With cedram.org   version française
Search for an article
Table of contents for this volume | Previous article
Aymeric Maury; Grégoire Allaire; François Jouve
Shape optimisation with the level set method for contact problems in linearised elasticity
SMAI-Journal of computational mathematics, 3 (2017), p. 249-292, doi: 10.5802/smai-jcm.27
Article PDF
Class. Math.: 74P05, 75P10, 74P15, 74M10, 74M15, 49Q10, 49Q12, 35J85
Keywords: Shape and topology Optimisation; Level set method; Unilateral contact problems; Frictional contact; Penalisation and Regularisation


This article is devoted to shape optimisation of contact problems in linearised elasticity, thanks to the level set method. We circumvent the shape non-differentiability, due to the contact boundary conditions, by using penalised and regularised versions of the mechanical problem. This approach is applied to five different contact models: the frictionless model, the Tresca model, the Coulomb model, the normal compliance model and the Norton-Hoff model. We consider two types of optimisation problems in our applications: first, we minimise volume under a compliance constraint, second, we optimise the normal force, with a volume constraint, which is useful to design compliant mechanisms. To illustrate the validity of the method, 2D and 3D examples are performed, the 3D examples being computed with an industrial software.


[1] G. Allaire, Conception optimale de structures, Mathématiques & Applications [Mathematics & Applications] 58, Springer-Verlag, Berlin, 2007
[2] G. Allaire, F. Jouve & A.-M. Toader, “Structural optimization using sensitivity analysis and a level-set method”, Journal of computational physics 194 (2004) no. 1, p. 363-393 Article
[3] A. Amassad, D. Chenais & C. Fabre, “Optimal control of an elastic contact problem involving Tresca friction law”, Nonlinear Anal. 48 (2002) no. 8, Ser. A: Theory Methods, p. 1107-1135 Article
[4] J. Andersson, “Optimal regularity and free boundary regularity for the Signorini problem”, Algebra i Analiz 24 (2012) no. 3, p. 1-21 Article
[5] J. Andersson, “Optimal regularity for the Signorini problem and its free boundary”, Invent. Math. 204 (2016) no. 1, p. 1-82 Article
[6] V. Barbu, Optimal control of variational inequalities, Research Notes in Mathematics 100, Pitman (Advanced Publishing Program), Boston, MA, 1984
[7] P. Beremlijski, J. Haslinger, M. Kočvara & J.V. Outrata, “Shape optimization in contact problems with Coulomb friction”, SIAM J. Optim. 13 (2002) no. 2, p. 561-587 Article
[8] P. Beremlijski, J. Haslinger, J.V. Outrata & R. Pathó, “Shape optimization in contact problems with Coulomb friction and a solution-dependent friction coefficient”, SIAM J. Control Optim. 52 (2014) no. 5, p. 3371-3400 Article
[9] P. Boieri, F. Gastaldi & D. Kinderlehrer, “Existence, uniqueness, and regularity results for the two-body contact problem”, Appl. Math. Optim. 15 (1987) no. 3, p. 251-277 Article
[10] J. Céa, “Conception optimale ou identification de formes: calcul rapide de la dérivée directionnelle de la fonction coût”, RAIRO Modél. Math. Anal. Numér. 20 (1986) no. 3, p. 371-402 Article
[11] W.-H. Chen & C.-R. Ou, “Shape optimization in contact problems with desired contact traction distribution on the specified contact surface”, Computational Mechanics 15 (1995), p. 534-545 Article
[12] J.E. Jr. Dennis & R.B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Classics in Applied Mathematics 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996, Corrected reprint of the 1983 original Article
[13] B. Desmorat, “Structural rigidity optimization with frictionless unilateral contact”, Internat. J. Solids Structures 44 (2007) no. 3-4, p. 1132-1144 Article
[14] S. Drabla, M. Sofonea & B. Teniou, “Analysis of a frictionless contact problem for elastic bodies”, Ann. Polon. Math. 69 (1998) no. 1, p. 75-88 Article
[15] G. Duvaut & J.L. Lions, Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques 21, Dunod, Paris, 1972
[16] C. Eck, J. Jarusek & M. Krbec, Unilateral contact problems, Variational methods and existence theorems, Pure and Applied Mathematics (Boca Raton) 270, Chapman & Hall/CRC, Boca Raton, FL, 2005 Article
