Shape and topology optimization methods have found application in many areas of engineering, ranging from conceptual layout of high-rise buildings to the design of patient-specific craniofacial bone replacements. Several fundamental difficulties arise when dealing with optimal shape problems. For example, these problems are in general ill-posed in that they do not admit solutions in the classical sense. Shapes with fine features are naturally favored, which leads to nonconvergent minimizing sequences that exhibit rapid oscillations. To address this issue, one either enlarges the space of admissible geometries allowing for generalized micro-perforated shapes, an approach known as “relaxation,” or alternatively places additional constraints to limit the complexity of the shapes, a strategy commonly referred to as “restriction.” Moreover, usual finite-element approximation schemes may suffer from numerical instabilities, necessitating a careful choice of approximation spaces. The resulting discrete optimization problems often are large-scale, nonlinear and nonconvex. After a brief survey of applications of optimal shape design, we discuss the issue of existence of solutions and present key elements of a suitable well-posed restriction formulation. The main premise is that these theoretical considerations have certain implications for the numerical solution schemes, and a closer examination sheds light on the appropriate algorithmic choices. We proceed to describe a consistent strategy for parameterization of shape and discretization of the governing state equation within this restriction framework.
