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

Abstract

SUMMARYThe theory for integrating the element matrices for rectangular, triangular and quadrilateral ÿnite elements for the solution of the Helmholtz equation for very short waves is presented. A numerical integration scheme is developed. Samples of Maple and Fortran code for the evaluation of integration absciss and weights are made available. The results are compared with those obtained using large numbers of Gauss-Legendre integration points for a range of testing wave problems. The results demonstrate that the method gives correct results, which gives conÿdence in the procedures, and show that large savings in computation time can be achieved.

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

“…In our programming implementation, H is symmetrized by averaging itself and its transpose. If the plane wave functions are employed in u d and the element is subparametric, H and G can be analytically evaluated [19,21,38,39]. Variation of (16) with respect to β yields:…”

“…In previous computational models that employ the planewave solution, ϕ l s are often picked at equal interval [17][18][19][20][21]24,29]. However, if one simply picks three fixed equispaced directions such as ϕ 1 = 0, ϕ 2 = 2π/3 and 4π/3, the element would be rotational variant.…”

“…In some cases, the error ratios even drop to one-tenth. Furthermore, when the element geometry is subparametric, the H-and G-matrices of TP (see (22)) can be evaluated by analytical integration as in the plane-wave basis elements [19,21,38,39]. Regarding computational cost, the proposed elements possess 6 nodal dofs.…”

“…In the plane-wave basis method, the plane-wave solutions are employed as the nodal enrichment functions in the context of the partition of unity finite element method [17][18][19][20][21]. The value of the Helmholtz variable at a node is the sum of the plane-wave modes propagating along different directions.…”

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

“…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].…”