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A TFETI domain decomposition solver for elastoplastic problems. (English) Zbl 1410.74061

Summary: We propose an algorithm for the efficient parallel implementation of elastoplastic problems with hardening based on the so-called TFETI (Total Finite Element Tearing and Interconnecting) domain decomposition method. We consider an associated elastoplastic model with the von Mises plastic criterion and the linear isotropic hardening law. Such a model is discretized by the implicit Euler method in time and the consequent one time step elastoplastic problem by the finite element method in space. The latter results in a system of nonlinear equations with a strongly semismooth and strongly monotone operator. The semismooth Newton method is applied to solve this nonlinear system. Corresponding linearized problems arising in the Newton iterations are solved in parallel by the above mentioned TFETI domain decomposition method. The proposed TFETI based algorithm was implemented in Matlab parallel environment and its performance was illustrated on a 3D elastoplastic benchmark. Numerical results for different time discretizations and mesh levels are presented and discussed and a local quadratic convergence of the semismooth Newton method is observed.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
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[1] Alberty, J.; Carstensen, C.; Funken, S. A.; Klose, R., Matlab implementation of the finite element method in elasticity, Computing, 69, 3, 239-263, (2002) · Zbl 1239.74092
[2] Alberty, J.; Carstensen, C.; Zarrabi, D., Adaptive numerical analysis in primal elastoplasticity with hardening, Comput. Methods Appl. Mech. Eng., 171, 3-4, 175-204, (1999) · Zbl 0956.74049
[3] Alberty, J.; Carstensen, C., Discontinuous Galerkin time discretization in elastoplasticity: motivation, numerical algorithms, and applications, Comput. Methods Appl. Mech. Eng., 191, 43, 4949-4968, (2002) · Zbl 1018.74040
[4] Badea, L.; Gilormini, P., Application of a domain decomposition method to elastoplastic problems, Int. J. Solids Struct., 31, 5, 643-656, (1994) · Zbl 0794.73076
[5] Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numerica, (2005), Cambridge University Press, pp. 1-137 · Zbl 1115.65034
[6] Blaheta, R., Numerical methods in elasto-plasticity, Documenta Geonica 1998, (1999), PERES Publishers Prague
[7] Carstensen, C.; Brokate, M.; Valdman, J., A quasi-static boundary value problem in multi-surface elastoplasticity. I: analysis, Math. Methods Appl. Sci., 14, 27, 1697-1710, (2004) · Zbl 1074.74013
[8] Carstensen, C.; Brokate, M.; Valdman, J., A quasi-static boundary value problem in multi-surface elastoplasticity. II: numerical solution, Math. Methods Appl. Sci., 8, 28, 881-901, (2005) · Zbl 1112.74007
[9] Brokate, M.; Sprekels, J., Hysteresis and phase transitions, (1996), Springer · Zbl 0951.74002
[10] Brzobohatý, T.; Dostál, Z.; Kovář, P.; Kozubek, T.; Markopoulos, A., Cholesky decomposition with fixing nodes to stable evaluation of a generalized inverse of the stiffness matrix of a floating structure, Int. J. Numer. Methods Eng., 88, 5, 493-509, (2011) · Zbl 1242.74235
[11] Carstensen, C.; Klose, R., Elastoviscoplastic finite element analysis in 100 lines of Matlab, J. Numer. Math., 10, 3, 157-192, (2002) · Zbl 1099.74544
[12] M. Čermák, J. Haslinger, S. Sysala, Numerical solutions of perfect plastic problems with contact: part II - numerical realization, in: E. Onate, D.R.J. Owen, D. Peric, B. Suarez (Eds.), Proceedings of the XII International Conference on Computational Plasticity Fundamentals and Applications, Barcelona, Spain, 2013, pp. 999-1009.
[13] M. Čermák, T. Kozubek, A. Markopoulos, An efficient FETI based solver for elasto-plastic problems of mechanics, in: Computational Plasticity XI - Fundamentals and Applications, COMPLAS XI, 2011, pp. 1330-1341.
