A variational-inequality approach to stochastic boundary value problems with inequality constraints and its application to contact and elastoplasticity

Arnst, Maarten; Ghanem, Roger

doi:10.1002/nme.3307

Article (Scientific journals)

A variational-inequality approach to stochastic boundary value problems with inequality constraints and its application to contact and elastoplasticity

Arnst, Maarten; Ghanem, Roger

2011 • In International Journal for Numerical Methods in Engineering

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https://hdl.handle.net/2268/103158

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10.1002/nme.3307

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Keywords :

variational methods; probabilistic methods; finite element methods

Abstract :

[en] This paper is concerned with stochastic boundary value problems (SBVPs) whose formulation involves inequality constraints. A class of stochastic variational inequalities (SVIs) is defined, which is well adapted to characterize the solution of specified inequality-constrained SBVPs. A methodology for solving such SVIs is proposed, which involves their discretization by projection onto polynomial chaos and collocation of the inequality constraints, followed by the solution of a finite-dimensional constrained optimization problem. Simulation studies in contact and elastoplasticity are provided to demonstrate the proposed framework.

Disciplines :

Mechanical engineering

Author, co-author :

Arnst, Maarten ; University of Southern California > Department of Civil and Environmental Engineering

Ghanem, Roger; University of Southern California > Department of Civil and Environmental Engineering

Language :

English

Title :

A variational-inequality approach to stochastic boundary value problems with inequality constraints and its application to contact and elastoplasticity

Publication date :

2011

Journal title :

International Journal for Numerical Methods in Engineering

ISSN :

0029-5981

eISSN :

1097-0207

Publisher :

John Wiley & Sons, Inc, Chichester, United Kingdom

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBi :

