Abstract
The problem of optimizing the distribution of contact forces between a rigid obstacle and a discretized linear elastic body is considered. The design variables are the initial gaps between the potential contact nodal points and the obstacle. Two different cost functionals are investigated: the first reflects the objective of minimizing the maximum contact force; the second is the equilibrium potential energy. Contrary to what has been claimed in the literature, it is shown that these cost functionals do not give, in general, the same optimal design. However, it is also shown that, if a certain frequently realized assumption is met by the system flexibility matrix, then this equality does hold.
The min-max cost functional is nonconvex and nondifferentiable, and Clarke's theory of nonsmooth optimization is used to establish a sufficient optimality condition. Investigating its consequences, both necessary and sufficient optimality conditions can be given. The equilibrium potential energy cost functional, on the other hand, turns out to have the remarkable porperties of differentiability and convexity.
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References
Conry, T. F., andSeireg, A.,A Mathematical Programming Method for Design of Elastic Bodies in Contact, Journal of Applied Mechanics, Vol. 2, pp. 387–392, 1971.
Haug, E. J., andKwak, B. M.,Contact Stress Minimization by Contour Design, International Journal for Numerical Methods in Engineering, Vol. 12. pp. 917–930, 1978.
Benedict, R. L., andTaylor, J. E.,Optimal Design for Elastic Bodies in Contact, Optimization of Distributed-Parameter Structures, Edited by E. J. Haug and J. Cea, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, pp. 1553–1599, 1981.
Benedict, R. L.,Maximum Stiffness Design for Elastic Bodies in Contact, Journal of Mechanical Design, Vol. 104, pp. 825–830, 1982.
Kikuchi, N., andTaylor, J. E.,Shape Optimization for Unilateral Elastic Contact Problems, Numerical Methods in Coupled Problems, Proceeding of the International Conference at University College, Swansea, Wales, pp. 430–441, 1981.
Benedict, B., Sokolowski, J., andZolesio, J. P.,Shape Optimization for Contact Problems, System Modelling and Optimization, Edited by P. Thoft-Christensen, Springer-Verlag, Berlin, Germany, pp. 790–799, 1984.
Bendsøe, M. P., andSokolowski, J.,Sensitivity Analysis and Optimal Design of Elastic Plates with Unilateral Point Supports, Mechanics of Structures and Machines, Vol. 15, pp. 383–393, 1987.
Haslinger, J., andNeittaanmäki, P.,Finite-Element Approximation for Optimal Shape Design, Theory and Applications, John Wiley and Sons, Chichester, England, 1988.
Klarbring, A.,Quadratic Programs in Frictionless Contact Problems, International Journal of Engineering Science, Vol. 24, pp. 1207–1217, 1986.
Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, New York, 1983.
Kikuchi, N.,A Class of Rigid Punch Problems Involving Forces and Moments by Reciprocal Variational Inequalities, Journal of Structural Mechanics, Vol. 7, pp. 273–295, 1979.
Demkowicz, L.,On Some Results Concerning the Reciprocal Formulation for the Signorini's Problem, Computers and Mathematics with Applications, Vol. 8, pp. 57–75, 1982.
Haslinger, J., andPanagiotopoulos, P. D.,The Reciprocal Variational Approach to the Signorini's Problem with Friction. Approximation Results, Proceedings of the Royal Society of Edinburgh, Vol. 98A, pp. 365–383, 1984.
Bendsøe, M. P., andSokolowski, J.,Design Sensitivity Analysis of Elastic-Plastic Analysis Problems, Mechanics of Structures and Machines, Vol. 16, pp. 81–102, 1988.
Murty, K. G.,Linear Complementarity, Linear and Nonlinear Programming, Helderman-Verlag, Berlin, Germany, 1988.
Cottle, R. W.,Monotone Solutions of the Parametric Complementarity Problem, Mathematical Programming, Vol. 3, pp. 210–224, 1974.
Ibaraki, T.,Complementarity Programming, Operations Research, Vol. 19, pp. 1523–1528, 1971.
Kaneko, I.,On Some Recent Engineering Applications of Complementarity Problems, Mathematical Programming Study, Vol. 17, pp. 111–125, 1982.
Haslinger, J., andRoubíček, T.,Optimal Control of Variational Inequalities. Approximation Theory and Numerical Realization, Applied Mathematics and Optimization, Vol. 14, pp. 187–201, 1986.
Ciarlet, P. G.,Introduction to Numerical Linear Algebra and Optimization, Cambridge University Press, Cambridge, England, 1989.
Ekeland, I., andTemam, R.,Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976.
Kikuchi, N., andOden, J. T.,Contact Problems in Elasticity: A Study of Variational Inequalities and Finite-Element Methods, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1988.
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Communicated by E. J. Haug
This work was supported by The Center for Industrial Information Technology (CENHT), Linköping Institute of Technology, Linköping, Sweden.
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Klarbring, A. On the problem of optimizing contact force distributions. J Optim Theory Appl 74, 131–150 (1992). https://doi.org/10.1007/BF00939896
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DOI: https://doi.org/10.1007/BF00939896