Shape Sensitivity Analysis and Optimization of Skeletal Structures and Geometrically Nonlinear Solids

Formulations and computational schemes for shape design sensitivity analysis and optimization have been developed for both skeletal structures and geometrically nonlinear elastic solids. The continuum approach, which is based on the weak variational form of the governing differential equation and the concept of the material derivative, plays a central role in such a development. In the first part of this work, the eigenvalue and eigenvector sensitivity equations for skeletal structures are derived with respect to configuration variables of joint and support locations. This derivation is done by the domain method as well as the boundary method. The discrete approach for the eigenvalue and eigenvector sensitivity analysis is also presented for the purpose of numerical comparison. The resultant sensitivity equations are first validated by a cantilever beam for eigenvalue sensitivity analysis and a simply-supported beam for eigenvector sensitivity analysis. The analytical solutions can be easily obtained for both examples. Moreover, the investigation of numerical accuracy and computational efficiency of these sensitivity equations is done with examples of several skeletal structures. The results show that the domain method has an advantage to be both computationally accurate and efficient. Finally, a design optimization of a vibrating beam is presented to investigate the effects of including the support locations and the support stiffness constants as design variables on the design. It is concluded that the support locations and the support stiffness constants are important to improve the quality of design. The second part of this thesis explores the possibility using the Eulerian formulation as the foundation for shape sensitivity analysis and optimization of a new class of design problems in which the performance criteria are defined in the deformed configuration of a geometrically nonlinear elastic solid. The displacement and rotation of this nonlinear elastic solid are assumed to be large while its strain is assumed to be small. Shape sensitivity equations are derived based upon the Eulerian formulation as well as the total Lagrangian formulation for a general functional. A prismatic bar is evaluated analytically to validate these sensitivity equations. A design optimization scheme is then established which uses the Eulerian formulation for analysis as well as sensitivity analysis, to design the shape of a uniformly loaded beam to minimize the area subjected to geometric and stress constraints. The results show that the proposed sensitivity equations and the design scheme work well for this example