Solving Contact Mechanics Problems with PERMON

  • Vaclav Hapla
  • David Horak
  • Lukas Pospisil
  • Martin Cermak
  • Alena Vasatova
  • Radim Sojka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9611)

Abstract

PERMON makes use of theoretical results in quadratic programming algorithms and domain decomposition methods. It is built on top of the PETSc framework for numerical computations. This paper describes its fundamental packages and shows their applications. We focus here on contact problems of mechanics decomposed by means of a FETI-type non-overlapping domain decomposition method. These problems lead to inequality constrained quadratic programming problems that can be solved by our PermonQP package.

References

  1. 1.
  2. 2.
    Amestoy, P., et al.: MUMPS web pages (2015). http://mumps.enseeiht.fr/index.php?page=home
  3. 3.
    Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zhang, H.: PETSc web pages (2015). http://www.mcs.anl.gov/petsc
  4. 4.
    Brzobohatý, T., Dostál, Z., Kozubek, T., Kovář, P., Markopoulos, A.: Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure. Int. J. Numer. Methods Eng. 88(5), 493–509 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dostál, Z., Horák, D., Kučera, R.: Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Commun. Numer. Methods Eng. 22(12), 1155–1162 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dostál, Z., Kozubek, T., Markopoulos, A., Menšík, M.: Cholesky decomposition of a positive semidefinite matrix with known kernel. Appl. Math. Comput. 217(13), 6067–6077 (2011)MathSciNetMATHGoogle Scholar
  7. 7.
    Dostál, Z., Kozubek, T., Vondrák, V., Brzobohatý, T., Markopoulos, A.: Scalable TFETI algorithm for the solution of multibody contact problems of elasticity. Int. J. Numer. Methods Eng. 82(11), 1384–1405 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    Dostál, Z.: Optimal Quadratic Programming Algorithms, with Applications to Variational Inequalities. SOIA, vol. 23. Springer, New York (2009)MATHGoogle Scholar
  9. 9.
    Dostál, Z., Horák, D.: Theoretically supported scalable FETI for numerical solution of variational inequalities. SIAM J. Numer. Anal. 45(2), 500–513 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dostál, Z., Horák, D., Kučera, R., Vondrák, V., Haslinger, J., Dobiáš, J., Pták, S.: FETI based algorithms for contact problems: scalability, large displacements and 3D coulomb friction. Comput. Methods Appl. Mech. Eng. 194(2–5), 395–409 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dostál, Z., Schöberl, J.: Minimizing quadratic functions subject to bound constraints. Comput. Optim. Appl. 30(1), 23–43 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Farhat, C., Mandel, J., Roux, F.X.: Optimal convergence properties of the FETI domain decomposition method. Comput. Methods Appl. Mech. Eng. 115, 365–385 (1994)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Farhat, C., Roux, F.X.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32(6), 1205–1227 (1991)CrossRefMATHGoogle Scholar
  14. 14.
    Farhat, C., Roux, F.X.: An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems. SIAM J. Sci. Stat. Comput. 13(1), 379–396 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Friedlander, A., Martínez, J.M., Raydan, M.: New method for large-scale box constrained convex quadratic minimization problems. Optim. Methods Softw. 5(1), 57–74 (1995)CrossRefGoogle Scholar
  16. 16.
    Gosselet, P., Rey, C.: Non-overlapping domain decomposition methods in structural mechanics. Arch. Comput. Methods Eng. 13(4), 515–572 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hapla, V., et al.: PERMON (Parallel, Efficient, Robust, Modular, Object-oriented, Numerical) web pages (2015). http://industry.it4i.cz/en/products/permon/
  18. 18.
    Hapla, V., et al.: PermonQP web pages (2015). http://industry.it4i.cz/en/products/permon/qp/
  19. 19.
    Hensinger, D.M., Drake, R.R., Foucar, J.G., Gardiner, T.A.: Pamgen, a library for parallel generation of simple finite element meshes. Technical report SAND2008-1933, Sandia National Laboratories Technical Report (2008)Google Scholar
  20. 20.
    Jolivet, P., Hecht, F., Nataf, F., Prud’homme, C.: Scalable domain decomposition preconditioners for heterogeneous elliptic problems. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, SC 2013, pp. 80:1–80:11. ACM, New York, NY, USA (2013)Google Scholar
  21. 21.
    Jolivet, P., et al.: HPDDM high-performance unified framework for domain decomposition methods. https://github.com/hpddm/hpddm
  22. 22.
    Kozubek, T., Vondrák, V., Menšík, M., Horák, D., Dostál, Z., Hapla, V., Kabelíková, P., Čermák, M.: Total FETI domain decomposition method and its massively parallel implementation. Adv. Eng. Softw. 60–61, 14–22 (2013)CrossRefGoogle Scholar
  23. 23.
    Kruis, J.: Domain Decomposition Methods for Distributed Computing. Saxe-Coburg Publications, Stirling (2006)Google Scholar
  24. 24.
    Kruis, J.: The FETI method and its applications: a review. In: Topping, B., Iványi, P. (eds.) Parallel, Distributed and Grid Computing for Engineering, vol. 21, pp. 199–216. Saxe-Coburg Publications, Stirling (2009)CrossRefGoogle Scholar
  25. 25.
    Li, X.S., et al.: SuperLU. http://acts.nersc.gov/superlu/
  26. 26.
    Markopoulos, A., Hapla, V., Cermak, M., Fusek, M.: Massively parallel solution of elastoplasticity problems with tens of millions of unknowns using PermonCube and FLLOP packages. Appl. Math. Comput. 267, 698–710 (2015)MathSciNetGoogle Scholar
  27. 27.
    Raback, P., et al.: Elmer web pages (2015). http://www.csc.fi/english/pages/elmer/
  28. 28.
    Šístek, J., et al.: The Multilevel BDDC solver library (BDDCML). http://users.math.cas.cz/~sistek/software/bddcml.html
  29. 29.
    Sousedík, B., Šístek, J., Mandel, J.: Adaptive-multilevel BDDC and its parallel implementation. Computing 95(12), 1087–1119 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    The Trilinos Project: PAMGEN web pages (2015). http://trilinos.org/packages/pamgen/
  31. 31.
    Vašatová, A., Čermák, M., Hapla, V.: Parallel implementation of the FETI DDM constraint matrix on top of PETSc for the PermonFLLOP package. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K., Kitowski, J., Wiatr, K. (eds.) Parallel Processing and Applied Mathematics. LNCS, vol. 9573, pp. 150–159. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  32. 32.
    Čermák, M., Hapla, V., Horák, D., Merta, M., Markopoulos, A.: Total-FETI domain decomposition method for solution of elasto-plastic problems. Adv. Eng. Softw. 84, 48–54 (2015)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Vaclav Hapla
    • 1
    • 2
  • David Horak
    • 1
    • 2
  • Lukas Pospisil
    • 1
    • 2
  • Martin Cermak
    • 1
  • Alena Vasatova
    • 1
    • 2
  • Radim Sojka
    • 1
    • 2
  1. 1.IT4Innovations National Supercomputing CenterVSB - Technical University of OstravaOstravaCzech Republic
  2. 2.Department of Applied MathematicsVSB - Technical University of OstravaOstravaCzech Republic

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