[17] ESI group, SYSTUS: a multiphysics simulation software
[18] R. Glowinski, J.L. Lions & R. Trémolières, Analyse numérique des inéquations variationnelles. Tome 1, Théorie générale premières applications, Méthodes Mathématiques de l’Informatique 5, Dunod, Paris, 1976
[19] H. Goldberg, W. Kampowsky & F. Tröltzsch, “On Nemytskij operators in $L_p$-spaces of abstract functions”, Math. Nachr. 155 (1992), p. 127-140 Article
[20] W. Han, “On the numerical approximation of a frictional contact problem with normal compliance”, Numer. Funct. Anal. Optim. 17 (1996) no. 3-4, p. 307-321 Article
[21] J. Haslinger, “Approximation of the Signorini problem with friction, obeying the Coulomb law”, Math. Methods Appl. Sci. 5 (1983) no. 3, p. 422-437 Article
[22] J. Haslinger, Shape optimization in contact problems, in Equadiff 6 (Brno, 1985), Univ. J. E. Purkyně, Brno, 1986, p. 445-450
[23] J. Haslinger, “Signorini problem with Coulomb’s law of friction. Shape optimization in contact problems”, Internat. J. Numer. Methods Engrg. 34 (1992) no. 1, p. 223-231, The Second World Congress of Computational Mechanics, Part I (Stuttgart, 1990) Article
[24] J. Haslinger & A. Klarbring, “Shape optimization in unilateral contact problems using generalized reciprocal energy as objective functional”, Nonlinear Anal. 21 (1993) no. 11, p. 815-834 Article
[25] J. Haslinger & P. Neittaanmäki, “On the existence of optimal shapes in contact problems”, Numer. Funct. Anal. Optim. 7 (1984/85) no. 2-3, p. 107-124 Article
[26] J. Haslinger & P. Neittaanmäki, “Shape optimization in contact problems. Approximation and numerical realization”, RAIRO Modél. Math. Anal. Numér. 21 (1987) no. 2, p. 269-291 Article
[27] J. Haslinger, P. Neittaanmäki & T. Tiihonen, “Shape optimization in contact problems based on penalization of the state inequality”, Apl. Mat. 31 (1986) no. 1, p. 54-77
[28] A. Henrot & M. Pierre, Variation et optimisation de formes, une analyse géométrique. [A geometric analysis], Mathématiques & Applications [Mathematics & Applications] 48, Springer, Berlin, 2005 Article
[29] J Herskovits, A Leontiev, G Dias & G Santos, “Contact shape optimization: a bilevel programming approach”, Structural and multidisciplinary optimization 20 (2000) no. 3, p. 214-221 Article
[30] P. Hild, “Two results on solution uniqueness and multiplicity for the linear elastic friction problem with normal compliance”, Nonlinear Anal. 71 (2009) no. 11, p. 5560-5571 Article
[31] D. Hilding, A. Klarbring & J. Petersson, “Optimization of structures in unilateral contact”, ASME Appl Mech Rev 52 (1999) no. 4, p. 1-4 Article
[32] T. Iwai, A. Sugimoto, T. Aoyama & H. Azegami, “Shape optimization problem of elastic bodies for controlling contact pressure”, JSIAM Lett. 2 (2010), p. 1-4 Article
[33] J. Jarušek & J.V. Outrata, “On sharp necessary optimality conditions in control of contact problems with strings”, Nonlinear Anal. 67 (2007) no. 4, p. 1117-1128 Article
[34] N.H. Kim, K.K. Choi & J.S. Chen, “Shape Design Sensitivity Analysis and Optimization of Elasto-Plasticity with Frictional Contact”, AIAA Journal 38 (2000) no. 9, p. 1742-1753 Article
[35] N.H. Kim, K.K. Choi, J.S. Chen & Y.H. Park, “Meshless shape design sensitivity analysis and optimization for contact problem with friction”, Computational Mechanics 25 (2000), p. 157-168 Article
[36] D. Kinderlehrer & G. Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980
[37] A. Klarbring, “On the problem of optimizing contact force distributions”, J. Optim. Theory Appl. 74 (1992) no. 1, p. 131-150 Article
[38] A. Klarbring, A. Mikelić & M. Shillor, “On friction problems with normal compliance”, Nonlinear Anal. 13 (1989) no. 8, p. 935-955 Article
[39] A. Klarbring, A. Mikelić & M. Shillor, “Optimal shape design in contact problems with normal compliance and friction”, Appl. Math. Lett. 5 (1992) no. 2, p. 51-55 Article
[40] D. Knees & A. Schröder, “Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints”, Math. Methods Appl. Sci. 35 (2012) no. 15, p. 1859-1884 Article
