# Code: Lid driven cavity using pressure free velocity form

### From CFD-Wiki

m (→Theory) |
(→Lid-driven cavity using pressure-free velocity formulation) |
||

Line 1: | Line 1: | ||

==Lid-driven cavity using pressure-free velocity formulation== | ==Lid-driven cavity using pressure-free velocity formulation== | ||

+ | |||

+ | This sample code uses four-node simple-cubic finite elements and simple iteration. | ||

===Theory=== | ===Theory=== | ||

Line 33: | Line 35: | ||

The algebraic equations to be solved are simple to set up, but of course are non-linear, requiring iteration of the linearized equations. | The algebraic equations to be solved are simple to set up, but of course are non-linear, requiring iteration of the linearized equations. | ||

- | The finite elements we will use here are apparently due to Melosh [3], but can also be found in Zienkiewitz [4]. These simple cubic-complete elements have three degrees-of-freedom at each of the four nodes. In the sample code we use this Hermite element for the pressure, and the modified form obtained by interchanging derivatives and the sign of one of them (though a simple bilinear element could be used for the pressure as well). The degrees-of-freedom are the pressure and pressure gardient, and the stream function and components of the solenoidal velocity for the modified element. | + | The finite elements we will use here are apparently due to Melosh [3], but can also be found in Zienkiewitz [4]. These simple cubic-complete elements have three degrees-of-freedom at each of the four nodes. In the sample code we use this Hermite element for the pressure, and the modified form obtained by interchanging derivatives and the sign of one of them (though a simple bilinear element could be used for the pressure as well). The degrees-of-freedom are the pressure and pressure gardient, and the stream function and components of the solenoidal velocity for the modified element. The normal component of the velocity is continuous at element interfaces as is required, but the tangential velocity component may not be continuous. |

The code implementing the lid-driven cavity problem is written for Matlab. The script below is problem-specific, and calls problem-independent functions to evaluate the element diffusion and convection matricies and evaluate the pressure from the resulting velocity field. These three functions accept general quadrilateral elements with straight sides as well as the rectangular elements used here. Other functions are a GMRES iterative solver using ILU preconditioning and incorporating the essential boundary conditions, and a function to produce non-uniform nodal spacing for the problem mesh. | The code implementing the lid-driven cavity problem is written for Matlab. The script below is problem-specific, and calls problem-independent functions to evaluate the element diffusion and convection matricies and evaluate the pressure from the resulting velocity field. These three functions accept general quadrilateral elements with straight sides as well as the rectangular elements used here. Other functions are a GMRES iterative solver using ILU preconditioning and incorporating the essential boundary conditions, and a function to produce non-uniform nodal spacing for the problem mesh. | ||

- | This "educational code" is a simplified version of the code used in [1]. The user interface is the code itself. The user can experiment with changing the mesh, the Reynolds number, and the number of nonlinear iterations performed, as well as the relaxation factor. There are suggestions in the code regarding near-optimum choices for this factor as a function of Reynolds number. These values are given in the paper as well. | + | This "educational code" is a simplified version of the code used in [1]. The user interface is the code itself. The user can experiment with changing the mesh, the Reynolds number, and the number of nonlinear iterations performed, as well as the relaxation factor. There are suggestions in the code regarding near-optimum choices for this factor as a function of Reynolds number. These values are given in the paper as well. For larger Reynolds numbers, a smaller relaxation factor speeds up convergence by smoothing the velocity factor <math>(\mathbf{v}\cdot\nabla)</math> in the convection term, but will impede convergence if made too small. |

The output consists of graphic plots of contour levels of the stream function and the pressure levels. | The output consists of graphic plots of contour levels of the stream function and the pressure levels. |

## Revision as of 17:10, 4 July 2011

## Lid-driven cavity using pressure-free velocity formulation

This sample code uses four-node simple-cubic finite elements and simple iteration.

### Theory

The incompressible Navier-Stokes equation is a differential algebraic equation, having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process. We show a simple, natural way of doing this.

The incompressible Navier-Stokes equation is composite, the sum of two orthogonal equations,

- ,
- ,

where and are solenoidal and irrotational projection operators satisfying and and are the nonconservative and conservative parts of the body force. This result follows from the Helmholtz Theorem . The first equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity and is related to the pressure Poisson equation. The explicit functional forms of the projection operator in 2D and 3D are found from the Helmholtz Theorem, showing that these are integro-differential equations, and not particularly convenient for numerical computation.

