# Finite Element Modeling For Calculating System Behavior

### Key Takeaways

• Finite element analysis is a fundamental numerical technique for solving differential equations in complex geometries.

• These complex systems may not be solvable by hand, and there may be coupling between different physical phenomena, especially in electronics.

• The best FEM systems will take data directly from your PCB layout and create multiphysics simulations with standard numerical algorithms.

Finite element modeling gives you intuitive multiphysics simulation results for complex electronics systems.

Differential equations are the primary mathematical tool used to describe physical phenomena. From Newton’s laws to the Schrodinger equation, any physical system has a differential equation that governs its behavior. Finding solutions to ordinary and partial differential equations is a long-standing area of applied mathematics research, and many computational packages have been developed as part of this effort.

Highly accurate numerical techniques can be implemented with today’s modern computing capabilities. These techniques would have been intractable and prone to huge errors in the past, but today, they can be executed in a reasonable amount of time with everyday computers. The results from methods like finite element modeling are also known to be very accurate when examining steady-state systems. If you need to model the behavior of complex electronics systems, you may want to use finite element modeling to examine what happens in the steady-state.

## What is Finite Element Modeling?

Finite element modeling (FEM) involves a class of methods used to solve a range of discretized differential equations using an iterative solution algorithm. Within this method, a system is broken down and discretized in small elements, and the solution to a differential equation governing the system is determined at each point in space.

To get started with finite element modeling, the differential equation governing a phenomenon of interest is discretized within the boundaries of the system. This involves entering a certain set of inputs into a numerical solver when setting up a finite element modeling simulation:

• Boundary conditions. The value of the solution to the differential equation of interest must be defined at the system boundary. Dirichlet, Neumann, and mixed problems can be addressed in these systems. A functional boundary condition can also be applied.

• Mesh creation. Building a mesh involves selecting a basis function that defines connections between neighboring points in the mesh. These are then used to define iterative relations between derivatives in the system’s differential equation and are ultimately linked back to the boundary conditions.

• Initial condition (for problems iterated in time). Time-dependent problems require an initial condition; these problems are solved using space-time finite element methods.

### Uses of Finite Element Modeling

Typical problems of interest include structural analysis, heat transfer/diffusion, fluid flow, mass transport, and electromagnetic potential (static field and voltage distribution). Coupled multiphysics problems can be handled as long as the problems are Poincare’ stable in time. In addition, nonlinear problems can be addressed with certain solver routines, which allows the treatment of CFD problems.

Finite element modeling results for a waveguide structure.

### FEM vs. FVM

When modeling fluid dynamics problems, the finite volume method (FVM) is more popular than FEM. In particular, this method can be used when the governing differential equation in the system uses a differential operator that is non-self-adjoint. This method solves an integral form of the governing equations so that the local continuity property required in finite element/finite difference methods can be ignored.

Consider Maxwell’s equations in an integral form, which can be written as follows:

Integral form of Maxwell’s equations.

These equations contain definite integrals that are evaluated within specific volumes in the system, thus the name “finite volume method.” FVM simply pieces together different regions in the system in the same way one would in finite element modeling, but the mesh is a 3D volume rather than a set of connected points.

### Finite-Difference Time-Domain

This numerical modeling method is normally used with time-domain problems, where the system is discretized directly, i.e., without the use of a basis function. Effectively, points anywhere in the system can be used to convert derivatives in the governing equations to finite differences.

## Tips for Setting Up Your Finite Element Simulation

Before you start running finite element simulations, there are some simple steps you can take to simplify your system and speed up convergence without sacrificing accuracy.

• Identify symmetries and homogeneities. By spotting any regions or directions in the system where the solution is expected to be homogeneous, derivatives along that direction or in that region can be ignored. This reduces the number of dimensions in the solution and speeds up computation time.

• Choose your mesh density. Adaptive meshing is often used to apply different mesh resolution values to different regions of the system. High accuracy may only be required in certain portions of the system, thus a finer mesh can be applied in those regions.

• Determine static vs. dynamic coefficients. Differential equations with variable or nonlinear components can be treated with finite element modeling. However, these coefficients may vary slowly compared to the solution, so in some cases, it is appropriate to approximate a coefficient as constant.

Once you’ve reduced the system to fewer dimensions (if possible) and determined the mesh density you need, the best field solver tools can take your PCB or IC layout and convert it into a meshed model for use in multiphysics problems. Automatic mesh generation for finite element modeling is possible with a number of powerful software tools. You can then select the particular numerical algorithm to use for your simulation in order to achieve the desired accuracy.

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