Multi-minimisations for shape control of fully free-form deformation features (δ-F4)

Fully free form deformation features (δ-F⁴) have been proposed to overcome the limits of low-level manipulations of free form surfaces. They correspond to shapes obtained by deformation of a surface part according to geometric constraints. In our approach, a δ-F⁴ is a result of the indirect manipulation of external forces applied to the nodes of a bar network coupled to the control polyhedron of a B-spline surface. The solution of the equation system corresponding to the constraint specifications, often under-constrained, requires the definition of an optimisation problem where an additional objective function has to be minimised. In this paper, we propose a new formulation of this optimisation problem where the proposed objective functions can be defined as a multiple combination of various local quantities. They can be related either to the geometry of the bar network (e.g. the length of a bar or the displacement of a node), or to its mechanical magnitudes (e.g. the external force applied at a node or a bar deformation energy). Different types of combinations are also proposed and classified according to the induced level of multi-minimisations. In this way, the shape of a δ-F⁴ can be controlled globally, with a unique minimisation, or locally with different minimisations applied to sub-domains of the surface.