On the problem of optimizing contact force distributions

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.This work was supported by The Center for Industrial Information Technology (CENHT), Linköping Institute of Technology, Linköping, Sweden.