Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multi-domain FEM-BEM coupling for acoustic scattering

Marcella Bonazzoli, Xavier Claeys

TL;DR

This work develops a comprehensive multi-domain FEM-BEM framework for time-harmonic acoustic scattering where the object comprises piecewise-homogeneous parts and an arbitrarily heterogeneous region ${\Omega_\Sigma}$. Building on Costabel’s coupling and the multi-trace formalism, it extends to configurations with cross-points and varying wavenumber by introducing single-trace and multi-trace formulations, along with regularized combined-field variants to guarantee stability (via a generalized Garding inequality) and injectivity. The paper proves that the resulting variational problems are well-posed and, where necessary, constructs immune-to-spurious-resonance versions (CFIE-type) using a regularizing operator $\mathsf{M}$ and trace-transformations. It also analyzes both two-domain and multi-domain geometries, derives equivalence with the underlying transmission problem, and provides explicit frameworks (STF/STF-CFIE and MTF/MTF-CFIE) that are suitable for robust discretization and preconditioning in the presence of cross-points. Overall, the contributions enable stable, accurate simulation of heterogeneous acoustic scattering with complex domain decompositions and arbitrary interfacial topology.

Abstract

We model time-harmonic acoustic scattering by an object composed of piece-wise homogeneous parts and an arbitrarily heterogeneous part. We propose and analyze new formulations that couple, adopting a Costabel-type approach, boundary integral equations for the homogeneous subdomains with volume variational formulations for the heterogeneous subdomain. This is an extension of the Costabel FEM-BEM coupling to a multi-domain configuration, with cross-points allowed, i.e. points where three or more subdomains are adjacent. While generally just the exterior unbounded subdomain is treated with the BEM, here we wish to exploit the advantages of BEM whenever it is applicable, that is, for all the homogeneous parts of the scattering object. Our formulation is based on the multi-trace formalism, which initially was introduced for acoustic scattering by piece-wise homogeneous objects. Instead, here we allow the wavenumber to vary arbitrarily in a part of the domain. We prove that the bilinear form associated with the proposed formulation satisfies a Gårding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances.

Multi-domain FEM-BEM coupling for acoustic scattering

TL;DR

This work develops a comprehensive multi-domain FEM-BEM framework for time-harmonic acoustic scattering where the object comprises piecewise-homogeneous parts and an arbitrarily heterogeneous region . Building on Costabel’s coupling and the multi-trace formalism, it extends to configurations with cross-points and varying wavenumber by introducing single-trace and multi-trace formulations, along with regularized combined-field variants to guarantee stability (via a generalized Garding inequality) and injectivity. The paper proves that the resulting variational problems are well-posed and, where necessary, constructs immune-to-spurious-resonance versions (CFIE-type) using a regularizing operator and trace-transformations. It also analyzes both two-domain and multi-domain geometries, derives equivalence with the underlying transmission problem, and provides explicit frameworks (STF/STF-CFIE and MTF/MTF-CFIE) that are suitable for robust discretization and preconditioning in the presence of cross-points. Overall, the contributions enable stable, accurate simulation of heterogeneous acoustic scattering with complex domain decompositions and arbitrary interfacial topology.

Abstract

We model time-harmonic acoustic scattering by an object composed of piece-wise homogeneous parts and an arbitrarily heterogeneous part. We propose and analyze new formulations that couple, adopting a Costabel-type approach, boundary integral equations for the homogeneous subdomains with volume variational formulations for the heterogeneous subdomain. This is an extension of the Costabel FEM-BEM coupling to a multi-domain configuration, with cross-points allowed, i.e. points where three or more subdomains are adjacent. While generally just the exterior unbounded subdomain is treated with the BEM, here we wish to exploit the advantages of BEM whenever it is applicable, that is, for all the homogeneous parts of the scattering object. Our formulation is based on the multi-trace formalism, which initially was introduced for acoustic scattering by piece-wise homogeneous objects. Instead, here we allow the wavenumber to vary arbitrarily in a part of the domain. We prove that the bilinear form associated with the proposed formulation satisfies a Gårding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances.
Paper Structure (16 sections, 22 theorems, 168 equations, 4 figures)

This paper contains 16 sections, 22 theorems, 168 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The transmission problem
  3. Trace spaces and operators
  4. Review of potential and boundary integral operators
  5. Trace spaces for multi-domain scattering
  6. Review of the classical Costabel coupling
  7. Single-trace FEM-BEM formulation
  8. Spurious resonances
  9. Single-trace combined field FEM-BEM formulation
  10. Regularizing operator and trace transformation operator
  11. The formulation
  12. Multi-trace FEM-BEM formulation
  13. Derivation of the formulation
  14. Properties of the multi-trace FEM-BEM formulation
  15. Multi-trace combined field FEM-BEM formulation
  16. ...and 1 more sections

Key Result

Proposition 4.1

Let $U \in \mathrm{H}^1_\textup{loc}(\overline\Omega)$ satisfy $-\Delta U - \kappa^2U = 0$ in $\Omega$. If $\Omega$ is unbounded, assume in addition that $U$ is $\kappa$-outgoing radiating. Then we have the representation formula Similarly, let $V \in \mathrm{H}^1_\textup{loc}(\mathbb{R}^d \backslash \Omega)$ satisfy $-\Delta V - \kappa^2V = 0$ in $\mathbb{R}^d \backslash \overline \Omega$, as we

Figures (4)

  • Figure 1: Example of geometric setting: composite medium, with $\Omega_\Sigma$ arbitrarily heterogeneous. Cross-points (red dots) are allowed.
  • Figure 2: Geometric setting for the classical Costabel coupling.
  • Figure 3: Situation without spurious resonances.
  • Figure 4: Illustration of the gap idea (the gap is highlighted in orange)

Theorems & Definitions (27)

  • Proposition 4.1: Representation formulas
  • Proposition 4.2: Definition and characterization of Cauchy data
  • Proposition 4.3: Generalized Gårding inequality
  • Proposition 4.4
  • Proposition 5.1: Modified polarity identity
  • Lemma 5.2: Variational characterization of $\mathbb{X}(\Gamma)$
  • Lemma 5.3: Characterizations of transmission conditions
  • Remark 5.4
  • Example 6.1: Spurious resonances
  • Proposition 6.2: Equivalence
  • ...and 17 more