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Nonlinear Cauchy problem and identification in contact mechanics: a solving method based on Bregman-gap. (English) Zbl 1452.35250

Summary: This paper proposes a solution method for identification problems in the context of contact mechanics when overabundant data are available on a part \(\Gamma_m\) of the domain boundary while data are missing from another part of this boundary. The first step is then to find a solution to a Cauchy problem. The method used by the authors for solving Cauchy problems consists of expanding the displacement field known on \(\Gamma_m\) toward the inside of the solid via the minimization of a function that measures the gap between solutions of two well-posed problems, each one exploiting only one of the superabundant data. The key question is then to build an appropriate gap functional in strongly nonlinear contexts. The proposed approach exploits a generalization of the Bregman divergence, using the thermodynamic potentials as generating functions within the framework of generalized standard materials (GSMs), but also implicit GSMs in order to address Coulomb friction. The robustness and efficiency of the proposed method are demonstrated by a numerical bi-dimensional application dealing with a cracked elastic solid with unilateral contact and friction effects on the crack’s lips.

MSC:

35R30 Inverse problems for PDEs
35J86 Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators
35Q74 PDEs in connection with mechanics of deformable solids

Software:

SciPy; Code_Aster; Matlab
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Full Text: DOI HAL

References:

[1] Haslinger J, Hlavácek I and Necas J 1996 Numerical methods for unilateral problems in solid mechanics Finite Element Methods (Part 2), Numerical Methods for Solids (Part 2)(Handbook of Numerical Analysis vol 4) (Amsterdam: Elsevier) pp 313-485 · Zbl 0873.73079 · doi:10.1016/S1570-8659(96)80005-6
[2] Wriggers P 2006 Computational Contact Mechanics (Berlin: Springer) · Zbl 1104.74002 · doi:10.1007/978-3-540-32609-0
[3] Halphen B and Nguyen Q S 1975 Sur les matériaux standards généralisés J. Mec.14 39-63 · Zbl 0308.73017
[4] Amontons G 1699 De la résistance causée dans les machines Mémoires de l’Académie Royale A 257-82
[5] Coulomb C A 1821 Théorie des machines simples (Paris: Bachelier)
[6] Fremond M 1988 Contact with Adhesion (Vienna: Springer) pp 177-221 · Zbl 0656.73051
[7] Raous M, Cangémi L and Cocu M 1999 A consistent model coupling adhesion, friction, and unilateral contact Comput. Methods Appl. Mech. Eng.177 383-99 · Zbl 0949.74008 · doi:10.1016/s0045-7825(98)00389-2
[8] Sutton A, Orteu J J and Schreier H 2009 Image Correlation for Shape, Motion and Deformation Measurements Basic Concepts,Theory and Applications (Berlin: Springer)
[9] Avril S et al 2008 Overview of identification methods of mechanical parameters based on full-fields measurements Exp. Mech.48 381-402 · doi:10.1007/s11340-008-9148-y
[10] Hadamard J 1923 Lectures on Cauchy’s Problem in Linear Partial Differential Equations (New York: Dover) · JFM 49.0725.04
[11] Fichera G 1963 Sul problema elastostatico di Signorini con ambigue condizioni al contorno Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat.34 138-42 · Zbl 0128.18305
[12] Kozlov V A, Maz’ya V G and Fomin A F 1992 An iterative method for solving the Cauchy problem for elliptic equations USSR Computational Mathematics and Mathematical Physics31 45-52 http://www.ams.org/mathscinet-getitem?mr=1099360 · Zbl 0774.65069
[13] Baumeister J and Leitão A 2001 On iterative methods for solving ill-posed problems modeled by partial differential equations J. Inverse Ill-Posed Problems9 13-29 · Zbl 0980.35166 · doi:10.1515/jiip.2001.9.1.13
[14] Marin L and Lesnic D 2002 Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition Comput. Methods Appl. Mech. Eng.191 3257-70 · Zbl 1045.74050 · doi:10.1016/s0045-7825(02)00262-1
