A Discontinuous Galerkin Augmented Electric Field Integral Equation (DG-AEFIE) for Low-Frequency Electromagnetic Scattering Analysis

Transcription

1 Forum for Electromagnetic Research Methods and Application Technologies (FERMAT) A Discontinuous Galerkin Augmented Electric Field Integral Equation (DG-AEFIE) for Low-Frequency Electromagnetic Scattering Analysis Yibei Hou, Xuezhe Tian and Gaobiao Xiao Dept. of Electronic Engineering, Shanghai Jiao Tong University yibhou@sjtu.edu.cn October , Okinawa, Japan

2 Copyright The use of this work is restricted solely for academic purposes. The author of this work owns the copyright and no reproduction in any form is permitted without written permission by the author. 2

3 Abstract In this presentation, a discontinuous Galerkin augmented electric field integral equation (DG-AEFIE) method based on the domain decomposition is proposed for full-wave solution of multiscale targets. The common surface integral equation based discontinuous Galerkin method allowing both conformal and nonconformal discretizations for multiscale structures suffers from the low-frequency breakdown. By augmenting DG-EFIE with current continuity equation, the proposed scheme can alleviate the low-frequency breakdown. In the augmented system, the EFIE and the current continuity equation are discretizated by using hybrid basis functions including RWG and half RWG basis functions. Since the half RWG basis is not divergenceconforming, line charge degrees of freedom on the adjoining edge are introduced in this work. It is observed that the resulting linear system is well-conditioned at low frequencies, which leads to a rapid convergence over wide frequency band. Keywords: Discontinuous Galerkin, electric field integral equation, low-frequency breakdown, multiscale problems. 3

4 Biography Yibei Hou received the B.S. degree from University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2014, and is currently pursuing the Ph.D. degree in electronic engineering at Shanghai Jiao Tong University, Shanghai, China. His research interests include computational electromagnetics and its application in scattering and radiation problems. Xuezhe Tian received his B.S. degree from Harbin Engineering University in 2011 and the Ph.D. degree in electrical engineering from the Shanghai Jiao Tong University in Since Aug. 2016, he has been working as a Post- Doctoral Researcher in the ElectroScience Laboratory of the Ohio State University. His research interests are in the scientific computing, focusing on the computational electromagnetics and its application in various scattering and radiation problems. 4

5 Biography Gaobiao Xiao received the M.S. degree from Huazhong University of Science and Technology, Wuhan, China, in 1988, the B.S. degree from the National University of Defense Technology, Changsha, China, in 1991, and the Ph.D. degree from Chiba University, Chiba, Japan, in He worked in Hunan University, Changsha, China, from 1991 to Since April 2004, he has been a faculty member in the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China. His research interests are numerical methods in electro- magnetic fields, coupled thermoelectromagnetic analysis, microwave filter designs, fiberoptic filter designs, and inverse scattering problems. 5

6 Outline Background and Formulation Discontinuous Galerkin Augmented Electric Field Integral Equation Numerical Results Conclusions

7 Background-Multiscale Problems Domain Decomposition:Divide and Conquer [1] Z. Peng, K. H. Lim, J.-F. Lee, Nonconformal domain decomposition methods for solving large multiscale electromagnetic scattering problems, Proceedings of the IEEE, vol.101,no.2, pp ,

8 7 Background-EFIE Boundary Condition EFIE Current expansion Galerkin Testing Excitation

9 5 Non-Conformal Mesh RWG basis functions Background-Basis functions Half-RWG

10 Background-(DG-EFIE) Impedance Matrix of EFIE (Half RWG) Unbounded integral when the observation point resides on the source wire Interior penalty (IP) term [1] Z. Peng, K. -H. Lim, and J. -F. Lee, A discontinuous Galerkin surface integral equation method for electromagnetic wave scattering from nonpenetrable targets, IEEE Trans. Antennas and Propagation, vol. 61, no. 7, pp , [2] G. B. Xiao and Y. B. Hou, Intuitive formula of discontinuous Galerkin surface integral equations 3 for electromagnetic scattering problems, IEEE Trans. Antennas Propag. (minor revision) 6

11 Background-(DG-EFIE) Non-conformal discretization Different basis functions can be seamlessly integrated hp-adaption: higher order bases (p refinement), refinement of the mesh (h refinement) Low frequency breakdown 1]W. C. Chew and L. J. Jiang, Overview of Large-Scale Computing: The past, the present, and the future, 3 Proceedings of IEEE, vol. 101, no. 2, pp , Feb

