Unification of parametric and implicit methods for shape sensitivity analysis and optimization with fixed mesh

Parametric and implicit methods are traditionally thought to be two irrelevant approaches in structural shape optimization. Parametric method works as a Lagrangian approach and often uses the parametric boundary representation (B-rep) of curves/surfaces, for example, Bezier and B-splines in combination with the conformal mesh of a finite element model, while implicit method relies upon level-set functions, that is, implicit functions for B-rep, and works as an Eulerian approach in combination with the fixed mesh within the scope of extended finite element method or finite cell method. The original contribution of this work is the unification of both methods. First, a new shape optimization method is proposed by combining the features of the parametric and implicit B-reps. Shape changes of the structural boundary are governed by parametric B-rep on the fixed mesh to maintain the merit in computer-aided design modeling and avoid laborious remeshing. Second, analytical shape design sensitivity is formulated for the parametric B-rep in the framework of fixed mesh of finite cell method by means of the Hamilton–Jacobi equation. Numerical examples are solved to illustrate the unified methodology.