Shape design sensitivity analysis and optimization of general shape arches

A method is presented for the shape design sensitivity analysis as applied to general arches whose shapes cannot be mapped on one plane. The shape design sensitivity formulation with respect to the perturbation in the direction normal to the middle surface of the shallow arch curve is derived using the material derivative and the adjoint variable method with the system variational equation expressed in a Cartesian coordinate system. A general shape arch is subdivided into segments each of which can be considered as a shallow arch. On each subdivision of the arch, a Cartesian coordinate system is installed and the shape design sensitivity for the shallow arch is applied. For numerical implementation, the finite element method is adopted and each finite element can be considered as such a subdivision. Numerical examples for sensitivity analysis and optimization are presented to illustrate the finite element sensitivity method proposed.