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Shape optimisation with the level set method for contact problems in linearised elasticity
Aymeric Maury1; Grégoire Allaire2; François Jouve1
1 Laboratoire J.L. Lions (UMR CNRS 7598), University Paris Diderot, Paris, France
2 CMAP (UMR CNRS 7641), Ecole Polytechnique, Palaiseau, France
The SMAI Journal of computational mathematics, Volume 3 (2017), pp. 249-292.

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.

Published online:
DOI: 10.5802/smai-jcm.27
Classification: 74P05, 75P10, 74P15, 74M10, 74M15, 49Q10, 49Q12, 35J85
Keywords: Shape and topology Optimisation; Level set method; Unilateral contact problems; Frictional contact; Penalisation and Regularisation
Aymeric Maury 1; Grégoire Allaire 2; François Jouve 1

1 Laboratoire J.L. Lions (UMR CNRS 7598), University Paris Diderot, Paris, France
2 CMAP (UMR CNRS 7641), Ecole Polytechnique, Palaiseau, France
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Shape optimisation with the level set method for contact problems in linearised elasticity},
     journal = {The SMAI Journal of computational mathematics},
     pages = {249--292},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Aymeric Maury; Grégoire Allaire; François Jouve. Shape optimisation with the level set method for contact problems in linearised elasticity. The SMAI Journal of computational mathematics, Volume 3 (2017), pp. 249-292. doi : 10.5802/smai-jcm.27. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.27/

[1] G. Allaire Conception optimale de structures, Mathématiques & Applications [Mathematics & Applications], 58, Springer-Verlag, Berlin, 2007, xii+278 pages | MR | Zbl

[2] G. Allaire; F. Jouve; A.-M. Toader Structural optimization using sensitivity analysis and a level-set method, Journal of computational physics, Volume 194 (2004) no. 1, pp. 363-393 | DOI | MR | Zbl

[3] A. Amassad; D. Chenais; C. Fabre Optimal control of an elastic contact problem involving Tresca friction law, Nonlinear Anal., Volume 48 (2002) no. 8, Ser. A: Theory Methods, pp. 1107-1135 | DOI | MR | Zbl

[4] J. Andersson Optimal regularity and free boundary regularity for the Signorini problem, Algebra i Analiz, Volume 24 (2012) no. 3, pp. 1-21 | DOI | MR | Zbl

[5] J. Andersson Optimal regularity for the Signorini problem and its free boundary, Invent. Math., Volume 204 (2016) no. 1, pp. 1-82 | DOI | MR | Zbl

[6] V. Barbu Optimal control of variational inequalities, Research Notes in Mathematics, 100, Pitman (Advanced Publishing Program), Boston, MA, 1984, iv+298 pages | MR | Zbl

[7] P. Beremlijski; J. Haslinger; M. Kočvara; J.V. Outrata Shape optimization in contact problems with Coulomb friction, SIAM J. Optim., Volume 13 (2002) no. 2, pp. 561-587 | DOI | MR | Zbl

[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., Volume 52 (2014) no. 5, pp. 3371-3400 | DOI | MR | Zbl

[9] P. Boieri; F. Gastaldi; D. Kinderlehrer Existence, uniqueness, and regularity results for the two-body contact problem, Appl. Math. Optim., Volume 15 (1987) no. 3, pp. 251-277 | DOI | MR

[10] J. Céa Conception optimale ou identification de formes: calcul rapide de la dérivée directionnelle de la fonction coût, RAIRO Modél. Math. Anal. Numér., Volume 20 (1986) no. 3, pp. 371-402 | DOI | Numdam | MR | Zbl

[11] W.-H. Chen; C.-R. Ou Shape optimization in contact problems with desired contact traction distribution on the specified contact surface, Computational Mechanics, Volume 15 (1995), pp. 534-545 | DOI | Zbl

[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, xvi+378 pages (Corrected reprint of the 1983 original) | DOI | MR

[13] B. Desmorat Structural rigidity optimization with frictionless unilateral contact, Internat. J. Solids Structures, Volume 44 (2007) no. 3-4, pp. 1132-1144 | DOI | MR | Zbl

