Abstact
Boundary element methods provide a powerful tool for solving boundary value problems of linear elastostatics, especially in complicated three–dimensional structures. In contrast to the standard Galerkin approach leading to dense stiffness matrices, in fast boundary element methods such as the fast multipole method the application of matrix–vector products can be realized with almost linear complexity. Since all boundary integral operators of linear elastostatics can be reduced to those of the Laplacian, the discretization of the corresponding single and double layer potentials of the Laplace operator has to be employed only. This technique results in a fast multipole method which is an efficient tool for the simulation of elastic stress fields in engineering and industrial applications.
Similar content being viewed by others
References
Bebendorf, M., Rjasanow, S.: Adaptive low–rank approximation of collocation matrices. Computing 70, 1–24 (2003)
Costabel, M.: Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)
Costabel, M., Stephan, E.P. Integral equations for transmission problems in linear elasticity. J. Integral Equations 2, 211–223 (1990)
Fu, Y., Klimkowski, K.J. Rodin, G.J. Berger, E., Browne, J.C., Singer, J.K., van de Geijn, R.A., Vemaganti, K.S.: A fast solution method for three-dimensional many-particle problems of linear elasticity. Int. J. Numer. Methods Engrg. 42, 1215-1229 (1998)
Gatica, G.N., Hsiao, G.C.: Boundary-field equation methods for a class of nonlinear problems. Pitman Research Notes in Mathematics Series. 331. Harlow: Longman House. New York, NY: Wiley (1995)
Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348 (1987)
Greengard, L.: The Rapid Evaluation of Potential Fields in Particle Simulation. The MIT Press (1987)
Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54, 463–491 (1989)
Han, H.: The boundary integro–differential equations of three–dimensional Neumann problem in linear elasticity. Numer. Math. 68, 269–281 (1994)
Hayami, K., Sauter, S.: A panel clustering method for 3-D elastostatics using spherical harmonics. In: Bertram, B. et al. (ed.): Integral methods in science and engineering, Proceedings of the 5th international conference, IMSE'98, Chapman Hall/CRC Res. Notes Math. 418, 179-184 (2000)
Hsiao, G.C. Schnack, E., Wendland, W.L.: A hybrid coupled finite-boundary element method in elasticity. Comput. Methods Appl. Mech. Eng. 173, 287–316 (1999)
Hsiao, G.C., Wendland, W.L.: Boundary element methods: foundation and error analysis. In: Stein, E., de Borst, R., Huges, T.J.R. (eds.). Encyclopedia of Computational Mechanics, Volume I: Fundamentals (John Wiley and Sons 2004), 339–373
Kupradze, V.D.: Three–dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity North Holland Publishing Company, Amsterdam (1979)
McLean, W., Steinbach, O.: Boundary element preconditioners for a hypersingular boundary integral equation on an interval. Adv. Comput. Math. 11, 271–286 (1999)
Nabors, K., Korsmeyer, F.T. Leighton, F.T., White, J.: Preconditioned, adaptive, multipole-accelerated iterative methods for three-dimensional first-kind integral equations of potential theory. SIAM J. Sci. Comput. 15, 713–735 (1994)
Nédélec, J.: Integral equations with non integrable kernels. Int. Eq. Operat. Th. 5, 562–572 (1982)
Newman, J.N.: Distributions of sources and normal dipoles over a quadrilateral panel. J. Engrg. Math. 20, 113–126 (1986)
Of, G.: A fast multipole boundary element method for the symmetric boundary integral formulation in linear elastostatics. In: Bathe, K.J. (ed.). Computational Fluid and Solid Mechanics (Elsevier 2003) 540–543 (2003)
Of, G., Steinbach, O.: A fast multipole boundary element method for a modified hypersingular boundary element method. In: Wendland, W.L., Efendiev, M. (eds.). Analysis and Simulation of Multifield Problems (Springer, Heidelberg 2003) 163–169.
Of, G., Steinbach, O., Wendland, W.L.: The fast multipole method for the symmetric boundary integral formulation. IMA J. Numer. Anal., published online
Perez–Jorda, J.M. Yang, W.: A concise redefinition of the solid spherical harmonics and its use in the fast multipole methods. J. Chem. Phys. 104, 8003–8006 (1996)
Popov, V., Power, H.: An O(N) Taylor series multipole boundary element method for three-dimensional elasticity problems. Eng. Anal. Bound. Elem. 25, 7–18 (2001)
Sirtori, S.: General stress analysis method by means of integral equations and boundary elements. Meccanica 14, 210–218 (1979)
Steinbach, O.: Artificial Multilevel Boundary Element Preconditioners. Proc. Appl. Math. Mech. 3, 539-542 (2003)
Steinbach, O.: A robust boundary element method for nearly incompressible linear elasticity. Numer. Math. 95, 553–562 (2003)
Steinbach, O., Wendland, W.L.: The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9, 191–216 (1998)
White, C.A., Head–Gordon, M.: Derivation and efficient implementation of the fast multipole method. J. Chem. Phys. 101, 6593–6605 (1994)
Yoshida, K., Nishimura, N., Kobayashi, S.: Application of fast multipole Galerkin boundary integral equation method to elastic crack problems in 3D. J. Numer. Methods Eng. 50, 525–547 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been supported by the German Research Foundation DFG under the Grant SFB 404 Multifield Problems in Continuum Mechanics.
Dedicated to George C. Hsiao on the occasion of his 70th birthday.
Rights and permissions
About this article
Cite this article
Of, G., Steinbach, O. & Wendland, W.L. Applications of a fast multipole Galerkin in boundary element method in linear elastostatics. Comput. Visual Sci. 8, 201–209 (2005). https://doi.org/10.1007/s00791-005-0010-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00791-005-0010-9