Elcon, nn2vft, nn2tft) % BMAT4424SW - Rectangular(Hermite simple-cubic)element thermal bouyancy matrix % for segregated solution. % % Quartic-complete, conforming, solenoidal, Hermite velocity basis % weights on 4-node rectangular elements with 6 DOF per node % and cubic Hermite temperature elements 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. % % Usage: % [Bm,RowNdx,ColNdx]=BMat4424SA(Xe, Elcon, nn2nft) % Xe(1,:) - x-coordinates of 4 corner nodes of element. % Xe(2,:) - y-coordinates of 4 corner nodes of element. % Elcon(4) - connectivity matrix for this element, list of nodes. % nn2vft(1,n) - global freedom number for velocity at node n. % nn2vft(2,n) - global freedom type for node n. % nn2tft(1,n) - global freedom number for temperature at node n. % nn2tft(2,n) - global freedom type for node n. % % Jonas Holdeman, December 2008, Revised July 2011 % Constants and fixed data nn=[-1 -1; 1 -1; 1 1; -1 1]; % defines local nodal order nnd = 4; % nnd = number of nodes in element nT = 3; nfT = nnd*nT; % nT = number of T dofs per node, nfT = number T dofs. nV = 6; nfV = nnd*nV; % nV = number of V dofs per node, nfV = number V dofs. NDT = 1:nT; NDV = 1:nV; % Define 5-point quadrature data once, on first call. % Gaussian weights and absissas to integrate 9th degree polynomials exactly. global GQ5; if (isempty(GQ5)) % Has 5-point quadrature data been defined? If not, define arguments & weights. Aq=[-.906179845938664,-.538469310105683, .0, .538469310105683, .906179845938664]; Hq=[ .236926885056189, .478628670499366, .568888888888889, .478628670499366, .236926885056189]; GQ5.size=25; GQ5.xa=[Aq;Aq;Aq;Aq;Aq]; GQ5.ya=GQ5.xa'; wt=[Hq;Hq;Hq;Hq;Hq]; GQ5.wt=wt.*wt'; end xa=GQ5.xa; ya=GQ5.ya; wt=GQ5.wt; Nq=GQ5.size; global ZS4424b; global ZG4424b; if (isempty(ZS4424b)|size(ZS4424b,2)~=Nq) % Evaluate and save/cache the set of shape functions at quadrature pts. ZS4424b=cell(nnd,Nq); ZG4424b=cell(nnd,Nq); for k=1:Nq for m=1:nnd ZS4424b{m,k}= Sr(nn(m,:),xa(k),ya(k)); ZG4424b{m,k}= gr(nn(m,:),xa(k),ya(k)); end end end % --------------- End fixed data ---------------- Ti=cell(nnd); Tgi=cell(nnd); for m=1:nnd % Loop over corner nodes 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; Tgi{m} = blkdiag(1,Jt'); end % Loop m % Jt=[x_q, x_r; y_q, y_r]; Bm=zeros(nfV,nfT); S=zeros(2,nfV); g=zeros(1,nfT); % Preallocate 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 Jtd=Jt/Det; % Initialize functions and derivatives at the quadrature point (xa,ya). mt = 0; mv = 0; for m=1:nnd g(1,mt+NDT)= ZG4424b{m,k}*Tgi{m}; mt = mt + nT; S(:,mv+NDV)=Jtd*ZS4424b{m,k}*Ti{m}; mv = mv + nV; end % loop m % Label rows by the test (weight) function index, columns by trial function index? % Submatrix ordered by psi,u,v Bm = Bm + S(2,:)'*g*wt(k); % Sy'*g end % loop k Bm = Bm*Det; gfr=zeros(nfV,1); gfc=zeros(1,nfT); mv = 0; mt=0; for m=1:nnd % m, mv gfr(mv+NDV)=(nn2vft(1,Elcon(m))-1)+NDV; mv = mv + nV; % get row dofs (V) gfc(mt+NDT)=(nn2tft(1,Elcon(m))-1)+NDT; mt = mt + nT; % get col dofs (T) end % loop on k %gfc=gfr'+3; RowNdx=repmat(gfr,1,nfT); ColNdx=repmat(gfc,nfV,1); RowNdx=reshape(RowNdx,nfV*nfT,1); ColNdx=reshape(ColNdx,nfV*nfT,1); Bm=reshape(Bm,nfV*nfT,1); return; % ------------------------------------------------------------------------------ function g=gr(ni,q,r) %G Cubic Hermite basis function for simple-cubic temperature. qi=ni(1); q0=q*ni(1); ri=ni(2); r0=r*ni(2); % Scalar simple-cubic Hermite g=[1/8*(1+q0)*(1+r0)*(2+q0*(1-q0)+r0*(1-r0)), -1/8/qi*(1+q0)*(1+r0)*(1-q^2),... -1/8/ri*(1+q0)*(1+r0)*(1-r^2)]; return; function S=Sr(ni,q,r) %SR Array of quartic 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. % 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;