Code: Lid driven cavity using pressure free velocity form

==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,
:<math>\frac{\partial\mathbf{v}}{\partial t}=\Pi^S(-\mathbf{v}\cdot\nabla\mathbf{v}+\nu\nabla^2\mathbf{v})+\mathbf{f}^S </math>,
:<math>\rho^{-1}\nabla p=\Pi^I(-\mathbf{v}\cdot\nabla\mathbf{v}+\nu\nabla^2\mathbf{v})+\mathbf{f}^I </math>,
where <math>\Pi^S</math> and <math>\Pi^I</math> are solenoidal and irrotational projection operators satisfying <math>\Pi^S+\Pi^I=1</math> and
<math>\mathbf{f}^S</math> and <math>\mathbf{f}^I</math> 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
:<math>(\mathbf{w},\frac{\partial\mathbf{v}}{\partial t})=-(\mathbf{w},\mathbf{v}\cdot\nabla\mathbf{v})-\nu(\nabla\mathbf{w}: \nabla\mathbf{v})+(\mathbf{w},\mathbf{f}^S)</math>,
:<math>(\mathbf{g}_i,\nabla p)=-(\mathbf{g}_i,\mathbf{v}\cdot\nabla\mathbf{v}_j)-\nu(\nabla\mathbf{g}_i: \nabla\mathbf{v}_j)+(\mathbf{g}_i,\mathbf{f}^I)\,</math>,

for divergence-free test functions <math>\mathbf{w}</math> and irrotational test functions <math>\mathbf{g}</math> 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 &mdash; 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,
:<math>\nabla\phi = \left[\frac{\partial \phi}{\partial x},\,\frac{\partial \phi}{\partial y}\right]^T, \quad
\nabla\times\phi = \left[\frac{\partial \phi}{\partial y},\,-\frac{\partial \phi}{\partial x}\right]^T. </math>

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][5]. 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 <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. 

A 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.

This modified version posted 3/15/2013 updates the code to MatLab version R2012b. The element properties are now defined in classes, and the code has been modified to accept class definitions for additional quartic and quintic divergence-free elements on quadrilaterals and quadratic and quartic divergence-free elements on triangles.

===Lid-driven cavity Matlab script===
<pre>
%LDC3W            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:
%   ELS3412r     - class of velocity basis functions
%   DMatW        - to evaluate element diffusion matrix 
%   CMatW        - to evaluate element convection matrix
%   GetPresW     - to evaluate the pressure 
%   regrade      - to regrade the mesh 
% Uses
%   ilu          - incomplete LU preconditioner
%   gmres        - iterative solver
% Indirectly uses:
%    Gquad2      - Gauss integraion rules for rectangle
%    ELG3412r    - class of pressure basis functions
%
% Jonas Holdeman   August 2007, revised March 2013
 ..............
</pre>

===[[PFV program script | Program script for pressure-free velocity method ('''LDC3W.m''')]]===

===[[PFV diffusion matrix | Diffusion matrix for pressure-free velocity method ('''DMatW.m''')]]===

===[[PFV convection matrix | Convection matrix for pressure-free velocity method ('''CMatW.m''')]]===

===[[PFV cubic velocity class| Modified-cubic velocity element class for pressure-free velocity method ('''ELS3412f.m''')]]===

===[[PFV cubic pressure class| Simple-cubic pressure element class for pressure-free velocity method ('''ELG3412f.m''')]]===

===[[PFV get pressure | Consistent pressure for pressure-free velocity method ('''GetPresW.m''')]]===

===[[PFV quadature rules| Gauss quadrature rules on rectangle ('''GQuad2.m''')]]===

===[[PFV mesh regrade | Grade node spacing ('''regrade.m''')]]===

==references==
[1] {{reference-paper |author = Holdeman, J. T. |year = 2010 | title = A Hermite finite element method for incompressible fluid flow | rest = Int. J. Numer. Meth. Fluids, '''64''': 376-408 }}

[2] {{reference-paper |author = Holdeman, J. T. and Kim, J.W. |year = 2010 | title = Computation of incompressible thermal flows using Hermite finite elements | rest = Comput. Methods Appl. Mech. Engr., '''199''': 3297-3304 }}

[3] {{reference-paper |author = Melosh, R. J. | year = 1963 | title = Basis of derivation of matricies for the direct stifness method | rest = J.A.I.A.A., '''1''': 1631-1637 }}

[4] {{reference-book | author = Zienkiewicz, O. C. | year = 1971 | title = The Finite Element Method in Engineering Science | rest = McGraw-Hill, London}}

[5] {{reference-paper |author = Holdeman, J. T. |year = 2012 | title = A velocity-stream function method for three-dimensional incompressible fluid flow | rest = Comput. Methods Appl. Mech. Engr., '''209-212''': 66-73}}