Shape optimization with topological changes and parametric control

Abstract Recent advances in shape optimization rely on free‐form implicit representations, such as level sets, to support boundary deformations and topological changes. By contrast, parametric shape optimization is formulated directly in terms of meaningful geometric design variables, but usually does not support free‐form boundary and topological changes. We propose a novel approach to shape optimization that combines and retains the advantages of the earlier optimization techniques. The shapes in the design space are represented implicitly as level sets of a higher‐dimensional function that is constructed using B‐splines (to allow free‐form deformations), and parameterized primitives combined with R ‐functions (to support desired parametric changes). Our approach to shape design and optimization offers great flexibility because it provides explicit parametric control of geometry and topology within a large space of free‐form shapes. The resulting method is also general in that it subsumes most other types of shape optimization as special cases. We describe an implementation of the proposed technique with attractive numerical properties. The explicit construction of an implicit representation supports straightforward sensitivity analysis that can be used with most gradient‐based optimization methods. Furthermore, our implementation does not require any error‐prone polygonization or approximation of level sets (isocurves and isosurfaces). The effectiveness of the method is demonstrated by several numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.