Advertisement

Mesh Generation for Symmetrical Geometries

  • Krister Åhlander
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3480)

Abstract

Symmetries are not only fascinating, but they can also be exploited when designing numerical algorithms and data structures for scientific engineering problems in symmetrical domains.

Geometrical symmetries are studied, particularly in the context of so called equivariant operators, which are relevant to a wide range of numerical applications.

In these cases, it is possible to exploit the symmetry via the generalized Fourier transform, thereby considerably reducing not only storage requirement but also computational cost.

The aim of this paper is to introduce group theoretical aspects of symmetry, to point out the benefits of using these concepts when designing numerical algorithms, and to state the implications for the generation of the mesh.

Keywords

Noncommutative Fourier analysis equivariant operators block diagonalization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allgower, E.L., Georg, K., Miranda, R., Tausch, J.: Numerical exploitation of equivariance. Zeitschrift für Angewandte Mathematik und Mechanik 78, 185–201 (1998)MathSciNetGoogle Scholar
  2. 2.
    Bossavit, A.: Symmetry, groups, and boundary value poblems. a progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry. Comput. Methods Appl. Mech. and Engrg. 56, 167–215 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Åhlander, K., Munthe-Kaas, H.: On Applications of the Generalized Fourier Transform in Numerical Linear Algebra. Technical Report 2004-029, Department of Information Technology, Uppsala University (2004)Google Scholar
  4. 4.
    Åhlander, K., Munthe-Kaas, H.: Eigenvalues for Equivariant Matrices. Journal of Computational and Applied Mathematics 14 (2005) (to appear)Google Scholar
  5. 5.
    Serre, J.P.: Linear Representations of Finite Groups. Springer, Heidelberg (1977) ISBN 0387901906.zbMATHGoogle Scholar
  6. 6.
    Maslen, D., Rockmore, D.: Generalized FFTs - a survey of some recent results. Technical Report PCS-TR96-281, Dartmouth College, Department of Computer Science, Hanover, NH (1996)Google Scholar
  7. 7.
    Rockmore, D.: Some applications of generalized FFTs. In: Finkelstein, L., Kantor, W. (eds.) Proceedings of the 1995 DIMACS Workshop on Groups and Computation, pp. 329–369 (1997)Google Scholar
  8. 8.
    Allgower, E.L., Böhmer, K., Georg, K., Miranda, R.: Exploiting symmetry in boundary element methods. SIAM J. Numer. Anal. 29, 534–552 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lomont, J.S.: Applications of Finite Groups. Academic Press, New York (1959)zbMATHGoogle Scholar
  10. 10.
    Bonnet, M.: Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations. Intl. J. Numer. Meth. Engng. 57, 1053–1083 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bossavit, A.: Boundary value problems with symmetry and their approximation by finite elements. SIAM J. Appl. Math. 53, 1352–1380 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Tausch, J.: Equivariant preconditioners for boundary element methods. SIAM Sci. Comp. 17, 90–99 (1996)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Krister Åhlander
    • 1
  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden

Personalised recommendations