V.M. Aushev1, V.O. Militsin2

1, 2 Bauman Moscow State Technical University (Moscow, Russia)
1 aushevvm@gmail.com, 2 vmilitsin@t1.ru

This article presents a sophisticated and highly efficient computational framework for tackling large-scale electromagnetic scattering problems involving complex, multiscale structures composed of both dielectric materials and perfect electric conductors (PECs). The authors base their approach on the method of moments (MoM), discretizing a system of robust Combined Field Integral Equations – specifically the JMCFIE for dielectrics and the CFIE for PECs – using a Galerkin scheme with RWG basis functions. This formulation is strategically selected for its superior properties, including stability across a wide frequency range, immunity to the problem of internal resonances that plague other formulations, and generally well-conditioned system matrices, which are crucial for iterative solvers.

The central innovation of the work is the development of a novel geometrically-adaptive preconditioner designed explicitly to address the significant challenge of multiscale geometries. In such problems, the computational mesh contains elements whose linear sizes differ by several orders of magnitude, severely hindering the convergence of iterative solvers. Unlike standard geometric preconditioners that use a fixed interaction radius, the proposed method defines a variable radius for each mesh element based on its local topological connections and geometric characteristics. This adaptive approach ensures that the preconditioner captures the essential near-field interactions efficiently, even in highly non-uniform meshes. The authors demonstrate that this preconditioner dramatically reduces the number of iterations required for convergence compared to conventional diagonal, block-diagonal, and fixed-radius geometric preconditioners, particularly for challenging geometries like a drone with composite materials.

The numerical solution is accelerated using an adaptive Multilevel Fast Multipole Algorithm (MLFMM), which hybridizes low-frequency multipole expansions with high-frequency diagonal translation operators to reduce the complexity of the dense matrix-vector multiplication from O(N²) to nearly O(N log N). The resulting software complex is rigorously verified against the canonical problem of scattering from a sphere, where it demonstrates expected second-order convergence and excellent agreement with the known analytical Mie series solution.

Extensive performance benchmarks against a leading commercial software package on complex real-world geometries – an aircraft and a quadcopter drone with composite material properties – showcase the practical superiority of the proposed methods. The results indicate a substantial computational speedup (by a factor of 1.5 to over 10) and, in some cases, reduced memory consumption, all while maintaining a comparable and high level of solution accuracy. Finally, the authors conclusively demonstrate the method's capability for cutting-edge simulation by solving massive problems with tens of millions of unknowns at frequencies up to 18 GHz, effectively modeling objects with electrical sizes exceeding 600 wavelengths. This confirms the framework's significant practical value for solving a wide class of modern electromagnetics problems in radar cross-section analysis and aerospace engineering.

Aushev V.M., Militsin V.O. Effective solution of multiscale scattering problems including dielectrics and perfect conductors with method of moments. Achieve­ments of modern radioelectronics. 2026. V. 80. № 4. P. 72–87. DOI: https://doi.org/10.18127/j20700784-202604-08 [in Russian]

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