A fast direct solution of the electric current volume integral equation (JVIE) with the Sherman-Morrison-Woodbury (SMW) formula-based algorithm is presented to analyze electromagnetic scattering from inhomogeneous dielectric objects. The JVIE is discretized with the nonconformal face-based Schaubert-Wilton-Glisson (SWG) basis functions. Compared with conformal discretization that is advantageous to discrete homogeneous regions, the nonconformal discretization provides a more flexible and efficient scheme to separately handle the inhomogeneous subdomains depending on local parameters. Moreover, to take full use of both discretization methods, the mixture discretization is adopted. With the increase of object size, the impedance matrix equation arising from the JVIE becomes too large to solve and store for direct solution. In this paper, the SMW formula-based algorithm is adopted, leading to remarkable reduction on the computational complexity and memory requirement in contrast with conventional direct solution. This algorithm compresses the impedance matrix into a product of block diagonal submatrices, which can be inversed rapidly in direct way. Numerical results are given to demonstrate the efficiency and accuracy of the proposed method.
2. Peng, Z., K.-H. Lee, and J.-F. Lee, "A discontinuous Galerkin surface integral equation method for electromagnetic wave scattering from nonpenetrable targets," IEEE Trans. Antennas Propag., Vol. 61, No. 7, 3617-3628, 2013.
3. Cai, Q.-M., et al., "Nonconformal discretization of electric current volume integral equation with higher order hierarchical vector basis functions," IEEE Trans. Antennas Propag., Vol. 65, No. 8, 4155-4169, 2017.
4. Nair, N. and B. Shanker, "Generalized method of moments: A novel discretization technique for integral equation," IEEE Trans. Antennas Propag., Vol. 59, No. 6, 2280-2293, 2011.
5. Botha, M. M., "Solving the volume integral equations of electromagnetic scattering," J. Comput. Phys., Vol. 218, No. 1, 141-158, 2006.
6. Markkanen, J., C.-C. Lu, X. Cao, and P. Ylä-oijala, "Analysis of volume integral equation formulations for scattering by high-contrast penetrable objects," IEEE Trans. Antennas Propag., Vol. 60, No. 5, 2367-2374, 2012.
7. Zhang, L.-M. and X.-Q. Sheng, "A discontinuous Galerkin volume integral equation method for scattering from inhomogeneous objects," IEEE Trans. Antennas Propag., Vol. 63, No. 12, 5661-5667, 2015.
8. Heldring, A., J. M. Rius, J. M. Tamayo, J. Parrón, and E. Ubeda, "Fast direct solution of method of moments linear system," IEEE Trans. Antennas Propag., Vol. 55, No. 2, 3220-3228, 2007.
9. Heldring, A., J. M. Rius, J. M. Tamayo, J. Parrón, and E. Ubeda, "Multiscale compressed block decomposition for fast direct solution of method of moments linear system," IEEE Trans. Antennas Propag., Vol. 59, No. 2, 526-536, 2011.
10. Chen, X.-L., C.-Q. Gu, Z. Li, and Z. Niu, "Accelerated direct solution of electromagnetic scattering via characteristic basis function method with Sherman-Morrison-Woodbury formula-based algorithm," IEEE Trans. Antennas Propag., Vol. 64, No. 10, 4482-4486, 2016.
11. Fang, X.-X., Q.-S. Cao, Y. Zhou, and Y. Wang, "Multiscale compressed and spliced Sherman-Morrison-Woodbury algorithm with characteristic basis function method," IEEE Trans. Electromagn. Compat., Vol. 60, No. 3, 716-724, 2018.
12. Schaubert, D. H., D. R. Wilton, and A. W. Glisson, "A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies," IEEE Trans. Antennas Propag., Vol. 32, No. 1, 77-85, 1984.
13. Zhang, L.-M. and X.-Q. Sheng, "Discontinuous Galerkin volume integral equation solution of scattering from inhomogeneous dielectric objects by using the SWG basis function," IEEE Trans. Antennas Propag., Vol. 65, No. 3, 1500-1504, 2017.