Boundary integral equation for tangential derivative of flux in Laplace and Helmholtz equations

Abstract

SUMMARYIn this paper, the boundary integral equations (BIEs) for the tangential derivative of flux in Laplace and Helmholtz equations are presented. These integral representations can be used in order to solve several problems in the boundary element method (BEM): cubic solutions including degrees of freedom in flux's tangential derivative value (Hermitian interpolation), nodal sensitivity, analytic gradients in optimization problems, or tangential derivative evaluation in problems that require the computation of such variable (elasticity problems in BEM). The analysis has been developed for 2D formulation. Kernels for tangential derivative of flux lead to high-order singularities (O(1/r 3 )). The limit to the boundary analysis has been carried out. Based on this analysis, regularization formulae have been obtained in order to use such BIE in numerical codes. A set of numerical benchmarks have been carried out in order to validate theoretical and practical aspects, by considering known analytic solutions for the test problems. The results show that the tangential BIEs have been properly developed and implemented.

A Two-Dimensional Acoustic Tangential Derivative Boundary Element Method Including Viscous and Thermal Losses

“…The kernel description can for example be found in the work by Gallego and Martínez-Castro. 21 The viscothermal implementation presented here relies on an adaptive integration scheme for the evaluation of near-singular integrals arising in the case of narrow gaps, 22 but also for the evaluation of the CPV integrals. Initial convergence studies have shown that this approach is feasible for the evaluation of the tangential kernels.…”