12 Outline Background and Formulation Discontinuous Galerkin Augmented Electric Field Integral Equation (DG-AEFIE) Numerical Results Conclusions

13 DG-AEFIE Formulation Augmented electric filed integral equation (AEFIE) Electric filed integral equation (EFIE) Current continuity equation DG-AEFIE DD of PEC surface Choice of basis functions Half RWG basis functions + RWG basis functions Surface charge basis functions + Line charge basis functions [1]Z. -G. Qian and W. C. Chew, Fast full-wave surface integral equation solver for multiscale structure modeling, IEEE Trans. Antennas Propag., vol. 57, pp , Nov

14 Surface current expansion DG-AEFIE Formulation Charge basis functions Divergence of RWG basis functions are the currents along the boundary contour line. Matrix form of current continuity equation 10

15 DG-AEFIE Formulation Matrix form of DG-EFIE Vector potential matrix Scalar potential matrix IP stabilization function 11

16 DG-AEFIE Formulation Linear system DG-AEFIE with charge neutrality DG-AEFIE with preconditioner Left preconditioned linear system 12

17 Approximation of impedance matrix DG-AEFIE Formulation Preconditioner schur complement 13

18 Outline Background and Formulation Discontinuous Galerkin Augmented Electric Field Integral Equation (DG-AEFIE) Numerical Results Conclusions

19 Sphere-Accuracy Plane wave with amplitude of 1V/m,along the -z axis, Generalized conjugate residual (GCR) and LU direct method, Relative residual error (RSS) of Discretization of Sphere: 1458 triangles, 731 nodes and 2187 edges. Condition number of the impedance matrix and relative error of current as the stabilization parameter varies at 300 MHz. RCS (db) Mie EFIE_LS DG-AEFIE DG-EFIE Observation Angle (Deg.) RCS of the PEC sphere computed by DG- AEFIE with preconditioner and Mie series solution at Hz (electrically size λ) 15

20 Sphere-Conditioning Discretization of Sphere: 1458 triangles, 731 nodes and 2187 edges. Condition number of the impedance matrix of five linear systems. Surface current density calculated by using EFIE-LS 15 and DG-AEFIE (electrically size λ) 16

21 Cone-Nonconformal Mesh 8 Average length λ Average length λ 8 Average length λ 7.39* * Hz 65 iterations 17

22 Chip-Nonconformal Mesh Plane wave excitation 7 7 Main body λ ; pin λ Voltage source excitation excitation 18

23 Outline Background and Formulation Discontinuous Galerkin Augmented Electric Field Integral Equation (DG-AEFIE) Numerical Results Conclusions

24 Conclusion DG-AEFIE can handle complicate structures with nonconformal mesh, DG-AEFIE is well-conditioned from the low frequencies to high frequencies, DG-AEFIE allows local refinement of the mesh.

25 References [1] S. M. Rao, D. R. Wilton, and A. W. Glisson, Electromagnetic scattering by surfaces of arbitrary shape, IEEE Trans. Antennas Propagat., vol. 30, no. 3, pp , May [2] Z.-G. Qian and W. C. Chew, Fast full-wave surface integral equation solver for multiscale structure modeling, IEEE Trans. Antennas Propag., vol. 57, pp , Nov [3] Z.-G. Qian and W. C. Chew, Enhanced A-EFIE with perturbation method, IEEE Trans. Antennas Propag., vol. 58, no. 10, pp , Oct [4] Z. Peng, K. H. Lee, and J. F. Lee, A discontinuous Galerkin surface integral equation method for electromagnetic wave scattering from nonpenetrable targets, IEEE Trans. Antennas Propagat., vol. 61, no. 7, pp , Jul [5] G. B. Xiao and Y. B. Hou, Intuitive formulation of discontinuous Galerkin surface integral equations for electromagnetic scattering problems, IEEE Trans. Antennas Propagat., doi: /tap [6] Z. Peng, K. H. Lim, J.-F. Lee, Nonconformal domain decomposition methods for solving large multiscale electromagnetic scattering problems, Proceedings of the IEEE, vol. 101, no. 2, pp , [7] Z. Peng, K.-H. Lim, and J.-F. Lee, Computations of electromagnetic wave scattering from penetrable composite targets using a surface integral equation method with multiple traces, IEEE Trans. Antennas Propag.,vol. 61, no. 1, pp , Jan [8] M. A. E. Bautista, F. Vipiana, M. A. Francavilla, J. A. T. Vasquez and G. Vecchi, A non-conformal domain decomposition scheme for the analysis of multi-scale structures, IEEE. Trans. Antennas Propagat., vol.63, no.8, pp , Aug