[15] Marin L and Lesnic D 2004 The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity Int. J. Solids Struct.41 3425-38 · Zbl 1071.74055 · doi:10.1016/j.ijsolstr.2004.02.009
[16] Belgacem F B and Fekih H E 2005 On Cauchy’s problem: I. A variational Steklov-Poincaré theory Inverse Problems21 1915 · Zbl 1112.35054 · doi:10.1088/0266-5611/21/6/008
[17] Azaïez M, Belgacem F B and Fekih H E 2006 On Cauchy’s problem: II. Completion, regularization and approximation Inverse Problems22 1307-36 · Zbl 1113.35046 · doi:10.1088/0266-5611/22/4/012
[18] Young D L, Tsai C C, Chen C W and Fan C M 2008 The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation Comput. Math. Appl.55 1189-200 · Zbl 1143.65087 · doi:10.1016/j.camwa.2007.05.015
[19] Hon Y C and Wei T 2001 Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation Inverse Problems17 261 · Zbl 0980.35167 · doi:10.1088/0266-5611/17/2/306
[20] Cimetière A, Delvare F, Jaoua M and Pons F 2001 Solution of the Cauchy problem using iterated Tikhonov regularization Inverse Problems17 553-70 · Zbl 0986.35128 · doi:10.1088/0266-5611/17/3/313
[21] Bourgeois L 2005 A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation Inverse Problems21 1087-104 · Zbl 1071.35120 · doi:10.1088/0266-5611/21/3/018
[22] Bourgeois L 2006 Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace’s equation Inverse Problems22 413 · Zbl 1094.35134 · doi:10.1088/0266-5611/22/2/002
[23] Egger H and Leitão A 2009 Nonlinear regularization methods for ill-posed problems with piecewise constant or strongly varying solutions Inverse Problems25 115014 · Zbl 1186.65126 · doi:10.1088/0266-5611/25/11/115014
[24] Klüger P and Leitão A 2003 Mean value iterations for nonlinear elliptic Cauchy problems Numer. Math.96 269-93 · Zbl 1081.65104 · doi:10.1007/s00211-003-0477-6
[25] Andrieux S and Baranger T N 2008 An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity Comput. Methods Appl. Mech. Eng.197 902-20 · Zbl 1169.74335 · doi:10.1016/j.cma.2007.08.022
[26] Baranger T N and Andrieux S 2008 An optimization approach for the Cauchy problem in linear elasticity Struct. Multidiscip. Optim.35 141-52 · Zbl 1273.74107 · doi:10.1007/s00158-007-0123-5
[27] Baranger T N and Andrieux S 2009 Data completion for linear symmetric operators as a Cauchy problem: an efficient method via energy like error minimization Vietnam J. Mech.31 247-61 · doi:10.15625/0866-7136/31/3-4/5652
[28] Andrieux S and Baranger T N 2016 On the determination of missing boundary data for solids with nonlinear material behaviors, using displacement fields measured on a part of their boundaries J. Mech. Phys. Solids97 140-55 · Zbl 1451.74111 · doi:10.1016/j.jmps.2016.02.008
[29] Andrieux S and Baranger T N 2015 Solution of nonlinear Cauchy problem for hyperelastic solids Inverse Problems31 115003 · Zbl 1333.35269 · doi:10.1088/0266-5611/31/11/115003
[30] Baranger T N, Andrieux S and Dang T B T 2015 The incremental Cauchy problem in elastoplasticity: general solution method and semi-analytic formulae for the pressurised hollow sphere C. R. Méc.343 331-43 · doi:10.1016/j.crme.2015.04.002
[31] Andrieux S and Baranger T N 2013 Three-dimensional recovery of stress intensity factors and energy release rates from surface full-field displacements Int. J. Solids Struct.50 1523-37 · doi:10.1016/j.ijsolstr.2013.01.002
[32] Andrieux S and Baranger T N 2012 Emerging crack front identification from tangential surface displacements C. R. Méc.340 565-74 · doi:10.1016/j.crme.2012.06.002
[33] Baranger T N and Andrieux S 2011 Constitutive law gap functionals for solving the Cauchy problem for linear elliptic PDE Appl. Math. Comput.218 1970-89 · Zbl 1269.65094 · doi:10.1016/j.amc.2011.07.009
[34] Rischette R, Baranger T N and Debit N 2011 Numerical analysis of an energy-like minimization method to solve the Cauchy problem with noisy data J. Comput. Appl. Math.235 3257-69 · Zbl 1216.65149 · doi:10.1016/j.cam.2010.12.019
