211 research outputs found
Accurate computation of Galerkin double surface integrals in the 3-D boundary element method
Many boundary element integral equation kernels are based on the Green's
functions of the Laplace and Helmholtz equations in three dimensions. These
include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's
equations. Integral equation formulations lead to more compact, but dense
linear systems. These dense systems are often solved iteratively via Krylov
subspace methods, which may be accelerated via the fast multipole method. There
are advantages to Galerkin formulations for such integral equations, as they
treat problems associated with kernel singularity, and lead to symmetric and
better conditioned matrices. However, the Galerkin method requires each entry
in the system matrix to be created via the computation of a double surface
integral over one or more pairs of triangles. There are a number of
semi-analytical methods to treat these integrals, which all have some issues,
and are discussed in this paper. We present novel methods to compute all the
integrals that arise in Galerkin formulations involving kernels based on the
Laplace and Helmholtz Green's functions to any specified accuracy. Integrals
involving completely geometrically separated triangles are non-singular and are
computed using a technique based on spherical harmonics and multipole
expansions and translations, which results in the integration of polynomial
functions over the triangles. Integrals involving cases where the triangles
have common vertices, edges, or are coincident are treated via scaling and
symmetry arguments, combined with automatic recursive geometric decomposition
of the integrals. Example results are presented, and the developed software is
available as open source
Efficient Exact Quadrature of Regular Solid Harmonics Times Polynomials Over Simplices in
A generalization of a recently introduced recursive numerical method for the
exact evaluation of integrals of regular solid harmonics and their normal
derivatives over simplex elements in is presented. The original
Quadrature to Expansion (Q2X) method achieves optimal per-element asymptotic
complexity, however, it considered only constant density functions over the
elements. Here, we generalize this method to support arbitrary degree
polynomial density functions, which is achieved in an extended recursive
framework while maintaining the optimality of the complexity. The method is
derived for 1- and 2- simplex elements in and can be used for
the boundary element method and vortex methods coupled with the fast multipole
method
Layer potential quadrature on manifold boundary elements with constant densities for Laplace and Helmholtz kernels in
A method is proposed for evaluation of single and double layer potentials of
the Laplace and Helmholtz equations on piecewise smooth manifold boundary
elements with constant densities. The method is based on a novel two-term
decomposition of the layer potentials, derived by means of differential
geometry. The first term is an integral of a differential 2-form which can be
reduced to contour integrals using Stokes' theorem, while the second term is
related to the element curvature. This decomposition reduces the degree of
singularity and the curvature term can be further regularized by a polar
coordinate transform. The method can handle singular and nearly singular
integrals. Numerical results validating the accuracy of the method are
presented for all combinations of single and double layer potentials, for the
Laplace and Helmholtz kernels, and for singular and nearly singular integrals
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