PFV get pressure

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Matlab function GetPresW.m to retrieve consistent pressure from velocity.

```function [P,Px,Py] = GetPresW(NumNod,NodNdx,Elcon,nn2nft,Xgrid,Ygrid,Q,EBCPr,nu)
%GETPRESW - Compute continuous simple cubic pressure and derivatives from (simple-cubic)
%  velocity field on general quadrilateral grid (bilinear geometric mapping).
%
% Inputs
%  NumNod - number of nodes
%  NodNdx - nodal index into Xgrid and Ygrid
%  Elcon  - element connectivity, nodes in element
%  nn2nft - global number and type of (non-pressure) DOF at each node
%  Xgrid  - array of nodal x-coordinates
%  Ygrid  - array of nodal y-coordinates
%  Q      - array of DOFs for cubic velocity elements
%  EBCp   - essential pressure boundary condition structure
%    EBCp.nodn - node number of fixed pressure node
%    EBCp.val  - value of pressure
%  nu - diffusion coefficient
% Outputs
%  P  - pressure
%  Px - x-derivative of pressure
%  Py - y-derivative of pressure
% Uses
%  ilu_gmres_with_EBC - to solve the system with essential/Dirichlet BCs
%  GQ3, GQ4, GQ5  - quadrature rules.

% Jonas Holdeman,   January 2007, revised June 2011

% Constants and fixed data
nn=[-1 -1; 1 -1; 1 1; -1 1];  % defines local nodal order
nnd = 4;                      % Number of nodes in elements
nd = 3;  ND=1:nd;             % Number DOFs in velocity fns (bicubic-derived)
np = 3;                       % Number DOFs in pressure fns (simple cubic)
% Parameters for GMRES solver
GMRES.Tolerance=1.e-9;
GMRES.MaxIterates=8;
GMRES.MaxRestarts=6;
DropTol = 1e-7;                  % Drop tolerence for ilu preconditioner

% Define 3-point quadrature data once, on first call (if needed).
% Gaussian weights and absissas to integrate 5th degree polynomials exactly.
global GQ3;
if (isempty(GQ3))       % Define 3-point quadrature data once, on first call.
Aq=[-.774596669241483, .000000000000000,.774596669241483]; %Abs
Hq=[ .555555555555556, .888888888888889,.555555555555556]; %Wts
GQ3.size=9; GQ3.xa=[Aq;Aq;Aq]; GQ3.ya=GQ3.xa';
wt=[Hq;Hq;Hq]; GQ3.wt=wt.*wt';
end
% Define 4-point quadrature data once, on first call (if needed).
% 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';     % 4x4
end
% Define 5-point quadrature data once, on first call (if needed).
% 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';     % 5x5
end
% -------------- end fixed data ------------------------

NumEl=size(Elcon,2);            % Number of elements
[NumNy,NumNx]=size(Xgrid);
NumNod=NumNy*NumNx;             % Number of nodes
MxVDof=nd*NumNod;               % Max number velocity DOFs
MxPDof=np*NumNod;               % Max number pressure DOFs

% We can use the same nodal connectivities (Elcon) for pressure elements,
%  but need new pointers from nodes to pressure DOFs
nn2nftp=zeros(2,NumNod); % node number -> pressure nf & nt
nf=-np+1;  nt=1;
for n=1:NumNod
nf=nf+np;               % all nodes have 3 dofs
nn2nftp(:,n)=[nf;nt];   % dof number & type (all nodes type 1)
end;

% Allocate space for pressure matrix, velocity data
pMat = spalloc(MxPDof,MxPDof,30*MxPDof);   % allocate P mass matrix
