 # The Finite Element Method

Scalar finite element methods have been used by civil and mechanical engineers to analyze material and structural problems since the 1940s. However it wasn't until the 1960s that FEM codes were developed to solve problems in electromagnetics. Some of the pioneers in this field were Silvester , Zienkiewicz , and Wexler . Initial FEM-based CEM modeling codes were applied to problems in electrostatics and magnetostatics. Later they were used to solve high-frequency problems in 2 dimensions. Practical 3-dimensional codes did not appear until the 1980s due largely to problems with vector parasites [4, 5] and unreliable absorbing boundary conditions . Unwanted reflections from absorbing boundaries continue to be a problem with full-wave 3D FEM codes even today.

Like BEM techniques, finite element methods can be based on different formulations (even the method of moments). However BEM techniques always solve an integral equation and FEM techniques always solve a differential equation. Every FEM code divides the entire problem domain into small elements. For 2D problems the elements are usually triangles or rectangles. For 3D problems, the elements are usually tetrahedra (4 faces) or bricks (6 faces). The domain must be finite and bounded. Modeling an unbounded (e.g. radiation) problem requires that the problem domain be bounded with special elements that absorb all incident energy. These elements are called ABC (Absorbing Boundary Condition) elements.

The unknowns in scalar FEM codes are the three orthogonal components of the field at the "nodes" (vertices) of each element. The unknowns in vector FEM codes are the field components along the edges of each element. Scalar codes are conceptually simpler, but they are unsuitable for full-wave modeling, because they are susceptible to spurious solutions that can cause significant and unpredictable errors in the solution. Vector FEM codes are much less likely to exhibit these parasitics.

To form a linear system of equations, the governing differential equation and associated boundary conditions are converted to an integro-differential form using either a variational method or a weighted-residual (moment) method. Variational methods solve for the unknown quantity by minimizing an energy functional. Weighted-residual methods multiply a weak form of Maxwell's equations by a weighting function and integrate over each element. Ultimately, a matrix equation is generated in the form, (1)

where [X] is a vector of the unknown field quantities, [B] is a vector of source terms, and [A] is a sparse matrix whose only non-zero values correspond to positions in the matrix corresponding to edges that share an element.

Generally, the matrices generated by FEM codes are must larger than the matrices generated by BEM codes applied to similar geometries. This is because gridding an entire problem volume requires many more elements than gridding just the material interfaces. However, because FEM matrices are very sparse, they do not necessarily require more storage or computing resources to solve than the small, but dense, matrices generated by BEM codes.

As indicated previously, modeling unbounded problems requires special absorbing elements (ABCs). Many formulations of these elements have been proposed [6-13]. The ABCs that have been developed for 2D FEM codes work very well; however 3D FEM ABCs work well only at prescribed angles of incidence resulting in the need to locate the boundaries sufficiently far from other structures. Hybrid FEM/BEM codes terminate open surfaces of the FEM volume with a BEM surface negating the need for ABCs . Unfortunately, the BEM portion of the resulting matrix is dense, which can significantly increase the amount of computational resources required.

Perhaps the most attractive feature of the finite element method is its ability to model configurations that have complicated geometries and incorporate various materials. The electrical properties of each element are defined independently and elements can be as small or as large as needed to facilitate the analysis.

Table 4 lists various strengths and weakness of FEM modeling techniques. Note that the capabilities of any particular modeling software depend on the specific formulation, the matrix solver and any optimization techniques employed.

Table 4: Strengths and Weaknesses of the Finite Element Method

 FEM Modeling Strengths FEM Modeling Weaknesses -        Excels at modeling inhomogeneous or complex materials -        Excels at modeling problems that combine small detailed geometries with larger objects -        Excels at modeling structures in resonant cavities or waveguides -        Absorbing boundary required for modeling unbounded (radiation) problems -        Difficult to model thin wires accurately

Table 5 lists various CEM modeling codes that are based on the finite element method. The codes listed and the comments in Table 5 are based on the information available to the authors as of the publication date of this report.

