Optimal design for the worst case scenario

Abstract

The talk discusses a problem of robust optimal design of elastic structures when the loading is unknown, and only an integral constraint for the loading is given [1]. The optimal design problem is formulated as minimization of the principal compliance of the domain equal to the maximum of the stored energy over all admissible loadings. The principal compliance is the maximal compliance under the extreme, worst possible applied force [2]. The robust optimal design is a min-max problem for the energy stored in the structure. The minimum is taken over the design parameters, while the maximum of the energy is chosen over the constrained class of loadings. It is shown that the problem for the extreme loading is reduced to an elasticity problem with mixed nonlinear boundary conditions; the last problem may have multiple stationary solutions. The optimization takes into account the possible multiplicity of extreme loadings and designs the structure to equally resist all of them. Continuous change of the loading constraint causes bifurcation of the solution of the optimization problem. It is shown that an invariance of the constraints under a symmetry transformation leads to a symmetry of the optimal design. Examples of optimal design are investigated; symmetries and bifurcations of the solutions are discussed.