Multilevel gradient method with Bézier parametrisation for aerodynamic shape optimisation

Since the evaluation of the cost functional requires the solution of the governing flow equations (0.2), each optimisation iteration is characterised by a high computational cost if complex flow modelling is considered. In this context, the improvement of optimisation algorithm efficiency is crucial. In [2] a multilevel strategy has been defined to make the convergence rate of a gradient-based method almost independent of the number of control parameters. More particularly, considering shape grid-point coordinates as design variables, a hierarchical parametrisation was defined considering different subsets of parameters extracted from the complete parameterisation, which can be prolongated to the higher level by linear mapping. Thus, let us consider, at level l, a linear application P (l) and its associated matrix M (l) ∈ IRp×nl : P (l) : IRl −→ IR α 7−→ P (α) =M α