Computational procedures for topological shape design

This research describes a method for the design of general shapes that is able to change the shape topology during the design process and can handle geometric and behavioral constraints expressed as algebraic or partial differential equations. The proposed method uses an iterative process that alternates between formulating and solving a sequence of numerical optimization problems. At each iteration a design parameterization is generated and a corresponding domain discretization is then derived automatically from this design parameterization. A numerical optimization problem is then formulated by expressing the objective and the problem constraints in terms of the design variables. The results of this optimization update the shape and the resulting shape then initiates a new design iteration. The process continues until convergence. The parameterization proposed is based on a geometric abstraction called the ’skeleton‘ and leads to a relatively small number of design variables. The parameterization is adaptive to shape evolutions, guarantees shape integrity during the design, and allows the redefinition of the objective and constraints in terms of adaptively generated design variables. The skeleton also forms the basis for automatic domain discretization parameterized by the design variables. This parameterized discretization provides efficient means for evaluating PDE constraints and computing their gradients. The method overcomes some of the problems associated with current shape design procedures including boundary variation methods and homogeniz ation methods. Computational examples with geometric and PDE constraints are presented to demonstrate the capabilities of the proposed shape design method in generating general geometries and topologies.