Parallel Generalized Finite Element Method for Magnetic Multiparticle Problems

  • Achim Basermann
  • Igor Tsukerman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3402)

Abstract

A parallel version of the Generalized Finite Element Method is applied to multiparticle problems. The main advantage of the method is that only a regular hexahedral grid is needed; the particles do not have to be meshed and are represented by special basis functions approximating the field behavior near the particles. A general-purpose parallel Schur complement solver with incomplete LU preconditioning (A. Basermann) showed excellent performance for the varying problem size, number of processors and number of particles. In fact, the scaling of the computational time with respect to the number of processors was slightly superlinear due to cache effects. Future research plans include parallel implementation of the new Flexible Local Approximation MEthod (FLAME) that incorporates desirable local approximating functions (e.g. dipole harmonics near particles) into the difference scheme.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Babuška, I., Melenk, J.M.: The partition of unity method. Intl. J. for Numer. Methods in Engineering 40(4), 727–758 (1997)MATHCrossRefGoogle Scholar
  2. 2.
    Babuška, I., Caloz, G., Osborn, J.E.: Special finite-element methods for a class of 2nd-order elliptic problems with rough coefficients. SIAM Journal on Numerical Analysis 31(4), 945–981 (1994)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Basermann, A.: QMR and TFQMR methods for sparse non-symmetric problems on massively parallel systems. In: The Mathematics of Numerical Analysis, series: Lectures in Applied Mathematics, vol. 32, pp. 59–76 (1996)Google Scholar
  4. 4.
    Basermann, A.: Parallel Block ILUT/ILDLT Preconditioning for Sparse Eigenproblems and Sparse Linear Systems. Num Linear Algebra with Applications 7, 635–648 (2000)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Basermann, A., Jaekel, U., Hachiya, K.: Preconditioning parallel sparse iterative solvers for circuit simulation. In: Proceedings of The 8th SIAM Conference on Appl. Lin. Alg., Williamsburg, VA, USA (July 2003) http://www.siam.org/meetings/la03/proceedings/basermaa.pdf
  6. 6.
    Belytschko, T., et al.: Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering 139(1-4), 3–47 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Duarte, C.A., Babuška, I., Oden, J.T.: Generalized finite element methods for three-dimensional structural mechanics problems. Computers & Structures 77(2), 215–232 (2000)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Freund, R.W., Nachtigal, N.M.: QMR: A quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik 60, 315–339 (1991)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Grant, J.A., Pickup, B.T., Nicholls, A.: A smooth permittivity function for Poisson–Boltzmann solvation methods. J. Comp. Chem. 22(6), 608–640 (2001) (and references therein)CrossRefGoogle Scholar
  10. 10.
    Griebel, M., Schweitzer, M.A.: A particle-partition of unity method for the solution of elliptic, parabolic and hyperbolic PDE. SIAM J. Sci. Comp. 22(3), 853–890 (2000)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Griebel, M., Schweitzer, M.A.: A particle-partition of unity method-Part II: efficient cover construction and reliable integration. SIAM J. Sci. Comp. 23(5), 1655–1682 (2002)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Griebel, M., Schweitzer, M.A.: A particle-partition of unity method-Part III: a multilevel solver. SIAM J. Sci. Comp. 24(2), 377–409 (2002)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Greengard, L., Rokhlin, V., Cheng, H.: A fast adaptive multipole algorithm in three dimensions. J. of Comp. Phys. 155(2), 468–498 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Melenk, J.M., Babuška, I.: The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139, 289–314 (1996)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Müller, W.: Comparison of different methods of force calculation. IEEE Trans. Magn. 25(2), 1058–1061 (1990)CrossRefGoogle Scholar
  16. 16.
    Plaks, A., Tsukerman, I., Friedman, G., Yellen, B.: Generalized Finite Element Method for magnetized nanoparticles. IEEE Trans. Magn. 39(3), 1436–1439 (2003)CrossRefGoogle Scholar
  17. 17.
    Proekt, L., Tsukerman, I.: Method of overlapping patches for electromagnetic computation. IEEE Trans. Magn. 38(2), 741–744 (2002)CrossRefGoogle Scholar
  18. 18.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)MATHCrossRefGoogle Scholar
  19. 19.
    Saad, Y., Sosonkina, M.: Distributed Schur complement techniques for general sparse linear systems. SISC 21, 1337–1356 (1999)MathSciNetGoogle Scholar
  20. 20.
    Strouboulis, T., Babuška, I., Copps, K.L.: The design and analysis of the Generalized Finite Element Method. Comp. Meth. in Appl. Mech. & Engrng. 181(1-3), 43–69 (2000)MATHCrossRefGoogle Scholar
  21. 21.
    Strouboulis, T., Copps, K., Babuska, I.: The generalized finite element method. Computer Methods in Applied Mech. and Engineering 190(32-33), 4081–4193 (2001)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Tsukerman, I., Proekt, L.: Generalized scalar and vector elements for electromagnetic computation. In: XI Intern Symposium on Theoretical Electr. Eng., Linz, Austria (August 2001)Google Scholar
  23. 23.
    Proekt, L., Yuferev, S., Tsukerman, I., Ida, N.: Method of overlapping patches for electromagnetic computation near imperfectly conducting cusps and edges. IEEE Trans. Magn. 38(2), 649–652 (2002)CrossRefGoogle Scholar
  24. 24.
    Tsukerman, I.A.: Approximation of conservative fields and the element ‘edge shape matrix’. IEEE Trans. Magn. 34, 3248–3251 (1998)CrossRefGoogle Scholar
  25. 25.
    Tsukerman, I.A., Plaks, A.: Comparison of accuracy criteria for approximation of conservative fields on tetrahedra. IEEE Trans. Magn. 34, 3252–3255 (1998)CrossRefGoogle Scholar
  26. 26.
    Tsukerman, I.A.: Accurate computation of ‘ripple solutions’ on moving finite element meshes. IEEE Trans. Magn. 31(3), 1472–1475 (1995)CrossRefGoogle Scholar
  27. 27.
    Tsukerman, I.: Flexible local approximation method for electro- and magnetostatics. IEEE Trans. Magn. 40(2), 941–944 (2004)CrossRefGoogle Scholar
  28. 28.
    Van der Vorst, H.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 13, 631–644 (1992)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Washio, T.: NEC Europe, private communicationGoogle Scholar
  30. 30.
    Yellen, B.B., Friedman, G.: Programmable assembly of heterogeneous colloidal particles using magnetic micro-well templates. Langmuir 20, 2553–2559 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Achim Basermann
    • 1
  • Igor Tsukerman
    • 2
  1. 1.C&C Research LaboratoriesNEC Europe Ltd.Sankt AugustinGermany
  2. 2.Department of Electrical & Computer EngineeringThe University of AkronUSA

Personalised recommendations