Equivalent weak or variational forms of the equations, proved to produce the same velocity solution as the Navier-Stokes equation are

- ,
- ,

for divergence-free test functions and irrotational test functions satisfying appropriate boundary conditions. Here, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces. The discrete form of this is emminently suited to finite element computation of divergence-free flow.

In the discrete case, it is desirable to choose basis functions for the velocity which reflect the essential feature of incompressible flow — the velocity elements must be divergence-free. While the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz Theorem. Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line integral of the tangential component of the vector potential around the channel in 3D, the flow being given by Stokes' Theorem. This leads naturally to the use of Hermite stream function (in 2D) or velocity potential elements (in 3D).

Involving, as it does, both stream function and velocity degrees-of-freedom, the method might be called a **velocity-stream function** or **stream function-velocity** method.

We now restrict discussion to 2D continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw a large number of candidate triangular and rectangular elements from the plate-bending literature. These elements have derivatives as components of the gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions,

Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many families of stream function elements.

Taking the curl of the scalar stream function elements gives divergence-free velocity elements [1][2]. The requirement that the stream function elements be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence on these interfaces.

Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces. Stream function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries [1], though consistent values may be used with some problems. These are all Dirichlet conditions.

The algebraic equations to be solved are simple to set up, but of course are non-linear, requiring iteration of the linearized equations.

The finite elements we will use here are apparently due to Melosh [3], but can also be found in Zienkiewitz [4]. These simple cubic-complete elements have three degrees-of-freedom at each of the four nodes. In the sample code we use this Hermite element for the pressure, and the modified form obtained by interchanging derivatives and the sign of one of them (though a simple bilinear element could be used for the pressure as well). The degrees-of-freedom are the pressure and pressure gardient, and the stream function and components of the solenoidal velocity for the modified element. The normal component of the velocity is continuous at element interfaces as is required, but the tangential velocity component may not be continuous.

The code implementing the lid-driven cavity problem is written for Matlab. The script below is problem-specific, and calls problem-independent functions to evaluate the element diffusion and convection matricies and evaluate the pressure from the resulting velocity field. These three functions accept general quadrilateral elements with straight sides as well as the rectangular elements used here. Other functions are a GMRES iterative solver using ILU preconditioning and incorporating the essential boundary conditions, and a function to produce non-uniform nodal spacing for the problem mesh.

This "educational code" is a simplified version of the code used in [1]. The user interface is the code itself. The user can experiment with changing the mesh, the Reynolds number, and the number of nonlinear iterations performed, as well as the relaxation factor. There are suggestions in the code regarding near-optimum choices for this factor as a function of Reynolds number. These values are given in the paper as well. For larger Reynolds numbers, a smaller relaxation factor speeds up convergence by smoothing the velocity factor in the convection term, but will impede convergence if made too small.

The output consists of graphic plots of contour levels of the stream function and the pressure levels.

This simplified version for this Wiki resulted from removal of computation of the vorticity, a restart capability, area weighting for the error, and production of publication-quality plots from one of the research codes used with the paper.

