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1.1 Finite Element Method for Electromagnetics

by Jose Alves

The finite element method (FEM) is based on a mathematical technique called weak or variational formulation which transforms local expressions of partial differential equations in a globalised expression where the solution is searched in mathematical function spaces with a specific regularity Jin [Jin15] Smajic [Sma16]. The approach seeks to enforce conservation of the given expression in a global manner instead of at each point of space. Hence the name weak formulation. A first advantage of this method is that the requirements on the regularity of the solution can be much weaker than for other methods. A second advantage is that the weak formulation can be used for Galerkin internal approximation: the discretized solution is sought in a discretized version of the continuous mathematical function space mentioned earlier.

This method is very powerful as it enables great flexibility regarding the geometrical and material complexity of the objects that can be analysed. It can naturally account for heterogeneous distribution of points and local properties on the mathematical object, the mesh, on which the computation is performed. The method is mathematically guaranteed to converge to arbitrary precisions when the mesh size is reduced at the cost of the computational requirements. Nevertheless, convergence can be mathematically proven for constant and homogeneous material properties only; For the general non-linear cases, convergence should be proven numerically.

One of the advantages of the method resides in its scalability for parallel computing. The resulting numerical implementation usually leads to sparse and diagonally dominant linear systems in which the information propagation happens at the local patch around points. This implies that excellent performance is to be expected on High Performance Computing (HPC) clusters. At the same time, the same fact that information is propagated through local patches of space implies that the method is not necessarily optimal when dealing with largely heterogeneous sized domains such as in antenna applications. The method requires the meshing of all objects involved as well as the media in between, which depending on the implementation of the numerical strategy can be quite expensive. FEM is used in a wide range of applications in CEM, ranging from low-to-medium frequency ranges (f<1 MHz) up to very high frequencies (f>100 GHz).

Some examples of where FEM is used are:

• Prediction of electric performance and radiation of 3D IC and antenna structures,

• Electromagnetic Processing of materials: Induction Heating and Hardening, Magnetic Pulse Forming/Welding/Crimping, Electromagnetic Stirring or Breaking Müller and Bühler [MullerBuhler01],

• Electric Motor Design.

[Jin15] Jian-Ming Jin. The finite element method in electromagnetics. John Wiley & Sons, 2015.

[MullerBuhler01] Ulrich Müller and Leo Bühler. Magnetofluiddynamics in channels and containers. Springer Science & Business Media, 2001.

[Sma16] J. Smajic. How to perform Electromagnetic Finite Element Analysis? NAFEMS, 2016.

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