A discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid/solid problems

Abstract

An evanescent wave is produced when a propagating incident wave impinges on an interface between two media or materials at a sub-critical angle. The sub-scale nature of such a wave makes it difficult to be captured computationally. In this paper, the Discontinuous Enrichment Method (DEM) developed in [C. Farhat, I. Harari, L.P. Franca, The discontinuous enrichment method, Comput. Methods Appl. Mech. Engrg. 190 (2001) 6455–6479; C. Farhat, I. Harari, U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Comput. Methods Appl. Mech. Engrg. 192 (2003) 1389–1419; C. Farhat, I. Harari, U. Hetmaniuk, The discontinuous enrichment method for multiscale analysis, Comput. Methods Appl. Mech. Engrg. 192 (2003) 3195–3210; C. Farhat, R. Tezaur, P. Weidemann-Goiran, Higher-order extensions of a discontinuous Galerkin method for mid-frequency Helmholtz problems, Int. J. Numer. Methods Engrg. 61 (2004) 1938–1956; C. Farhat, P. Wiedemann-Goiran, R. Tezaur, A discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of short wave exterior Helmholtz problems on unstructured meshes. New computational methods for wave propagation, Wave Motion 39(4) (2004) 307–317; L. Zhang, R. Tezaur, C. Farhat, The discontinuous enrichment method for elastic wave propagation in the medium-frequency regime, Int. J. Numer. Methods Engrg. 66 (2006) 2086–2114; R. Tezaur, C. Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, Int. J. Numer. Methods Engrg. 66 (2006) 796–815] is extended to a class of evanescent wave problems. New DEM elements are constructed by enriching polynomial approximations with free-space evanescent solutions in order to achieve high accuracy for problems with fluid/fluid or fluid/solid interfaces on which evanescent waves may occur. The new DEM elements are applied to the solution of various two-dimensional model problems with evanescent waves. Their performance is observed to be better than that of the basic Helmholtz type DEM elements, and superior to that of the classical higher-order polynomial finite element method.