Free-form optimization of shell structures for maximizing elastic buckling load

Abstract

This paper presents a free-form optimization method for maximizing the elastic buckling load of a shell structure. A shell structure is assumed to be movable in the normal or the tangent direction to the surface for the shape optimization. The 1^st buckling load factor is maximized under a volume constraint. The 1^st eigenvalue constraint is employed to avoid the repeated eigenvalues problem in this max-min problem. This problem is formulated as a distributed-parameter shape optimization problem, and the shape gradient function for this problem is theoretically derived using the Lagrange multiplier method, the adjoint variable method and the formulae of the material derivative. The shape gradient function derived is applied to the free-form optimization method for shells, a gradient method in the Hilbert space, where the optimal shape variation is calculated as the displacement field of the linear elastic analysis of the fictitious shell model. With this method, the optimal free-form surface or boundary is determined without shape parameterization. The effectiveness of the 1st eigenvalue constraint is also studied. The results show the validity of this method to determine the optimal free-form of shell structures for the elastic buckling design problem.