[14] Dohrmann, C. R., A preconditioner for substructuring based on constrained energy minimization, SIAM J. Sci. Comput., 25, 246-258, (2003) · Zbl 1038.65039
[15] Dostál, Z., Optimal quadratic programming algorithms, with applications to variational inequalities, SOIA, vol. 23, (2009), Springer US · Zbl 1401.90013
[16] 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) · Zbl 1107.65104
[17] Dostál, Z.; Kozubek, T.; Markopoulos, A.; Brzobohatý, T.; Vondrák, V.; Horyl, P., Theoretically supported scalable TFETI algorithm for the solution of multibody 3D contact problems with friction, Comput. Methods Appl. Mech. Eng., 205, 110-120, (2012) · Zbl 1239.74064
[18] 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, 1384-1405, (2010) · Zbl 1188.74054
[19] Farhat, C.; Roux, F.-X., A method of finite element tearing and interconnecting and its parallel solution algorithm, Int. J. Numer. Methods Eng., 32, 1205-1227, (1991) · Zbl 0758.65075
[20] 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, 379-396, (1992) · Zbl 0746.65086
[21] 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)
[22] Farhat, C.; Lesoinne, M.; LeTallec, P.; Pierson, K.; Rixen, D., FETI-DP: a dual-primal unified FETI method, part i: a faster alternative to the two-level FETI method, Int. J. Numer. Methods Eng., 50, 1523-1544, (2001) · Zbl 1008.74076
[23] Farhat, C.; Lesoinne, M.; Pierson, K., A scalable dual-primal domain decomposition method, Numer. Linear Algebra Appl., 7, 687-714, (2000) · Zbl 1051.65119
[24] Fragakis, Y.; Papadrakakis, M., The mosaic of high performance domain decomposition methods for structural mechanics: formulation, interrelation and numerical efficiency of primal and dual methods, Comput. Methods Appl. Mech. Eng., 192, 3799-3830, (2003) · Zbl 1054.74069
[25] Fučík, S.; Kufner, A., Nonlinear differential equation, (1980), Elsevier · Zbl 0647.35001
[26] Gruber, P.; Kienesberger, J.; Langer, U.; Schöeberl, J.; Valdman, J., Fast solvers and a posteriori error estimates in elastoplasticity, (Langer, Ulrich; etal., Numerical and Symbolic Scientific Computing. Progress and Prospects, Texts & Monographs in Symbolic Computation, (2012), Springer), 45-63 · Zbl 06082197
[27] Gruber, P.; Valdman, J., Solution of one-time step problems in elastoplasticity by a slant Newton method, SIAM J. Sci. Comput., 31, 1558-1580, (2009) · Zbl 1186.74025
[28] Han, W.; Reddy, B. D., Plasticity: mathematical theory and numerical analysis, (1999), Springer · Zbl 0926.74001
[29] Hofinger, A.; Valdman, J., Numerical solution of the two-yield elastoplastic minimization problem, Computing, 81, 1, 35-52, (2007) · Zbl 1177.74167
[30] Johnson, C., Existence theorems for plasticity problems, J. Math. Pures Appl., 55, 431-444, (1976) · Zbl 0351.73049
[31] Johnson, C., On plasticity with hardening, J. Math. Anal. Appl., 62, 325-336, (1978) · Zbl 0373.73049
[32] Justino, M. R.; Park, K. C.; Felippa, C. A., An algebraically partitioned FETI method for parallel structural analysis: implementation and numerical performance evaluation, Int. J. Numer. Methods Eng., 40, 2739-2758, (1997) · Zbl 0888.73061
[33] J. Kienesberger, U. Langer, J. Valdman, On a robust multigrid-preconditioned solver for incremental plasticity problems, in: Proceedings of IMET 2004 - Iterative Methods, Preconditioning & Numerical PDEs, Prague.
[34] Klawonn, A.; Rheinbach, O., A parallel implementation of dual-primal FETI methods for three dimensional linear elasticity using a transformation of basis, SIAM J. Sci. Comput., 28, 1886-1906, (2006) · Zbl 1124.74049
[35] Klawonn, A.; Rheinbach, O., Highly scalable parallel domain decomposition methods with an application to biomechanics, Z. Angew. Math. Mech., 90, 1, 5-32, (2010) · Zbl 1355.65169
[36] Klawonn, A.; Widlund, O. B., FETI and Neumann-Neumann iterative substructuring methods: connections and new results, Commun. Pure Appl. Math., 54, 57-90, (2001) · Zbl 1023.65120
[37] A. Klawonn, O.B. Widlund, Dual and dual-primal FETI methods for elliptic problems with discontinuous coefficients in three dimensions, in: Domain Decomposition Methods, Proceedings of the 12th International Conference on Domain Decomposition Methods, Chiba, Japan, October 1999, (DDM.org, Augsburg, Germany, 2001).