Vdata = zeros(MxPDof,1);       % allocate for velocity data (RHS)
Qp=zeros(MxPDof,1);       % Nodal pressure DOFs

% Begin essential boundary conditions, allocate space
MaxPBC = 1;
EBCp.Mxdof=MxPDof;
% Essential boundary condition for pressure
EBCp.dof = nn2nftp(1,EBCPr.nodn(1));  % Degree-of-freedom index
EBCp.val = EBCPr.val;                         % Pressure Dof value

% partion out essential (Dirichlet) dofs
p_vec = [1:EBCp.Mxdof]';         % List of all dofs
EBCp.p_vec_undo = zeros(1,EBCp.Mxdof);
% form a list of non-diri dofs
EBCp.ndro = p_vec(~ismember(p_vec, EBCp.dof));	% list of non-diri dofs
% calculate p_vec_undo to restore Q to the original dof ordering
EBCp.p_vec_undo([EBCp.ndro;EBCp.dof]) = [1:EBCp.Mxdof]; %p_vec';

Qp(EBCp.dof(1))=EBCp.val(1);

Vdof = zeros(nd,nnd);             % Nodal velocity DOFs
Xe = zeros(2,nnd);

% BEGIN GLOBAL MATRIX ASSEMBLY
for ne=1:NumEl
for k=1:4
ki=NodNdx(:,Elcon(k,ne));
Xe(:,k)=[Xgrid(ki(2),ki(1));Ygrid(ki(2),ki(1))];
end
% Get stream function and velocities
for n=1:nnd
Vdof(ND,n)=Q((nn2nft(1,Elcon(n,ne))-1)+ND); % Loop over local nodes of element
end
[pMmat,Rndx,Cndx] = pMassMat(Xe,Elcon(:,ne),nn2nftp);     % Pressure "mass" matrix
pMat=pMat+sparse(Rndx,Cndx,pMmat,MxPDof,MxPDof);  % Global mass assembly

[CDat,RRndx] = PCDat(Xe,Elcon(:,ne),nn2nftp,Vdof);   % Convective data term
Vdata([RRndx]) = Vdata([RRndx])-CDat(:);

[DDat,RRndx] = PDDatL(Xe,Elcon(:,ne),nn2nftp,Vdof);   % Diffusive data term
Vdata([RRndx]) = Vdata([RRndx]) + nu*DDat(:); % +nu*DDat;
end;   % Loop ne
% END GLOBAL MATRIX ASSEMBLY

Vdatap=Vdata(EBCp.ndro)-pMat(EBCp.ndro,EBCp.dof)*EBCp.val;
pMatr=pMat(EBCp.ndro,EBCp.ndro);
Qs=Qp(EBCp.ndro);            % Non-Dirichlet (active) dofs

Pr=ilu_gmres_with_EBC(pMatr,Vdatap,[],GMRES,Qs,DropTol);

Qp=[Pr;EBCp.val];         % Augment active dofs with esential (Dirichlet) dofs
Qp=Qp(EBCp.p_vec_undo);      % Restore natural order
Qp=reshape(Qp,np,NumNod);
P= reshape(Qp(1,:),NumNy,NumNx);
Px=reshape(Qp(2,:),NumNy,NumNx);
Py=reshape(Qp(3,:),NumNy,NumNx);
return;
% >>>>>>>>>>>>> End pressure recovery <<<<<<<<<<<<<

% -------------------- function pMassMat ----------------------------

function [MM,Rndx,Cndx]=pMassMat(Xe,Elcon,nn2nftp)
% Simple cubic gradient element "mass" matrix
% -------------- Constants and fixed data -----------------
global GQ4;
xa=GQ4.xa; ya=GQ4.ya; wt=GQ4.wt; Nq=GQ4.size;
nn=[-1 -1; 1 -1; 1 1; -1 1]; % defines local nodal order
nnd=4;
np=3; np4=nnd*np; NP=1:np;
%
global ZG3412pm;
if (isempty(ZG3412pm)|size(ZG3412pm,2)~=Nq)
% Evaluate and save/cache the set of shape functions at quadrature pts.
ZG3412pm=cell(nnd,Nq);
for k=1:Nq
for m=1:nnd
ZG3412pm{m,k}=Gr(nn(m,:),xa(k),ya(k));
end
end
end  % if(isempty(*))
% --------------------- end fixed data -----------------

TGi=cell(nnd);
for m=1:nnd   % Loop over corner nodes
J=(Xe*GBL(nn(:,:),nn(m,1),nn(m,2)))'; % GBL is gradient of bilinear function
TGi{m} = blkdiag(1,J);
end  % Loop m

MM=zeros(np4,np4);  G=zeros(2,np4);   % Preallocate arrays
for k=1:Nq
% Initialize functions and derivatives at the quadrature point (xa,ya).
