An approximation theory for optimum sheets in unilateral contact

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. The original saddle-point problem is replaced by a sequence of approximating problems, the solutions of which are shown to converge weakly to exact solutions. In the case that complete material removal is not admissible, the state variable is shown to converge strongly in the (Hl )2-norm to the unique exact state solution. We consider a particular finite-element discretization that fits into the general theory and present the mathematical programming problem that results from it.

[1]  Maximum Stiffness Design for Elastic Bodies in Contact , 1982 .

[2]  Joakim Petersson.  On stiffness maximization of variable thickness sheet with unilateral contact , 1996 .

[3]  The rigid punch problem with friction , 1991 .

[4]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[5]  Jaroslav Haslinger.  Finite element analysis for unilateral problems with obstacles on the boundary , 1977 .

[6]  Martin P. Bendsøe,et al.  Topology design of structures , 1993 .

[7]  Martin P. Bendsøe,et al.  Optimization of Structural Topology, Shape, And Material , 1995 .

[8]  Kazimierz Malanowski,et al.  An Example of a Max-Min Problem in Partial Differential Equations , 1970 .

[9]  O. Sigmund,et al.  Checkerboard patterns in layout optimization , 1995 .

[10]  J. Taylor.  Maximum Strength Elastic Structural Design , 1969 .

[11]  William Prager,et al.  Problems of Optimal Structural Design , 1968 .

[12]  J. E. Taylor,et al.  A Finite Element Method for the Optimal Design of Variable Thickness Sheets , 1973 .

[13]  M. Patriksson,et al.  Topology Optimization of Sheets in Contact by a Subgradient Method , 1997 .

[14]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

[15]  J. Haslinger,et al.  Finite Element Approximation for Optimal Shape Design: Theory and Applications , 1989 .

[16]  J. Oden,et al.  Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , 1987 .