Semismooth Newton methods for variational problems with inequality constraints

Inequality constraints occur in many different fields of application, e.g., in structural mechanics, flow processes in porous media or mathematical finance. In this paper, we make use of the mathematical structure of these conditions in order to obtain an abstract computational framework for problems with inequality conditions. The constraints are enforced locally by means of Lagrange multipliers which are defined with respect to dual basis functions. The reformulation of the inequality conditions in terms of nonlinear complementarity functions leads to a system of semismooth nonlinear equations that is solved by a generalized version of Newton's method for semismooth problems. By this, both nonlinearities in the pde model and inequality constraints are treated within a single Newton iteration which converges locally superlinear. The scheme can efficiently be implemented in terms of an active set strategy with local static condensation of the non-essential variables. Numerical examples from different fields of application illustrate the generality and the robustness of the method (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

[1]  Lawrence C. Evans,et al.  An Introduction to Variational Inequalities and Their Applications (D. Kinderlehrer and G. Stampacchia) , 1981 .

[2]  D. Lamberton,et al.  Variational inequalities and the pricing of American options , 1990 .

[3]  G. Saxcé,et al.  New Inequality and Functional for Contact with Friction: The Implicit Standard Material Approach∗ , 1991 .

[4]  Michael A. Celia,et al.  Dynamic Effect in the Capillary Pressure–Saturation Relationship and its Impacts on Unsaturated Flow , 2002 .

[5]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[6]  W. Hager.  Review: R. Glowinski, J. L. Lions and R. Trémolières, Numerical analysis of variational inequalities , 1983 .

[7]  P. W. Christensen.  A nonsmooth Newton method for elastoplastic problems , 2002 .

[8]  Todd Arbogast,et al.  A Multiscale Mortar Mixed Finite Element Method , 2007, Multiscale Model. Simul..

[9]  Peter Deuflhard,et al.  Newton Methods for Nonlinear Problems , 2004 .

[10]  Todd Arbogast,et al.  Mixed Finite Element Methods on Nonmatching Multiblock Grids , 2000, SIAM J. Numer. Anal..

[11]  P. Alart,et al.  A mixed formulation for frictional contact problems prone to Newton like solution methods , 1991 .

[12]  P. Deuflhard.  Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms , 2011 .

[13]  Barbara I. Wohlmuth,et al.  A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier , 2000, SIAM J. Numer. Anal..

[14]  Barbara Wohlmuth,et al.  Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D , 2007 .

[15]  Jong-Shi Pang,et al.  NE/SQP: A robust algorithm for the nonlinear complementarity problem , 1993, Math. Program..

[16]  Barbara I. Wohlmuth,et al.  Discretization Methods and Iterative Solvers Based on Domain Decomposition , 2001, Lecture Notes in Computational Science and Engineering.

[17]  R. Helmig.  Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems , 2011 .

[18]  B. M. Fulk.  MATH , 1992 .

[19]  Peter Wriggers,et al.  Computational Contact Mechanics , 2002 .

[20]  J. Bear,et al.  Introduction to Modeling of Transport Phenomena in Porous Media , 1990 .

[21]  Barbara I. Wohlmuth,et al.  An Optimal A Priori Error Estimate for Nonlinear Multibody Contact Problems , 2005, SIAM J. Numer. Anal..

[22]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[23]  Weimin Han,et al.  Numerical approximations of problems in plasticity: error analysis and solution algorithms , 1997 .

[24]  Andrew G. Glen,et al.  APPL , 2001 .

[25]  Christian Kanzow,et al.  Theorie und Numerik restringierter Optimierungsaufgaben , 2002 .

[26]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[27]  Barbara I. Wohlmuth,et al.  A Primal-Dual Active Set Algorithm for Three-Dimensional Contact Problems with Coulomb Friction , 2008, SIAM J. Sci. Comput..

[28]  Barbara Wohlmuth,et al.  Nonlinear complementarity functions for plasticity problems with frictional contact , 2009 .

[29]  P. Wriggers.  Computational contact mechanics , 2012 .

[30]  王东东,et al.  Computer Methods in Applied Mechanics and Engineering , 2004 .

[31]  Rainer Helmig,et al.  Variational inequalities for modeling flow in heterogeneous porous media with entry pressure , 2009 .

[32]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[33]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[34]  T. Laursen.  Computational Contact and Impact Mechanics , 2003 .

[35]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[36]  M. J. De Neef,et al.  Modelling Capillary Effects in Heterogeneous Porous Media , 2000 .

[37]  E. C. Childs.  Dynamics of fluids in Porous Media , 1973 .

[38]  Yves Achdou,et al.  Numerical Procedure for Calibration of Volatility with American Options , 2005 .

[39]  William G. Gray,et al.  Thermodynamic basis of capillary pressure in porous media , 1993 .

[40]  J. Oden,et al.  Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , 1987 .

[41]  P. Wilmott,et al.  Option pricing: Mathematical models and computation , 1994 .

[42]  Philip John Binning,et al.  Practical implementation of the fractional flow approach to multi-phase flow simulation , 1999 .

[43]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[44]  Christian Wieners,et al.  Nonlinear solution methods for infinitesimal perfect plasticity , 2007 .

[45]  Christof Eck,et al.  Unilateral Contact Problems: Variational Methods and Existence Theorems , 2005 .

[46]  Rainer Helmig,et al.  Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media , 2000 .

[47]  W. Han,et al.  Plasticity: Mathematical Theory and Numerical Analysis , 1999 .

[48]  J. Molenaar,et al.  The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media , 1995 .