A review of finite-element methods for time-harmonic acoustics

Abstract

State-of-the-art finite element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the field are specifically addressed: the effective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, infinite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales, and requiring a large computational effort; efficient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems including sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite element methods are described. Stabilized, multiscale and other wavebased discretization methods developed to reduce this error are reviewed. A review of finite element methods for acoustic inverse problems and shape optimization is also given.Running Title: Finite element methods for acoustics

An efficient GPU-based time domain solver for the acoustic wave equation

“…There is a large body of existing work on solving the wave equation developed over the past few decades. These methods may be roughly classified into finite element method (FEM) [10], boundary element method (BEM) [11], finite-difference time-domain (FDTD) [5] and spectral methods [12].…”

A damping preconditioner for time-harmonic wave equations in fluid and elastic material

“…It has successfully been used for interior scattering problems like acoustic scattering in a car cabin [1] as well as for exterior problems. A review [2] gives an overview of recent research on finite element methods for acoustic problems. Since the paper [3] the research on the construction of absorbing boundary conditions and absorbing layers at the truncation boundary of the exterior domain has been active; see [2] and references therein.…”

“…A review [2] gives an overview of recent research on finite element methods for acoustic problems. Since the paper [3] the research on the construction of absorbing boundary conditions and absorbing layers at the truncation boundary of the exterior domain has been active; see [2] and references therein. The size of the scattering problems is often limited in high frequency problems because the methods become ineffective as the frequency grows.…”

Applicability of an explicit time-domain finite-element method on room acoustics simulation

“…Also, a time discretization error must be considered in a time-domain analysis. These requirements engender the solution of large-scale problems with many degrees of freedom (DOF) and many time steps for a simulation at high frequencies.Many dispersion reduction methods have been developed to increase the computational efficiency [1][2][3][4][5][6]. The authors have also proposed some dispersion-reduced TD-FEM based on an implicit method to solve the large-scale problems efficiently [1,2].…”