J=(Xe*GBL(nn(:,:),xa(k),ya(k)))';         % transpose of Jacobian J at (xa,ya)
Det=J(1,1)*J(2,2)-J(1,2)*J(2,1);          % Determinant of J
Ji=[J(2,2),-J(1,2); -J(2,1),J(1,1)]/Det;  % inverse of J

mm = 0;
for m=1:nnd
G(:,mm+NP) = Ji*ZG3412pm{m,k}*TGi{m};
mm = mm+np;
end  % loop m
MM = MM + G'*G*(wt(k)*Det);
end        % end loop k (quadrature pts)

gf=zeros(np4,1);          % array of global freedoms
m=0;
for n=1:4                 % Loop over element nodes
gf(m+NP)=(nn2nftp(1,Elcon(n))-1)+NP; % Get global freedoms
m=m+np;
end

Rndx=repmat(gf,1,np4);     % Row indices
Cndx=Rndx';                % Column indices

MM = reshape(MM,1,np4*np4);
Rndx=reshape(Rndx,1,np4*np4);
Cndx=reshape(Cndx,1,np4*np4);
return;

% --------------------- funnction PCDat -----------------------

function [PC,Rndx]=PCDat(Xe,Elcon,nn2nftp,Vdof)
% Evaluate convective force data
% ----------- Constants and fixed data ---------------
global GQ5;
xa=GQ5.xa; ya=GQ5.ya; wt=GQ5.wt; Nq=GQ5.size;
nn=[-1 -1; 1 -1; 1 1; -1 1]; % defines local nodal order
nnd=4;    % number of nodes
np = 3;  np4=4*np;  NP=1:np;
nd = 3;  nd4=4*nd;  ND=1:nd;
%
global ZS3412pc; global ZSX3412pc; global ZSY3412pc; global ZG3412pc;
if (isempty(ZS3412pc)|size(ZS3412pc,2)~=Nq)
% Evaluate and save/cache the set of shape functions at quadrature pts.
ZS3412pc=cell(nnd,Nq); ZSX3412pc=cell(nnd,Nq);
ZSY3412pc=cell(nnd,Nq); ZG3412pc=cell(nnd,Nq);
for k=1:Nq
for m=1:nnd
ZS3412pc{m,k} =Sr(nn(m,:),xa(k),ya(k));
ZSX3412pc{m,k}=Sxr(nn(m,:),xa(k),ya(k));
ZSY3412pc{m,k}=Syr(nn(m,:),xa(k),ya(k));
ZG3412pc{m,k}=Gr(nn(m,:),xa(k),ya(k));
end  % loop m over nodes
end  % loop k over Nq
end  % if(isempty(*))
% ----------------- end fixed data ------------------

Ti=cell(nnd);  TGi=cell(nnd);
for m=1:nnd   % Loop over corner nodes
J=(Xe*GBL(nn(:,:),nn(m,1),nn(m,2)))';   % Jacobian at (xa,ya)
Det=J(1,1)*J(2,2)-J(1,2)*J(2,1);      % Determinant of J & Jt
JiD=[J(2,2),-J(1,2); -J(2,1),J(1,1)]; % inv(J)*det(J)
Ti{m} = blkdiag(1,JiD');
TGi{m} = blkdiag(1,J);
end  % Loop m over corner nodes

PC=zeros(np4,1);
S=zeros(2,nd4);  Sx=zeros(2,nd4);  Sy=zeros(2,nd4);  G=zeros(2,np4);

for k=1:Nq      % Loop over quadrature points
J=(Xe*GBL(nn(:,:),xa(k),ya(k)))';       % Jacobian at (xa,ya)
Det=J(1,1)*J(2,2)-J(1,2)*J(2,1);      % Determinant of J & Jt
Jtbd=(J/Det)';                        % transpose(J/D)
JiD=[J(2,2),-J(1,2); -J(2,1),J(1,1)]; % inv(J)*det(J)
Ji=JiD/Det;                           % inverse(J)
for m=1:4         % Loop over element nodes
mm=nd*(m-1);
S(:,mm+ND) =Jtbd*ZS3412pc{m,k}*Ti{m};
Sx(:,mm+ND)=Jtbd*(Ji(1,1)*ZSX3412pc{m,k}+Ji(1,2)*ZSY3412pc{m,k})*Ti{m};
Sy(:,mm+ND)=Jtbd*(Ji(2,1)*ZSX3412pc{m,k}+Ji(2,2)*ZSY3412pc{m,k})*Ti{m};
mm=np*(m-1);
G(:,mm+NP)=Ji*ZG3412pc{m,k}*TGi{m};
end    % end loop over element nodes

% Compute the fluid velocities at the quadrature point.
