A Stable Boundary Element Method for Reliable Long-Time Industrial Sound Emission

Jul 16, 2026

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A stable space-time formulation is presented for long-time industrial sound emission based on the acoustic wave equation in R3. The approach uses a well-posed Galerkin formulation in both space and time and relies on a hypersingular boundary integral operator. Numerical experiments validate that the resulting time-stepping scheme is stable and accurate for complex industrial geometries, and it is benchmarked against alternative schemes. Results show strong efficiency and close agreement with physical acoustic measurements.

LanguageEnglish

CategoryResearch & Report

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Pages26

Reading Time1 h 6 min

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Abstract

Introduction

Background and motivation for time-domain sound emission

Problem setup and boundary integral reduction

Numerical experiments and validation approach

Related work in frequency-domain and time-dependent BEM

arXiv :2607 .02308v1 [math .NA] 2 Jul 2026 A Stable Boundary Element Method for Reliable Long-Time Industrial Sound Emission Simon Schneider∗‡ Ceyhun Özdemir† Heiko Gimperlein† Karsten Urban‡ Bernd Graf∗ Abstract In this paper we investigate a stable space-time formulation for long-time industrial sound emission problems. To this end, we use a well-posed Galerkin formulation in space and time of the acoustic wave equation in R3 , involving a hypersingular boundary integral operator. Our numerical experiments confirm that the resulting time stepping scheme is stable and accurate for complex acoustic problems in industrial geometries, in contrast to alternative well-known schemes. The proposed method is shown to be efficient for real-world problems, and we obtain very good agreement with physical acoustic measurements. 1 Introduction Sound emission and propagation are of central interest in areas such as room acoustics, damage detection, and the prevention of noise pollution. The increasing stringency of noise regulations has increased the impact of Noise, Vibration, and Harshness (NVH) analysis, focused on detecting and mitigating excessive sound radiation caused by structural vibrations, [8, 25] . Beyond traditional simulations in the frequency-domain, there is growing interest in time-domain approaches, as they are essential to treat transient situations involving a broad spectrum of frequencies, such as the sequence of acceleration and braking events. To predict the sound sources in an NVH analysis in the time-domain, there are stable, accurate and efficient tools, like multibody simulations. However, the calculation of sound radiation in the time-domain has remained largely unresolved, especially in large or unbounded computational domains where the surface area is small compared to the volume. In the frequency-domain, boundary element methods (BEMs) provide established numerical methods. They require computations only on the vibrating surface, thereby reducing the spatial dimension and ∗ Structural Mechanics and Acoustics, Ulm University of Applied Sciences, Germany †Engineering Mathematics, University of Innsbruck, Austria ‡Institute of Numerical Mathematics, University of Ulm, Germany The authors are grateful to A. Aimi and C. Guardasoni for discussions concerning the stability of second-kind boundary integral formulations. avoiding the need to construct a mesh of the computational domain. In the timedomain, however, standard BEMs used by engineers are often unstable, limiting their adoption by practitioners. This work considers a stable BEM for realworld industrial sound emission problems involving complex geometries, longtime intervals and a broad spectrum of frequencies. The formulation employs a first-kind boundary integral equation involving the hypersingular operator W. The weak coercivity of W assures stability of the associated space-time Galerkin approximations. This article aims to study the performance of this approach in the demanding conditions of real-world engineering problems, including stability, convergence, efficiency and accurate prediction of measurements. To be specific, we consider the sound radiation of a bounded Lipschitz domain Ω ⊂ R3 resulting from vibrations acting on the boundary Γ := ∂Ω . The radiated sound pressure u : R+ × R3 \Ω → R satisfies the acoustic wave equation (in the strong form) 1 ∂2 u ~~ ~~ c2 ∂τ 2 (τ, x) − ∆u(τ, x) = 0, (τ, x) ∈ R+ × R3 \Ω, (1a) u (τ, x) = uτ (τ, x) = 0, (τ, x) ∈ R−0 × R3 \Ω , (1b) un (τ, x) = f(τ, x), (τ, x) ∈ R+ × Γ . (1c) Here, x ∈ R3 \Ω or x ∈ Γ is the spatial variable, τ ∈ R+ denotes the time and c ∈ R+ the speed of sound. The Neumann data f in (1c) couple the structural vibrations and the acoustic sound emission: ∂2 sn f (τ, x) = −ρ ~~ ~~ (τ, x)∂τ 2 , where ρ is the density of air and sn the displacement of Γ in the outer normal direction, whose second time derivative represents the structure-borne noise. Problem (1

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How are the numerical results verified?

The method’s predictions for gearbox housings are compared with measurements from a test bench in an anechoic chamber, showing close agreement.

Jul 16, 2026

Upload by Maeve