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Understanding the Differences Between FEM, FDM, and FVM in Engineering Simulations

Finite-element analysis of an air-filled shell and tube heat exchanger with water flowing in the inner tubes. Simulation results available from COMSOL Multiphysics software reveal flow velocity, temperature distribution, and pressure within the vessel.

One reason for the finite element method’s success in multi-physics analysis is that it is a very general method. Solving the resulting equation systems are the same or very similar to well-known and efficient methods used for structural and electromagnetics analysis. Another reason for the method’s success is that it makes it easy to “increase the order of the elements” so that the physics fields can be approximated very accurately. This typically corresponds to locally approximating the physics fields with polynomials of “higher order,” such as second- and third-degree polynomials, or higher. This technique is often critical, for example, in the case of accurate stress analysis.

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If we consider the example of stress analysis, it is quite common that there are important stress concentrations close to some of the corners of a mechanical part. In this case, the finite-element method allows for two different ways of increasing the accuracy of the solution around this corner. One way is to increase the order of the elements, as described earlier. Another method is to locally refine the mesh close to that corner; the element density increases locally. The finer the mesh (i.e., more elements), the more accurate approximation one gets for the stress field around the corner of interest. Both techniques are used in finite-element software and are frequently made automatic from the user’s perspective. This is known as “adaptive mesh refinement.”

Another advantage with the finite-element method, which is particularly important for multi-physics analysis, is that you can combine different kinds of functions that approximate the solution within each element. This is called mixed formulations. This is important, for example, in the case of electromagnetic heating. The physics and mathematics require one type of function for the electromagnetic field and another type of function for heat transfer; they both need to be tightly coupled to get an accurate solution and for the solution to converge. Mixed formulations are straightforward to handle the finite-element method, but difficult or impossible with other methods.

The benefits with both the finite-element method and the finite-volume method are that curved and irregular CAD geometries are handled in a natural way.

However, the mathematics behind the finite-element method is quite advanced and thus the method requires mathematical expertise for its implementation. Implementations of finite-difference and finite-volume methods are comparatively straightforward.

For certain time-dependent simulations, one needs to use so-called explicit solvers for reasons of efficiency. Implementing such solver techniques is more difficult for the finite-element method than for the finite-difference and finite-volume methods. However, this has successfully been commercialized in some cases, such as in crash simulations.

The finite-difference method is defined dimension per dimension; this makes it easy to increase the “element order” to get higher-order accuracy. If you can fit the simulation in a rectangular or box-shaped geometry using a regular grid, efficient implementations are much easier than for finite-element and finite-volume methods. Regular grids are useful for very-large-scale simulations on supercomputers often used in, as mentioned before, meteorological, seismological, and astrophysical simulations.

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With the finite-difference method, you may easily run into problems handling curved boundaries for the purpose of defining the boundary conditions. Boundary conditions are needed to truncate the computational domain. They represent communication with the surrounding world, which is the part that you do not want included in your simulation. If one can overcome the boundary-condition problem on curved boundaries, the method gives very efficient and high quality results.

For computations that need high accuracy, the extra effort in making boundary-fitted meshes and the associated complications of such meshes for the implementation may be worth it. Examples include Formula 1 car computational-fluid-dynamic (CFD) simulations and space-shuttle CFD simulations. The finite-difference method is more difficult to use for handling material discontinuities. In addition, it does not lend itself for local grid refinement or anything similar to “adaptive mesh refinement.” This may be needed to resolve local rapid variations in solutions such as around a corner of a complex shape, as described earlier.

The finite-volume method is a natural choice for CFD problems, since the partial differential equations you have to solve for CFD are conservation laws. However, both finite differences and finite elements can also be used for CFD. Efficient technology for CFD with the finite-element method has become increasingly popular over the last 10 to 15 years. Techniques for CFD with the finite-difference and finite-volume method have been known and used much longer.

The finite-volume method’s strength is that it only needs to do flux evaluation for the cell boundaries. This also holds for nonlinear problems, which makes it extra powerful for robust handling of (nonlinear) conservation laws appearing in transport problems.

The local accuracy of the finite-volume method, such as close to a corner of interest, can be increased by refining the mesh around that corner, similar to the finite-element method. However, the functions that approximate the solution when using the finite-volume method cannot be easily made of higher order. This is a disadvantage of the finite-volume method compared to the finite-element and finite-difference methods.

All kinds of structural analysis, heat transfer, chemical engineering, electromagnetics (including electrostatics, magnetostatics, low-frequency electromagnetics, and frequency-domain high-frequency electromagnetic waves), multi-physics, and CFD.

Weather calculations, astrophysics, seismology, physical realism in computer graphics, and special effects.

CFD, heat transfer, and chemical engineering.

All of these methods are frequently used today in commercial software, as well as in academic environments. The finite-element method is usually most taxing on a computer system, but it depends on the type of analysis. 

Bjorn Sjodin is VP of Product Management for COMSOL Inc.

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