### Lid-driven cavity Matlab script

%LDCW LID-DRIVEN CAVITY % Finite element solution of the 2D Navier-Stokes equation using 4-node, 12 DOF, % (3-DOF/node), simple-cubic-derived rectangular Hermite basis for % the Lid-Driven Cavity problem. % % This could also be characterized as a VELOCITY-STREAM FUNCTION or % STREAM FUNCTION-VELOCITY method. % % Reference: "A Hermite finite element method for incompressible fluid flow", % Int. J. Numer. Meth. Fluids, 64, P376-408 (2010). % % Simplified Wiki version % The rectangular problem domain is defined between Cartesian % coordinates Xmin & Xmax and Ymin & Ymax. % The computational grid has NumEx elements in the x-direction % and NumEy elements in the y-direction. % The nodes and elements are numbered column-wise from the % upper left corner to the lower right corner. % %This script calls the user-defined functions: % regrade - to regrade the mesh % DMatW - to evaluate element diffusion matrix % CMatW - to evaluate element convection matrix % GetPresW - to evaluate the pressure % ilu_gmres_with_EBC - to solve the system with essential/Dirichlet BCs % % Jonas Holdeman August 2007, revised June 2011 clear all; disp('Lid-driven cavity'); disp(' Four-node, 12 DOF, simple-cubic stream function basis.'); % ------------------------------------------------------------- nd = 3; nd2=nd*nd; % Number of DOF per node - do not change!! % ------------------------------------------------------------- ETstart=clock; % Parameters for GMRES solver GMRES.Tolerance=1.e-14; GMRES.MaxIterates=15; GMRES.MaxRestarts=6; % Optimal relaxation parameters for given Reynolds number % (see IJNMF reference) % Re 100 1000 3200 5000 7500 10000 12500 % RelxFac: 1.04 1.11 .860 .830 .780 .778 .730 % ExpCR1 1.488 .524 .192 .0378 -- -- -- % ExpCRO 1.624 .596 .390 .331 .243 .163 .133 % CritFac: 1.82 1.49 1.14 1.027 .942 .877 .804 % Define the problem geometry, set mesh bounds: Xmin = 0.0; Xmax = 1.0; Ymin = 0.0; Ymax = 1.0; % Set mesh grading parameters (set to 1 if no grading). % See below for explanation of use of parameters. xgrd = .75; ygrd=.75; % (xgrd = 1, ygrd=1 for uniform mesh) % Set " RefineBoundary=1 " for additional refinement at boundary, % i.e., split first element along boundary into two. RefineBoundary=1; % DEFINE THE MESH % Set number of elements in each direction NumEx = 18; NumEy = NumEx; % PLEASE CHANGE OR SET NUMBER OF ELEMENTS TO CHANGE/SET NUMBER OF NODES! NumNx=NumEx+1; NumNy=NumEy+1; % Define problem parameters: % Lid velocity Vlid=1.; % Reynolds number Re=1000.; % factor for under/over-relaxation starting at iteration RelxStrt RelxFac = 1.; % % Number of nonlinear iterations MaxNLit=20; % %-------------------------------------------------------- % Viscosity for specified Reynolds number nu=Vlid*(Xmax-Xmin)/Re; % Grade the mesh spacing if desired, call regrade(x,agrd,e). % if e=0: refine both sides, 1: refine upper, 2: refine lower % if agrd=xgrd|ygrd is the parameter which controls grading, then % if agrd=1 then leave array unaltered. % if agrd<1 then refine (make finer) towards the ends % if agrd>1 then refine (make finer) towards the center. % % Generate equally-spaced nodal coordinates and refine if desired. if (RefineBoundary==1) XNc=linspace(Xmin,Xmax,NumNx-2); XNc=[XNc(1),(.62*XNc(1)+.38*XNc(2)),XNc(2:end-1),(.38*XNc(end-1)+.62*XNc(end)),XNc(end)]; YNc=linspace(Ymax,Ymin,NumNy-2); YNc=[YNc(1),(.62*YNc(1)+.38*YNc(2)),YNc(2:end-1),(.38*YNc(end-1)+.62*YNc(end)),YNc(end)]; else