A method for shape design sensitivity analysis for arches and axisymmetric shells of general shapes is developed. The basic approach is to divide the structures into many segments. For each of the segments, the formula for a shallow arch or shell can be applied and the results assembled. To interconnect those segments, the existing sensitivity formula, obtained for a variation only in the direction perpendicular to the plane on which the structure is mapped, has been extended to include a variation normal to the middle surface. The method follows the adjoint variable approach based on the material derivative concept as established in the literature. Explicit formulas are derived for functionals in terms of displacement, stress and compliance.
Problems with nonhomogeneous displacement boundary conditions are also considered. A formulation based on the Lagrange multiplier is suitable for functional with reaction forces such as appeared in the compliance. Another approach is to utilize a transformed homogeneous problem by introducing a known function that satisfies the nonhomogeneity.
For numerical implementation of formulas derived, the structural shapes and design perturbations are parameterized using cubic splines with natural boundary conditions. An existing finite element code is used to calculate the responses of the original and the adjoint structures. Several examples are taken to illustrate the method and compare the results with those by finite differencing. It is shown that the results are good in general, but sizable discrepancies are observed for those structures with a large variation of curvature. The cause is not obvious and needs further study.
The shape design sensitivity formulas derived are next used for optimization. The design of an elliptic arch of minimum compliance is considered. The objective functional has been reduced by 40 times of the initial value. An omega seal applicable to the connection of two thick cylindrical shells in a power plant is designed for minimum compliance. It has been reduced to one half of the initial. At the same time, the maximum deflection and maximum stress have been reduced significantly. It is shown that the proposed approach of segmentation of general shape structures works well and is applicable to practical design problems.
