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
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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.
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)
- J. Petersson, On stiffness maximization of variable thickness sheet with unilateral contact, Quart. Appl. Math. 54 (1996), no. 3, 541–550. MR 1402408, DOI https://doi.org/10.1090/qam/1402408
J. E. Taylor, Maximum strength elastic structural design, J. Engrg. Mech. Div., Proc. ASCE 95(EM3), 653–663 (1969)
W. Prager and J. E. Taylor, Problems of optimal structural design, Trans. J. Appl. Mech. ASME 35 (1), 102–106 (1968)
R. L. Benedict, Maximum stiffness design for elastic bodies in contact, J. Mech. Design 104, 825–830 (1982)
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
- 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
- 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
- Martin P. Bendsøe, Optimization of structural topology, shape, and material, Springer-Verlag, Berlin, 1995. MR 1350791
- 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
- 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, DOI https://doi.org/10.1016/0020-7225%2891%2990104-B
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- J. Haslinger and P. Neittaanmäki, Finite element approximation for optimal shape design, John Wiley & Sons, Ltd., Chichester, 1988. Theory and applications. MR 982710
I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovišek, Solution of Variational Inequalities in Mechanics, Springer-Verlag, New York, 1988
- 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
- Jaroslav Haslinger, Finite element analysis for unilateral problems with obstacles on the boundary, Apl. Mat. 22 (1977), no. 3, 180–188 (English, with Russian and Czech summaries). MR 440956
- Roland Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 737005
- 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, DOI https://doi.org/10.1002/%28SICI%291097-0207%2819970415%2940%3A7%3C1295%3A%3AAID-NME115%3E3.3.CO%3B2-G
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
A. Diaz and O. Sigmund, Checkerboard patterns in layout optimization, Struct. Optim. 10, 40–45 (1995)
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)
J. Petersson, On stiffness maximization of variable thickness sheet with unilateral contact, Quart. Appl. Math. 54, 541–550 (1996)
J. E. Taylor, Maximum strength elastic structural design, J. Engrg. Mech. Div., Proc. ASCE 95(EM3), 653–663 (1969)
W. Prager and J. E. Taylor, Problems of optimal structural design, Trans. J. Appl. Mech. ASME 35 (1), 102–106 (1968)
R. L. Benedict, Maximum stiffness design for elastic bodies in contact, J. Mech. Design 104, 825–830 (1982)
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
J. Céa and K. Malanowski, An example of a max-min problem in partial differential equations, SIAM J. Control 8, 305–316 (1970)
M. P. Bendsøe and C. A. Mota Soares (editors), Topology Design of Structures, Kluwer Academic Publishers, Dordrecht, 1993
M. P. Bendsøe, Optimization of Structural Topology, Shape, and Material, Springer-Verlag, Berlin, 1995
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991
A. Klarbring, A. Mikelić, and M. Shillor, The rigid punch problem with friction, Internat. J. Engrg. Sci. 29, 751–768 (1991)
N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988
J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design, John Wiley and Sons, London, 1988
I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovišek, Solution of Variational Inequalities in Mechanics, Springer-Verlag, New York, 1988
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, Amsterdam, 1978
J. Haslinger, Finite element analysis for unilateral problems with obstacles on the boundary, Apl. Mat. 22, 180–187 (1977)
R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984
J. Petersson and M. Patriksson, Topology optimization of sheets in contact by a subgradient method, Internat. J. Numer. Methods Engrg. 40, 1295–1321 (1997)
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
A. Diaz and O. Sigmund, Checkerboard patterns in layout optimization, Struct. Optim. 10, 40–45 (1995)
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© Copyright 1998
American Mathematical Society
