Symmetric Galerkin boundary integral formulation for interface and multi-zone problems

Gray, L. J.; Paulino, Gláucio H.

doi:10.1002/(sici)1097-0207(19970830)40:16<3085::aid-nme194>3.0.co;2-u

Cited by 87 publications

(38 citation statements)

References 20 publications

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“…The symmetric formulation has existed in the BEM community for some time [7,[23][24][25], and the single-layer potential formulation has been used for solving elasticity problems [6,7]. However, to the best of our knowledge, neither the symmetric approach nor the single-layer formulation have so far been applied to the EEG problem.…”

Section: G Existing Workmentioning

confidence: 99%

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A common formalism for the Integral formulations of the forward EEG problem

Kybic

Clerc

Abboud

et al. 2005

IEEE Trans. Med. Imaging

393

334

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Abstract-The forward electro-encephalography (EEG) problem involves finding a potential V from the Poisson equation ∇ · (σ∇V ) = f , in which f represents electrical sources in the brain, and σ the conductivity of the head tissues. In the piecewise constant conductivity head model, this can be accomplished by the Boundary Element Method (BEM) using a suitable integral formulation. Most previous work uses the same integral formulation, corresponding to a double-layer potential. In this article we present a conceptual framework based on a well-known theorem (Theorem 1) that characterizes harmonic functions defined on the complement of a bounded smooth surface. This theorem says that such harmonic functions are completely defined by their values and those of their normal derivatives on this surface. It allows us to cast the previous BEM approaches in a unified setting and to develop two new approaches corresponding to different ways of exploiting the same theorem. Specifically, we first present a dual approach which involves a single-layer potential. Then, we propose a symmetric formulation, which combines single and double-layer potentials, and which is new to the field of EEG, although it has been applied to other problems in electromagnetism. The three methods have been evaluated numerically using a spherical geometry with known analytical solution, and the symmetric formulation achieves a significantly higher accuracy than the alternative methods. Additionally, we present results with realistically shaped meshes. Beside providing a better understanding of the foundations of BEM methods, our approach appears to lead also to more efficient algorithms.

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Section: G Existing Workmentioning

confidence: 99%

“…It is based on the classical theory of Newtonian potentials as described in chapter 2 of [26], the work of Nédélec [7] and is also closely related to algorithms in [23,24]. However, as far as we know, it has so far never been described for the EEG problem.…”

Section: G Symmetric Approachmentioning

confidence: 99%

A common formalism for the Integral formulations of the forward EEG problem

Kybic

Clerc

Abboud

et al. 2005

IEEE Trans. Med. Imaging

393

334

View full text Add to dashboard Cite

show abstract

“…The multipole expansion of T i k can then found by means of a straightforward differentiation of the expansion of U i k , while B ikqs can be rewritten in a form similar to (22). Now, let Γ(x 0 ) ⊂ ∂Ω andΓ(x 0 ) ⊂ ∂Ω denote two subsets of ∂Ω such that (21) holds for any x ∈ Γ(x 0 ) andx ∈Γ(x 0 ) (such subsets are said to be well-separated).…”

Section: Fast Multipole Methods For Elastostatic Sgbemmentioning

confidence: 99%

“…This in turn requires C 1,α regularity of the density (usually the displacement discontinuity) at the collocation points and specially designed singular integration procedures, making the implementation of traction CBIEs quite involved. In contrast, the symmetric Galerkin BIE (SGBIE) approach to crack problems only requires H 1/2 densities for the continuous problem, making C 0 boundary element interpolations suitable (see [12] for an early implementation of the latter approach for plane cracks, and also [21,22,19,8,28], among many references on this treatmnent). Moreover, SGBEM is based on a variational (weak) version of the integral equations, thus with n denoting the outward unit normal to Ω.…”

Section: Introductionmentioning

confidence: 99%

Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems

Pham¹,

Mouhoubi²,

Bonnet³

et al. 2012

Engineering Analysis with Boundary Elements

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“…69 for two-dimensional static problems, and discussed on a variational basis in Ref. 100 for elastodynamics.…”

Section: Coupling Of Bem With Femmentioning

confidence: 99%

Symmetric Galerkin Boundary Element Method

Bonnet

Maier²,

Polizzotto³

2008

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This review concerns a methodology for solving numerically, to engineering purposes, boundary and initial-boundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of single-layer and double-layer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form; the discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that the variables in the approximate formulation are generalized variables in Prager's sense. As main consequences of the above provisions, symmetry is exhibited by matrices with a key role in the algebraized versions, some quadratic forms have a clear energy meaning, variational properties characterize the solutions and other results, invalid in traditional boundary element methods, enrich the theory underlying the computational applications. The present survey outlines recent theoretical and computational developments of the title methodology with particular reference to linear elasticity, elastoplasticity, fracture mechanics, time-dependent problems, variational approaches, singular integrals, approximation issues, sensitivity analysis, coupling of boundary and finite elements, computer implementations. Areas and aspects which at present require further research are identified and comparative assessments are attempted with respect to traditional boundary integral-element methods.

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Symmetric Galerkin boundary integral formulation for interface and multi-zone problems

Cited by 87 publications

References 20 publications

A common formalism for the Integral formulations of the forward EEG problem

A common formalism for the Integral formulations of the forward EEG problem

Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems

Symmetric Galerkin Boundary Element Method

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