Reply to Comments by R. Piltner on “Development of two-dimensional elements with a central circular hole” [Comput. Methods Appl. Mech. Engrg., 188 (2000) 431–440

Abstract

In this paper, four- and eight-node quadrilateral finite element models which can readily be incorporated into the standard finite element program framework are devised for plane Helmholtz problems. In these models, frame (boundary) and domain approximations are defined. The former is obtained by nodal interpolation and the latter is truncated from Trefftz solution sets. The equality of the two approximations are enforced along the element boundary. Both the Bessel and plane wave solutions are employed to construct the domain approximation. For full rankness, a minimal of four and eight domain modes are required for the four- and eight-node elements, respectively. By using local coordinates and directions, rank sufficient and invariant elements with minimal and close to minimal numbers of domain approximation modes are devised. In most tests, the proposed hybrid-Trefftz elements with the same number of nodes yield close solutions. In absolute majority of the tests, the proposed elements are considerably more accurate than their single-field counterparts.