# Finite Element Method (FEM) vs. Finite Volume Method (FVM) in Field Solvers for Electronics

January 3, 2020 If you are like me, you have been in a situation in which there was a preferred method of doing a particular task. This preferred method was somewhat in opposition to another way of approaching a task. However, whenever there is a preferred method, i.e., one approach versus another, it is usually accompanied by these words of wisdom: “If it ain’t broke, don’t fix it.”

When I say wisdom, in this case, it is heavily laced with sarcasm. Mainly because I know that there is usually a better way to accomplish a given task, and at the very least, there is a more efficient way of doing it. However, the term versus still epitomizes the essence of choice.

The majority of the time, one method versus another is dependent on several parameters that uniquely make whichever method you choose, the right fit for your particular circumstances, for example, Finite Element Method (FEM) versus the Finite Volume Method (FVM). Both methods are similar in the fact that they are systematic numerical methods for solving partial differential equations (PDEs). However, it still requires a more in-depth understanding of what each offer and what your requirements are to make an accurate decision as to which fits your needs.

## What is the Finite Element Method (FEM)?

The finite element method (FEM) is a systematic numerical method for solving problems of engineering and mathematical physics, more specifically PDEs. The FEM generally addresses issues in heat transfer, structural analysis, fluid flow, electromagnetic potential, and mass transport. Also, the analytical nature of the solutions of these issues typically requires the solution to boundary value problems for PDEs.

Furthermore, the FEM formulation of the problem will result in a system of algebraic equations. The FEM also appraises the unknown function over the domain. Thus, to solve the problem, it subdivides a large system into smaller, simpler parts that are called finite elements. After which, these simple equations that model the finite elements are then compiled into a larger system of equations that models the entire problem. The FEM will then use variational methods from the calculus of variations to estimate a solution by minimizing a related error function.

Lately, the FEM is in use in applications for simulating quantum effects in low dimensional systems like carbon nanotubes, metallic nanoparticles, quantum wells, quantum dots, monolayer transition metal dichalcogenides, and artificial molecules. Modeling complicated structures are vital processes for advanced electronics designs.

## What is the Finite Volume Method (FVM)?

The Finite volume method (FVM) is a widely used numerical technique. The fundamental conservation property of the FVM makes it the preferable method in comparison to the other methods, i.e., FEM, and finite difference method (FDM). Also, the FVM’s approach is comparable to the known numerical methods like FEM and FDM, which means that its evaluation of volumes is at discrete places over a meshed geometry.

Furthermore, the FVM transforms the set of partial differential equations into a system of linear algebraic equations. Although the discrete approximation procedure in use in the FVM is distinctive, it also utilizes two basic steps. Firstly, it transforms and integrates the PDEs into balance equations over an element. The process incorporates the changing of the surface and volume integrals into discrete algebraic relations over elements as well as their surfaces using an integration quadrature of a specified order of accuracy. This will result in a set of semi-discrete equations.

Secondly, in this next step, the interpolation profiles are chosen to estimate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations. In regards to the two steps in the FVM process, your approximation selection affects the overall accuracy of the subsequent numeric.

## The Finite Element Method (FEM) vs. Finite Volume Method (FVM)

With FEM and FVM, both methods share some similarities, since they both represent a systematic numerical method for solving PDEs. However, one crucial difference is the ease of implementation. Among the majority of engineers, the prevailing opinion is that the FDM is the easiest to implement, while FEM is the most difficult, leaving FVM somewhere in the middle.

Overall, there has always been a debate among engineers working with fluid flow about the suitability of FEM for Computational fluid dynamics (CFD). Furthermore, there are some engineers who firmly believe that FVM is vastly superior to FEM. However, there is no corroborating evidence to substantiate either claim. The consensus here is, each method is suitable; it just depends on the requirements of the issue you are trying to solve.

Therefore, I will cover some of the advantages and disadvantages of both to clarify which might be best suited for your issue.

The FEM is successful in Multiphysics analysis because it is a very general method. Meaning that it is similar to already established methods in use for electromagnetics and structural analysis. Also, the FEM allows for natural increases in the order of the elements, which permits very accurate approximations of physics fields. In general, this corresponds to locally approximating the physics fields with (higher-order) polynomials.

Another advantage of the FEM is the ability to combine different kinds of functions that estimate the solution within each element. We commonly refer to this as mixed formulations. This is a critical advantage for the FEM because mixed formulations are straightforward with the FEM, whereas it is difficult or near impossible with other methods.

Although the FEM has the benefit of naturally handling both curved and irregular CAD geometries, the mathematics behind the FEM is relatively advanced, and thus requires mathematical expertise for its implementation. This is in contrast to the FVM since it is comparatively straightforward. Finally, for particular time-dependent simulations, one must utilize explicit solvers for reasons of efficiency. However, due to the FEM’s difficulty in the implementation of such techniques, its use is not advisable.

The FVM is a natural choice for solving CFD issues because the PDEs you have to resolve for CFD are conservation laws. However, you can also use both FDM and FEM for CFD, as well.

The FVM’s most significant advantage is that it only needs to do flux evaluation for the cell boundaries. Also, this is true for nonlinear issues as well, which, in turn, makes it an excellent choice for the handling of (nonlinear) conservation laws.

The accuracy of the FVM, for example, close to a corner of interest, can be increased by refining the mesh around that corner, similar to the FEM. However, unlike the FEM, the functions to estimate the solution is not easily made for higher-order. Therefore, this is a disadvantage of the FVM versus the FEM. Utilizing multiphysics solvers and algorithms will be technically complicated for technically complicated designs.

Both methods are extremely useful analysis tools. However, the decision to use one method versus the other should strictly depend on your individual issue needs. Just like in life, one size or method does not fit all.

Be sure to have your designers and production teams working together towards implementing the use of the FEM or the FVM with Cadence’s suite of design and analysis tools. With Cadence’s Clarity you can find the analysis solutions you’re looking for, that can then provide your layout optimization in Allegro with all necessary information.

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