[35] Rischette R, Baranger T N and Andrieux S 2013 Regularization of the noisy cauchy problem solution approximated by an energy-like method Int. J. Numer. Methods Eng.95 271-87 · Zbl 1352.65443 · doi:10.1002/nme.4501
[36] Rischette R, Baranger T N and Debit N 2014 Numerical analysis of an energy-like minimization method to solve a parabolic Cauchy problem with noisy data J. Comput. Appl. Math.271 206-22 · Zbl 1321.65147 · doi:10.1016/j.cam.2014.03.024
[37] Baranger T N, Andrieux S and Rischette R 2014 Combined energy method and regularization to solve the Cauchy problem for the heat equation Inverse Probl. Sci. Eng.22 199-212 · Zbl 1304.65212 · doi:10.1080/17415977.2013.836191
[38] Baranger T N, Johansson B T and Rischette R 2013 On the alternating method for Cauchy problems and its finite element discretisation Applied Inverse Problems(Springer Proceedings in Mathematics & Statistics vol 48) ed L Beilina (New York: Springer) pp 183-97 · Zbl 1278.65150 · doi:10.1007/978-1-4614-7816-4_11
[39] Escriva X, Baranger T N and Tlatli N H 2007 Leak identification in porous media by solving the Cauchy problem C. R. Méc.335 401-6 · doi:10.1016/j.crme.2007.04.001
[40] Andrieux S, Baranger T N and Abda A B 2006 Solving Cauchy problems by minimizing an energy-like functional Inverse Problems22 115-33 · Zbl 1089.35084 · doi:10.1088/0266-5611/22/1/007
[41] Andrieux S, Ben Abda A and Nouri Baranger T 2005 Data completion via an energy error functional C. R. Méc.333 171-7 · Zbl 1223.80004 · doi:10.1016/j.crme.2004.10.005
[42] Cocu M 1984 Existence of solutions of Signorini problems with friction Int. J. Eng. Sci.22 567-75 · Zbl 0554.73096 · doi:10.1016/0020-7225(84)90058-2
[43] Bregman L M 1967 The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming USSR Comput. Math. Math. Phys.7 200-17 · Zbl 0186.23807 · doi:10.1016/0041-5553(67)90040-7
[44] Censor Y and Lent A 1981 An iterative row-action method for interval convex programming J. Optim. Theory Appl.34 321-53 · Zbl 0431.49042 · doi:10.1007/bf00934676
[45] Banerjee A, Merugu S, Dhillon I S and Ghosh J 2005 Clustering with Bregman divergences Journal of Machine Learning Research6 1705-49 · Zbl 1190.62117
[46] Frigyik A B, Srivastava S and Gupta M R 2008 Functional Bregman divergence and Bayesian estimation of distributions IEEE Trans. Inf. Theory54 5130-9 · Zbl 1319.62137 · doi:10.1109/tit.2008.929943
[47] De Saxce G and Feng Z Q 1991 New inequality and functional for contact with friction: the implicit standard material approach∗ Mech. Struct. Mach.19 301-25 · doi:10.1080/08905459108905146
[48] Laborde P and Renard Y 2007 Fixed point strategies for elastostatic frictional contact problems Math. Methods Appl. Sci.31 415-41 · Zbl 1132.74032 · doi:10.1002/mma.921
[49] Khenous H, Pommier J and Renard Y 2006 Hybrid discretization of the signorini problem with coulomb friction. theoretical aspects and comparison of some numerical solvers Appl. Numer. Math.56 163-92 · Zbl 1089.74046 · doi:10.1016/j.apnum.2005.03.002
[50] EDF Electricité de France 2020 Finite element Code_aster, analysis of structures and thermomechanics for studies and research 1989-2018 Open source on www.code-aster.org
[51] Jones E, Oliphant T, Peterson P et al 2020 SciPy: Open Source Scientific Tools for Python 2001-18
[52] Mathwork 2018 Matlab Version R2018a
[53] Colton D and Kress R 1998 Inverse Acoustic and Electromagnetic Scattering Theory (Berlin: Springer) · Zbl 0893.35138 · doi:10.1007/978-3-662-03537-5
[54] Bui H D 2006 Fracture Mechanics: Inverse Problems and Solutions (Berlin: Springer) · Zbl 1108.74002
[55] Rudin L I, Osher S and Fatemi E 1992 Nonlinear total variation based noise removal algorithms Physica D 60 259-68 · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-f
[56] Belgacem F B 2007 Why is the Cauchy problem severely ill-posed? Inverse Problems23 823-36 · Zbl 1118.35060 · doi:10.1088/0266-5611/23/2/020
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