An accurate treatment of non-homogeneous boundary conditions for development of the BEM

Abstract

This paper proposes an enhancement of the treatment of non-homogeneous boundary conditions to improve the boundary element method (BEM) formulation. The standard formulation is modified by introducing the boundary conditions in the integral kernels. The boundary conditions are implicitly defined through known parameters depending on the geometry, rather than by prescribing nodal values as is done in the standard formulation. The main advantage of this procedure is that the right-hand side of the system of equations is integrated taking the exact distribution of loads into account. This approach is implemented in the Bézier–Bernstein space to yield a geometry-independent field approximation. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The application of the proposed method covers the resolution of complex boundary value problems as optimization with uncertain data, material modelling with graded impedance, and the definition of general boundary constraints. The performance of the proposed method is shown by solving the Helmholtz equation in two dimensions. The proposed method is numerically compared to the standard BEM formulation in two benchmark problems. Finally, the application of complex impedance boundary conditions is analysed in a numerical example.