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KEY WORDS

Boundary Element Method , Fast Multipole Method, Reciprocity Based FEM.

ABSTRACT

The boundary element method (BEM) is a numerical method for solving boundaryvalue or initial-value problems formulated by use of boundary integral equations. In  the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

CITATION INFORMATION

Acta Mechanica Slovaca. Volume 15, Issue 3, Pages 30 – 37, ISSN 1335-2393

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  Application of Fast Multipole Boundary Method in Elastostatic Problems

REFERENCES

[1] Trebuňa, F., Šimčák, F., Bocko, J., Pástor, M., Stress analysis of casting padestal supporting Structure, Acta Mechanica Slovaca, vol. 13, No.1, p. 36-43.

[2] Horyl, P., Kozoubek, T., Markopoulos, A., Brzobohatý, T., Demonstration of the Matsol library development for the efficient solution of contact problems and its comparison with ANSYS, Acta Mechanica Slovaca, Vol. 14, No.1, 2010, p. 36-42.
[3] Sládek J., Sládek V., Jakubovičová L., Application of Boundary Element Methods in Fracture Mechanics, University of Žilina, Faculty of Mechanical Engineering, Žilina, 2002.
[4] Greengard L. F., Kropinski M.C., Mayo A. , Integral equation methods for Stokes flow and isotropic elasticity in the plane, Journal of Computational Physics 1996, vol. 125,

p. 403–414.
[5] Peirce A., P., Napier, J.A.L., A spectral multipole method for efficient solution of large-scale boundary element models in elastostatics, Int J Numer Meth Engng, 38,pp.4009-4034.

[6] Kompiš, V., Štiavnický, M., Kompiš, M., Žmindák, M.: Trefftz interpolation based multi-domain boundary point method, Engineering Analysis with Boundary Elements 29 (2005), p.391-396.
[7] Kompiš V., Štiavnický M., Žmindák M.:, Murčínková Z., Trefftz radial basis functions (TRBF), Computer Assisted Mechanics and Engineering Sciences, vol. 15, 2008, p. 239-249.

[8] Sládek, J., Sládek, V., Staňák, P., Analysis of thermo-piezoelectricity problems by meshless methods, Acta Mechanica Slovaca, Vol. 14, No. 4, 2010, p.16-27.
[9] Chen, Y., Lee, J.D., Eskandrian, A., Meshless Methods in Solid Mechanics, SpringerScience+Business media, Inc., 2006.
[10] Liu Y., A new fast multipole boundary element method for solving large-scale two dimensional elastostatic problems. Int. J. Num. Meth. Engng., vol.65, p. 863-881.
[11] Liu Y., Fast Multipole Boundary Element Method, Theory and Applications in Engineering, Cambrige University Press, 2009.
[12] Timoshenko S.P, Goodier J.N. Theory of Elasticity (3rd edn), McGraw-Hill: New York, 1987.

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ams 2 2016

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