Semicoercive Variational Inequalities: From Existence to Numerical Solution of Nonmonotone Contact Problems

In this paper, we present a novel numerical solution procedure for semicoercive hemivariational inequalities. As a concrete example, we consider a unilateral semicoercive contact problem with nonmonotone friction modeling the deformation of a linear elastic block in a rail, and provide numerical results for benchmark tests.

[1]  Jean-Paul Penot,et al.  Noncoercive problems and asymptotic conditions , 2006, Asymptot. Anal..

[2]  SOLUTION OF SEMICOERCIVE SIGNORINI PROBLEM BASED ON A DUALITY SCHEME WITH MODIFIED LAGRANGIAN FUNCTIONAL , 2012 .

[3]  Daniel Goeleven,et al.  Variational and Hemivariational Inequalities : Theory, Methods and Applications , 2003 .

[4]  P. Panagiotopoulos Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions , 1985 .

[5]  Daniel Goeleven,et al.  Semicoercive variational hemivariational inequalities , 1995, J. Glob. Optim..

[6]  Vicenţiu D. Rădulescu,et al.  Hartman–Stampacchia results for stably pseudomonotone operators and non-linear hemivariational inequalities , 2010 .

[7]  Jen-Chih Yao,et al.  Regularized Equilibrium Problems with Application to Noncoercive Hemivariational Inequalities , 2004 .

[8]  Zaki Chbani,et al.  Recession methods for equilibrium problems and applications to variational and hemivariational inequalities , 1998 .

[9]  O. Chadli,et al.  Applications of Equilibrium Problems to a Class of Noncoercive Variational Inequalities , 2007 .

[10]  J. Gwinner A Discretization Theory for Monotone Semicoercive Problems and Finite Element Convergence for p-Harmonic Signorini Problems , 1994 .

[11]  J. Haslinger,et al.  Contact problems with nonmonotone friction: discretization and numerical realization , 2007 .

[12]  Joachim Gwinner,et al.  hphp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics , 2013, J. Comput. Appl. Math..

[13]  Anne Kuefer Quasidifferentiability And Nonsmooth Modelling In Mechanics Engineering And Economics , 2016 .

[14]  Panagiotis D. Panagiotopoulos,et al.  Hemivariational Inequalities: Applications in Mechanics and Engineering , 1993 .

[15]  P. Panagiotopoulos Inequality problems in mechanics and applications , 1985 .

[16]  A. Matei,et al.  Contact models leading to variational–hemivariational inequalities , 2012 .

[17]  Joachim Gwinner,et al.  Nichtlineare Variationsungleichungen mit Anwendungen , 1978 .

[18]  P. Panagiotopoulos,et al.  An existence result on noncoercive hemivariational inequalities , 1997 .

[19]  Giuseppe Buttazzo,et al.  Compatibility conditions for nonlinear Neumann problems , 1991 .

[20]  P. Panagiotopoulos,et al.  Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications , 1999 .

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

[22]  Ivan P. Gavrilyuk,et al.  Variational analysis in Sobolev and BV spaces , 2007, Math. Comput..

[23]  Samir Adly,et al.  Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions , 2004, J. Glob. Optim..

[24]  P. D. Panagiotopoulos,et al.  Mathematical Theory of Hemivariational Inequalities and Applications , 1994 .

[25]  M. Schatzman Problèmes aux limites non linéaires, non coercifs , 1973 .

[26]  Error estimates for the approximation of semicoercive variational inequalities , 1994 .

[27]  Zdzislaw Naniewicz,et al.  Semicoercive Variational-hemivariational Inequalities with Unilateral Growth Conditions , 2000, J. Glob. Optim..

[28]  D. Goeleven Noncoercive variational problems and related results , 1996 .

[29]  P. D. Panagiotopoulos,et al.  Coercive and semicoercive hemivariational inequalities , 1991 .

[30]  F. Clarke Optimization And Nonsmooth Analysis , 1983 .

[31]  Zdeněk Dostál,et al.  Solution of Coercive and Semicoercive Contact Problems by FETI Domain Decomposition , 1998 .

[32]  Jen-Chih Yao,et al.  Existence of Solutions for Noncoercive Hemivariational Inequalities by an Equilibrium Approach Under Pseudomonotone Perturbation , 2014, J. Optim. Theory Appl..

[33]  Joachim Gwinner,et al.  A Study of Regularization Techniques of Nondifferentiable Optimization in View of Application to Hemivariational Inequalities , 2014, J. Optim. Theory Appl..

[34]  Dominikus Noll,et al.  Convergence of Non-smooth Descent Methods Using the Kurdyka–Łojasiewicz Inequality , 2014, J. Optim. Theory Appl..

[35]  Samir Adly,et al.  STABILITY OF THE SOLUTION SET OF NON-COERCIVE VARIATIONAL INEQUALITIES , 2002 .

[36]  Samir Adly,et al.  A discretization theory for a class of semi-coercive unilateral problems , 2000, Numerische Mathematik.

[37]  N. Ovcharova Regularization Methods And Finite Element Approximation Of Hemivariational Inequalities With Applications To Nonmonotone Contact Problems , 2017 .

[38]  R. Glowinski Lectures on Numerical Methods for Non-Linear Variational Problems , 1981 .

[39]  Samir Adly,et al.  Stability of linear semi-coercive variational inequalities in Hilbert spaces: application to the Signorini-Fichera problem , 2006 .

[40]  J. Gwinner,et al.  Discretization of semicoercive variational inequalities , 1991 .

[41]  Franco Tomarelli,et al.  Some existence results on noncoercive variational inequalities , 1986 .

[42]  A. Kaplan,et al.  Stable methods for ill-posed variational problems : prox-regularization of elliptic variational inequalities and semi-infinite problems , 1994 .

[43]  Jaroslav Haslinger,et al.  Finite Element Method for Hemivariational Inequalities , 1999 .

[44]  Zhen-Hai Liu,et al.  Elliptic variational hemivariational inequalities , 2003, Appl. Math. Lett..

[45]  Samir Adly,et al.  Recession mappings and noncoercive variational inequalities , 1996 .

[46]  Karl Kunisch,et al.  Obstacle Problems with Cohesion: A Hemivariational Inequality Approach and Its Efficient Numerical Solution , 2011, SIAM J. Optim..

[47]  J. Gwinner,et al.  From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics , 2015 .

[48]  Z. Chbani,et al.  Some existence results for coercive and noncoercive hemivariational inequalities , 1998 .

[49]  Guido Stampacchia VARIATIONAL INEQUALITIES by Guido STAMP ACCHIA , 2017 .

[50]  I. Gavrilyuk Book Review: Variational analysis in Sobolev and BV spaces , 2007 .

[51]  J. Gwinner,et al.  A note on pseudomonotone functions, regularization, and relaxed coerciveness , 1997 .

[52]  V. F. Demʹi︠a︡nov,et al.  Constructive nonsmooth analysis , 1995 .

[53]  Haim Brezis,et al.  Équations et inéquations non linéaires dans les espaces vectoriels en dualité , 1968 .

[54]  J. Gwinner,et al.  On Semicoerciveness, a Class of Variational Inequalities, and an Application to von Kármán Plates , 2002 .

[55]  Nan-Jing Huang,et al.  Existence theorems of the variational-hemivariational inequalities , 2013, J. Glob. Optim..