A non-parametric free-form optimization method for shell structures

This paper presents a numerical shape optimization method for the optimum free-form design of shell structures. It is assumed that the shell is varied in the out-of-plane direction to the surface to determine the optimal free-form. A compliance minimization problem subject to a volume constraint is treated here as an example of free-form design problem of shell structures. This problem is formulated as a distributed-parameter, or non-parametric, shape optimization problem. The shape gradient function and the optimality conditions are theoretically derived using the material derivative formulae, the Lagrange multiplier method and the adjoint variable method. The negative shape gradient function is applied to the shell surface as a fictitious distributed traction force to vary the shell. Mathematically, this method is a gradient method with a Laplacian smoother in the Hilbert space. Therefore, this shape variation makes it possible both to reduce the objective functional and to maintain the mesh regularity simultaneously. 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 the optimum free-form design of shell structures.