Solving electromagnetic integral equations: from high-order discretization to H-matrix compression

Mathematics, information, scientific, software Solving electromagnetic integral equations: from high-order discretization to H-matrix compression H/F The simulation of electromagnetic (EM) wave problems plays a key role in many fields, ranging from object characterization and inspection to radar stealth. A common approach consists in transforming the initial 3D volumetric problem into a 3D surface integral equation defined at material interfaces and discretizing it using a Boundary Element Method (BEM). High-order (HO) discretization schemes accelerate approximation convergence and thus provide a significant gain in accuracy for a given number of mesh elements, but complicates the assembly of the linear system. In particular, the calculation of singular and quasi-singular integrals becomes challenging, while the integration of HO methods into fast compression algorithms based on the hierarchical matrix formalism (HMAT) raises questions about the overall efficiency of the BEM. This postdoctoral position aims to address these in order to obtain a solution that is both efficient and robust for the targeted EM applications. The analysis (and, where possible, removal) of these potential operational limitations is the subject of a collaboration between two CEA laboratories that use BEM codes (already HO-compatible) for distinct EM applications: one in HPC for radar cross-section computation in the defence sector (CESTA site), and the other in software dedicated to non-destructive testing (Saclay site, where the postdoctoral researcher will be based). Within this collaborative project, the postdoctoral researcher will contribute to the project as a whole and will focus on the following activities.Development of a dedicated library for computing the elementary matrices associated with near, singular and quasi-singular interactions using advanced integration methods from the literature [2–6], or variants to be defined as part of the postdoctoral project. This demonstrator (Python, C++, Fortran or other language) will be interfaced with CEA codes for a given pair of source/receiver elements (curved quadrilateral/triangle), a given type of HO finite element (e.g., Raviart–Thomas), and a given integral kernel (e.g., EFIE/MFIE) provided as input.Development of a methodology to evaluate and analyze the performance of these integration methods, typically based on use cases of increasing complexity: presence of geometric singularities, degraded mesh quality, nature of the kernel...Study on how does increasing the approximation order modify the compressibility of BEM matrices, with evaluation on standard EM configurations (e.g., Cobra cavity, NASA almond...). The objective will be to identify any structural limitations inherent in standard compression algorithms and variants (to be prototyped from a CEA HMAT demonstrator) that limit the reduction of memory footprint in HO. This work will contribute to the preparation of an internship topic (based at CESTA) for the third semester of the postdoctoral project.Dissemination of this work through a scientific contribution (international conference, publication) is expected.Practical information. The candidate must have obtained a PhD thesis within the last three years and have solid experience in scientific computing and its implementation. Expertise in BEM, high-order approximation and/or integration methods, and/or H-matrix compression will be a key asset. The postdoctoral researcher will be based at the CEA Saclay site (a few assignments at the CESTA site are envisaged during the postdoctoral period – EU nationality required); the contract duration is 18 months.