XNc=linspace(Xmin,Xmax,NumNx); YNc=linspace(Ymax,Ymin,NumNy); end if xgrd ~= 1 XNc=regrade(XNc,xgrd,0); end; % Refine mesh if desired if ygrd ~= 1 YNc=regrade(YNc,ygrd,0); end; [Xgrid,Ygrid]=meshgrid(XNc,YNc);% Generate the x- and y-coordinate meshes. % Allocate storage for fields psi0=zeros(NumNy,NumNx); u0=zeros(NumNy,NumNx); v0=zeros(NumNy,NumNx); %--------------------Begin grid plot----------------------- % ********************** FIGURE 1 ************************* % Plot the grid figure(1); clf; orient portrait; orient tall; subplot(2,2,1); hold on; plot([Xmax;Xmin],[YNc;YNc],'k'); plot([XNc;XNc],[Ymax;Ymin],'k'); hold off; axis([Xmin,Xmax,Ymin,Ymax]); axis equal; axis image; title([num2str(NumNx) 'x' num2str(NumNy) ... ' node mesh for Lid-driven cavity']); pause(.1); %-------------- End plotting Figure 1 ---------------------- %Contour levels, Ghia, Ghia & Shin, Re=100, 400, 1000, 3200, ... clGGS=[-.1175,-.1150,-.11,-.1,-.09,-.07,-.05,-.03,-.01,-1.e-4,-1.e-5,-1.e-7,-1.e-10,... 1.e-8,1.e-7,1.e-6,1.e-5,5.e-5,1.e-4,2.5e-4,5.e-4,1.e-3,1.5e-3,3.e-3]; CL=clGGS; % Select contour level option if (Vlid<0) CL=-CL; end NumNod=NumNx*NumNy; % total number of nodes MaxDof=nd*NumNod; % maximum number of degrees of freedom EBC.Mxdof=nd*NumNod; % maximum number of degrees of freedom nn2nft=zeros(2,NumNod); % node number -> nf & nt NodNdx=zeros(2,NumNod); % Generate lists of active nodal indices, freedom number & type ni=0; nf=-nd+1; nt=1; % ________ for nx=1:NumNx % | | for ny=1:NumNy % | | ni=ni+1; % |________| NodNdx(:,ni)=[nx;ny]; nf=nf+nd; % all nodes have 3 dofs nn2nft(:,ni)=[nf;nt]; % dof number & type (all nodes type 1) end; end; %NumNod=ni; % total number of nodes nf2nnt=zeros(2,MaxDof); % (node, type) associated with dof ndof=0; dd=[1:nd]; for n=1:NumNod for k=1:nd nf2nnt(:,ndof+k)=[n;k]; end ndof=ndof+nd; end NumEl=NumEx*NumEy; % Generate element connectivity, from upper left to lower right. Elcon=zeros(4,NumEl); ne=0; LY=NumNy; for nx=1:NumEx for ny=1:NumEy ne=ne+1; Elcon(1,ne)=1+ny+(nx-1)*LY; Elcon(2,ne)=1+ny+nx*LY; Elcon(3,ne)=1+(ny-1)+nx*LY; Elcon(4,ne)=1+(ny-1)+(nx-1)*LY; end % loop on ny end % loop on nx % Begin essential boundary conditions, allocate space MaxEBC = nd*2*(NumNx+NumNy-2); EBC.dof=zeros(MaxEBC,1); % Degree-of-freedom index EBC.typ=zeros(MaxEBC,1); % Dof type (1,2,3) EBC.val=zeros(MaxEBC,1); % Dof value X1=XNc(2); X2=XNc(NumNx-1); nc=0; for nf=1:MaxDof ni=nf2nnt(1,nf); nx=NodNdx(1,ni); ny=NodNdx(2,ni); x=XNc(nx); y=YNc(ny); if(x==Xmin | x==Xmax | y==Ymin) nt=nf2nnt(2,nf); switch nt; case {1, 2, 3} nc=nc+1; EBC.typ(nc)=nt; EBC.dof(nc)=nf; EBC.val(nc)=0; % psi, u, v end % switch (type) elseif (y==Ymax) nt=nf2nnt(2,nf); switch nt; case {1, 3} nc=nc+1; EBC.typ(nc)=nt; EBC.dof(nc)=nf; EBC.val(nc)=0; % psi, v case 2 nc=nc+1; EBC.typ(nc)=nt; EBC.dof(nc)=nf; EBC.val(nc)=Vlid; % u end % switch (type) end % if (boundary) end % for nf EBC.num=nc; if (size(EBC.typ,1)>nc) % Truncate arrays if necessary EBC.typ=EBC.typ(1:nc); EBC.dof=EBC.dof(1:nc); EBC.val=EBC.val(1:nc); end % End ESSENTIAL (Dirichlet) boundary conditions % partion out essential (Dirichlet) dofs p_vec = [1:EBC.Mxdof]'; % List of all dofs EBC.p_vec_undo = zeros(1,EBC.Mxdof); % form a list of non-diri dofs EBC.ndro = p_vec(~ismember(p_vec, EBC.dof)); % list of non-diri dofs % calculate p_vec_undo