[41] P. Laborde & Y. Renard, “Fixed point strategies for elastostatic frictional contact problems”, Math. Methods Appl. Sci. 31 (2008) no. 4, p. 415-441 Article
[42] W. Li, Q. Li, G. P Steven & Y.M. Xie, “An evolutionary shape optimization for elastic contact problems subject to multiple load cases”, Computer methods in applied mechanics and engineering 194 (2005) no. 30, p. 3394-3415 Article
[43] N.D. Mankame & G.K. Ananthasuresh, “Topology optimization for synthesis of contact-aided compliant mechanisms using regularized contact modeling”, International Conference on Modeling, Simulation and Optimization for Design of Multi-disciplinary Engineering Systems 24-26 September, Goa, India (2004)
[44] F. Mignot, “Contrôle dans les inéquations variationelles elliptiques”, Journal of Functional Analysis 22 (1976) no. 2, p. 130-185 Article
[45] I. Milne, R.O. Ritchie & B. Karihaloo, Comprehensive structural integrity, Elsevier Science, 2003
[46] F. Murat & J. Simon, “Etudes de problèmes d’optimal design”, Lecture Notes in Computer Science, Springer Verlag, Berlin 41 (1976), p. 54-62 Article
[47] J. T. Oden & J. A. C. Martins, “Models and computational methods for dynamic friction phenomena”, Comput. Methods Appl. Mech. Engrg. 52 (1985) no. 1-3, p. 527-634, FENOMECH ’84, Part III, IV (Stuttgart, 1984) Article
[48] S. Osher & R. Fedkiw, Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences 153, Springer-Verlag, New York, 2003
[49] S. Osher & J.A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations”, J. Comput. Phys. 79 (1988) no. 1, p. 12-49 Article
[50] J.V. Outrata, “On the numerical solution of a class of Stackelberg problems”, Z. Oper. Res. 34 (1990) no. 4, p. 255-277 Article
[51] J.V. Outrata, J. Jarušek & J. Stará, “On optimality conditions in control of elliptic variational inequalities”, Set-Valued Var. Anal. 19 (2011) no. 1, p. 23-42 Article
[52] I. Paczelt & T. Szabo, “Optimal shape design for contact problems”, Structural Optimization 7 (1994), p. 66-75 Article
[53] O. Pironneau, Optimal shape design for elliptic systems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984 Article
[54] R. Schumann, “Regularity for Signorini’s problem in linear elasticity”, Manuscripta Math. 63 (1989), p. 255-291 Article
[55] Scilab Enterprises, Scilab: Le logiciel open source gratuit de calcul numérique, Scilab Enterprises, Orsay, France, 2012
[56] J.A. Sethian, Level set methods and fast marching methods, Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge Monographs on Applied and Computational Mathematics 3, Cambridge University Press, Cambridge, 1999
[57] J. Simon, “Differentiation with respect to the domain in boundary value problems”, Numer. Funct. Anal. Optim. 2 (1980) no. 7-8, p. 649-687 (1981) Article
[58] J. Sokolowski & J-P. Zolesio, Introduction to shape optimization, shape sensitivity analysis, Springer Series in Computational Mathematics 16, Springer-Verlag, Berlin, 1992 Article
[59] N. Strömberg & A. Klarbring, “Topology Optimization of Structures with Contact Constraints by using a Smooth Formulation and a Nested Approach”, 8th World Congress on Structural and Multidisciplinary Optimization (2009)
[60] N. Strömberg & A. Klarbring, “Topology optimization of structures in unilateral contact”, Struct. Multidiscip. Optim. 41 (2010) no. 1, p. 57-64 Article
[61] S. Stupkiewicz, J. Lengiewicz & J. Korelc, “Sensitivity analysis for frictional contact problems in the augmented Lagrangian formulation”, Comput. Methods Appl. Mech. Engrg. 199 (2010) no. 33-36, p. 2165-2176 Article
[62] N. Tardieu & A. Constantinescu, “On the determination of elastic coefficients from indentation experiments”, Inverse Problems 16 (2000) no. 3, p. 577-588 Article
[63] F. Tröltzsch, Optimal control of partial differential equations, Theory, methods and applications, Graduate Studies in Mathematics 112, American Mathematical Society, Providence, RI, 2010 Article
[64] M.Y. Wang, X. Wang & D. Guo, “A level set method for structural topology optimization”, Comput. Methods Appl. Mech. Engrg. 192 (2003) no. 1-2, p. 227-246 Article