U = S*Vdof(:);
Ux = Sx*Vdof(:);
Uy = Sy*Vdof(:);
UgU = U(1)*Ux+U(2)*Uy;   % U dot grad U
PC = PC + G'*UgU*(wt(k)*Det);
end    % end loop over Nq quadrature points

gf=zeros(1,np4);          % array of global freedoms
m=0;
for n=1:4                 % Loop over element nodes
gf(m+NP)=(nn2nftp(1,Elcon(n))-1)+NP; % Get global freedoms
m=m+np;
end
Rndx=gf;
PC = reshape(PC,1,np4);
return;

% ----------------- function PDDatL -------------------------

function [PD,Rndx]=PDDatL(Xe,Elcon,nn2nftp,Vdof)
% Evaluate diffusive force data (Laplacian form)
% --------------- Constants and fixed data -------------------
global GQ3;
xa=GQ3.xa; ya=GQ3.ya; wt=GQ3.wt; Nq=GQ3.size;
nnd=4;    % number of nodes
nn=[-1 -1; 1 -1; 1 1; -1 1]; % defines local nodal order
np = 3;  npdf=nnd*np;  NP=1:np;
nd = 3;  nd4=nnd*nd;  ND=1:nd;
global ZSXX3412pd; global ZSXY3412pd; global ZSYY3412pd; global ZG3412pd;
if (isempty(ZSXX3412pd)|size(ZSXX3412pd,2)~=Nq)
% Evaluate and save/cache the set of shape functions at quadrature pts.
ZSXX3412pd=cell(nnd,Nq); ZSXY3412pd=cell(nnd,Nq);
ZSYY3412pd=cell(nnd,Nq);  ZG3412pd=cell(nnd,Nq);
global ZSYY3412pd;
for k=1:Nq
for m=1:nnd
ZSXX3412pd{m,k}=Sxxr(nn(m,:),xa(k),ya(k));
ZSXY3412pd{m,k}=Sxyr(nn(m,:),xa(k),ya(k));
ZSYY3412pd{m,k}=Syyr(nn(m,:),xa(k),ya(k));
ZG3412pd{m,k}=Gr(nn(m,:),xa(k),ya(k));
end  % loop m over nodes
end  % loop k over Nq
end  % if(isempty(*))
% ------------ end fixed data -------------------
%
Ti=cell(nnd);  TGi=cell(nnd);
for m=1:nnd   % Loop over corner nodes
J=(Xe*GBL(nn(:,:),nn(m,1),nn(m,2)))';   % Jacobian at (xa,ya)
Det=J(1,1)*J(2,2)-J(1,2)*J(2,1);      % Determinant of J & Jt
JiD=[J(2,2),-J(1,2); -J(2,1),J(1,1)]; % inv(J)*det(J)
Ti{m} = blkdiag(1,JiD');
TGi{m} = blkdiag(1,J);
end  % Loop m over corner nodes

PD=zeros(npdf,1);
Sxx=zeros(2,nd4);  Syy=zeros(2,nd4);  G=zeros(2,npdf);
for k=1:Nq          % Loop over quadrature points
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;
JiD=[Jt(2,2),-Jt(1,2); -Jt(2,1),Jt(1,1)];
Ji=JiD/Det;
for m=1:nnd        % Loop over element nodes
mm=nd*(m-1);    % This transform is approximate !!
