A surrogate model based on Non-Uniform Rational B-Splines hypersurfaces

Abstract

This study aims at providing an original metamodeling technique based on the Non-Uniform Rational B-Splines (NURBS) formalism. The proposed approach is able to fit general non-convex sets of target points (TPs) by extending the NURBS formalism to the N-dimensional (N-D) case, getting in this way a general NURBS hypersurface. The shape of such a hypersurface is tuned by several parameters: the number of control points (CPs), their coordinates and related weights, the degrees of the blending functions and the knot-vector components defined along each direction. The goal of the proposed strategy is to find the best NURBS hypersurface approximating a given set of TPs. To this purpose the problem is formulated as an unconstrained least-square distance problem wherein the optimisation variables are all the parameters tuning the shape of the NURBS hypersurface. Nevertheless, when the number of CPs and the degrees of the basis functions are included among the design variables the resulting problem is defined over a space having a variable dimension. To deal with this aspect, a special genetic algorithm, able to solve problems characterised by a variable number of design variables, is considered to determine automatically (i.e. without the user’s intervention) the optimum value of both the design space size (related to the integer variables of the NURBS hypersurface) and the NURBS hypersurface continuous parameters. The effectiveness of the proposed approach is proven by means of a meaningful benchmark.