to restore Q to the original dof ordering EBC.p_vec_undo([EBC.ndro;EBC.dof]) = [1:EBC.Mxdof]; %p_vec'; Q=zeros(MaxDof,1); % Allocate space for solution (dof) vector % Initialize fields to boundary conditions for k=1:EBC.num Q(EBC.dof(k))=EBC.val(k); end; errpsi=zeros(NumNy,NumNx); % error correct for iteration MxNL=max(1,MaxNLit); np0=zeros(1,MxNL); % Arrays for convergence info nv0=zeros(1,MxNL); Qs=[]; Dmat = spalloc(MaxDof,MaxDof,36*MaxDof); % to save the diffusion matrix Vdof=zeros(nd,4); Xe=zeros(2,4); % coordinates of element corners NLitr=0; ND=1:nd; while (NLitr<MaxNLit), NLitr=NLitr+1; % <<< BEGIN NONLINEAR ITERATION tclock=clock; % Start assembly time <<<<<<<<< % Generate and assemble element matrices Mat=spalloc(MaxDof,MaxDof,36*MaxDof); RHS=spalloc(MaxDof,1,MaxDof); %RHS = zeros(MaxDof,1); Emat=zeros(1,16*nd2); % Values 144=4*4*(nd*nd) % BEGIN GLOBAL MATRIX ASSEMBLY for ne=1:NumEl for k=1:4 ki=NodNdx(:,Elcon(k,ne)); Xe(:,k)=[XNc(ki(1));YNc(ki(2))]; end if NLitr == 1 % Fluid element diffusion matrix, save on first iteration [DEmat,Rndx,Cndx] = DMatW(Xe,Elcon(:,ne),nn2nft); Dmat=Dmat+sparse(Rndx,Cndx,DEmat,MaxDof,MaxDof); % Global diffusion mat end if (NLitr>1) % Get stream function and velocities for n=1:4 Vdof(ND,n)=Q((nn2nft(1,Elcon(n,ne))-1)+ND); % Loop over local element nodes end % Fluid element convection matrix, first iteration uses Stokes equation. [Emat,Rndx,Cndx] = CMatW(Xe,Elcon(:,ne),nn2nft,Vdof); Mat=Mat+sparse(Rndx,Cndx,-Emat,MaxDof,MaxDof); % Global convection assembly end end; % loop ne over elements % END GLOBAL MATRIX ASSEMBLY Mat = Mat -nu*Dmat; % Add in cached/saved global diffusion matrix disp(['(' num2str(NLitr) ') Matrix assembly complete, elapsed time = '... num2str(etime(clock,tclock)) ' sec']); % Assembly time <<<<<<<<<<< pause(1); Q0 = Q; % Save dof values % Solve system tclock=clock; %disp('start solution'); % Start solution time <<<<<<<<<<<<<< RHSr=RHS(EBC.ndro)-Mat(EBC.ndro,EBC.dof)*EBC.val; Matr=Mat(EBC.ndro,EBC.ndro); Qs=Q(EBC.ndro); Qr=ilu_gmres_with_EBC(Matr,RHSr,[],GMRES,Qs); Q=[Qr;EBC.val]; % Augment active dofs with esential (Dirichlet) dofs Q=Q(EBC.p_vec_undo); % Restore natural order stime=etime(clock,tclock); % Solution time <<<<<<<<<<<<<< % ****** APPLY RELAXATION FACTOR ********************* if(NLitr>1) Q=RelxFac*Q+(1-RelxFac)*Q0; end % **************************************************** % Compute change and copy dofs to field arrays dsqp=0; dsqv=0; for k=1:MaxDof ni=nf2nnt(1,k); nx=NodNdx(1,ni); ny=NodNdx(2,ni); switch nf2nnt(2,k) % switch on dof type case 1 dsqp=dsqp+(Q(k)-Q0(k))^2; psi0(ny,nx)=Q(k); errpsi(ny,nx)=Q0(k)-Q(k); case 2 dsqv=dsqv+(Q(k)-Q0(k))^2; u0(ny,nx)=Q(k); case 3 dsqv=dsqv+(Q(k)-Q0(k))^2; v0(ny,nx)=Q(k); end % switch on dof type end % for np0(NLitr)=sqrt(dsqp); nv0(NLitr)=sqrt(dsqv); if (np0(NLitr)<=1e-15|nv0(NLitr)<=1e-15) MaxNLit=NLitr; np0=np0(1:MaxNLit); nv0=nv0(1:MaxNLit); end; disp(['(' num2str(NLitr) ') Solution time for linear system = '... num2str(etime(clock,tclock)) ' sec']); % Solution time <<<<<<<<<<<< %---------- Begin plot of intermediate results ---------- % ********************** FIGURE 2 ************************* figure(1); % Stream function (intermediate) subplot(2,2,3); contour(Xgrid,Ygrid,psi0,8,'k'); % Plot contours (trajectories) axis([Xmin,Xmax,Ymin,Ymax]); title(['Lid-driven cavity, Re=' num2str(Re)]); axis equal; axis image; % Plot convergence