In this paper, we present a flexible and efficient ternary parallelization approach for the multilevel fast multipole algorithm (MLFMA) for the solution of extremely large 3D scattering problems with over 10 billion unknowns. In the ternary parallelization approach, a binary clustering tree of all the message passing interface (MPI) processes is built firstly. Then, the MLFMA tree is categorized into plane wave partitioning, hierarchical-structure partitioning and box partitioning levels via a top-down approach to match the process clustering tree. Compared with the bottom-up approach, inter-process communications in the far interaction are reduced. An inner-group transition level is designed for switching partitions on the intermediate level between the hierarchical-structure partitioning and box partitioning levels. The ternary strategy can realize as high parallel efficiency as the hierarchical partitioning strategy while maintaining flexibility in selecting the total number of processes. An auxiliary-tree-based parallel mesh refinement technique and a hybrid octree storage strategy are designed to facilitate the realization of fast full-wave simulation on a scale of over 10 billion unknowns with the ternary MLFMA. The accuracy of the solutions is validated by comparing the radar cross-sections (RCSs) of a sphere of diameter 3042 wavelengths with 10,773,415,728 unknowns via MLFMA and the Mie series, which is the largest number of unknowns to be computed via full wave numerical methods to date. Furthermore, the solutions for complicated objects, namely, a ship and an aircraft model, both of which have maximum lengths that exceed 10 thousand wavelengths and over 10 billion unknowns, are presented.

In this article, a massively parallel approach of the multilevel fast multipole algorithm (PMLFMA) on graphics processing unit (GPU) heterogeneous platform, noted as GPU-PMLFMA, is presented for solving extremely large electromagnetic scattering problems involving tens of billions of unknowns, In this approach, the flexible and efficient ternary partitioning scheme is employed at first to partition the MLFMA octree among message-passing interface (MPI) processes. Then, the computationally intensive parts of the PMLFMA on each MPI process, matrix filling, aggregation and disaggregation, and so on are accelerated by using the GPU. Different parallelization strategies in coincidence with the ternary parallel MLFMA approach are designed for GPU to ensure high computational throughput. Special memory usage strategy is designed to improve computational efficiency and benefit data reusing. The CPU/GPU asynchronous computing pattern is designed with the OpenMP and compute unified device architecture (CUDA), respectively, for accelerating the CPU and GPU execution parts and computation time overlapped. GPU architecture-based optimization strategies are implemented to further improve the computational efficiency. Numerical results demonstrate that the proposed GPU-PMLFMA can achieve over three times speedup, compared with the eight-threaded conventional PMLFMA. Solutions of scattering by electrically large and complicated objects with about 24 000 wavelengths and over 41.8 billion unknowns are presented.

A flexible and efficient resistive sheet boundary condition (RSBC) based hybrid finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) is presented for computing electromagnetic scattering from inhomogeneous objects with microwave-absorbing honeycomb structures. In the proposed algorithm, each non-magnetic and high lossy material coated unit cell wall of the honeycomb is first approximated by the multilayered RSBC as a zero-thickness resistive sheet to eliminate the computational burden due to the extremely thin and multilayered characteristics of the coated unit cell wall. Then, the RSBC is incorporated into the FE-part of the FE-BI-MLFMA formulation. To further reduce the burden of meshing complicated objects involving cellular structures after RSBC approximation, the hybrid conformal and nonconformal domain decomposition method (DDM) of the FE-BI-MLFMA, which integrates the nonconformal Schwarz DDM-FE and the simplified discontinuous Galerkin (S-DG), is employed to bring significant flexibility and versatility in geometry modeling and mesh generating. Effective block low-rank multifrontal solver-based domain decomposition FEM-ABC preconditioner is constructed to speed up the solution of the FE-BI equations by using locally approximated integral operators for the BI part. Numerical examples are given to demonstrate the accuracy, capability and performance of the proposed algorithm, including a high-definition complicated fighter model with antenna array, multilayer dielectric radome and microwave-absorbing honeycomb structures.

The mechanism of each term in the discontinuous Galerkin (DG) method is analyzed and studied numerically. A simplified DG self-dual integral equation (SDIE) formulation is proposed for solving electromagnetic scattering from large-scale objects with impedance boundary condition (IBC). Numerical results show that the proposed formulation is more flexible and memory saving than the conventional DG formulations, especially for implementing the multilevel fast multipole algorithm (MLFMA). Moreover, a massively parallel strategy of the MLFMA is employed to further strength its capability for electrically large problems. Numerical experiments demonstrate the accuracy and efficiency of the proposed formulation for analyzing electromagnetic scattering problems of IBC objects with billions of unknowns.

In this communication, we present flexible and efficient solutions of large-scale electromagnetic scattering problems using the parallel discontinuous Galerkin (DG) surface integral equation (SIE) method. A simplified formulation of the conventional DG is used by removing the interior penalty term, thereby avoiding computational burden of the contour integral in the conventional DG. It is demonstrated numerically that the simplification causes convergence deficiency and changes the accuracy convergence rate of the solution for objects with sharp corners. The former is overcome by constructing preconditioners using the near-field matrix of the whole region, and the latter is proved to have a neglectable influence on the accuracy of the final solutions as long as normal mesh density requirement is satisfied. The well-scaling ternary parallelization approach of the multilevel fast multipole algorithm (MLFMA) is incorporated into the DG domain decomposition framework to reduce the complexity of the solution and strength its capability for large-scale problems, with the distribution of the near-field-related calculation adjusted to adapt the nonuniform meshes for a better workload balance. Numerical results are included to validate the accuracy and demonstrate the scalability and versatility of the proposed method. In addition, we demonstrated the effectiveness of our algorithm by solving a high-definition complicated aircraft model with inlets and exhausts involving over one billion unknowns.

This paper presents a new self-dual integral equation (SDIE) for electromagnetic scattering from arbitrarily IBC objects including partly coated objects. The proposed SDIE is constructed by using the combined field integral equation (CFIE) and impedance boundary condition (IBC), shorted for C-SDIE. To overcome the difficulty of discontinuous surface impedance from nonuniform IBC/partly coated objects, the discontinuous Galerkin (DG) method is applied to discretize the C-SDIE. Numerical experiments confirm that the DG-C-SDIE has promising numerical performance in terms of accuracy and efficiency. Furthermore, the domain decomposition preconditioning based on DG is employed to further enhance the proposed DG-C-SDIE for large-scale, multi-scale objects. Numerical results demonstrate the capability of the proposed DG-C-SDIE.