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A comparison of solvers for linear complementarity problems arising from large-scale masonry structures

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Abstract

We compare the numerical performance of several methods for solving the discrete contact problem arising from the finite element discretisation of elastic systems with numerous contact points. The problem is formulated as a variational inequality and discretised using piecewise quadratic finite elements on a triangulation of the domain. At the discrete level, the variational inequality is reformulated as a classical linear complementarity system. We compare several state-of-art algorithms that have been advocated for such problems. Computational tests illustrate the use of these methods for a large collection of elastic bodies, such as a simplified bidimensional wall made of bricks or stone blocks, deformed under volume and surface forces.

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References

  1. A. Berman, R. J. Plemmons: Nonnegative Matrices in the Mathematical Sciences. Computer Science and Scientific Computing Series. Academic Press, New York, 1979.

    MATH  Google Scholar 

  2. S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, Vol. 15. Springer-Verlag, New York, 1994.

    MATH  Google Scholar 

  3. G. Duvaut, J.-L. Lions: Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften, Vol. 219. Springer-Verlag, Berlin-Heidelberg-New York, 1976.

    Google Scholar 

  4. G. Fichera: Encyclopedia of physics. Existence Theorems in Elasticity-Boundary Value Problems of Elasticity with Unilateral Constraints, Volume VIa/2 (S. Flügge, ed.). Springer-Verlag, Berlin, 1972, pp. 347–427.

    Google Scholar 

  5. P. E. Gill, W. Murray, and M. H. Wright: Practical Optimization. Academic Press, London, 1981.

    MATH  Google Scholar 

  6. R. Glowinski, J.-L. Lions, and R. Trémolières: Numerical Analysis of Variational Inequalities. Studies in Mathematics and its Applications, Vol. 8. North-Holland, Amsterdam-New York-Oxford, 1981, English version edition.

    MATH  Google Scholar 

  7. R. L. Graves: A principal pivoting simplex algorithm for linear and quadratic programming. Oper. Res. 15 (1967), 482–494.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Hintermüller, K. Ito, and K. Kunisch: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2003), 865–888.

    Article  MATH  Google Scholar 

  9. M. Hintermüller, V. A. Kovtunenko, and K. Kunisch: The primal-dual active set method for a crack problem with non-penetration. IMA J. Appl. Math. 69 (2004), 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Hintermüller, V. A. Kovtunenko, and K. Kunisch: Generalized Newton methods for crack problems with nonpenetration condition. Numer. Methods Partial Differential Equations 21 (2005), 586–610.

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovíšek: Solution of Variational Inequalities in Mechanics. Applied Mathematical Sciences, Vol. 66. Springer-Verlag, Berlin-Heidelberg-New York, 1988.

    MATH  Google Scholar 

  12. I. Hlaváček, J. Nedoma: On a solution of a generalized semi-coercive contact problem in thermo-elasticity. Math. Comput. Simul. 60 (2002), 1–17.

    Article  MATH  Google Scholar 

  13. N. Kikuchi, J.T. Oden: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. Studies in Applied Mathematics, Vol. 8. SIAM, Philadelphia, 1988.

    MATH  Google Scholar 

  14. C. L. Lawson, R. J. Hanson: Solving Least Squares Problems. Series in Automatic Computation. Prentice-Hall, Englewood Cliffs, 1974.

    Google Scholar 

  15. K. G. Murty: Complementarity, Linear and Nonlinear Programming. Heldermann-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  16. L.F. Portugal, J. J. Judice, and L. N. Vicente: A comparison of block pivoting and interior-point algorithms for linear least squares problems with nonnegative variables. Math. Comput. 63 (1994), 625–643.

    Article  MathSciNet  MATH  Google Scholar 

  17. V.V. Prasolov: Problems and Theorems in Linear Algebra. Translations of Mathematical Monographs, Vol. 134. AMS, Providence, 1994.

    MATH  Google Scholar 

  18. S. J. Wright: Primal-Dual Interior-Point Methods. SIAM, Philadelphia, 1997.

    MATH  Google Scholar 

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Authors and Affiliations

  1. University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland, UK

    Mark Ainsworth & L. Angela Mihai

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  1. Mark Ainsworth
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  2. L. Angela Mihai
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This work was supported by the Engineering and Physical Science Research Council of Great Britain under grant GR/S35101, and the first author was supported by a fellowship from the Royal Society of Edinburgh.

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Ainsworth, M., Mihai, L.A. A comparison of solvers for linear complementarity problems arising from large-scale masonry structures. Appl Math 51, 93–128 (2006). https://doi.org/10.1007/s10492-006-0008-8

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  • DOI: https://doi.org/10.1007/s10492-006-0008-8

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