Abstract
In this paper a general framework for topology optimization of structures in unilateral contact is developed. A linear elastic structure that is unilaterally constrained by rigid supports is considered. The supports are modeled by Signorini’s contact conditions which in turn are treated by the augmented Lagrangian approach as well as by a smooth approximation. The latter approximation must not be confused with the well-known penalty approach. The state of the system, which is defined by the equilibrium equation and the two different contact formulations, is solved by a Newton method. The design parametrization is obtained by using the SIMP-model. The minimization of compliance for a limited value of volume is considered. The optimization problems are solved by SLP. This is done by using a nested approach where the state equations are linearized and sensitivities are calculated by the adjoint method. In order to avoid mesh-dependency the sensitivities are filtered by Sigmund’s filter. The final LP-problem is solved by an interior point method that is available in Matlab. The implementation is done for a general design domain in 2D as well as in 3D by using fully integrated isoparametric elements. The implementation seems to be very efficient and robust.
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References
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, New York
Christensen P, Klarbring A (1999) Newton’s method for frictional contact problems. In: The proceedings of European conference on computational mechanics, München, 31 August–3 September 1999
Christensen PW, Klarbring A (2009) An introduction to structural optimization. Springer, New York
Facchinei F, Jiang H, Qi L (1999) A smoothing method for mathematical programs with equilibrium constraints. Math Program 85:107–134
Fancello EA (2006) Topology optimization of minimum mass design considering local failure constraints and contact boundary conditions. Struct Multidiscipl Optim 32:229–240
Hilding D (2000) A heuristic smoothing procedure for avoiding local optima in optimization of structures subjected to unilateral constraints. Struct Multidiscipl Optim 20:29–36
Hilding D, Klarbring A, Petersson J (1999) Optimization of structures i unilateral contact. Appl Mech Rev 52:139–160
Ireman P, Klarbring A, Strömberg N (2002) Finite element algorithms for thermoelastic wear problems. Eur J Mech A Solids 36:423–440
Johansson L (2001) A Newton method for rigid body frictional impact with multiple simultaneous impact points. Comput Methods Appl Mech Eng 191:239–254
Klarbring A, Rönnqvist M (1995) Nested approach to structural optimization in nonsmooth mechanics. Struct Multidiscipl Optim 10:79–86
Klarbring A, Lundvall O, Strömberg N (2004) A flexible multi-body approach for frictional contact in spur gears. J Sound Vib 278:479–499
Mangasarian OL (1976) Equivalence of the complementarity problem to a system of non-linear equations. SIAM J Appl Math 31:89–92
Mankame ND, Ananthasuresh GK (2004) Topology optimization for synthesis of contact-aided compliant mechanisms using regularized contact modeling. Comput Struct 82:1267–1290
Mehrotra S (1992) On the implementation of a primal-dual interior point method. SIAM J Optim 2:575–601
Petersson J, Patriksson M (1997) Topology optimization of sheets in contact by a subgradient method. Int J Numer Methods Eng 40:1295–1321
Sigmund O (2001) A 99 linetopology optimization code written in Matlab. Struct Multidisc Optim 21:120–127
Strömberg N (1997) An augmented Lagrangian method for fretting problems. Eur J Mech A Solids 16:573–593
Strömberg N (1999) A Newton method for three-dimensional fretting problems. Int J Solids Struct 36:2075–2090
Strömberg N (2003) A method for structural dynamic contact problems with friction and wear. Int J Numer Methods Eng 58:2371–2385
Strömberg N (2005) An implicit method for frictional contact, impact and rolling. Eur J Mech A Solids 24:1016–1029
Strömberg N (2006) Frictional contact/impact between a hyperelastic body and moving rigid obstacles. In: The proceedings of the 3rd European conference on computational mechanics, Lisbon, 5–9 June 2006
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Strömberg, N., Klarbring, A. Topology optimization of structures in unilateral contact. Struct Multidisc Optim 41, 57–64 (2010). https://doi.org/10.1007/s00158-009-0407-z
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DOI: https://doi.org/10.1007/s00158-009-0407-z