Isogeometric locally-conformal perfectly matched layer for time-harmonic acoustics

Abstract

This paper presents an isogeometric formulation of the perfectly matched layer (PML) for time-harmonic acoustic simulations. The new formulation is a generalization of the conventional locally-conformal PML, in which the NURBS patch supporting the PML domain is transformed from real space to complex space. This is achieved by complex coordinate stretching, based on a stretching vector field indicating the directions in which incident sound waves are absorbed. To solve the difficulty that stretching vectors cannot be defined for control points not lying on the patch, an interpolation method is proposed in which the control-point stretching vectors are calculated by interpolating those at a set of integration points. An automatically matched layer (AML) is also formulated in the isogeometric context. Due to its form-invariant nature, the Helmholtz equation which governs time-harmonic acoustics is kept unchanged in the complex space, except that the Laplacian operator and the sound pressure should be transformed to their complex forms. The performance of the isogeometric PML formulation is discussed through several numerical examples, spanning one to three dimensions and involving both sound radiation and sound scattering problems. It is found that the proposed method presents superior computational accuracy, high geometric adaptivity, and good robustness against challenging geometric features. The geometry-preserving ability inherent in the isogeometric framework brings extra benefits by eliminating geometric errors that are unavoidable in the conventional PML. Meanwhile, these properties are not sensitive to the location of the sound source or the depth of the PML domain. The implementation of the isogeometric PML only requires the replacement of real control-point coordinates by complex ones, making it possible to incorporate the proposed method to various existing numerical routines.