PFV diffusion matrix

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Latest revision as of 15:58, 15 March 2013

Function DMatW.m for pressure-free velocity diffusion matrix

function [Dm,RowNdx,ColNdx]=DMatW(eu,Xe,Elcon,nn2nft)
%DMatW - Returns the element diffusion 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. 
%
% Usage:
%   [Dm,Rndx,Cndx] = DMatW(Xe,Elcon,nn2nft,es)
%   es    - reference for basis function definitions
%   Xe(1,:) -  x-coordinates of corner nodes of element.  
%   Xe(2,:) -  y-coordinates of corner nodes of element.  
%   Elcon - connectivity matrix for this element. 
%   nn2nft - global DOF and type of DOF at each node 
%
% Indirectly may use (handle passed by eu):
%   GQuad2   - function providing 2D rectangle quadrature rules.
%   TQuad2   - function providing 2D triangle quadrature rules.
%
% Jonas Holdeman, January 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;       % nndofs = number of dofs per node, 
nn = eu.nn;          % defines local nodal order, [-1 -1; 1 -1; 1 1; -1 1]

% ------------------------------------------------------------------------
persistent QQDM4; 
if isempty(QQDM4) 
     QRorder = 2*(eu.mxpowr-1)+1; % =9
    [QQDM4.xa, QQDM4.ya, QQDM4.wt, QQDM4.nq] = eu.hQuad(QRorder);
end  % if isempty...
xa = QQDM4.xa; ya = QQDM4.ya; wt = QQDM4.wt; Nq = QQDM4.nq;
% ------------------------------------------------------------------------

persistent ZZ_SXd; persistent ZZ_SYd; 
if (isempty(ZZ_SXd)||isempty(ZZ_SYd)||size(ZZ_SXd,2)~=Nq)
% Evaluate and save/cache the set of shape functions at quadrature pts. 
   ZZ_SXd=cell(nnodes,Nq); ZZ_SYd=cell(nnodes,Nq); 
   for k=1:Nq
      for m=1:nnodes
      [ZZ_SXd{m,k},ZZ_SYd{m,k}]=eu.DS(nn(m,:),xa(k),ya(k));
      end
   end
end  % if(isempty(*))
% -------------------------- End fixed data ------------------------------

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   % Loop over corner nodes 
     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; ...
        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));  % 2nd cross derivatives
        TT(5,2)= Bxy(2);
        TT(5,3)=-Bxy(1);
     end
     Ti{m}=TT;
   end  % Loop m 
end

% Allocate arrays  
Dm=zeros(nedofs,nedofs); Sx=zeros(2,nedofs); Sy=zeros(2,nedofs); 
 ND=1:nndofs;
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);
    Sx(:,mm+ND)=Jtd*(Ji(1,1)*ZZ_SXd{m,k}+Ji(1,2)*ZZ_SYd{m,k})*Ti{m};
    Sy(:,mm+ND)=Jtd*(Ji(2,1)*ZZ_SXd{m,k}+Ji(2,2)*ZZ_SYd{m,k})*Ti{m};
  end  % loop m
   
  Dm = Dm+(Sx'*Sx+Sy'*Sy)*(wt(k)*Det);
   
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
 
Dm = reshape(Dm,nedofs*nedofs,1);
RowNdx=reshape(RowNdx,nedofs*nedofs,1);
ColNdx=reshape(ColNdx,nedofs*nedofs,1);   

return;