Mixed-form Fast Multipole Algorithm Research Articles

An Integral Equation Modeling of Lossy Conductors With the Enhanced Augmented Electric Field Integral Equation

The formulation of the enhanced augmented electric field integral equation for dielectrics is generalized to conductor problems in this paper. The conductive region is simulated as a lossy dispersive medium using a full wave solver. In order to calculate the method of moments matrix elements in the conductive region accurately, we investigate the evaluations of the integrals of Green’s function in lossy media. After comparing with some other integration methods, we propose a new method to evaluate such integrals. This method turns out to improve the accuracy and efficiency. Moreover, the proposed formulation can be regarded as a generalized impedance boundary condition (IBC). This generalized IBC will become global if the skin depth is comparable to the size of the structures/details. The mixed-form fast multipole algorithm is employed for the simulations. Numerical examples of complex circuit structures are given to demonstrate the accuracy and capabilities of the proposed method.

The Error Control of Mixed-Form Fast Multipole Algorithm Based on the High-Order Multipole Rotation

In this letter, the mixed-form fast multipole algorithm (MF-FMA) with feasible error controllability is implemented through high-order multipole rotation that can make the translation matrices very sparse. The saving of both storage and CPU time makes it attainable to increase the number of harmonics in the middle frequency regime, thus shifting up the transition region of MF-FMA. Consequently, the broadband accuracy of the MF-FMA can be improved significantly.

An Enhanced Augmented Electric-Field Integral Equation Formulation for Dielectric Objects

A full-wave surface integral equation (SIE) method based on the augmented electric-field integral equation (A-EFIE) for dielectric objects with low-frequency stability is presented in this paper. Motivated by the A-EFIE formulation for perfect electric conductor (PEC), the internal and external problems are both augmented with the current continuity equation and renormalized to eliminate the low-frequency breakdown. Although the magnetic-field integral equation operator $\mathcal{K}$ is free of low-frequency breakdown, its matrix form is ill-conditioned and unsolvable if the traditional Rao–Wilton–Glisson (RWG) basis function is used as the testing and basis functions. As a remedy, the Buffa–Christiansen (BC) basis function is introduced to alleviate this testing issue. After this treatment, the matrix form of operator $\mathcal{K}$ is well conditioned. To solve problems with a large number of unknowns, a preconditioning scheme is introduced to accelerate the convergence and the mixed-form fast multipole algorithm (FMA) is adopted to accelerate the matrix vector product.

Fast Full-Wave Surface Integral Equation Solver for Multiscale Structure Modeling

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We describe a full-wave solver to model large-scale and complex multiscale structures. It uses the augmented electric field integral equation (A-EFIE), which includes both the charge and the current as unknowns to avoid the imbalance between the vector potential and the scalar potential in the conventional EFIE. The formulation proves to be stable in the low-frequency regime with the appropriate frequency scaling and the enforcement of charge neutrality. To conquer large-scale and complex problems, we solve the equation using iterative methods, design an efficient constraint preconditioning, and employ the mixed-form fast multipole algorithm (FMA) to accelerate the matrix-vector product. Numerical tests on various examples show high accuracy and fast convergence. At last, complex interconnect and packaging problems with over one million integral equation unknowns can be solved without the help of a parallel computer. </para>

Generalized Impedance Boundary Condition for Conductor Modeling in Surface Integral Equation

A generalized impedance boundary condition is developed to rigorously model on-chip interconnects in the full-wave surface integral equation by a two-region formulation. It is a combination of the electric-field integral equation for the exterior region and the magnetic-field integral equation for the interior conductive region. The skin effect is, therefore, well captured. A novel integration technique is proposed to evaluate the Green's function integrals in the conductive medium. Towards tackling large-scale problems, the mixed-form fast multipole algorithm and the multifrontal method are incorporated. A new scheme of the loop-tree decomposition is also used to alleviate the low-frequency breakdown for the formulation. Numerical examples show the accuracy and reduced computation cost.

A mixed-form fast multipole algorithm

The fast multipole algorithm manifests in two very different forms at low frequencies and at mid frequencies. Each can operate in their respective regimes, but are not tenable in the other regimes. The paper reports on a way to factorize the Green's function for fast algorithm using a mixed form. The low-frequency fast multipole algorithm (LF-FMA) will be used at low frequencies or the long-wavelength regime, and the multilevel fast multipole algorithm (MLFMA) will be used for the mid frequencies or the shorter-wavelength regime. For object modeling where both long-wavelength and wave physics are important, we propose a mixed-form fast multipole algorithm (MF-FMA). This algorithm has no low frequency break down, and it can work seamlessly from static (where circuit physics is important) to dynamic (where wave physics is important).