[14] S. Drabla; M. Sofonea; B. Teniou Analysis of a frictionless contact problem for elastic bodies, Ann. Polon. Math., Volume 69 (1998) no. 1, pp. 75-88 | DOI | MR | Zbl

[15] G. Duvaut; J.L. Lions Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques, 21, Dunod, Paris, 1972, xx+387 pages | MR | Zbl

[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, x+398 pages | DOI | MR | Zbl

[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, xii+268 pages | MR | Zbl

[19] H. Goldberg; W. Kampowsky; F. Tröltzsch On Nemytskij operators in L p -spaces of abstract functions, Math. Nachr., Volume 155 (1992), pp. 127-140 | DOI | MR | Zbl

[20] W. Han On the numerical approximation of a frictional contact problem with normal compliance, Numer. Funct. Anal. Optim., Volume 17 (1996) no. 3-4, pp. 307-321 | DOI | MR | Zbl

[21] J. Haslinger Approximation of the Signorini problem with friction, obeying the Coulomb law, Math. Methods Appl. Sci., Volume 5 (1983) no. 3, pp. 422-437 | DOI | MR | Zbl

[22] J. Haslinger, Equadiff 6 (Brno, 1985) (1986), pp. 445-450 | MR

[23] J. Haslinger Signorini problem with Coulomb’s law of friction. Shape optimization in contact problems, Internat. J. Numer. Methods Engrg., Volume 34 (1992) no. 1, pp. 223-231 The Second World Congress of Computational Mechanics, Part I (Stuttgart, 1990) | DOI | MR | Zbl

[24] J. Haslinger; A. Klarbring Shape optimization in unilateral contact problems using generalized reciprocal energy as objective functional, Nonlinear Anal., Volume 21 (1993) no. 11, pp. 815-834 | DOI | MR | Zbl

[25] J. Haslinger; P. Neittaanmäki On the existence of optimal shapes in contact problems, Numer. Funct. Anal. Optim., Volume 7 (1984/85) no. 2-3, pp. 107-124 | DOI | MR | Zbl

[26] J. Haslinger; P. Neittaanmäki Shape optimization in contact problems. Approximation and numerical realization, RAIRO Modél. Math. Anal. Numér., Volume 21 (1987) no. 2, pp. 269-291 | DOI | Numdam | MR | Zbl

[27] J. Haslinger; P. Neittaanmäki; T. Tiihonen Shape optimization in contact problems based on penalization of the state inequality, Apl. Mat., Volume 31 (1986) no. 1, pp. 54-77 | MR | Zbl

[28] A. Henrot; M. Pierre Variation et optimisation de formes, une analyse géométrique. [A geometric analysis], Mathématiques & Applications [Mathematics & Applications], 48, Springer, Berlin, 2005, xii+334 pages | DOI | MR | Zbl

[29] J Herskovits; A Leontiev; G Dias; G Santos Contact shape optimization: a bilevel programming approach, Structural and multidisciplinary optimization, Volume 20 (2000) no. 3, pp. 214-221 | DOI

[30] P. Hild Two results on solution uniqueness and multiplicity for the linear elastic friction problem with normal compliance, Nonlinear Anal., Volume 71 (2009) no. 11, pp. 5560-5571 | DOI | MR | Zbl

[31] D. Hilding; A. Klarbring; J. Petersson Optimization of structures in unilateral contact, ASME Appl Mech Rev, Volume 52 (1999) no. 4, pp. 1-4 | DOI

[32] T. Iwai; A. Sugimoto; T. Aoyama; H. Azegami Shape optimization problem of elastic bodies for controlling contact pressure, JSIAM Lett., Volume 2 (2010), pp. 1-4 | DOI | MR | Zbl

[33] J. Jarušek; J.V. Outrata On sharp necessary optimality conditions in control of contact problems with strings, Nonlinear Anal., Volume 67 (2007) no. 4, pp. 1117-1128 | DOI | MR | Zbl

[34] N.H. Kim; K.K. Choi; J.S. Chen Shape Design Sensitivity Analysis and Optimization of Elasto-Plasticity with Frictional Contact, AIAA Journal, Volume 38 (2000) no. 9, pp. 1742-1753 | DOI

[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, Volume 25 (2000), pp. 157-168 | DOI