info subplot(2,2,2); semilogy(1:NLitr,nv0(1:NLitr),'k-+',1:NLitr,np0(1:NLitr),'k-o'); xlabel('Nonlinear iteration number'); ylabel('Nonlinear correction'); axis square; title(['Iteration conv., Re=' num2str(Re)]); legend('U','Psi'); % Plot nonlinear iteration correction contours subplot(2,2,4); contour(Xgrid,Ygrid,errpsi,8,'k'); % Plot contours (trajectories) axis([Xmin,Xmax,Ymin,Ymax]); axis equal; axis image; title(['Iteration correction']); pause(1); % ********************** END FIGURE 2 ************************* %---------- End plot of intermediate results --------- if (nv0(NLitr)<1e-15) break; end % Terminate iteration if non-significant end; % <<< (while) END NONLINEAR ITERATION format short g; disp('Convergence results by iteration: velocity, stream function'); disp(['nv0: ' num2str(nv0)]); disp(['np0: ' num2str(np0)]); % >>>>>>>>>>>>>> BEGIN PRESSURE RECOVERY <<<<<<<<<<<<<<<<<< % Essential pressure boundary condition % Index of node to apply pressure BC, value at node PBCnx=fix((NumNx+1)/2); % Apply at center of mesh PBCny=fix((NumNy+1)/2); PBCnod=0; for k=1:NumNod if (NodNdx(1,k)==PBCnx & NodNdx(2,k)==PBCny) PBCnod=k; break; end end if (PBCnod==0) error('Pressure BC node not found'); else EBCp.nodn = [PBCnod]; % Pressure BC node number EBCp.val = [0]; % set P = 0. end % Cubic pressure [P,Px,Py] = GetPresW(NumNod,NodNdx,Elcon,nn2nft,Xgrid,Ygrid,Q,EBCp,nu); % ******************** END PRESSURE RECOVERY ********************* % ********************** CONTINUE FIGURE 1 ************************* figure(1); % Stream function (final) subplot(2,2,3); [CT,hn]=contour(Xgrid,Ygrid,psi0,CL,'k'); % Plot contours (trajectories) clabel(CT,hn,CL([1,3,5,7,9,10,11,19,23])); hold on; plot([Xmin,Xmin,Xmax,Xmax,Xmin],[Ymax,Ymin,Ymin,Ymax,Ymax],'k'); hold off; axis([Xmin,Xmax,Ymin,Ymax]); axis equal; axis image; title(['Stream lines, ' num2str(NumNx) 'x' num2str(NumNy) ... ' mesh, Re=' num2str(Re)]); % Plot pressure contours (final) subplot(2,2,4); CPL=[-.002,0,.02,.05,.07,.09,.11,.12,.17,.3]; [CT,hn]=contour(Xgrid,Ygrid,P,CPL,'k'); % Plot pressure contours clabel(CT,hn,CPL([3,5,7,10])); hold on; plot([Xmin,Xmin,Xmax,Xmax,Xmin],[Ymax,Ymin,Ymin,Ymax,Ymax],'k'); hold off; axis([Xmin,Xmax,Ymin,Ymax]); axis equal; axis image; title(['Simple cubic pressure contours, Re=' num2str(Re)]); % ********************* END FIGURE 1 ************************* disp(['Total elapsed time = '... num2str(etime(clock,ETstart)/60) ' min']); % Elapsed time from start <<<

### Diffusion matrix for pressure-free velocity method (**DMatW.m**)

### Convection matrix for pressure-free velocity method (**CMatW.m**)

### Consistent pressure for pressure-free velocity method (**GetPresW.m**)

### GMRES solver with ILU preconditioning and Essential BC (**ilu_gmres_with_EBC.m**)

### Grade node spacing (**regrade.m**)

## references

[1] **Holdeman, J. T. (2010)**, "A Hermite finite element method for incompressible fluid flow", Int. J. Numer. Meth. Fluids, **64**: 376-408.

[2] **Holdeman, J. T. and Kim, J.W. (2010)**, "Computation of incompressible thermal flows using Hermite finite elements", Comput. Methods Appl. Mech. Engr., **199**: 3297-3304.

[3] **Melosh, R. J. (1963)**, "Basis of derivation of matricies for the direct stifness method", J.A.I.A.A., **1**: 1631-1637.

[4] **Zienkiewicz, O. C. (1971)**, *The Finite Element Method in Engineering Science*, McGraw-Hill, London.