The Boundary Element Method for Acoustic Transmission with Nonconforming Grids

TL;DR

The paper tackles the challenge of efficiently solving acoustic transmission problems with boundary element methods at high frequency and high material contrast. It develops a nonconforming BEM framework that uses independent triangular surface meshes at the interface and couples them through mortar projections, enabling reduced degrees of freedom while supporting multiple direct, indirect, and high-contrast formulations. The authors demonstrate substantial computational gains through comprehensive benchmarks, including FEM-BEM coupling and operator preconditioning, and validate the approach on a large-scale acoustic foam model. The work significantly broadens the practical applicability of BEM to heterogeneous and complex geometries, though projection-induced errors at nonconforming interfaces set practical limits for high-accuracy needs and future research directions are outlined for curved surfaces and multi-subdomain junctions.

Abstract

Acoustic wave propagation through a homogeneous material embedded in an unbounded medium can be formulated as a boundary integral equation and accurately solved with the boundary element method. The computational efficiency deteriorates at high frequencies due to the increase in mesh size with a fixed number of elements per wavelength and also at high material contrasts due to the ill-conditioning of the linear system. This study presents the design of boundary element methods feasible for nonconforming surface meshes at the material interface. The nonconforming algorithm allows for independent grid generation, improves flexibility, and reduces the degrees of freedom. It works for different boundary integral formulations for Helmholtz transmission problems, operator preconditioning, and coupling with finite element solvers. The extensive numerical benchmarks at canonical configurations and an acoustic foam model confirm the significant improvements in computational efficiency when employing the nonconforming grid coupling in the boundary element method.

Figures (7)

• Figure 1: An example of two nonconforming triangular grids generated on the same rectangular domain.
• Figure 2: The error of the mortar projection between nonconforming meshes.
• Figure 3: The accuracy of the BEM with mesh refinement. The relative error is the difference with respect to the conforming exterior PMCHWT formulation with 49,098 vertices in the mesh.
• Figure 4: The efficiency statistics of the nonconforming BEM, for all formulations in Table \ref{['table:formulations']}.
• Figure 5: The real part of the acoustic field in the $(x,y)$-plane at $z=0.5$ for an incident plane wave travelling in the positive $x$-direction. The black square depicts the boundary of the unit cube. The FEM-BEM coupled system uses conforming (left) or nonconforming (right) grids.