Towards a space reduction approach for efficient structural shape optimization

Shape optimization frequently works with geometries involving several dozen design variables. The high dimensionality itself can be an impediment to efficient optimization. Moreover, a possibly high number of explicit/implicit constraints restrict the design space. Traditional CAD geometric parameterization methods present serious difficulties in expressing these constraints leading to a high failure rate of generating admissible shapes. In this paper, we discuss shape interpolation between admissible instances of finite element/CFD meshes. We present an original approach to automatically generate a hyper-surface locally tangent to the manifold of admissible shapes in a properly chosen linearized space. This permits us to reduce the size of the optimization problem while allowing us to morph exclusively between feasible shapes. To this end, we present a two-level a posteriori mesh parameterization approach for the design domain geometry. We use Principal Component Analysis and Diffuse Approximation to replace the geometry-based variables with the smallest set of variables needed to represent an admissible shape for a chosen precision. We demonstrate this approach in two typical shape optimization problems.