Minimax optimization problem of structural design

The paper discusses a problem of robust optimal design of elastic structures when the loading is unknown. It is assumed that only an integral constraint for the loading is given. We suggest to minimize 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 loading. Hence the robust design should optimize the behavior of the structure in the worst possible scenario, which itself depends on the structure and is subject of optimization. We formulate the problem of robust optimal design as a min–max problem for the energy stored in the structure. The maximum of the energy is chosen over the constrained class of loadings, while the minimum is taken over the set of design parameters. We show that the problem for the extreme loading can be reduced to an elasticity problem with mixed nonlinear boundary condition; this problem may have multiple solutions. The optimization with respect to the designed structure takes into account the possible multiplicity of extreme loadings so that in the optimal design the strong material is distributed to equally resist to all extreme loadings. Continuous change of the loading constraint causes bifurcation of the solution of the optimization problem. We show that an invariance of the constraints under a symmetry transformation leads to a symmetry of the optimal design. Examples of robust optimal design are demonstrated.