An Alternative Approach to Boundary Element Methods via the Fourier Transform
Fabian M. E. Duddeck

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 16, No. 1, pp. 1-14, 2006
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Keywords Boundary Element Methods, Fourier Transform, Fundamental Solutions, Kirchhoff Plates.
Abstract In general, the use of Boundary Element Methods (BEM) is restricted to physical cases for which a fundamental solution can be obtained. For simple differential operators (e.g. isotropic elasticity) these special solutions are known in their explicit form. Hence, the realization of the BEM is straight forward. For more com\discretionary {-}{}{}pli\discretionary {-}{}{}cated problems (e.g. anisotropic materials), we can only construct the fundamental solution numerically. This is normally done before the actual problem is tackled; the values of the fundamental solutions are stored in a table and all values needed later are interpolated from these entries. The drawbacks of this approach lie in the high amount of storage capacity, which is required, and in numerically errors due to interpolation especially near the singularity of the fundamental solution. Hence, an alternative BEM, the Fourier BEM, was proposed in \relax \begingroup \catcode `\ 12\relax \catcode `\\12\relax \catcode `\$12\relax \catcode `\&12\relax \catcode `\#12\relax \catcode `\^12\relax \catcode `\_12\relax \catcode `\%12\relax \catcode `\~12\relax \endgroup \relax \cite *{duddeck1} which is based on boundary integral equations (BIE) obtained via Fourier transform. It can be applied to all problems as long as the differential operator is linear and has constant coefficients. \\ The first step to derive the Fourier transformed BIE con\discretionary {-}{}{}sists in a rigorous mathematical formulation via distribution theory, which was developed by \relax \begingroup \catcode `\ 12\relax \catcode `\\12\relax \catcode `\$12\relax \catcode `\&12\relax \catcode `\#12\relax \catcode `\^12\relax \catcode `\_12\relax \catcode `\%12\relax \catcode `\~12\relax \endgroup \relax \cite *{schwartz} at the end of the 1940's and which is still the mathematical basis for the treatment of partial differential equations, e.g. \relax \begingroup \catcode `\ 12\relax \catcode `\\12\relax \catcode `\$12\relax \catcode `\&12\relax \catcode `\#12\relax \catcode `\^12\relax \catcode `\_12\relax \catcode `\%12\relax \catcode `\~12\relax \endgroup \relax \cite *{hoermander}. In the context of BEM, this theory offers a straightforward approach towards the discussion of singularities normally encountered in the BIE. Distribution theory is able to handle all kind of singularities (jumps, weak, strong and hyper singular values) occurring in the BEM formulations and it is the adequate approach for the discussion of the corresponding integrations. In fact it can be shown by this approach, cf. \relax \begingroup \catcode `\ 12\relax \catcode `\\12\relax \catcode `\$12\relax \catcode `\&12\relax \catcode `\#12\relax \catcode `\^12\relax \catcode `\_12\relax \catcode `\%12\relax \catcode `\~12\relax \endgroup \relax \cite *{duddeck1}, that all strong and hyper singular components are vanishing. In addition, the distribution theory enlarges the applicability of Fourier transform leading to alternative formulations for linear differential equations with constant coefficients. All differentiations are converted to multiplications; the differential operator becomes a simple algebraic expression, which can easily be inverted. This inverse differential operator is the Fourier transform of the fundamental solution. \\ In the approach discussed here, this Fourier fundamental solution and not the fundamental solution itself is taken for the computation of all entries to the BEM-matrices. Based on Parseval's formula, which states the equivalence of energy expressions in the Fourier and the original space, alternative BIE can be derived in the Fourier space leading to the same entries for the matrices. Thus for the Fourier BEM, every term should be established in the Fourier space. Because a Galerkin approach leads to symmetric matrices and does not require a second integration in the Fourier BEM, this approach was preferred to the conventional collocation BEM. The trial and the test functions can be easily transformed to the Fourier space as long as they are defined on straight ele\discretionary {-}{}{}ments. Otherwise a numerical approach can be selected. \\ In this paper, the method is applied to thin plate problems according to Kirchhoff's theory. The differential operator is of fourth order lea\discretionary {-}{}{}ding to highly singular integral equations. Although these singularities are quite complex, it can be shown easily that all strong and hyper singular terms are va\discretionary {-}{}{}nishing in both, the original and the Fourier transformed space. In the small example, all integrals were solved analytically, thus -- in contrast to other publications, e.g. \relax \begingroup \catcode `\ 12\relax \catcode `\\12\relax \catcode `\$12\relax \catcode `\&12\relax \catcode `\#12\relax \catcode `\^12\relax \catcode `\_12\relax \catcode `\%12\relax \catcode `\~12\relax \endgroup \relax \cite *{maucher99} -- no numerical errors, i.e. artificial oscillations, are oc\discretionary {-}{}{}cur\discretionary {-}{}{}ring at the corners of a rectangular plate.
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