Nonconvex bundle method with application to a delamination problem

Delamination is a typical failure mode of composite materials caused by weak bonding. It arises when a crack initiates and propagates under a destructive loading. Given the physical law characterizing the properties of the interlayer adhesive between the bonded bodies, we consider the problem of computing the propagation of the crack front and the stress field along the contact boundary. This leads to a hemivariational inequality, which after discretization by finite elements we solve by a nonconvex bundle method, where upper-$$C^1$$C1 criteria have to be minimized. As this is in contrast with other classes of mechanical problems with non-monotone friction laws and in other applied fields, where criteria are typically lower-$$C^1$$C1, we propose a bundle method suited for both types of nonsmoothness. We prove its global convergence in the sense of subsequences and test it on a typical delamination problem of material sciences.

[1]  Pierre Apkarian,et al.  Nonsmooth H∞ synthesis , 2005, IEEE Trans. Autom. Control..

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

[3]  J. Czepiel.  Proximal Bundle Method for simplied unilateral adhesion contact problem of elasticity , 2011 .

[4]  V. A. Kovtunenko,et al.  A hemivariational inequality in crack problems , 2011 .

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

[6]  Warren Hare,et al.  Computing proximal points of nonconvex functions , 2008, Math. Program..

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

[8]  Jaroslav Haslinger,et al.  On numerical solution of hemivariational inequalities by nonsmooth optimization methods , 1995, J. Glob. Optim..

[9]  Anna Nagurney,et al.  Variational Inequalities , 2009, Encyclopedia of Optimization.

[10]  P. Wolfe,et al.  The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices , 1975 .

[11]  Dominikus Noll,et al.  Design of a flight control architecture using a non-convex bundle method , 2013, Math. Control. Signals Syst..

[12]  Jochem Zowe,et al.  A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results , 1992, SIAM J. Optim..

[13]  A. Daniilidis,et al.  Filling the Gap Between Lower-C 1 and Lower-C 2 Functions , 2005 .

[14]  R. Mifflin.  A modification and an extension of Lemarechal’s algorithm for nonsmooth minimization , 1982 .

[15]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

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

[17]  V. Mantič,et al.  A quasistatic mixed-mode delamination model , 2012 .

[18]  Alexander Shapiro,et al.  Differentiability and semismoothness properties of integral functions and their applications , 2005, Math. Program..

[19]  P. Neittaanmäki,et al.  Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control , 1992 .

[20]  M. M. MÄKELÄ,et al.  Comparing Nonsmooth Nonconvex Bundle Methods in Solving Hemivariational Inequalities , 1999, J. Glob. Optim..

[21]  Zdzisław Denkowski,et al.  Set-Valued Analysis , 2021 .

[22]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[23]  J. Gwinner,et al.  On the Regularization Method in Nondifferentiable Optimization Applied to Hemivariational Inequalities , 2014 .

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

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

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

[27]  Minh Ngoc Dao,et al.  Minimizing memory effects of a system , 2015, Math. Control. Signals Syst..

[28]  J. Zowe.  The BT-Algorithm for Minimizing a Nonsmooth Functional Subject to Linear Constraints , 1989 .

[29]  Krzysztof C. Kiwiel,et al.  An aggregate subgradient method for nonsmooth convex minimization , 1983, Math. Program..

[30]  A. Bagirov,et al.  Bundle Methods for Nonsmooth DC Optimization , 2018 .

[31]  D. Noll.  Cutting Plane Oracles to Minimize Non-smooth Non-convex Functions , 2010 .

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

[33]  A. Ruszczynski,et al.  Nonlinear Optimization , 2006 .

[34]  E. Stephan,et al.  Numerical solution of an adhesion problem with FEM and BEM , 2012 .

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

[36]  M. Kocvara,et al.  A Rate-Independent Approach to the Delamination Problem , 2006 .

[37]  M. Miettien.  On numerical solution of hemivariational inequlities by nonsmooth optimization , 1995, J. Glob. Optim..

[38]  J. Spingarn.  Submonotone subdifferentials of Lipschitz functions , 1981 .

[39]  Jan Vlcek,et al.  A bundle-Newton method for nonsmooth unconstrained minimization , 1998, Math. Program..

[40]  Claude Lemaréchal,et al.  Variable metric bundle methods: From conceptual to implementable forms , 1997, Math. Program..

[41]  D. Noll.  Bundle Method for Non-Convex Minimization with Inexact Subgradients and Function Values , 2013 .

[42]  P. Apkarian,et al.  Nonsmooth H ∞ synthesis , 2005 .

[43]  K. Kiwiel.  Methods of Descent for Nondifferentiable Optimization , 1985 .

[44]  Pierre Apkarian,et al.  A Trust Region Spectral Bundle Method for Nonconvex Eigenvalue Optimization , 2008, SIAM J. Optim..

[45]  M. Pullan.  CONVEX ANALYSIS AND MINIMIZATION ALGORITHMS Volumes I and II (Comprehensive Studies in Mathematics 305, 306) , 1995 .

[46]  J. Haslinger,et al.  Mathematical modeling of delamination and nonmonotone friction problems by hemivariational inequalities , 2005 .

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

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

[49]  Krzysztof C. Kiwiel,et al.  A Proximal Bundle Method with Approximate Subgradient Linearizations , 2006, SIAM J. Optim..

[50]  Pierre Apkarian,et al.  Robust eigenstructure clustering by non-smooth optimisation , 2015, Int. J. Control.

[51]  R. Rockafellar,et al.  Prox-regular functions in variational analysis , 1996 .

[52]  Joachim Gwinner,et al.  Finite-element convergence for contact problems in plane linear elastostatics , 1992 .

[53]  Warren Hare,et al.  A proximal bundle method for nonsmooth nonconvex functions with inexact information , 2015, Computational Optimization and Applications.

[54]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[55]  J. Holtmannspötter,et al.  Sensitivity of double cantilever beam test to surface contamination and surface pretreatment , 2013 .

[56]  Aris Daniilidis,et al.  Approximate convexity and submonotonicity , 2004 .

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

[58]  R. Mifflin.  Semismooth and Semiconvex Functions in Constrained Optimization , 1977 .

[59]  Aude Rondepierre,et al.  A Proximity Control Algorithm to Minimize Nonsmooth and Nonconvex Functions , 2008 .