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A mixed variational formulation for shape optimization of solids with contact conditions

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Abstract

This paper is concerned with the development of a mixed variational formulation and computational procedure for the shape optimization problem of linear elastic solids in possible contact with a rigid foundation. The objective is to minimize the maximum value of the von Mises equivalent stress in a body (non-differentiable objective function), subject to a constraint on its volume and bound constraints on the design. For design purposes, the contact boundary is considered fixed.

A finite element model that is appropriate for the mixed formulation is utilized in the discretization of the state and adjoint state equations. An elliptical mesh generator was used to generate the finite element mesh at each new design. The computational model is tested in several example problems.

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References

  • Benedict, R.L.; Taylor, J.E. 1981:Optimal design for elastic bodies in contact. In: Haug, E.J.; Cea, J. (eds.)Optimization of distributed parameter structures II, pp. 1553–1599. Amsterdam: Sijthoff and Noordhoff

    Google Scholar 

  • Bertsekas, D.P. 1982:Constrained optimization and Lagrange multiplier methods. New York: Academic Press

    Google Scholar 

  • Braibant, V.; Fleury, C. 1984: Shape optimal design using Bsplines.Comp. Meth. Appl. Mech. Eng. 44, 247–267

    Google Scholar 

  • Campos, L.T. 1981:A numerical analysis of a class of contact problems with friction in elastostatics. M.Sc. Thesis, The University of Texas, Austin, TX

    Google Scholar 

  • Choi, K.K.; Seong, H.G. 1986: A domain method for shape design sensitivity analysis of built-up structures.Comp. Meth. Appl. Mech. Eng. 57, 1–15

    Google Scholar 

  • Chung, K.Y. 1985:Shape optimization and free boundary value problems with grid adaptation. Ph.D. Thesis, The University of Michigan, Ann Arbor, MI

    Google Scholar 

  • Dems, K.; Mróz, Z. 1978: Multiparameter structural shape optimization by the finite element method.Int. J. Num. Meth. Eng. 13, 247–263

    Google Scholar 

  • Duvaut, G.; Lions, J.L. 1976:Inequalities in mechanics and physics. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Haber, R.B. 1987: Application of the Eulerian Lagrangian kinematic description to structural shape design. In: Mota Soares, C.A. (ed.)Computer aided optimal design: structural and mechanical systems, pp. 573–587. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Haslinger, J.; Neittaanmäki, P. 1988:Finite element approximation for optimal shape design. New York: John Wiley & Sons

    Google Scholar 

  • Haug, E.J.; Choi, K.K.; Komkov, V. 1986:Design sensitivity analysis of structural systems. New York: Academic Press

    Google Scholar 

  • Kikuchi, N. 1986: Adaptive grid-design methods for finite element analysis.Comp. Meth. Appl. Mech. Eng. 55, 129–160

    Google Scholar 

  • Kikuchi, N.; Oden, J.T. 1988:Contact problems in elasticity: a study of variational inequalities and finite element methods. Philadelphia: SIAM

    Google Scholar 

  • Murat, F.; Simon, J. 1976: Studies on optimal shape design problems. In: Cea, J. (ed.)Lectures notes in computer science 41, pp. 54–62. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Pedersen, P.; Laursen, C.L. 1983: Design for minimum stress concentration by finite elements and linear programming.J. Struct. Mech. 10, 375–391

    Google Scholar 

  • Pshenichny, B.N.; Danilin, Y.M. 1978:Numerical methods in extremal problems. Moscow: Mir

    Google Scholar 

  • Rasmussen, J. 1990: The structural optimization system CAOS.Struct. Optim. 2, 109–115

    Google Scholar 

  • Rodrigues, H.C. 1988:Shape optimal design of elastic bodies using a mixed variational formulation. Ph.D. Thesis, The University of Michigan, Ann Arbor, MI

    Google Scholar 

  • Rodrigues, H.C. 1989: Shape optimal design of elastic bodies using a mixed variational formulation.Comp. Meth. Appl. Mech. Eng. 69, 29–44

    Google Scholar 

  • Sokolowski, J. 1988: Sensitivity analysis of contact problems with prescribed friction.J. Appl. Math. Opt. 19, 99–118

    Google Scholar 

  • Sokolowski, J; Zolesio, J.P. 1987: Shape sensitivity analysis of unilateral problems.SIAM J. Math. Anal. 18, 1416–1437

    Google Scholar 

  • Taylor, J.E.; Bendsøe, M.P. 1984: An interpretation for min-max structural design problems including a method for relaxing constraints.Int. J. Solids & Struct. 20, 301–314

    Google Scholar 

  • Thompson, J.F.; Warsi, Z.U.A.; Mastin, C.W. 1985:Numerical grid generation foundations and applications. New York: North Holland

    Google Scholar 

  • Zienkiewicz, O.C.; Campbell, J.S. 1973: Shape optimization and sequential linear programming. In: Gallagher, R.H.; Zienkiewicz, O.C. (eds.)Optimum structural design, pp. 109–126. London: Wiley

    Google Scholar 

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  1. Instituto Superior Técnico, CEMUL, Av. Rovisco Pais, Lisbon, Portugal

    H. C. Rodrigues

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  1. H. C. Rodrigues
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Rodrigues, H.C. A mixed variational formulation for shape optimization of solids with contact conditions. Structural Optimization 6, 19–28 (1993). https://doi.org/10.1007/BF01743171

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  • DOI: https://doi.org/10.1007/BF01743171

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