The discontinuous enrichment method for multiscale and higher-frequency wave propagation problems.

Abstract

Wave propagation problems pertain to many technologies including sonar, radar, geophysical exploration, medical imaging, nondestructive testing, and structural design. In the medium frequency regime, the analysis of these problems by the standard finite element method is either computationally unfeasible or simply unreliable, because of the well-known pollution effect. In practice, the higher-order (or p-type) finite element method alleviates this effect, but only to some extent. Alternative approximation methods based on plane waves have recently emerged to address this issue. The discontinuous enrichment method (DEM) is such an alternative. It distinguishes itself from similar approaches by its ability to evaluate the important system matrices analytically, thereby bypassing the typical accuracy and cost issues associated with high-order quadrature rules. DEM also provides a unique multiscale approach to computation by employing fine scales that contain solutions of the underlying homogeneous partial differential equation in a discontinuous framework. The theoretical and computational underpinnings of this method will be overviewed. Then, recent applications to underwater acoustic and elastoacoustic scattering as well as automotive structural vibrations in the midfrequency regime will be discussed. Accuracy of 1 to 2 orders of magnitude and/or solution time improvement over the higher-order Galerkin method will be demonstrated in three dimensions.