Improvements for the ultra weak variational formulation

• Published in: International journal for numerical methods in engineering Vol. 94; no. 6; pp. 598 - 624
• Main Authors: Luostari, T., Huttunen, T., Monk, P.
• Format: Journal Article
• Language: English
• Published: Chichester Blackwell Publishing Ltd 11.05.2013; Wiley; Wiley Subscription Services, Inc
• Subjects: Acoustics; Basis functions; Classical and quantum physics: mechanics and fields; Classical mechanics of continuous media: general mathematical aspects; Computing time; element shapes; evanescent wave basis; Evanescent waves; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Helmholtz equation; Helmholtz equations; Linear acoustics; Mathematical models; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Physics; plane wave basis; Plane waves; Reproduction; Sciences and techniques of general use; ultra weak variational formulation; upwind discontinuous Galerkin method; Wave propagation; Waves and wave propagation: general mathematical aspects; Helmholtz equations; element shapes; ultra weak variational formulation; Triangular finite element; Upwind scheme; Finite element method; System with n degrees of freedom; upwind discontinuous Galerkin method; evanescent wave basis; Variational calculus; Modelling; Conditioning; plane wave basis; Discontinuity; Trefftz method; Curved interface; Helmholtz equation; Adjoint method; Experimental study; Bessel functions; Mixed method; Galerkin-Petrov method; Travelling waves; Weak solution; Plane waves; Wave functions; Evanescent wave; Preconditioning; Finite difference method
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.4469

SUMMARYIn this paper, we investigate strategies to improve the accuracy and efficiency of the ultra weak variational formulation (UWVF) of the Helmholtz equation. The UWVF is a Trefftz type, nonpolynomial method using basis functions derived from solutions of the adjoint Helmholtz equation. We shall consider three choices of basis function: propagating plane waves (original choice), Bessel basis functions, and evanescent wave basis functions. Traditionally, two‐dimensional triangular elements are used to discretize the computational domain. However, the element shapes affect the conditioning of the UWVF. Hence, we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom. Our results include the first tests of a plane wave method on meshes of mixed element types. In many modeling problems, evanescent waves occur naturally and are challenging to model. Therefore, we introduce evanescent wave basis functions for the first time in the UWVF to tackle rapidly decaying wave modes. The advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces.Copyright © 2013 John Wiley & Sons, Ltd.