International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 563436, 7 pages

http://dx.doi.org/10.1155/2015/563436

## Fast Integral Equation Solution of Scattering of Multiscale Objects by Domain Decomposition Method with Mixed Basis Functions

^{1}Department of Microwave Engineering, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China^{2}Department of Electronics and Photonics, Institute of High Performance Computing, 1 Fusionopolis Way, Singapore 138632

Received 29 March 2015; Accepted 30 May 2015

Academic Editor: Felipe Cátedra

Copyright © 2015 Ran Zhao 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

Nonconformal nonoverlapping domain decomposition method (DDM) with mixed basis functions is presented to realize fast integral equation solution of electromagnetic scattering of multiscale objects. The original multiscale objects are decomposed into several closed subdomains. The higher order hierarchical vector basis functions are used in the electrically large smooth subdomains to significantly reduce the number of unknowns, while traditional Rao-Wilton-Glisson basis functions are used for subdomains with tiny structures. A well-posed matrix is successfully derived by the present DDM. Besides, the nonconformal property of DDM allows flexible mesh generation for complicated objects. Numerical results are presented to validate the proposed method and illustrate its advantages.

#### 1. Introduction

As an effective full wave method, integral equation method (IEM) is one of the most widespread methods for solving electromagnetic (EM) scattering and radiation problems. Different from partial differential equations, integral equations satisfy the radiation condition automatically, and their numerical solution does not require an absorption boundary condition. Besides, the unknowns of IEM are only distributed on surface for perfect electric conductor (PEC), and the number of unknowns is much less than the one by finite element method and finite difference method. However, for electrically large structures, traditional IEM with low order basis like Rao-Wilton-Glisson (RWG) basis [1] still leads into a large number of unknowns, which makes it challenging to store and solve the matrix equation. To circumvent this difficulty, many fast methods were developed, including the multilevel fast multipole algorithm (MLFMA) [2], the adaptive integral method [3], integral equation fast Fourier transformation (IE-FFT) [4], hierarchical matrix methods [5, 6], and its parallel version [7].

These fast methods greatly reduced the computational complexity and memory requirements. Another kind of fast methods is to reduce the dimension of the matrix by using higher order basis functions. Higher order basis functions can be categorized into two kinds, the interpolatory type and the hierarchical type [8–10]. The higher order interpolatory basis functions interpolate the value of field on a large patch with many interpolation points. The limitation of this basis is the expansion order which must be kept constant for all patches. Besides, the interpolatory type basis requires conformal mesh. The higher order hierarchical vector (HOHV) basis function does not need constant expansion order on different patches. It defines both low order and higher order basis on the same patch, and actually the basis of order is a subset of basis of order .

With the HOHV basis function, a well-conditioned matrix can be obtained for conventional structures [9]. However, when modeling multiscale structures, the high contrast (electrically large and electrically small) sized meshes exist together, so the proper equilibration technique of MoM matrix equation is needed to decrease the condition number [11]. At the same time, ill-shaped meshes are inevitable and the orthogonality of HOHV basis function is affected. This makes the matrix equation ill-conditioned and causes convergence difficulty for Krylov subspace methods [12]. Therefore, HOHV basis function cannot be directly applied to model multiscale structures.

Recently, a novel nonconformal, nonoverlapping domain decomposition method (DDM) based on IEM has been developed by Peng et al. successfully [13]. This IE-DDM is based on the philosophy of “divide and conquer.” It divides original problems into several closed subdomains and enforces transmission conditions (TCs) on touching faces to maintain the continuity of currents across interfaces. Because of the nonconformal property of IE-DDM, each subdomain can be meshed independently, and high mesh quality can be easily realized in subdomains. In addition, IE-DDM also provides an effective preconditioner and makes the system matrix for multiscale problems well-posed.

In this paper, mixed basis functions are incorporated into the framework of IE-DDM (named as IE-DDM/MBFs) to further enhance the ability of IE-DDM for multiscale objects. Here, higher order hierarchical vector basis functions defined over quadrilaterals are used to expand surface current on electrically large smooth surfaces, and the traditional RWG basis functions defined over triangles are used to expand surface current on electrically small structures. Compared to the traditional higher order method, the present method realizes flexible mesh generation and fast convergence for multiscale EM problems.

This paper is organized as follows: the derivation of IE-DDM/MBFs is presented in Section 2. Numerical examples of EM scattering from PEC structures are presented in Section 3. Finally, conclusions are drawn in Section 4.

#### 2. Derivation of IE-DDM/MBFs

Consider EM scattering of a PEC object . is the exterior surface of . According to the surface equivalence principle, the surface equivalent current is represented as , where . Based on the relationship between current source and scattering field, the scattered field can be represented as follows:In (1), is the scalar Green function in free space, where is the distance between the source point and field point. is the intrinsic impedance in free space. is the wave number in free space. By using electric field boundary condition or magnetic field boundary condition, we can derive the electric field integral equation (EFIE) or the magnetic field integral equation (MFIE). Based on the linear combination of EFIE and MFIE, the combined field integral equation (CFIE) can be expressed asHere, is the principle value of , is the combination factor, and . As shown in Figure 1, IE-DDM decomposes the original domain into two closed nonoverlapping subdomains and , with . In Figure 1, and are the exterior surfaces of and , respectively. Furthermore, surfaces and can be decomposed as and , where , , and and are the touching faces between and . As shown in Figure 2, and are the surface current on touching faces and , respectively.