A TFETI Domain Decompositon Solver for Von Mises Elastoplasticity Model with Combination of Linear Isotropic-Kinematic Hardening
In this paper we present the efficient parallel
implementation of elastoplastic problems based on the TFETI (Total
Finite Element Tearing and Interconnecting) domain decomposition
method. This approach allow us to use parallel solution and compute
this nonlinear problem on the supercomputers and decrease the
solution time and compute problems with millions of DOFs. In
our approach we consider an associated elastoplastic model with
the von Mises plastic criterion and the combination of linear
isotropic-kinematic hardening law. This model is discretized by
the implicit Euler method in time and by the finite element
method in space. We consider the 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. The
implementation of this problem is realized in our in-house MatSol
packages developed in MatLab.
[1] R. Blaheta, Numerical methods in elasto-plasticity, Documenta Geonica
1998, PERES Publishers, Prague, 1999.
[2] T. Brzobohaty, Z. Dostal, P. Kovar, T. Kozubek, A. Markopoulos,
Cholesky decomposition with fxing 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.
[3] M. Cermak, T. Kozubek, An efficient TFETI based solver for
elasto-plastic problems of mechanics, Advances in Electrical and
Electronic Engineering 10 (1), 57–62, 2012.
[4] M. Cermak, T. Kozubek, S. Sysala, J. Valdman, A TFETI domain
decomposition solver for elastoplastic problems, Appl. Math. and
Comput. 231, 634–653, 2014.
[5] Z. Dost´al, D. Hor´ak, R. Kuˇcera, 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.
[6] C. Farhat, J. Mandel, F-X. Roux, Optimal convergence properties of the
FETI domain decomposition method, Comput. Meth. Appl. Mech. Eng.
115, 365–385, 1994.
[7] C. Farhat, F-X. Roux, A method of finite element tearing and
interconnecting and its parallel solution algorithm, Int. J. Numer.
Methods Eng. 32, 1205–1227, 1991.
[8] S. Fuˇc´ık, A. Kufner, Nonlinear Differential Equation, Elsevier, 1980.
[9] W. Han, B. D. Reddy, Plasticity: mathematical theory and numerical
analysis, Springer, 1999.
[10] V. Hapla et al: FLLOP Web Page. (Online). Available:
http://industry.it4i.cz/en/products/permon/
[11] A. Kossa, L. Szab´o, Exact integration of the von Mises elastoplasticity
model with combined linear isotropic-kinematic hardening, International
Journal of Plasticity 25, 1083–1106, 2009.
[12] T. Kozubek, A. Markopoulos, T. Brzobohat´y, R. Kuˇcera, V. Vondr´ak, Z.
Dost´al, MatSol - MATLAB efficient solvers for problems in engineering,
http://matsol.vsb.cz/
[13] J. Mandel, R. Tezaur, Convergence of a substructuring method with
Lagrange multipliers, Numer. Math. 73, 473–487, 1996.
[14] R. Mifflin, Semismoothness and semiconvex function in constraint
optimization, SIAM J. Cont. Optim. 15, 957–972, 1977.
[15] E. A. de Souza Neto, D. Peri´c, D. R. J. Owen, Computational methods
for plasticity: theory and application. Wiley, 2008.
[16] L. Qi, J. Sun, A nonsmooth version of Newton’ s method, Math. Progr.,
58, 353–367, 1993.
[17] B.F. Smith et al: PETSc Web page. (Online). Available:
http://www.mcs.anl.gov/petsc/
[18] S. Sysala, Application of a modified semismooth Newton method to some
elasto-plastic problems. Math. Comput. Simul. 82, 2004–2021, 2012.
[19] S. Sysala, Properties and simplifications of constitutive time-discretized
elastoplastic operators. Z. Angew. Math. Mech., 1–23, 2013.
[1] R. Blaheta, Numerical methods in elasto-plasticity, Documenta Geonica
1998, PERES Publishers, Prague, 1999.
