A numerical integration scheme for special quadrilateral finite elements for the Helmholtz equation

Abstract

SUMMARYThis paper is an extension to an earlier paper dealing with the general problem of integrating special wave elements and speciÿcally deals with quadrilateral elements, which have their own unique problems. The theory for integrating quadrilateral wave ÿnite elements for the solution of the Helmholtz equation for very short waves is presented. The results are compared with those obtained using large numbers of Gauss-Legendre integration points.

Partition of unity method for Helmholtz equation: q-convergence for plane-wave and wave-band local bases

“…Here, in the case of plane-waves, we will employ a semi-analytical scheme which follows from [13], [14].…”

Numerical modeling of undersea acoustics using a partition of unity method with plane waves enrichment

“…To this end, we use the semi-analytical quadratures developed by Bettess et al [36,37]. In these rules, the non-oscillatory term of the integrands is approximated by a set of n lp Lagrangian polynomials,…”

“…The key point in the rules proposed by Bettess et al [36,37] is to perform an analytical integration for these integration weights.…”

“…Later, Strouboulis and Hidajat applied the Filon's rule to evaluate the integrals over rectangular elements [35]. In this work, we follow the semi-analytical approach developed by Bettes, Sugimoto et al, which is valid for first order triangular and quadrilateral elements [36,37].…”

A numerical integration scheme for special finite elements for the Helmholtz equation

“…The values for Ei(x) obtained from the two codes were checked against each other and against values evaluated to arbitrary precision by Maple. Because 12 separate special cases arise, details are given elsewhere [30].…”