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Within the CFD community, there is a lot of debate about how to simulate fluid motion. What’s better, the finite-volume method (FVM) or the finite-element method (FEM)? Rita Schnipke looks into the issue.
Figure 1 – “stair step” approximation for a curved surface
The FVM is a variation of the finite-difference method (FDM). Early CFD researchers used the FDM for work on brick-shaped structured meshes. This structure provided well-defined grids that could be constructed to relate the nodes or vertexes of the volumes in the analysis domain.
From FDM to FVM
The FVM method considered the flux of the dependent variable from node to node instead of the mathematical expression used in the FDM. Since the fluid is flowing through the analysis domain, it was a natural and intuitive way to represent the physics of the fluid. With the FVM method, however, the fluxes must balance exactly in order to get a viable solution. The FVM transforms the analysis goal from being a solution at the nodes to a solution of balanced fluxes.
Early FVM algorithms relied on structured or brick-shaped analysis domains because the fluxes were more readily understood by developers and easier to implement in a CFD computer program. Some FVM CFD codes still rely on this structured mesh analysis domain.
But, the structured mesh has some problems if you’re modeling anything but bricks. With any type of organic or complex shape, you’ll not get a very accurate representation of your analysis geometry.
Stair steps and compute resources
Figure 1 shows the structured mesh representation of a curved surface. You can see the classic “stair-step” effect. As you can imagine, there are inherent inaccuracies associated with treating curved walls with this stair-step shape.
Another issue with the structured mesh is the amount of compute resources needed to carry a finely meshed area all the way through the model. Figure 2 shows the fine mesh that is required to model a two-dimensional cascade. Note that the fine mesh built around the foil shape has to extend to the boundaries. For three-dimensional bodies, the structured mesh must maintain the same number of elements in each of the three coordinate directions; the fine mesh must extend far beyond the region of high gradients.
Figure 2. Structured mesh on a two-dimensional cascade (http://www.iafr.eu/TESI/4.htm)
So, why don’t the FVM developers use an unstructured mesh?
Some do, but it requires a massive effort to convert the FVM to a completely unstructured mesh. Even then, numerical instabilities abound due to the nature of unstructured meshes. Compute resources are also taxed because of the tremendous amount of bookkeeping required to keep track of which fluxes should be attached to which nodes.
The FEM for CFD
In the early 1980s, CFD researchers began exploring the finite element method (FEM). Most of the early adopters were already having success using finite elements for structural analysis. In fact, the FEM has been the standard for structural analysis (FEA) for decades.
Extending FEA to the CFD world wasn’t easy, however. Early adopters of the FEM found it difficult to apply the lessons learned from the FVM work already done. By this time, the algorithms developed for the FVM were well-honed to be efficient for solving difficult CFD problems, but they resulted in inefficient meshes. Because the governing equations for CFD are coupled and non-linear, approximations were necessary. Early FEM researchers were not familiar with the approximation methods or found them too “un-mathematical.”
I was part of an early group in the mid-1990s at the University of Virginia that had a great deal of industrial experience developing FVM codes for the electric power industry. We decided to use the basic-weighted integral approach of the FEM to discretize the partial differential equations (PDEs) for fluid velocities and pressure. We combined this approach with the overall solution algorithms and other numerical approximations (such as upwinding) of the FVM.
The combination of these techniques resulted in a CFD code that incorporated the most computing-efficient aspects of the FVM, but could more accurately, robustly and efficiently model the difficult geometries of the real world. While the solution incorporates both methods, the basic foundation is the FEM, so we called it FEM-based CFD.
Best of both worlds, but mostly the FEM
FEM-based CFD was successfully demonstrated on some of the most difficult CFD problems, including turbomachinery, combustion, high-speed external flow and many more. It became the basis for the code used in CFdesign software.
From a CFD user’s perspective, there are critical advantages in the FEM-based approach:
- CFD solutions can be more easily obtained, since you are solving a FEM-like minimization problem.
- Meshing complex geometries is easier and more accurate with the tetrahedral meshes common to the FEM.
- Linking up CAD and other CAE analyses (such as FEA) is more readily accomplished with the FEM.
The principal advantage of the FVM is the more advanced physics that have been developed for this method due to its earlier adoption by CFD developers. This advantage, however, is diminishing in conjunction with the rapid progress of new and improved finite-element methods within the CFD community.
With their unique background of developing CFD code using both the FVM and FEM, CFdesign developers can pull the best from both worlds, but the core methodology is still solidly FEM.
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Rita Schnipke, PhD, has been a CFD software innovator for more than 30 years. She co-founded Blue Ridge Numerics Inc. with the express purpose of delivering a CFD tool suitable for mainstream product development processes.
Do you agree or disagree with this article? What do you have to add to the FVM vs. FEM debate? Comment here.
Very nice!…
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