Sxx(:,mm+ND)=Jtd*(Ji(1,1)^2*ZSXX3412pd{m,k}+2*Ji(1,1)*Ji(1,2)*ZSXY3412pd{m,k}+Ji(1,2)^2*ZSYY3412pd{m,k})*Ti{m};
Syy(:,mm+ND)=Jtd*(Ji(2,1)^2*ZSXX3412pd{m,k}+2*Ji(2,1)*Ji(2,2)*ZSXY3412pd{m,k}+Ji(2,2)^2*ZSYY3412pd{m,k})*Ti{m};
mm=np*(m-1);
G(:,mm+NP) =Ji*ZG3412pd{m,k}*TGi{m};
end    % end loop over element nodes

LapU = (Sxx+Syy)*Vdof(:);
PD = PD+G'*LapU*(wt(k)*Det);
end    % end loop over quadrature points

gf=zeros(1,npdf);          % array of global freedoms
m=0;  K=1:np;
for n=1:nnd                 % Loop over element nodes
nfn=nn2nftp(1,Elcon(n))-1;  % Get global freedom
gf(m+NP)=(nn2nftp(1,Elcon(n))-1)+NP;
m=m+np;
end
Rndx=gf;
PD = reshape(PD,1,npdf);
return;

% ------------------------------------------------------------------------------
function gv=Gr(ni,q,r)
qi=ni(1); q0=q*ni(1);
ri=ni(2); r0=r*ni(2);
gv=[1/8*qi*(1+r0)*(r0*(1-r0)+3*(1-q^2)), -1/8*(1+r0)*(1+q0)*(1-3*q0), ...
-1/8*qi/ri*(1-r^2)*(1+r0); ...
1/8*ri*(1+q0)*(q0*(1-q0)+3*(1-r^2)),  -1/8/qi*ri*(1-q^2)*(1+q0), ...
-1/8*(1+q0)*(1+r0)*(1-3*r0)];
return;

function gx=Gxr(ni,q,r)
qi=ni(1); q0=q*ni(1);
ri=ni(2); r0=r*ni(2);
% x-derivative of irrotational vector
gx=[-3/4*qi^2*q0*(1+r0), 1/4*qi*(1+r0)*(1+3*q0), 0; ...
1/8*qi*ri*(4-3*q0^2-3*r0^2), -1/8*ri*(1+q0)*(1-3*q0), -1/8*qi*(1+r0)*(1-3*r0)];
return;

function gy=Gyr(ni,q,r)
qi=ni(1); q0=q*ni(1);
ri=ni(2); r0=r*ni(2);
% y-derivative of irrotational vector
gy=[1/8*qi*ri*(4-3*q0^2-3*r0^2), -1/8*ri*(1+q0)*(1-3*q0), -1/8*qi*(1+r0)*(1-3*r0); ...
-3/4*ri^2*r0*(1+q0), 0, 1/4*ri*(1+q0)*(1+3*r0)];
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=[ .125*ri*q1*(q0*(1-q0)+3*(1-r^2)), .125*q1*r1*(3*r0-1),    .125*ri/qi*q1^2*(1-q0); ...
-.125*qi*r1*(r0*(1-r0)+3*(1-q^2)), .125*qi/ri*r1^2*(1-r0), .125*q1*r1*(3*q0-1)];
return;

function S=Sxr(ni,q,r)
%SXR  Array of x-derivatives 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=[.125*qi*ri*(4-3*q^2-3*r^2), .125*qi*r1*(3*r0-1), -.125*ri*q1*(3*q0-1); ...
.75*qi^2*r1*q0,              0,                   .25*qi*r1*(3*q0+1)];
return;

function s=Syr(ni,q,r)
%SYR  Array of y-derivatives 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=[-.75*ri^2*q1*r0,             .25*ri*q1*(3*r0+1),   0 ; ...
-.125*qi*ri*(4-3*q^2-3*r^2), .125*qi*r1*(1-3*r0), .125*ri*q1*(3*q0-1)];
return;

function s=Sxxr(ni,q,r)
%SXXR  Array of second x-derivatives 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=[-3/4*qi^2*ri*q0, 0, -1/4*ri*qi*(1+3*q0); 3/4*qi^3*r1, 0, 3/4*qi^2*r1 ];
return;

function s=Syyr(ni,q,r)
%SYYR  Array of second y-derivatives 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=[-3/4*ri^3*q1, 3/4*ri^2*q1, 0; 3/4*qi*ri^2*r0, -1/4*qi*ri*(1+3*r0), 0 ];
return;

function s=Sxyr(ni,q,r)
%SXYR  Array of second (cross) xy-derivatives 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=[-3/4*qi*ri^2*r0, 1/4*qi*ri*(1+3*r0), 0; 3/4*qi^2*ri*q0, 0, 1/4*qi*ri*(1+3*q0)];
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;
```