This paper presents a review of finite element open boundary techniques for the computation of static and quasistatic electromagnetic fields. The underlying assumptions and ideas behind these techniques are outlined and explained. The advantages and shortcomings of each technique are discussed. The goal of this review is to aid those working on numerical solutions of open boundary problems using the finite element method to gain perspectives on the variety of techniques presently available or being developed. This review is also intended to facilitate the communication of numerical techniques between the low-frequency field computation community and those in the high-frequency community.

A review of finite element open boundary techniques for static and quasi-static electromagnetic field problems

The accuracy of finite-difference analysis in electromagnetics can be qualitatively improved by employing arbitrary local approximating functions, not limited to Taylor expansion polynomials. In the proposed new class of flexible local approximation methods (FLAME), desirable local analytical approximations (such as harmonic polynomials, plane waves, and cylindrical or spherical harmonics) are directly incorporated into the finite-difference scheme. Although the method usually (but not necessarily) operates on regular Cartesian grids, it is in some cases much more accurate than the finite-element method with its complex meshes. This paper reviews the theory of FLAME and gives a tutorial-style explanation of its usage. While one motivation for the new approach is to minimize the notorious "staircase" effect at curved and slanted interface boundaries, it has much broader applications and implications. As illustrative examples, the paper examines the simulation of: 1) electrostatic fields of finite-size dielectric particles in free space or in a solvent with or without salt; 2) scattering of electromagnetic waves; 3) plasmon resonances; and 4) wave propagation in a photonic crystal.

Electromagnetic applications of a new finite-difference calculus

A quasi-multiple medium (QMM) method based on the direct boundary element method (BEM) is presented to extract the capacitance of three-dimensional (3-D) very large scale integration interconnects with multiple dielectrics. QMM decomposes each dielectric layer into a few fictitious medium blocks, and generates an overall coefficient matrix with high sparsity. With the storage technique of a sparse blocked matrix and iterative equation solver generalized minimal residual, the QMM can greatly reduce the CPU time and memory usage of large-scale direct BEM computation. Numerical examples of 3-D multilayered and multiconductor structures cut from actual layout show the efficiency of the QMM method for capacitance extraction. We also compared the QMM accelerated BEM with geometry independent measured equation of invariance (GIMEI) and Zhu's overlapping domain decomposition method (ODDM).

http://tiger.cs.tsinghua.edu.cn/Faculty/yu/papers/tmtt03.pdf

Fast capacitance extraction of actual 3-D VLSI interconnects using quasi-multiple medium accelerated BEM

A novel integral formulation of the measured equation of invariance method is derived from the reciprocity theorem and implemented for perfectly conducting (PEC) 2-D scattering problems. This formulation uses the electric and magnetic Green's functions of the environment to obtain a matrix equation for the induced surface current with the same number of unknowns as the conventional boundary element-method of moments (BE-MoM) approach. However, the matrix that must be inverted in the new formulation is sparse and circulant, with only three non-zero elements per row. Sample results for two-dimensional TM and TE problems with perfectly conducting scatterers show enormous CPU time and memory savings over the conventional BEM-MoM approach. The new formulation has important advantages over the original finite difference formulation of MEI, but also shares some of its limitations.

/pdf/integral-formulation-of-the-measured-equation-of-invariance-spy53ayeyz.pdf

Integral formulation of the measured equation of invariance: a novel sparse matrix boundary element method

The measured equation of invariance (MEI) has been previously introduced to efficiently and accurately handle the boundary truncation for finite methods. The present authors give a theoretical analysis that provides several important insights into the capabilities of the MEI. From the numerical study, they can explain why the MEI works better than one would expect. Both the theoretical and the numerical analyses demonstrate that the accuracy of the solution is dependent on the electrical size of the geometry as well as the distance between the mesh boundary and the geometry. From the analysis, the authors propose a new set of metrons that is less sensitive to the excitation than the previously proposed sinusoidal metrons. >

A theoretical and numerical analysis of the measured equation of invariance

Computations of electromagnetic fields are based either on differential equations or on integral equations. The differential equation approach using finite difference or finite element methods results in sparse matrices, which is an advantage, but has to cover large volumes, which is a disadvantage. The integral equation approach using the method of moments (MOM) limits the mesh to the surface of the object, which is an advantage, but results in full matrices, which is a disadvantage. It is noted that the ideal case would be to reduce the finite difference type equations close to the object surface and still preserve the sparsity of the matrices. The measured equation of invariance is a new concept in field computation capable of approaching this ideal situation. The mathematics and reasonings to reach a novel computational method based on this concept are presented. It is shown that the method is robust for both convex and concave objects, is much faster than the MOM, and uses a fraction of the memory. >

/pdf/the-measured-equation-of-invariance-a-new-concept-in-field-54i4ikgacl.pdf

The measured equation of invariance-a new concept in field computation

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The measured equation of invariance-a new concept in field computation