## Finite Element Method for Hemivariational Inequalities: Theory, Methods and ApplicationsHemivariational inequalities represent an important class of problems in nonsmooth and nonconvex mechanics. By means of them, problems with nonmonotone, possibly multivalued, constitutive laws can be formulated, mathematically analyzed and finally numerically solved. The present book gives a rigorous analysis of finite element approximation for a class of hemivariational inequalities of elliptic and parabolic type. Finite element models are described and their convergence properties are established. Discretized models are numerically treated as nonconvex and nonsmooth optimization problems. The book includes a comprehensive description of typical representants of nonsmooth optimization methods. Basic knowledge of finite element mathematics, functional and nonsmooth analysis is needed. The book is self-contained, and all necessary results from these disciplines are summarized in the introductory chapter. Audience: Engineers and applied mathematicians at universities and working in industry. Also graduate-level students in advanced nonlinear computational mechanics, mathematics of finite elements and approximation theory. Chapter 1 includes the necessary prerequisite materials. |

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### Contents

MATHEMATICAL PRELIMINARIES | 3 |

12 ELEMENTS OF NONSMOOTH ANALYSIS | 18 |

13 EQUATIONS AND INEQUALITIES WITH MONOTONE OPERATORS | 26 |

14 APPROXIMATION OF EQUATIONS AND INEQUALITIES OF MONOTONE TYPE | 48 |

References | 81 |

NONSMOOTH MECHANICS CONVEX AND NONCONVEX PROBLEMS | 83 |

22 NONLINEAR ELASTOSTATICS | 85 |

23 LITERATURE REVIEW | 98 |

TIME DEPENDENT CASE | 163 |

41 DISCRETIZATION | 166 |

42 CONVERGENCE ANALYSIS | 170 |

43 ALGEBRAIC REPRESENTATION | 194 |

44 CONSTRAINED HEMIVARIATIONAL INEQUALITIES | 196 |

References | 200 |

NONSMOOTH OPTIMIZATION METHODS | 203 |

NONSMOOTH OPTIMIZATION METHODS | 205 |

Finite Element Approximation of Hemivariational Inequalities | 101 |

APPROXIMATION OF ELLIPTIC HEMIVARIATIONAL INEQUALITIES | 103 |

31 AUXILIARY RESULTS | 107 |

32 DISCRETIZATION | 110 |

33 CONVERGENCE ANALYSIS | 115 |

34 CONSTRUCTION OF FINITE ELEMENT SPACES AND INTERPOLATION OPERATORS | 121 |

35 ALGEBRAIC REPRESENTATION | 134 |

36 CONSTRAINED HEMIVARIATIONAL INEQUALITIES | 139 |

37 APPROXIMATION OF VECTORVALUED HEMIVARIATIONAL INEQUALITIES | 151 |

References | 161 |

52 NONCONVEX CASE | 217 |

References | 225 |

Numerical Examples | 229 |

NUMERICAL EXAMPLES | 231 |

61 NONMONOTONE FRICTION AND CONTACT PROBLEMS | 232 |

62 DELAMINATION PROBLEM | 249 |

258 | |

259 | |

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### Common terms and phrases

algebraic algorithm analysis approximation assumptions Banach space bilinear form boundary conditions bounded bundle methods closed and convex component constrained hemivariational inequality contact nodes convergence result convex function convex set convex subset cutting plane defined definition denote derivative df(u differentiable discrete displacement elliptic equation equivalent example exists Find finite dimensional finite element finite element method follows formulation friction function f given Glowinski gradient Green's formula Hilbert space holds indicator function integral iteration Kiwiel l,oo Lemarechal Lemma limsup linearization error Lipschitz function mapping Ph mathematical matrix Miettinen minimization monotone multivalued nonconvex nondifferentiable nonempty nonlinear nonmonotone norm numerical Panagiotopoulos parabolic piecewise linear positive constant priori estimates problem relation Remark respectively satisfied scalar sequence Sobolev spaces solves subgradient subgradient method subsequence subspace substationary point superpotential uh,Eh unique solution variable metric variational inequalities vector