[38] Korneev, V. G.; Langer, U., Approximate solution of plastic flow theory problems, (1984), Teubner-Verlag Leipzig, vol. 69 · Zbl 0574.73047
[39] T. Kozubek, A. Markopoulos, T. Brzobohatý, R. Kučera, V. Vondrák, Z. Dostál, MatSol - MATLAB efficient solvers for problems in engineering. <http://matsol.vsb.cz/>.
[40] 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)
[41] P. Krejčí, Hysteresis, convexity and dissipation in hyperbolic equations, GAKUTO Int. Ser. Math. Sci. Appl. 8, 1996.
[42] Kučera, R.; Kozubek, T.; Markopoulos, A., On large-scale generalized inverses in solving two-by-two block linear systems, Linear Algebra Appl., 438, 7, 3011-3029, (2013) · Zbl 1264.65061
[43] Mandel, J., Balancing domain decomposition, Commun. Numer. Methods Eng., 9, 233-241, (1993) · Zbl 0796.65126
[44] Mandel, J.; Dohrmann, C. R., Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl., 10, 639-659, (2003) · Zbl 1071.65558
[45] Mandel, J.; Tezaur, R., Convergence of a substructuring method with Lagrange multipliers, Numer. Math., 73, 473-487, (1996) · Zbl 0880.65087
[46] Matthies, H., Existence theorems in thermoplasticity, J. Mec., 18, 4, 695-712, (1979)
[47] Matthies, H., Finite element approximations in thermo-plasticity, Numer. Funct. Anal. Optim., 1, 2, 145-160, (1979) · Zbl 0439.73071
[48] Mifflin, R., Semismoothness and semiconvex function in constraint optimization, SIAM J. Control Optim., 15, 957-972, (1977)
[49] MUMPS Web page, <http://graal.ens-lyon.fr/MUMPS/>.
[50] Nečas, J.; Hlaváček, I., Mathematical theory of elastic and elasto-plastic bodies. an introduction, (1981), Elsevier · Zbl 0448.73009
[51] Park, K. C.; Justino, M. R.; Felippa, C. A., An algebraically partitioned FETI method for parallel structural analysis: algorithm description, Int. J. Numer. Methods Eng., 40, 2717-2737, (1997) · Zbl 0889.73068
[52] Qi, L.; Sun, J., A nonsmooth version of newton’ s method, Math. Program., 58, 353-367, (1993) · Zbl 0780.90090
[53] Rixen, D. J.; Farhat, C., A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems, Int. J. Numer. Methods Eng., 44, 489-516, (1999) · Zbl 0940.74067
[54] Roux, F-X., Spectral analysis of interface operator, (Keyes, D. E.; etal., Proceedings of the Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, (1992), SIAM Philadelphia), 73-90
[55] Roux, F.-X.; Farhat, C., Parallel implementation of direct solution strategies for the coarse grid solvers in 2-level FETI method, Contemp. Math., 218, 158-173, (1998) · Zbl 0960.74068
[56] Sauter, M.; Wieners, C., On the superlinear convergence in computational elasto-plasticity, Comput. Methods Appl. Mech. Eng., 200, 3646-3658, (2011) · Zbl 1239.74014
[57] Simo, J. C.; Hughes, T. J.R., Computational inelasticity, Interdisciplinary Applied Mathematics, vol. 7, (1998), Springer-Verlag · Zbl 0934.74003
[58] de Souza Neto, E. A.; Perić, D.; Owen, D. R.J., Computational methods for plasticity: theory and application, (2008), Wiley
[59] Stein, E., Error-controlled adaptive finite elements in solid mechanics, (2003), Wiley
[60] Sysala, S., Application of a modified semismooth Newton method to some elasto-plastic problems, Math. Comput. Simul., 82, 2004-2021, (2012) · Zbl 1252.74059
[61] Sysala, S., Properties and simplifications of constitutive time-discretized elastoplastic operators, Z. Angew. Math. Mech., 1-23, (2013)
[62] Toselli, A.; Widlund, O. B., Domain decomposition methods - algorithms and theory, (2005), Springer · Zbl 1069.65138
[63] Wieners, C., A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing, Comput. Visual. Sci., 13, 4, 161-175, (2010) · Zbl 1216.65164
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