Principal Compliance and Robust Optimal Design

Abstract

The paper addresses a problem of robust optimal design of elastic structures when the loading is unknown and only an integral constraint for the loading is given. We propose 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. The robust optimal design is formulated 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 design parameters. It is shown that the problem for the extreme loading can be reduced to an elasticity problem with mixed nonlinear boundary conditions; the last problem may have multiple solutions. The optimization with respect to the designed structure takes into account the possible multiplicity of extreme loadings and divides resources (reinforced material) 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 revealed.