Table 5: CEM Modeling Codes that use the Finite Element Method

 Software Title Description Source Amaze (HiPhi/Magnum) 3D quasi-static Field Precision Analyst 2D and 3D full-wave STAAR Microwave Studio - FDS 3D full-wave CST EMDS 3D full-wave Agilent Technologies EMS 3D quasi-static ElectromagneticWorks, Inc. FEMM 2D quasi-static Dr. David Meeker Flux2D/3D 2D and 3D quasi-static Magsoft HFSS 3D full-wave Ansys / Ansoft JCMsuite optics JCMwave MagNet / ElecNet 3D quasi-static Infolytica Magneto / Oersted 2D quasi-static Integrated Eng. Software Maxwell 2D/3D quasi-static Ansys / Ansoft Opera 2D/3D 2D/3D quasi-static Vector Fields Q3D Extractor 2D/3D quasi-static Ansys / Ansoft QuickField 2D quasi-static Tera Analysis Tricomp EStat/PerMag 2D quasi-static Field Precision Tricomp WaveSim 2D high-frequency Field Precision

## References

 P. Silvester, "High-order finite element waveguide analysis (Program Descriptions)," IEEE Trans. on Microwave Theory and Tech., vol. 17, no. 8, pp. 651-652, Aug. 1969.

 O. C. Zienkiewicz, A. K. Bahrani, and P. L Arlett, "Numerical solution of 3-dimensional field problems," Proc. IEE (London), vol. 115, pp. 367-369, Feb. 1968.

 B. H. McDonald and A. Wexler, "Finite-element solution of unbounded field problems," IEEE Trans. on Microwave Theory and Tech., vol. 20, no. 12, pp. 841-847, Dec. 1972.

 D. R. Lynch and K. D. Paulsen, "Origin of Vector Parasites in Numerical Maxwell Solutions," IEEE Trans. on Microwave Theory and Techniques, vol. 39, no. 3, pp. 383-394, Mar. 1991.

 D. R. Lynch and K. D. Paulsen, "Elimination of Vector Parasites in Numerical Maxwell Solutions," IEEE Trans. on Microwave Theory and Techniques, vol. 39, no. 3, pp. 395-404, Mar. 1991.

 Y. Li and Z. Cendes, "High-accuracy absorbing boundary condition," IEEE Trans. on Magnetics, vol. 31, no. 3, pp. 1524-1529, May 1995.

 A. Bayliss, E. Turkel, "Radiation boundary conditions for wave-like equations," Communications on Pure and Applied Mathematics, vol. 33, pp. 707--725, 1980.

 A. Bayliss, M. Gunzburger, E. Turkel, "Boundary conditions for the numerical solution of elliptic equations in exterior regions," SIAM Journal of Applied Mathematics, vol. 42, pp. 430-451, 1982.

 T.G. Moore, J.G. Blaschak, A. Taflove, G.A. Kriegsmann, "Theory and application of radiation boundary operators," IEEE Trans. Antennas Propagat., vol. 36, pp. 1797-1812, 1988.

 J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, pp. 185-200, 1994.

 J. M. Jin, and W. C. Chew, "Combining PML and ABC for finite element analysis of scattering problems," Micro. Opt. Tech. Lett., vol. 12, pp. 192-197, 1996.

 Wu J-Y, Kingsland DM, Lee J-F, Lee R. , "A Comparison of anisotropic PML to Berenger's PML and its application to the finite-element method for EM scattering," IEEE Trans. on Antennas and Propagat., vol. 45, no.1, pp. 40-50, 1997.

 R. Mittra and M. Kuzuoglu, "A review of some recent advances in perfectly-matched absorbers for mesh truncation in FEM," IEEE Antennas and Propagation Society International Symposium, 1997 Digest, vol. 2, pp.1302 -1305, 1997.

 M. Ali, T. H. Hubing, and J. Drewniak, "A hybrid FEM/MOM technique for electromagnetic scattering and radiation from dielectric objects with attached wires," IEEE Trans. on Electromag. Compat., vol. 39, no. 4, pp. 304-314, Nov. 1997.