Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



An approximation theory for optimum sheets in unilateral contact

Authors: Joakim Petersson and Jaroslav Haslinger
Journal: Quart. Appl. Math. 56 (1998), 309-325
MSC: Primary 74P05; Secondary 74M15, 74S05
DOI: https://doi.org/10.1090/qam/1622499
MathSciNet review: MR1622499
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give an approximation theory for the optimum variable thickness sheet problem considered in [1] and [2]. This problem, which is a stiffness maximization of an elastic continuum in unilateral contact, admits complete material removal, i.e., the design variable is allowed to take zero values.

References [Enhancements On Off] (What's this?)

  • [1] M. P. Rossow and J. E. Taylor, A finite element method for the optimal design of variable thickness sheets, AIAA J. 11, 1566-1568 (1973)
  • [2] J. Petersson, On stiffness maximization of variable thickness sheet with unilateral contact, Quart. Appl. Math. 54 (1996), no. 3, 541–550. MR 1402408, https://doi.org/10.1090/qam/1402408
  • [3] J. E. Taylor, Maximum strength elastic structural design, J. Engrg. Mech. Div., Proc. ASCE 95(EM3), 653-663 (1969)
  • [4] W. Prager and J. E. Taylor, Problems of optimal structural design, Trans. J. Appl. Mech. ASME 35 (1), 102-106 (1968)
  • [5] R. L. Benedict, Maximum stiffness design for elastic bodies in contact, J. Mech. Design 104, 825-830 (1982)
  • [6] J. Petersson, Stiffness optimization of general structure in Signorini-type contact, Contact Mechanics, edited by M. Raous, M. Jean and J. J. Moreau, Plenum Press, New York, 1995, pp. 41-48
  • [7] Jean Céa and Kazimierz Malanowski, An example of a max-min problem in partial differential equations, SIAM J. Control 8 (1970), 305–316. MR 0274915
  • [8] Martin P. Bendsøe and Carlos A. Mota Soares (eds.), Topology design of structures, NATO Advanced Science Institutes Series E: Applied Sciences, vol. 227, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1250185
  • [9] Martin P. Bendsøe, Optimization of structural topology, shape, and material, Springer-Verlag, Berlin, 1995. MR 1350791
  • [10] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205
  • [11] A. Klarbring, A. Mikelić, and M. Shillor, The rigid punch problem with friction, Internat. J. Engrg. Sci. 29 (1991), no. 6, 751–768. MR 1107199, https://doi.org/10.1016/0020-7225(91)90104-B
  • [12] N. Kikuchi and J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 961258
  • [13] J. Haslinger and P. Neittaanmäki, Finite element approximation for optimal shape design, John Wiley & Sons, Ltd., Chichester, 1988. Theory and applications. MR 982710
  • [14] I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovišek, Solution of Variational Inequalities in Mechanics, Springer-Verlag, New York, 1988
  • [15] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • [16] Jaroslav Haslinger, Finite element analysis for unilateral problems with obstacles on the boundary, Apl. Mat. 22 (1977), no. 3, 180–188 (English, with Czech and loose Russian summaries). MR 0440956
  • [17] Roland Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 737005
  • [18] Joakim Petersson and Michael Patriksson, Topology optimization of sheets in contact by a subgradient method, Internat. J. Numer. Methods Engrg. 40 (1997), no. 7, 1295–1321. MR 1449228, https://doi.org/10.1002/(SICI)1097-0207(19970415)40:7<1295::AID-NME115>3.3.CO;2-G
  • [19] C. S. Jog and R. B. Haber, Checkerboard and other spurious modes in solutions to distributed-parameter and topology design problems, WCSMO-1: First World Congress of Structural and Multidisciplinary Optimization, edited by N. Olhoff and G. I. N. Rozvany, Elsevier Science Ltd., Oxford, 1995, pp. 237-242
  • [20] A. Diaz and O. Sigmund, Checkerboard patterns in layout optimization, Struct. Optim. 10, 40-45 (1995)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 74P05, 74M15, 74S05

Retrieve articles in all journals with MSC: 74P05, 74M15, 74S05

Additional Information

DOI: https://doi.org/10.1090/qam/1622499
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society