Analytical integrations in 3D BEM for elliptic problems: Evaluation and implementation

Abstract

SUMMARYThe present publication deals with 3D elliptic boundary value problems (potential, Stokes, elasticity) in the framework of linear, isotropic, and homogeneous materials. Numerical approximation of the unique solution is achieved by 3D boundary element methods (BEMs). Adopting polynomial test and shape functions of arbitrary degree on flat triangular discretizations, the closed form of integrals that are involved in the 3D BEMs is proposed and discussed. Analyses are performed for all operators (single layer, double layer, hypersingular).

Spherical dynamic models of top‐down tectonics

“…In our numerical scheme, the plate surfaces are discretised into triangular elements. On each triangle the integral is calculated using analytical integration (see Salvadori [2010] for a review on all strategies for performing such integrals for any elliptic problem). The is therefore said to be discretised in “Boundary Elements,” also called “Panels,” and the free model parameters (viscosity, density) are assumed constant on each panel in order to perform the analytical integration, and for this reason are sometimes called “Linear Boundary Elements.” It has been shown that the linear system arising from the discretised integrals is well‐conditioned and dense [ Zhu et al , 2006]; however, solving such system inverting its associated dense matrix is computationally inconvenient because the number of operations necessary to calculate the matrix itself scales as N 2 , where N is the number of Panels.…”

“…In our numerical scheme, the plate surfaces are discretised into triangular elements. On each triangle the integral is calculated using analytical integration (see Salvadori [2010] for a review on all strategies for performing such integrals for any elliptic problem). The equation (4) is therefore said to be discretised in "Boundary Elements," also called "Panels," and the free model parameters (viscosity, density) are assumed constant on each panel in order to perform the analytical integration, and for this reason are sometimes called "Linear Boundary Elements."…”

Analytical integrations for the approximation of 3D hyperbolic scalar boundary integral equations

“…In a nutshell, singularities inK rs A ðd,t,T k Þ due to a hyperbolic operator coincides with singularities in matrixK rs ðdÞ of the corresponding steady state operator (see [33]). As a consequence, for any t 40 such that w T k ðtÀr=cÞ ¼ 1, there exists a ball B y centered at y A G and with radius cðsupft A T k gÀtÞ such that 8xA B y ,K rs A ðd,t,T k Þ ¼K rs ðdÞ ð 176Þ…”

Cell‐based volume integration for boundary integral analysis

“…Note that to define F κ , interior values – at the grid points – of the function ϕ κ are required. These can be obtained from the interior point version of Equation , namely The evaluation of the boundary integral is, of course, straightforward, except for when P is close to the boundary; this is handled by employing analytic integration . The volume integral is treated as the preceding, using F 0, κ to split off a simple boundary integral, together with a volume integral that is computed based upon linear interpolation of cell vertex values.…”