Coupled BEM-FEM for the convected Helmholtz equation with non-uniform flow in a bounded domain

TL;DR

This paper introduces a novel coupled BEM-FEM approach for solving the convected Helmholtz equation with non-uniform flow, utilizing the Prandtl--Glauert transformation to facilitate efficient numerical solutions.

Contribution

It develops two new variational formulations for the convected Helmholtz equation, enabling effective BEM-FEM coupling in complex flow scenarios.

Findings

The first formulation is sensitive to resonant frequencies.

The second formulation avoids resonance issues.

Numerical simulations demonstrate the effectiveness of both methods.

Abstract

We consider the convected Helmholtz equation modeling linear acoustic propagation at a fixed frequency in a subsonic flow around a scattering object. The flow is supposed to be uniform in the exterior domain far from the object, and potential in the interior domain close to the object. Our key idea is the reformulation of the original problem using the Prandtl--Glauert transformation on the whole flow domain, yielding (i) the classical Helmholtz equation in the exterior domain and (ii) an anisotropic diffusive PDE with skew-symmetric first-order perturbation in the interior domain such that its transmission condition at the coupling boundary naturally fits the Neumann condition from the classical Helmholtz equation. Then, efficient off-the-shelf tools can be used to perform the BEM-FEM coupling, leading to two novel variational formulations for the convected Helmholtz equation. The first formulation involves one surface unknown and can be affected by resonant frequencies, while the second formulation avoids resonant frequencies and involves two surface unknowns. Numerical simulations are presented to compare the two formulations.