International Journal of Antennas and Propagation

Volume 2018, Article ID 5062021, 7 pages

https://doi.org/10.1155/2018/5062021

## Versatile Solver of Nonconformal Volume Integral Equation Based on SWG Basis Function

State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310058, China

Correspondence should be addressed to Hai Lin; nc.ude.ujz.dac@nil

Received 14 June 2018; Accepted 24 September 2018; Published 9 December 2018

Academic Editor: Paolo Baccarelli

Copyright © 2018 Chunbei Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, a versatile solver of a nonconformal volume integral equation based on the Schaubert-Wilton-Glisson (SWG) basis function is presented. Instead of using a piecewise constant function, the robust conventional SWG basis function is chosen and used directly for discontinuous boundaries. A new map method technique is proposed for constructing SWG pairs, which reduces the complexity from to compared with a brute-force method. The integral equation is solved by the method of moments (MoM) and further accelerated by the multilevel fast multipole algorithm (MLFMA). What’s more, the hybrid scheme of MLFMA and adaptive cross approximation (ACA) is developed to resolve the low-frequency (LF) breakdown when dealing with over-dense mesh objects. Numerical results show that when in analysis of radiation or scattering problems from inhomogeneous dielectric objects or in LF conditions, the proposed solver shows high efficiency without loss of accuracy, which demonstrates the versatile performance of the proposed method.

#### 1. Introduction

Electromagnetic (EM) scattering and radiation analysis of dielectric materials are attracting increasing attention for their potential for vast applications, such as in designing a microstrip dielectric antenna and scattering reduction of the coated layer in a stealth plane, dielectric radome, biological media, and plasma sheath. Generally, these problems can be solved by an integral equation using method of moments (MoM) [1] because it has lesser degrees of freedom than differential equation methods. Compared with a surface integral equation (SIE) such as a PMCHWT equation [2], a volume integral equation (VIE) is more flexible, robust, and accurate [3]; therefore, VIE is usually preferred or even the only option for complex dielectric anisotropic objects.

In [4], the discontinuous Galerkin (DG) VIE using the Schaubert-Wilton-Glisson (SWG) basis function is proposed and a hybrid discretization scheme that uses a mix of nonconformal and conformal meshes is adopted. In [4, 5], Zhang et al. found that the explicit enforcement of the continuity condition at the interface between two neighboring elements is not required in the DGVIE. This is because of the inherent quality of the Fredholm integral equation of the second kind, where the boundary conditions are naturally imbedded. Therefore, the conventional SWG basis function is a better choice to apply in nonconformal VIE than the piecewise constant function [6], since the SWG basis is more robust and accurate.

In this paper, we exploit a versatile solver of a nonconformal volume integral equation based on the SWG basis function. SWG pairs are used in the two neighboring elements sharing the common face, while half-SWG basis functions are adopted in the discontinuous boundary elements. Hence, before the simulation, it is important to find all the neighboring two elements (i.e., two neighboring tetrahedrons which share the common face) efficiently. In this paper a new map method technique is proposed to find SWG pairs faster than the brute-force method, which reduces the complexity of the computation time from to . To improve the efficiency of the proposed solver, the multilevel fast multipole algorithm (MLFMA) [7] is employed to accelerate the matrix vector multiplication (MVM) in the iterative solution process. In the analysis of dielectric objects containing different media, different scales of discrete elements are used for nonconformal VIE and this usually results in a dense mesh. However, when dealing with over-meshed objects, in other words, when the minimum box size is too small compared to the wavelength, the MLFMA suffers from a low-frequency (LF) breakdown problem [8, 9] limited by the addition theorem to calculate Green’s function. Another fast low-rank decomposition solver, namely adaptive cross approximation (ACA) [10], is less efficient compared to MLFMA, since it is a purely algebraic solver and is independent of Green’s function. Therefore, a hybrid scheme of MLFMA and ACA is developed to resolve the LF breakdown in this paper. For the different levels of an octree structure, different algorithms are used. This is because ACA does not exist in an LF breakdown and it is applied to accelerate MVM operations in the lowest levels, while MLFMA is more efficient, has lower complexity, and can be applied in higher levels. In summary, the contributions of this work can be listed as follows: (1)The nonconformal VIE expanded by the SWG basis function in discontinued boundaries is explained in detail, and the solver is accelerated by MLFMA(2)A new technique for constructing SWG pairs is introduced which reduces the complexity of the computation time from to . To the best of our knowledge, this part has never been published before(3)A hybrid scheme of MLFMA and ACA is developed which resolves the LF breakdown problem

#### 2. Formulation

First, the matrix system discretized by MoM based on the SWG basis function will be introduced, and the treatment of discontinued boundaries and the corresponding impedance matrix will be described in detail. Then, the brute-force method and map method for constructing SWG pairs are introduced and the complexities are analyzed. Finally, the hybrid MLFMA-ACA scheme is presented to settle the LF breakdown problem.

##### 2.1. Volume Electric Field Integral Equation

Consider an inhomogeneous dielectric object with a permittivity of and a permeability of residing in free space with EM parameters and . The object is excited by an incident electric field , and the volume electric field integral equation (VEFIE) associated with the vector potential and scalar potential can be expressed by [11] where is the angular frequency and is the wavenumber of free space. is the volume equivalent currents and is related to the electric flux density by

Discretizing the unknown vector by where is the SWG basis function [12] associated with the th face, is the corresponding unknown coefficient, and is the total number of unknowns. Figure 1 shows the definition of the SWG pair and the half-SWG. Applying Galerkin’s test, then (1) can be rewritten in the matrix equations form as follows: where , , and are the matrices corresponding to the first, second, and third terms of the left side of (1), the unknown vector relates with the unknown coefficient , and and are the impedance matrix and excitation vector, respectively.