Fast direct higher-order solution of complex large scale electromagnetic scattering problems via Locally Corrected Nystrom Discretization of CFIE

Solutions of large-scale electromagnetic scattering problems are typically obtained with Rao-Wilton-Glisson (RWG) Method of Moments (MoM) accelerated with Multi-Level-Fast-Multipole-Algorithm (MLFMA). Due to the low-order RWG MoM discretization and iterative nature of MLFMA such solutions present various challenges when realistic large-scale targets are considered. The most salient of these challenges are: 1) inefficient error control of the solution beyond 2-3 digits, 2) poor conditioning of the matrix resulting from multiscale discretization of the model and/or the presence of highly resonant phenomena, and 3) the necessity to repeat the solution process for each excitation. In this work, we present a novel computational framework, which efficiently overcomes these three challenges. Our method is based on Non-Uniform-Rational-B-Splines (NURBS) representation of the scatterer geometries and Higher-Order (HO) Locally Corrected Nystrom (LCN) discretization of the pertinent Combined Field Integral Equation (CFIE). Due to the HO nature of the LCN scheme, scattering solutions of desired precision are obtained with exponentially higher efficiency as compared to the RWG MoM (Jeffrey, et. al., IEEE AP Mag. pp. 294–308, vol. 55, no. 3, June 2013). This allows us to accurately reconstruct the fields in both lit and deeply shaded areas, eliminate lack of causality in time domain responses synthesized from broadband frequency domain solutions, and overcome various other problems stemming from insufficient solution accuracy. The second and third challenges are addressed through fast direct solution of the matrix equation resultant from LCN discretization of the CFIE. Our fast direct solution of the matrix equation is based on the block-LU decomposition aided with Adaptive-Cross-Approximation (ACA) compression of the blocks similar to Shaeffer's algorithm (Shaeffer, IEEE TAP, no. 8, pp. 2306–2313, 2008). To overcome the computational complexity of the fast direct solution, the block LU decomposition is performed on multiple Graphics Processing Units (GPUs) with fast Out-of-Core memory augmentation. A commodity workstation featuring two Xeon hexacore processors, 2 K10 GPUs, 256Gb of RAM, and SSD RAID array, affords us solutions for a broad range of complex scattering problems, previously demonstrated with over 1 million RWG degrees of freedom, which cannot be obtained using iterative algorithms. In the case of problems requiring solutions with multiple excitations such as antenna array modeling and scattering problems with many incidence angles, despite the large computational complexity of the direct block LU factorization, the overall solution time was seen to be substantially lower than in the case of a MLFMA accelerated iterative approach as the latter must be repeated for each excitation.