[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, xiv+313 pages | MR | Zbl

[37] A. Klarbring On the problem of optimizing contact force distributions, J. Optim. Theory Appl., Volume 74 (1992) no. 1, pp. 131-150 | DOI | MR | Zbl

[38] A. Klarbring; A. Mikelić; M. Shillor On friction problems with normal compliance, Nonlinear Anal., Volume 13 (1989) no. 8, pp. 935-955 | DOI | MR | Zbl

[39] A. Klarbring; A. Mikelić; M. Shillor Optimal shape design in contact problems with normal compliance and friction, Appl. Math. Lett., Volume 5 (1992) no. 2, pp. 51-55 | DOI | MR | Zbl

[40] D. Knees; A. Schröder Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints, Math. Methods Appl. Sci., Volume 35 (2012) no. 15, pp. 1859-1884 | DOI | MR | Zbl

[41] P. Laborde; Y. Renard Fixed point strategies for elastostatic frictional contact problems, Math. Methods Appl. Sci., Volume 31 (2008) no. 4, pp. 415-441 | DOI | MR | Zbl

[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, Volume 194 (2005) no. 30, pp. 3394-3415 | DOI | Zbl

[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, Volume 22 (1976) no. 2, pp. 130-185 | DOI | Zbl

[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, Volume 41 (1976), pp. 54-62 | DOI | Zbl

[47] J. T. Oden; J. A. C. Martins Models and computational methods for dynamic friction phenomena, Comput. Methods Appl. Mech. Engrg., Volume 52 (1985) no. 1-3, pp. 527-634 FENOMECH ’84, Part III, IV (Stuttgart, 1984) | DOI | MR | Zbl

[48] S. Osher; R. Fedkiw Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003, xiv+273 pages | MR | Zbl

[49] S. Osher; J.A. Sethian Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., Volume 79 (1988) no. 1, pp. 12-49 | DOI | MR | Zbl

[50] J.V. Outrata On the numerical solution of a class of Stackelberg problems, Z. Oper. Res., Volume 34 (1990) no. 4, pp. 255-277 | DOI | MR

[51] J.V. Outrata; J. Jarušek; J. Stará On optimality conditions in control of elliptic variational inequalities, Set-Valued Var. Anal., Volume 19 (2011) no. 1, pp. 23-42 | DOI | MR | Zbl

[52] I. Paczelt; T. Szabo Optimal shape design for contact problems, Structural Optimization, Volume 7 (1994), pp. 66-75 | DOI

[53] O. Pironneau Optimal shape design for elliptic systems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984, xii+168 pages | DOI | MR

[54] R. Schumann Regularity for Signorini’s problem in linear elasticity, Manuscripta Math., Volume 63 (1989), pp. 255-291 | DOI | MR | Zbl

[55] Scilab Enterprises Scilab: Le logiciel open source gratuit de calcul numérique (2012) http://www.scilab.org

[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, xx+378 pages | MR | Zbl

[57] J. Simon Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., Volume 2 (1980) no. 7-8, p. 649-687 (1981) | DOI | MR | Zbl

[58] J. Sokolowski; J-P. Zolesio Introduction to shape optimization, shape sensitivity analysis, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992, ii+250 pages | DOI | MR | Zbl

[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., Volume 41 (2010) no. 1, pp. 57-64 | DOI | MR | Zbl

[61] S. Stupkiewicz; J. Lengiewicz; J. Korelc Sensitivity analysis for frictional contact problems in the augmented Lagrangian formulation, Comput. Methods Appl. Mech. Engrg., Volume 199 (2010) no. 33-36, pp. 2165-2176 | DOI | MR | Zbl

[62] N. Tardieu; A. Constantinescu On the determination of elastic coefficients from indentation experiments, Inverse Problems, Volume 16 (2000) no. 3, pp. 577-588 | DOI | MR | Zbl

[63] F. Tröltzsch Optimal control of partial differential equations, Theory, methods and applications, Graduate Studies in Mathematics, 112, American Mathematical Society, Providence, RI, 2010, xvi+399 pages | DOI | MR | Zbl

[64] M.Y. Wang; X. Wang; D. Guo A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., Volume 192 (2003) no. 1-2, pp. 227-246 | DOI | MR

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