Influence of element types on numeric error for acoustic boundary elements

Continuous interpolation of the sound pressure is favored in most applications of boundary element methods for acoustics. Few papers are known to the authors in which discontinuous elements are applied. Mostly they were used because they guarantee C-1 continuity of the geometry at element edges. The effect of superconvergence is known for boundary element collocation on discontinuous elements. This effect is observed if the collocation points are located at the zeroes of orthogonal functions, e.g. at the zeroes of the Legendre polynomials. In this paper, we start with a review of related work. Then, formulation of discontinuous elements and position of nodes on the element are presented and discussed. One parameter controls the location of nodes on the element. The major part of the paper consists of the investigation of the computational example of a long duct. For that, the numeric solution is compared with the analytic solution of the corresponding one-dimensional problem. Error dependence in terms of frequency, element size and location of nodes on discontinuous elements is reported. It will be shown that the zeroes of the Legendre polynomials account for an optimal position of nodes. Similar results are observed for triangular elements. It can be seen that the error in the Euclidean norm changes by one or two orders of magnitude if the location of nodes is shifted over the element. It can be seen that the optimal location varies with the wave-number although remaining in the vicinity of the zeroes of orthogonal functions. The irregular mesh of a sedan cabin compartment accounts for the second example. Optimal choice of node position is confirmed for this example. One of the key results of this paper is that discontinuous boundary elements perform more efficiently than continuous ones, in particular for linear elements. This, however, further implies that nodes are located at the zeroes of orthogonal functions on the element.