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Description
The boundary element method (BEM) is one of the very interesting domain based approximation techniques that has been successfully applied to solve numerous physical problems. It solves a given partial differential equation (PDE) using only boundary discretization, which significantly alleviates the meshing effort, but also limits the possible PDEs to be solved to only linear ones. Furthermore, the BEM in its classical form cannot accurately capture the solution in the presence of the edges and corners and this problem is addressed in this work. The BEM requires a uniquely defined normal at each point of the computational domain and for this reason, a systematic approach is used to investigate this issue. The boundary integral equations (BIEs) for the potential problem are studied in terms of singularity orders – both singular and hypersingular BIEs are defined and regularization techniques based on [1] are considered. Galerkin and collocation methods for solving the BIEs are introduced and studied, followed by the investigation of special quadrature rules for singular integration based on the Duffy transformation [2]. By combining singular and hypersingular formulations, a special discontinuous discretization technique is introduced for both solution methods for solving Dirichlet, Neumann and mixed problems. Based on these developments, a simple potential problem is solved on cube geometry using both Lagrange and isogeometric discretization. The resulting combination of the proposed numerical methods allows the definition of a special patch test for the BEM, similar to that proposed for the well-known finite element method [3]. While the analytical solution to BIEs in principle does not exist in closed form due to the singularities in its kernels, it is possible to integrate such an integrals with almost machine precision accuracy following the fact that the BEM is as accurate as the singular integration scheme.
REFERENCES
[1] Liu, Y., Rudolphi, T. (1991). ”Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations”, Eng. Anal. Boundary Elem., 8(6), 301–311.
[2] Duffy, M. G. (1982). ”Quadrature over a pyramid or cube of integrands with a singularity at a vertex.”, SIAM J. Numer. Anal., 19(6), 1260–1262.
[3] B. M. Irons and A. Razzaque, ”Experience with the patch test for convergence of finite elements”, in A. K. Aziz (ed.), The Mathematical Foundations of the Finite Element Method with Applications to Partial Diferential Equations, Academic Press, New York, 1972
