301 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
Planewave density interpolation methods for 3D Helmholtz boundary integral equations
This paper introduces planewave density interpolation methods for the
regularization of weakly singular, strongly singular, hypersingular and nearly
singular integral kernels present in 3D Helmholtz surface layer potentials and
associated integral operators. Relying on Green's third identity and pointwise
interpolation of density functions in the form of planewaves, these methods
allow layer potentials and integral operators to be expressed in terms of
integrand functions that remain smooth (at least bounded) regardless the
location of the target point relative to the surface sources. Common
challenging integrals that arise in both Nystr\"om and boundary element
discretization of boundary integral equation, can then be numerically evaluated
by standard quadrature rules that are irrespective of the kernel singularity.
Closed-form and purely numerical planewave density interpolation procedures are
presented in this paper, which are used in conjunction with Chebyshev-based
Nystr\"om and Galerkin boundary element methods. A variety of numerical
examples---including problems of acoustic scattering involving multiple
touching and even intersecting obstacles, demonstrate the capabilities of the
proposed technique
direct evaluation of hypersingular galerkin surface integrals ii
Direct boundary limit algorithms for evaluating hypersingular Galerkin surface integrals have been successful in identifying and removing the divergent terms, leaving finite integrals to be evaluated. This paper is concerned with the numerical computation of these multi-dimensional integrals. The integrands contain a weakly singular logarithmic term that is difficult to evaluate directly using standard numerical techniques. Herein it is shown that analytic integration of these weakly singular terms can be accomlished by suitably re-ordering the parameter integrals. In addition to improved accuracy, this process also reduces the dimension of the numerical quadrature, and hence improves efficiency
Boundary element based formulations for crack shape sensitivity analysis
The present paper addresses several BIE-based or BIE-oriented formulations for sensitivity analysis of integral functionals with respect to the geometrical shape of a crack. Functionals defined in terms of integrals over the external boundary of a cracked body and involving the solution of a frequency-domain boundary-value elastodynamic problem are considered, but the ideas presented in this paper are applicable, with the appropriate modifications, to other kinds of linear field equations as well. Both direct differentiation and adjoint problem techniques are addressed, with recourse to either collocation or symmetric Galerkin BIE formulations. After a review of some basic concepts about shape sensitivity and material differentiation, the derivative integral equations for the elastodynamic crack problem are discussed in connection with both collocation and symmetric Galerkin BIE formulations. Building upon these results, the direct differentiation and the adjoint solution approaches are then developed. In particular, the adjoint solution approach is presented in three different forms compatible with boundary element method (BEM) analysis of crack problems, based on the discretized collocation BEM equations, the symmetric Galerkin BEM equations and the direct and adjoint stress intensity factors, respectively. The paper closes with a few comments
Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D
We present an effective harmonic density interpolation method for the
numerical evaluation of singular and nearly singular Laplace boundary integral
operators and layer potentials in two and three spatial dimensions. The method
relies on the use of Green's third identity and local Taylor-like
interpolations of density functions in terms of harmonic polynomials. The
proposed technique effectively regularizes the singularities present in
boundary integral operators and layer potentials, and recasts the latter in
terms of integrands that are bounded or even more regular, depending on the
order of the density interpolation. The resulting boundary integrals can then
be easily, accurately, and inexpensively evaluated by means of standard
quadrature rules. A variety of numerical examples demonstrate the effectiveness
of the technique when used in conjunction with the classical trapezoidal rule
(to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type
quadrature rule (to integrate over surfaces given as unions of non-overlapping
quadrilateral patches) in three-dimensions
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