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Fast Boundary Element Methods for Composite Materials

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Multi-scale Simulation of Composite Materials

Part of the book series: Mathematical Engineering ((MATHENGIN))

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Abstract

In this chapter, we construct numerical solutions to the problems in the field of solid mechanics by combining the Boundary Element Method (BEM) with interpolation by means of radial basis functions. The main task is to find an approximation to a particular solution of the corresponding elliptic system of partial differential equations. To construct the approximation, the differential operator is applied to a vector of radial basis functions. The resulting vectors are linearly combined to interpolate the function on the right-hand side. The solvability of the interpolation problem is established. Additionally, stability and accuracy estimates for the method are given. A fast numerical method for the solution of the interpolation problem is proposed. These theoretical results are then illustrated on several numerical examples related to the Lamé system.

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Notes

  1. 1.

    One seeks for the solution u in the subspace \(\left\{ v\in ( H^{1}( \varOmega ))^{3}:\text {div }\sigma _d( v,x) \in ( L^{2}( \varOmega ))^{3}\right\} \) of the Sobolev space \(( H^{1}( \varOmega ))^{3}\).

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Authors and Affiliations

  1. University of Saarland, 66123, Saarbrücken, Germany

    Richards Grzhibovskis, Christian Michel & Sergej Rjasanow

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  1. Richards Grzhibovskis
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  2. Christian Michel
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  3. Sergej Rjasanow
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Correspondence to Sergej Rjasanow .

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Editors and Affiliations

  1. Technische Mechanik, Universität des Saarlandes, Saarbrücken, Germany

    Stefan Diebels

  2. Universität des Saarlandes, Saarbrücken, Germany

    Sergej Rjasanow

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Grzhibovskis, R., Michel, C., Rjasanow, S. (2019). Fast Boundary Element Methods for Composite Materials. In: Diebels, S., Rjasanow, S. (eds) Multi-scale Simulation of Composite Materials. Mathematical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57957-2_5

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  • DOI: https://doi.org/10.1007/978-3-662-57957-2_5

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