×

zbMATH — the first resource for mathematics

A comparison of solvers for linear complementarity problems arising from large-scale masonry structures. (English) Zbl 1164.74355
Summary: We compare the numerical performance of several methods for solving the discrete contact problem arising from the finite element discretization of elastic systems with numerous contact points. The problem is formulated as a variational inequality and discretized 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.

MSC:
74B10 Linear elasticity with initial stresses
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
49J40 Variational inequalities
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] A. Berman, R. J. Plemmons: Nonnegative Matrices in the Mathematical Sciences. Computer Science and Scientific Computing Series. Academic Press, New York, 1979.
[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. · Zbl 0804.65101
[3] G. Duvaut, J.-L. Lions: Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften, Vol. 219. Springer-Verlag, Berlin-Heidelberg-New York, 1976. · Zbl 0331.35002
[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.
[5] P. E. Gill, W. Murray, and M. H. Wright: Practical Optimization. Academic Press, London, 1981.
[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.
[7] R. L. Graves: A principal pivoting simplex algorithm for linear and quadratic programming. Oper. Res. 15 (1967), 482–494. · Zbl 0154.19604
[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. · Zbl 1080.90074
[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. · Zbl 1084.49029
[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. · Zbl 1086.74045
[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.
[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. · Zbl 1021.74030
[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. · Zbl 0685.73002
[14] C. L. Lawson, R. J. Hanson: Solving Least Squares Problems. Series in Automatic Computation. Prentice-Hall, Englewood Cliffs, 1974. · Zbl 0860.65028
[15] K. G. Murty: Complementarity, Linear and Nonlinear Programming. Heldermann-Verlag, Berlin, 1988.
[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. · Zbl 0812.90124
[17] V.V. Prasolov: Problems and Theorems in Linear Algebra. Translations of Mathematical Monographs, Vol. 134. AMS, Providence, 1994.
[18] S. J. Wright: Primal-Dual Interior-Point Methods. SIAM, Philadelphia, 1997. · Zbl 0863.65031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.