Metric-based anisotropic mesh adaptation for 3D acoustic boundary element methods

Abstract

This paper details the extension of a metric-based anisotropic mesh adaptation strategy to the boundary element method for problems of 3D acoustic wave propagation. Traditional mesh adaptation strategies for boundary element methods rely on Galerkin discretizations of the boundary integral equations, and the development of appropriate error indicators. They often require the solution of further integral equations. These methods utilise the error indicators to mark elements where the error is above a specified tolerance and then refine these elements. Such an approach cannot lead to anisotropic adaptation regardless of how these elements are refined, since the orientation and shape of current elements cannot be modified. In contrast, the method proposed here is independent of the discretization technique (e.g., collocation, Galerkin). Furthermore, it completely remeshes at each refinement step, altering the shape, size, and orientation of each element according to an optimal metric based on a numerically recovered Hessian of the boundary solution. The resulting adaptation procedure is truly anisotropic and independent of the complexity of the geometry. We show via a variety of numerical examples that it recovers optimal convergence rates for domains with geometric singularities. In particular, a faster convergence rate is recovered for scattering problems with complex geometries.

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

“…The influence of the mesh discretization on the accuracy of a numerical solution still poses a challenge to the BEM community [9,5,20]. Anisotropic features of a solution (e.g., some elastic materials) as well as discontinuities near geometric singularities (e.g., corners and edges) are difficult to capture and ultimately diminish the regularity of the boundary solution and subsequent performance of a BEM.…”

“…We employ here a truly anisotropic metric-based boundary mesh adaptation recently introduced [20]. It does not use a Dörfler marking but rather generates a sequence of non-nested meshes.…”

“…From a practical point of view, we only need to recover a pointwise 3D Hessian at the nodes of the current surface mesh. The method we employ here is based on a local quadratic recovery and uses the boundary solution only, see [20] for more details.…”

“…The proposed strategy constructs adapted meshes that have been shown to recover optimal convergence rates for domains with corners and edges [20]. The main advantage of this strategy is to provide manageable system sizes for a desired accuracy (to achieve the same with a uniform mesh, one may require a discretization exceeding memory capabilities) to the FM-BEM solver.…”

“…In general, most adaptation techniques for BEMs have centered on isotropic techniques [29,5,9,44] that improve performance but do not always recover optimal convergence rates for 3D problems with anisotropic features [4]. In this work, we employ a recently introduced anisotropic mesh adaptation (AMA) strategy using a metric-based error estimator whose effectiveness in recovering convergence order was first demonstrated for volumetric (finite element) methods [51,52] and only recently for first-order BEMs [20]. The methodology is independent of discretization technique as well as the choice of partial differential equation (PDE) and integral equation formulation, iteratively constructing meshes refined in size, shape and orientation according to an "optimal" metric relying on a reconstructed Hessian of the boundary solution.…”