J0110101 Robust shape optimization method for shell structures with unknown loadings

Abstract

1. Abstract In this paper, we propose a robust shape optimization method for a shell structure with unknown loadings. The concept of the principal compliance minimization for minimizing the maximal compliance is applied to the shape optimization design of a shell structure. The principal compliance minimization problem can be transformed to the equivalent maximization problem of the fundamental eigenvalue of the stiffness term, and this problem is formulated as the distributed-parameter optimization problem based on the variational method. The derived shape gradient function is applied to the H 1 gradient method for shells to determine the optimal shape variation, or the optimal free-form. With this method, the optimal smooth curvature distribution of a shell structure can be determined without shape parameterization. The calculated results show the effectiveness of the proposed method for robust shape optimization of a shell with unknown loadings. 2. Keywords: Robust shape optimization, shell structure, loading uncertainties, principal compliance, H 1 gradient Method, 3. Introduction Structural optimization techniques are widely utilized in many structural design fields. In general optimum design problems, the boundary condition is treated deterministically, although the condition such as loading condition frequently contains uncertainties. A design problem we often encounter loading conditions from all directions or multiple loading conditions by sharing of parts in actual design problems, which is one of the design problems with unknown or uncertain loadings. As the optimal design is generally vulnerable to the variation of loading because the structural performances such as stiffness or strength are strongly influenced by loading, the reliability design is often introduced to the formulation of optimal design problems. Safety factor or probabilistic approach is a method of reliability design problems. However, too large factor often causes excessive performances, redundant structure and weight gain. Another approach to avoid the vulnerability of the optimally designed structure to variations of loading has been proposed by Cherkaev et al. [1], in which the concept of the principal compliance minimization is introduced, which is defined as the minimization of the maximal compliance under the worst possible loading. They formulated it as a min-max compliance problem, and showed that the principal compliance minimization problem can be transformed to the equivalent maximization problem of the fundamental eigenvalue of the stiffness term. They applied this idea to a simple size optimization problem. Takezawa et al. [2] applied the concept of the principle compliance to a topology optimization problem under the assumption that the loading domain is limited in a small sub-domain of the linear elastic domain to solve the full size linear elastic system efficiently. In this paper, we newly propose a robust shape optimization method for shell structures by employing this concept to the free-form optimization method for shells. This method is a parameter-free shape optimization method based on the variational method, which was proposed by one of the authors [3]. In this method, a shape optimization problem is formulated in the continuous system, and the optimal smooth curvature distribution of a shell structure is determined without any shape parameterization, although almost all shape optimization methods need shape parameterization. The principal compliance minimization problem is transformed to the equivalent maximization problem of the fundamental eigenvalue of the stiffness term based on this concept. The transformed objective functional is maximized under the volume constraint, and the shape gradient function is theoretically derived using the material derivative method and the adjoint variable method. The derived shape gradient function is applied to the H 1 gradient method for shells to determine the optimal shape variation, or the optimal free-form. We carried out a numerical example to verify the effectiveness of the proposed robust shape optimization method.