[2] T. Brzobohaty, Z. Dostal, P. Kovar, T. Kozubek, A. Markopoulos,
Cholesky decomposition with fxing 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.
[3] M. Cermak, T. Kozubek, An efficient TFETI based solver for
elasto-plastic problems of mechanics, Advances in Electrical and
Electronic Engineering 10 (1), 57–62, 2012.
[4] M. Cermak, T. Kozubek, S. Sysala, J. Valdman, A TFETI domain
decomposition solver for elastoplastic problems, Appl. Math. and
Comput. 231, 634–653, 2014.
[5] Z. Dost´al, D. Hor´ak, R. Kuˇcera, 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.
[6] C. Farhat, J. Mandel, F-X. Roux, Optimal convergence properties of the
FETI domain decomposition method, Comput. Meth. Appl. Mech. Eng.
115, 365–385, 1994.
[7] C. Farhat, F-X. Roux, A method of finite element tearing and
interconnecting and its parallel solution algorithm, Int. J. Numer.
Methods Eng. 32, 1205–1227, 1991.
[8] S. Fuˇc´ık, A. Kufner, Nonlinear Differential Equation, Elsevier, 1980.
[9] W. Han, B. D. Reddy, Plasticity: mathematical theory and numerical
analysis, Springer, 1999.
[10] V. Hapla et al: FLLOP Web Page. (Online). Available:
http://industry.it4i.cz/en/products/permon/
[11] A. Kossa, L. Szab´o, Exact integration of the von Mises elastoplasticity
model with combined linear isotropic-kinematic hardening, International
Journal of Plasticity 25, 1083–1106, 2009.
[12] T. Kozubek, A. Markopoulos, T. Brzobohat´y, R. Kuˇcera, V. Vondr´ak, Z.
Dost´al, MatSol - MATLAB efficient solvers for problems in engineering,
http://matsol.vsb.cz/
[13] J. Mandel, R. Tezaur, Convergence of a substructuring method with
Lagrange multipliers, Numer. Math. 73, 473–487, 1996.
[14] R. Mifflin, Semismoothness and semiconvex function in constraint
optimization, SIAM J. Cont. Optim. 15, 957–972, 1977.
[15] E. A. de Souza Neto, D. Peri´c, D. R. J. Owen, Computational methods
for plasticity: theory and application. Wiley, 2008.
[16] L. Qi, J. Sun, A nonsmooth version of Newton’ s method, Math. Progr.,
58, 353–367, 1993.
[17] B.F. Smith et al: PETSc Web page. (Online). Available:
http://www.mcs.anl.gov/petsc/
[18] S. Sysala, Application of a modified semismooth Newton method to some
elasto-plastic problems. Math. Comput. Simul. 82, 2004–2021, 2012.
[19] S. Sysala, Properties and simplifications of constitutive time-discretized
elastoplastic operators. Z. Angew. Math. Mech., 1–23, 2013.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:69531", author = "Martin Cermak and Stanislav Sysala", title = "A TFETI Domain Decompositon Solver for Von Mises Elastoplasticity Model with Combination of Linear Isotropic-Kinematic Hardening", abstract = "In this paper we present the efficient parallel
implementation of elastoplastic problems based on the TFETI (Total
Finite Element Tearing and Interconnecting) domain decomposition
method. This approach allow us to use parallel solution and compute
this nonlinear problem on the supercomputers and decrease the
solution time and compute problems with millions of DOFs. In
our approach we consider an associated elastoplastic model with
the von Mises plastic criterion and the combination of linear
isotropic-kinematic hardening law. This model is discretized by
the implicit Euler method in time and by the finite element
method in space. We consider the 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. The
implementation of this problem is realized in our in-house MatSol
packages developed in MatLab.
", keywords = "Isotropic-kinematic hardening, TFETI, domain
decomposition, parallel solution.", volume = "9", number = "4", pages = "581-6", }
