Construction of Fair Surfaces Over Irregular Meshes

Abstract

This paper describes the process of constructing a fair, open or closed C1 surface over a given irregular curve mesh. The input to the surface construction consists of point and/or curve data which are individually marked to be interpolated or approximated and are arranged according to an arbitrary irregular curve mesh topology (Fig. 1). The surface constructed from these data will minimize flexibly chosen fairness criteria. The set of available fairness criteria is able to measure surface characteristics related to curvature, variation of curvature, and higher order surface derivatives based on integral functionals of quadratic form derived from the second, third and higher order parametric derivatives of the surface. The choice is based on the desired shape character. The construction of the surface begins with a midpoint refinement decomposition of the irregular mesh into aggregates of patch complexes in which the only remaining type of building block is the quadrilateral Be´zier patch of degrees 4 by 4. The fairing process may be applied regionally or to the entire surface. The fair surface is built up either in a single global step or iteratively in a three stage local process, successively accounting for vertex, edge curve and patch interior continuity and fairness requirements. This surface fairing process will be illustrated by two main examples, a benchmark test performed on a topological cube, resulting in many varieties of fair shapes for a closed body, and a practical application to a ship hull surface for a modern container ship, which is subdivided into several local fairing regions with suitable transition pieces. The examples will demonstrate the capability of the fairing approach of contending with irregular mesh topologies, dealing with multiple regions, applying global and local fairing processes and will illustrate the influence of the choice of criteria upon the character of the resulting shapes.

Tightening

“…The optimization-based approaches appear in the CAD/CAM literature on problems such as ship hull and car body design [Celniker and Gossard 1991;Hahmann and Konz 1997;Moreton and Sequin 1992;Westgard and Nowacki 2001]. The objective of these methods is to deform a surface composed of curved patches to minimize an energy function.…”

C1 finite elements on non-tensor-product 2d and 3d manifolds

“…(This distinction between the valences near the regular case of n = 4 and higher valences is both geometrically motivated and relevant in practice where the majority of irregularities are of valence 3 and 5.) The construction in [15] differs from the work in [4,22,25] in that it does not require solving equations while constructing the physical domain: the domain and the elements are modeled analogous to splines in B-spline form. That is, control points carry the geometric information and evaluation and differentiation amount to explicit formulas in terms of the control points.…”

Matched Gk-constructions always yield Ck-continuous isogeometric elements

“…Such elements can readily be constructed from the rich literature on geometric surface constructions or be based on new constructions in that field. For example, the construction in Scott et al (2013) (see also Westgaard and Nowacki, 2001; Peters, 1995) could be used to build an isogeometric element rather than to just serve as a surface definition in the context of the boundary element method (there, linear combinations of trivariate Green's functions are to be determined, a setup quite different from the one discussed in the present paper).…”