On the formulation of a BEM in the Bézier–Bernstein space for the solution of Helmholtz equation

Abstract

• Novel boundary element approach to yield a geometry-independent field approximation. • Geometrically based on both computer aid design (CAD). • Field variables are independently approximated from the geometry. • Generalised Lagrange interpolation functions are used as element shape functions . • Applicability for solving Helmholtz equation in longitudinally invariant problems. This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain.