Reaction forces control design of shell structures in plastic range based on free-form optimization method

In this study, we propose an optimization method for the reaction forces control problem of shell structures in plastic range. The sum of squared error norms between reaction forces in the plastic range and the target forces are minimized under a volume constraint. The total strain theory is employed for simplicity under the assumption that the considered force or enforced displacement increases monotonically, so that the final deformation is path-independent. The optimum shape design problem is formulated as a distributed-parameter system, and we assume that a shell is varied in the out-of-plane direction to the surface whereas the thickness is not varied during the shape change. The shape gradient function and the optimality conditions for this problem are derived theoretically using the material derivative method and the adjoint variable method. The shape gradient function calculated from a non-linear finite-element analysis is applied to the H1 gradient method for shells proposed by one of the authors to determine optimum shape variation. The optimal shape of a shell structure can be obtained without shape parameterization while maintaining surface smoothness. Numerical examples are presented to demonstrate the validity and practical utility of the proposed free-form optimization method.