The partial differential equations shown earlier in this section must be discretized or transformed into a set of algebraic equations which can be solved digitally. There are numerous methods available to do this discretization. The three most popular (based on the number of commercial computational fluid dynamics (CFD) codes available) are:
In the finite difference method, the partial derivatives are replaced with a series expansion representation, usually a Taylor series. The series is truncated usually after 1 or 2 terms. The more terms you include, the more accurate the solution. However, more terms in the expansion causes the complexity and number of discrete points or nodes of the solution to increase dramatically. Applying this method to a regularly shaped geometry is straightforward. However, for irregularly shaped geometries, the equations must be transformed before the Taylor series can be applied. This transformation introduces all sorts of problems in terms of additional cross-coupling of equations, mesh generation and general convergence.
In the finite volume method, the governing equations are integrated over a volume or cell assuming a piece-wise linear variation of the dependent variables (u, v, w, p, T). Again the piece-wise linear variation determines both the accuracy and the complexity. Using these integrations, you essentially balance fluxes across the boundaries of the individual volumes. The flux is calculated at the mid-point between the discrete nodes in the domain. Hence, you must calculate a flux between all neighboring nodes in the domain. In a topologically regular mesh (same number of divisions in any one direction), this flux calculation is quite straightforward. In an irregular mesh (as in an automatically generated tetrahedral mesh), this calculation will lead to an excruciating amount of fluxes and a major bookkeeping effort to make sure all the fluxes have been calculated properly.
In the finite element method, Galerkin's method of weighted residuals is generally used. In this method, the governing partial differential equations are integrated over an element or volume after having been multiplied by a weight function. The dependent variables are represented on the element by a shape function, which is the same form as the weight function. The shape function may take any of several forms. Autodesk Simulation CFD uses linear for 2D triangular elements, bi-linear for 2D quadrilateral elements, linear for 3D tetrahedral elements, tri-linear for 3D hexahedral elements and a mix for the 3D 5 and 6 sided elements. The main advantage as well as the main disadvantage of finite elements is that it is a mathematical approach that is difficult to put any physical significance on the terms in the algebraic equations. In the finite volume method, you are always dealing with fluxes - not so with finite elements. However, the application of finite elements on any geometric shape is the same. Also, the boundary conditions which must be added after the fact for finite volume methods are an integral part of the discretized equations.
The table below summarizes the advantages and disadvantages of the various methods:
| Method | Advantages | Disadvantages |
|
Finite Element |
1. More mathematics involved 2. Natural boundary conditions (for fluxes) 3. Master element formulation 4. Any shaped geometry can be modeled with the same effort |
1. More mathematics involved - less physical significance |
|
Finite Volume and Finite Difference |
1. Fluxes have more physical significance |
1. Irregular geometries require far more effort |
For simple geometries, you can show that all 3 of these methods produce the exact same solution matrix or digital representation. All 3 methods can produce similar sets of discretized equations for the governing equations of fluid flow and heat transfer. You can use similar velocity-pressure algorithms (segregated, coupled, SIMPLE, SIMPLE-R,...) with any of the discretization methods [Autodesk Simulation CFD authors have successfully used SIMPLE variants in both finite element and finite volume codes].
Autodesk Simulation CFD uses the finite element method primarily because of its flexibility in modeling any geometric shape.
For fluid flow, there are special considerations. As seen earlier in this section, there are 5 equations for 5 unknowns (u, v, w, p, T). However, there are two problems with these equations which are specific to computational fluid dynamics (CFD).
First, the governing equations are not only coupled, but they have non-linear terms, namely the advection or inertia terms. The handling of these terms has been an ongoing research project for at least the last 40 years. In fact, Autodesk Simulation CFD is constantly re-evaluating the method we use to discretize these advection terms. If these terms are not modeled accurately enough, they will introduce an error known as “numerical diffusion”. As its name indicates, the errors can completely swamp any physical diffusion and mis-represent the physics of the real world problem. If you model the advection terms with the usual methods of obtaining high accuracy (central differences, standard Galerkin schemes), you introduce numerical dispersion errors where the numerical solution oscillates around the true solution. These dispersion errors can quite easily lead to divergent solutions, especially in turbulent flows. Most commercial finite volume and finite element methods have discretized these terms in some special way which is a compromise of accuracy and stability. Finite volume methods use techniques like skew upwinding and QUICK schemes. Successful finite element methods use some sort of streamline upwind element. (Yes, there are finite element CFD methods available which do not use this method, but they are not generally applicable). Autodesk Simulation CFD uses several variants of the streamline upwind discretization schemes to model the advection terms.
The second major difficulty with the governing partial differential equations is that no explicit equation for pressure exists for incompressible flows. For example, if we use the Navier-Stokes or momentum equations to solve for the velocities, we have only the continuity equation to solve for pressure. However, pressure does not appear in the continuity equation. This problem has been side-stepped by manipulating a combination of these equations. The most predominant method (commercially, that is) for solving this dilemma of the missing pressure equation was developed for finite volume methods and is known as SIMPLE or some variant of it. This method is well-explained in the book: Numerical Heat Transfer by Suhas V. Patankar (Hemisphere Publishing, 1980, ISBN 0-89116-522-3). Almost all of the commercial finite volume CFD codes use this method and the 2 most popular finite element CFD codes do as well. Albeit it is a special application of the method for finite elements. Autodesk Simulation CFD uses a variant of this tried and true pressure-velocity algorithm based on the SIMPLE-R technique described in Patankar's book.
While it is true that early finite element CFD methods struggled with modeling high speed flows, the application in Autodesk Simulation CFD of many of the successfully demonstrated finite volume techniques to the finite element discretization method has produced a highly robust means of predicting not only high speed turbulent flows, but compressible flows, as well. All of this has been accomplished with the strictest application of Galerkin's method of weighted residuals. Hence, the geometric flexibility inherent in finite elements has been maintained in Autodesk Simulation CFD.
For a more complete theoretical discussion of some of the methods used in Autodesk Simulation CFD, see: A Streamline Upwind Finite Element Method For Laminar And Turbulent Flow by Rita J Schnipke, Ph.D. Dissertation, University of Virginia, 1986 (available through University Microfilms in Ann Arbor, Michigan -www.umi.com).