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A parallel linear solver exploiting the physical properties of the underlying mechanical problem

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Abstract

The iterative solution of large systems of equations may benefit from parallel processing. However, using a straight-forward domain decomposition in “layered” geomechanical finite element models with significantly different stiffnesses may lead to slow or non-converging solutions. Physics-based domain decomposition is the answer to such problems, as explained in this paper and demonstrated on the basis of a few examples. Together with a two-level preconditioner comprising an additive Schwarz preconditioner that operates on the sub-domain level, an algebraic coarse grid preconditioner that operates on the global level, and additional load balancing measures, the described solver provides an efficient and robust solution of large systems of equations. Although the solver has been developed primarily for geomechanical problems, the ideas are applicable to the solution of other physical problems involving large differences in material properties.

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References

  1. Brinkgreve, R.B.J., Swolfs, W.M., Engin, E.: PLAXIS 3D 2011 User Manual. Plaxis BV. Delft, The Netherlands (2011)

  2. Cai, X.-C., Dryja, M., Sarkis, M.: Restricted additive schwarz preconditioners with harmonic overlap for symmetric positive definite linear systems. SIAM J. Numer. Anal. 41, 1209–1231 (2003)

    Article  Google Scholar 

  3. Dongarra, J.J., Duff, L.S., Sorensen, D.C., van der Vorst, H.A.: Numerical linear algebra for high-performance computers. SIAM, Philadelphia (1998)

  4. Eisenstat, S.C., Elman, H.C., Schultz, M.H.: Variational iterative methods for non-symmetric systems of linear equations. SIAM J. Num. Anal. 20, 345–357 (1983)

    Article  Google Scholar 

  5. George, J.A., Liu, J.W.-H.: Computer solution of large sparse positive definite systems Prentice-Hall (1981)

  6. Golub, G.H., van Loan, C.F.: Matrix computations. The John Hopkins University Press, Baltimore (1996)

  7. Jönsthövel, T.B., van Gijzen, M.B., Vuik, C., Kasbergen, C., Scarpas, A.: Preconditioned Conjugate Gradient method enhanced by deflation of rigid body modes applied to composite materials. Comput. Model. Eng. Sci. 47, 97–118 (2009)

  8. Karypis, G., Kumar, V.: Metis. A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices Version 4, Vol. 0. Technical report, University of Minnesota, Department of Computer Science / Army HPC Research Center, Minneapolis, USA (Sep 1998)

  9. Lingen, F.J.: Design of an object oriented finite element package for parallel computers. PhD thesis, Delft University of Technology. Delft, The Netherlands (2000)

  10. Lingen, F.J., Bonnier, P.G., Brinkgreve, R.B.J., van Gijzen, M.B., Vuik, C.: A parallel linear solver exploiting the physical properties of the underlying mechanical problem Technical Report, Vol. 12-12. Delft University of Technology, Delft, The Netherlands (Sep 2012)

  11. Meijerink, J.A., Van der Vorst, H.A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp. 31, 148–162 (1977)

    Google Scholar 

  12. Saad, Y.: Iterative methods for sparse linear systems, Second Edition. SIAM, Philadelphia (2003)

    Book  Google Scholar 

  13. Saad, Y., Schultz, M.H.: GMRES A Generalized Minimal Residual algorithm for solving non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    Article  Google Scholar 

  14. Saukh, S.: Incomplete Cholesky factorization in fixed memory with flexible drop-tolerance strategy (Sep 2003)

  15. Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with PARDISO. Journal of Future Generation Computer Systems 20 (3), 475–487 (2004)

    Article  Google Scholar 

  16. Tang, J.M., MacLachlan, S.P., Nabben, R., Vuik, C.: A comparison of two-level preconditioners based on multigrid and deflation. SIAM. J. Matrix Anal. and Appl. 31, 1715–1739 (2010)

    Article  Google Scholar 

  17. Toselli, A., Widlund, W.: Domain decomposition methods. Computational Mathematics, Vol. 34. Springer, Berlin (2005)

  18. van der Vorst, H.A.: High performance preconditioning. SIAM J. Sci. Stat. Comput. 10, 1174–1185 (1989)

    Article  Google Scholar 

  19. Vuik, C., Segal, A., Meijerink, J.A.: An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients. J. Comp. Phys. 152, 385–403 (1999)

    Article  Google Scholar 

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Authors and Affiliations

  1. Dynaflow Research Group, Houtsingel 95, 2719, EB, Zoetermeer, The Netherlands

    F. J. Lingen

  2. Plaxis bv, P.O. Box 572, 2600, AN, Delft, The Netherlands

    P. G. Bonnier & R. B. J. Brinkgreve

  3. Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600, GA, Delft, The Netherlands

    M. B. van Gijzen & C. Vuik

Authors
  1. F. J. Lingen
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  2. P. G. Bonnier
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  3. R. B. J. Brinkgreve
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  4. M. B. van Gijzen
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  5. C. Vuik
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Correspondence to C. Vuik.

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Lingen, F.J., Bonnier, P.G., Brinkgreve, R.B.J. et al. A parallel linear solver exploiting the physical properties of the underlying mechanical problem. Comput Geosci 18, 913–926 (2014). https://doi.org/10.1007/s10596-014-9435-x

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  • DOI: https://doi.org/10.1007/s10596-014-9435-x

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