since 15 November 2011

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Bibliography

Ghanem R, Spanos P. Stochastic Finite Elements: A Spectral Approach, Springer:New York, 1991.
Le Maître O, Knio O, Reagan M, Najm H, Ghanem R. A stochastic projection method for fluid flow. I: basic formulation. Journal of Computational Physics 2001;173:481-511, DOI: 10.1006/jcph.2001.6889.
Babuska I, Chatzipantelidis P. On solving elliptic stochastic partial differential equations. Computer Methods in Applied Mechanics and Engineering 2002;191:4093-4122, DOI: 10.1016/S0045-7825(02)00354-7.
Xiu D, Karniadakis G. Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics 2003;187:137-167, DOI: 10.1016/S0021-9991(03)00092-5.
Debusschere B, Najm H, Matta A, Knio O, Ghanem R, Le Maître O. Protein labeling reactions in electrochemical microchannel flow: numerical simulation and uncertainty propagation. Physics of Fluids 2003;15:2238-2250, DOI: 10.1063/1.1582857.
Soize C, Ghanem R. Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM Journal of Scientific Computing 2004;26:395-410, DOI: 10.1137/S1064827503424505.
Pellissetti M, Ghanem R. A method for the validation of predictive computations using a stochastic approach. ASME Journal of Offshore Mechanics and Arctic Engineering 2004;126:227-234, DOI: 10.1115/1.1782915.
Fichera G. Elastostatic Problems with Unilateral Constraints: The Signorini Problem with Ambiguous Boundary Conditions, Seminari dell'istituto Nazionale di Alta Matematica, 1964. (Available from:).
Lions J, Stampacchia G. Variational inequalities. Communications on Pure and Applied Mathematics 1967;20:493-519, DOI: 10.1002/cpa.3160200302.
Duvaut G, Lions J. Inequalities in Mechanics & Physics, Springer:Berlin, Heidelberg, New York, 1976.
Kinderlehrer D, Stampacchia G. An Introduction to Variational Inequalities and Their Applications, SIAM:Philadelphia, 1987.
Glowinski R, Lions J, Tremolières R. Numerical Analysis of Variational Inequalities, North-Holland Publishing Company:Amsterdam, New York, Oxford, 1981.
Kikuchi N, Oden J. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM:Philadelphia, 1995.
Nocedal J, Wright S. Numerical Optimization, Springer:New York, 2006.
Bertsekas D. Nonlinear Programming, Athena Scientific:Belmont, 1999.
Gill P, Murray W, Wright M. Numerical Linear Algebra and Optimization, Addison Wesley Publishing Company:Redwood City, 1990.
Luenberger D, Ye Y. Linear and Nonlinear Programming, Springer:New York, 2008.
Han W, Reddy B. Plasticity Mathematical Theory and Numerical Analysis, Springer:New York, Berlin, Heidelberg, 1999.
Simo J. Topics in the numerical analysis and simulation of plasticity, In Handbook of Numerical Analysis VI, Ciarlet P, Lions J (eds), North Holland, Amsterdam, 1998;183-499.
Wriggers P. Computational Contact Mechanics, Springer:New York, 2006.
Gwinner J. A class of random variational inequalities and simple random unilateral boundary value problems-existence, discretization, finite element approximation. Stochastic Analysis and Applications 2000;18:967-993, DOI: 10.1080/07362990008809706.
Gwinner J, Raciti F. On a class of random variational inequalities on random sets. Numerical Functional Analysis and Optimization 2006;27:619-636, DOI: 10.1080/01630560600790819.
Matthies H, Rosic B. Inelastic media under uncertainty: stochastic models and computational approaches, IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media, Cape Town, South Africa, 2008;185-194. DOI: 10.1007/978-1-4020-9090-5-17.
Rosic B, Matthies H. Computational approaches to inelastic media with uncertain parameters. Journal of the Serbian Society for Computational Mechanics 2008;2:28-43.
Rosic B, Matthies H, Zivkovic M. Mathematical formulation and numerical simulation of stochastic elastoplastic behaviour, ECCM European Conference on Computational Mechanics, Paris, France, 2010.
Anders M, Hori M. Stochastic finite element method for elasto-plastic body. International Journal for Numerical Methods in Engineering 1999;46:1897-1916, DOI: 10.1002/(SICI)1097-0207(19991220)46:11<1897::AID-NME758>3.0.CO;2-3.
Anders M, Hori M. Stochastic finite element method for elasto-plastic bodies. International Journal for Numerical Methods in Engineering 2001;51:449-478, DOI: 10.1002/nme.165.
Bensoussan A, Turi J. Stochastic variational inequalities for elasto-plastic oscillators. Comptes Rendus Mathematique 2006;343:339-406, DOI: 10.1016/j.crma.2006.08.008.
Jiang H, Xu H. Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Transactions on Automatic Control 2008;53:1462-1475, DOI: 10.1109/TAC.2008.925853.
Gurkan G, Ozge A, Robinson S. Sample-path solution of stochastic variational inequalities. Mathematical Programming 1999;84:313-333, DOI: 10.1007/s101070050024.
Robinson S. Analysis of sample-path optimization. Mathematics of Operations Research 1996;21:513-528, DOI: 10.1287/moor.21.3.513.
Fukushima M. A fixed point approach to certain convex programs with applications in stochastic programming. Mathematics of Operations Research 1983;8:517-524.
Lin G, Xu H, Fukushima M. Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints. Mathematics of Operations Research 2008;67:423-441, DOI: 10.1007/s00186-007-0201-x.
Bensoussan A, Lions J. Applications of Variational Inequalities in Stochastic Control, Elsevier:Amsterdam, New York, Oxford, 1982.
Forster R, Kornhuber R. A polynomial chaos approach to stochastic variational inequalities. Journal of Numerical Mathematics 2010;18:235-255, DOI: 10.1515/JNUM.2010.012.
Forster R. On the stochastic Richards equation, PhD Thesis, Freien Universität Berlin, 2011.
Ghanem R, Kruger R. Numerical solution of spectral stochastic finite element systems. Computer Methods in Applied Mechanics and Engineering 1996;129:289-303, DOI: 10.1016/0045-7825(95)00909-4.
Gautschi W. Orthogonal Polynomials: Computation and Approximation, Oxford University Press:New York, 2004.
Courant R. Methods of Mathematical Physics, John Wiley & Sons:Berlin, 1989.
Funaro D. Polynomial Approximation of Differential Equations, Springer:Berlin, Heidelberg, 1992.
Cameron R, Martin W. The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. The Annals of Mathematics 1947;48:385-392. Available from:.
Erdos P, Turan P. On interpolation. I. quadrature- and mean-convergence in the Lagrange-interpolation. Annals of Mathematics 1937;38:142-155. Available from:.
Babuska I, Nobile F, Tempone R. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Journal on Numerical Analysis 2007;45:1005-1034, DOI: 10.1137/050645142.
Nevai P. Mean convergence of Lagrange interpolation. Journal of Approximation Theory 1980;30:236-276. Available from:.