Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Space-time stochastic Galerkin boundary elements for acoustic scattering problems

Heiko Gimperlein, Fabian Meyer, Ceyhun Özdemir

TL;DR

This work develops a stochastic space-time boundary-element method for the exterior acoustic wave equation with uncertain sources and boundary data, combining a time-domain boundary integral formulation with a polynomial-chaos (SG) expansion in the stochastic variables. After deriving SD, SN, and acoustic boundary formulations on the scattering boundary, the authors prove stability and convergence of the SG discretizations in space and time and implement a marching-on-in-time scheme that preserves sparsity in the stochastic dimension. The approach is validated in three-space dimensions via model problems and a traffic-noise application, showing spectral convergence in stochastic DOFs and robust behavior with respect to space-time discretization. The results demonstrate the method’s practical potential for uncertainty quantification in engineering acoustics, enabling efficient analysis of mean and variance of acoustic fields in complex, real-world scenarios.

Abstract

Acoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space-time stochastic Galerkin boundary element method which is applied to sound-hard, sound-soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.

Space-time stochastic Galerkin boundary elements for acoustic scattering problems

TL;DR

This work develops a stochastic space-time boundary-element method for the exterior acoustic wave equation with uncertain sources and boundary data, combining a time-domain boundary integral formulation with a polynomial-chaos (SG) expansion in the stochastic variables. After deriving SD, SN, and acoustic boundary formulations on the scattering boundary, the authors prove stability and convergence of the SG discretizations in space and time and implement a marching-on-in-time scheme that preserves sparsity in the stochastic dimension. The approach is validated in three-space dimensions via model problems and a traffic-noise application, showing spectral convergence in stochastic DOFs and robust behavior with respect to space-time discretization. The results demonstrate the method’s practical potential for uncertainty quantification in engineering acoustics, enabling efficient analysis of mean and variance of acoustic fields in complex, real-world scenarios.

Abstract

Acoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space-time stochastic Galerkin boundary element method which is applied to sound-hard, sound-soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.
Paper Structure (18 sections, 6 theorems, 85 equations, 13 figures, 2 tables)

This paper contains 18 sections, 6 theorems, 85 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Stochastic boundary problems and integral formulations
  3. Dirichlet and Neumann boundary conditions
  4. Acoustic boundary conditions
  5. Stochastic space time boundary element method
  6. Stochastic Galerkin discretization
  7. Space-time discretization of the Stochastic Galerkin system of the stochastic Dirichlet problem
  8. Space-time discretization of the Stochastic Galerkin system of the stochastic acoustic problem
  9. Numerical experiments
  10. Numerical example for the single layer potential with stochastic Dirichlet data
  11. Numerical examples for an acoustic problem
  12. Application to traffic noise
  13. Conclusions
  14. Set-up of space-time anisotropic Sobolev spaces
  15. Algorithmic aspects of the stochastic problem
  16. ...and 3 more sections

Key Result

Theorem 2.1

The single layer and hypersingular operators are continuous for $r \in \mathbb{R}$:

Figures (13)

  • Figure 1: 2nd and 4th spatial refinement level of the sphere Section \ref{['subsec:singlelayerStochastic']}.
  • Figure 2: Plot of a) mean and b) variance of benchmark solution at time $T=2$, refinement level 5 and SG polynomial degree 12. Numerical experiment from Section \ref{['subsec:singlelayerStochastic']}.
  • Figure 3: Plot of relative energy errors for a) a fixed space-time mesh and b) a fixed SG polynomial degree. Numerical experiment from Section \ref{['subsec:singlelayerStochastic']}.
  • Figure 4: 2nd and 4th spatial refinement level of the unit cube Section \ref{['subsec:Acoustic']}.
  • Figure 5: Plot of a) Mean and b) variance of the first component of the solution at time $T=1$, refinement level 5 and SG polynomial degree 4 and $\alpha \sim \mathcal{U}(0.1,1.9)$ on $\Gamma$.
  • ...and 8 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 2 more