Least Degree G 1-refinable Multi-sided Surfaces Suitable for Inclusion into C 1 bi-2 Splines.

Geometrically smooth spline surfaces, generalized to include n-sided facets or configurations of n ≠ 4 quads, can exhibit a curious lack of additional degrees of freedom for modelling or engineering analysis when refined. This paper establishes a minimal polynomial degree for smooth constructions of multi-sided surfaces that guarantees more flexibility in all directions under refinement. Degree bi-4 is both necessary and sufficient for flexibility-increasing G 1-refinability within a bi-quadratic C 1 spline complex. Sufficiency is proven by two alternative flexibly G 1-refinable constructions exhibiting good highlight line distributions.

Low degree splines for locally quad-dominant meshes.

A mesh is locally quad-dominant (lqd) if all non-4-sided facets are surrounded by quadrilaterals. Lqd meshes allow for irregular nodes where n ≠ 4 quads meet and for multi-sided facets, called T-gons, that end quad-strips and so adjust mesh density. This paper introduces a new class of bi-cubic (bi-3) Geometric T-joint (GT) splines whose control nets are τ-nets, i.e. T-gons surrounded by quads. These GT-splines join smoothly with each other, bi-3 G-splines and regular C 1 bi-quadratic splines to form smooth surfaces of degree at most bi-3.

Smooth polar caps for locally quad-dominant meshes.

A polar configuration is a node surrounded by m triangles. Polar configurations are common to cap off cylinders and spheres. When the triangles, interpreted as quadrilaterals with one edge collapsed, are surrounded by a quad-strip then the extended polar configuration qualifies as part of a locally quad-dominant (lqd) mesh. Recent constructions, referred to as semi-structured splines, can use lqd meshes as control nets: multi-sided configurations that merge parameter directions are covered by G-spline; and T-junctions that transition from coarse and fine are covered by GT-splines. This paper complements existing semi-structured splines by providing the missing component for polar configurations. A spectrum of constructions of differing degree are introduced, tested and compared. Bi-2 C 1 splines are extended to polar configurations covered by C 1 surfaces consisting of (macro-)patches of degree as low as bi-2. Bi-3 C 2 splines are extended to polar configurations covered by surfaces that are C 2 except for a C 1 pole and consist of (macro-)patches of degree as low as bi-3.

A sharp degree bound on G 2-refinable multi-sided surfaces.

Refinement of a space of splines should yield additional degrees of freedom for modeling and engineering analysis, both along boundaries and in the interior. Yet such additional flexibility fails to materialize for multi-sided G 2 surface constructions when the polynomial degree is too low. This paper establishes a tight lower bound on the polynomial degree of flexibility-increasing refinable multi-sided G 2 surface constructions within a C 2 spline complex - by ruling out bi-5 constructions and by exhibiting a multi-sided bi-6 construction that yields good highlight line and curvature distributions. The bi-6 construction consists of one 2 × 2 macro-patch for each of the n sectors that join to form the multi-sided surface.

Fair free-form surfaces that are almost everywhere parametrically C 2.

Compared to G k continuity, C k continuity simplifies the construction of functions on surfaces and their refinement, e.g. to solve differential equations on the surface. The new class of almost everywhere parametrically C 2 free-form surfaces provide such a parameterization. For example, a new bi-6 construction combines a fast-contracting C 2 guided subdivision surface with a tiny multi-sided G 1 cap. The cap is chosen to be smaller than any refinement anticipated for geometric modeling or computing on surfaces. Fast contraction means that one subdivision step shrinks the remaining hole more than three steps of Catmull-Clark subdivision. This yields smooth surfaces consisting of a finite number of pieces that are suitable for engineering practice. Both the subdivision construction and the cap are guided by a reference surface. This guide conveys the basic shape, but has a different structure and lower smoothness.

Refinable smooth surfaces for locally quad-dominant meshes with T-gons.

Multi-sided facets in polyhedral models and meshes serve to connect regular submeshes (star-configurations) and to start or end quad-strips (T-configurations). Using the polyhedral mesh as control net, recursive subdivision algorithms often yield poor shape for these non-quad configurations. Polynomial surface constructions such as geometrically smooth splines (G-splines) do better, but lack subdivision-like refinability. Such refinability is useful for hierarchical modeling and engineering analysis. This paper introduces a new class of G-splines that generalizes bi-quadratic C 1 splines to polyhedral control nets with star- and T-configurations and that is refinable.

Localized G-splines for quad & T-gon meshes.

Enriching tensor-product B-spline control nets by allowing T-gons (where strips of quadrilaterals start or end) and irregular nodes (where n ≠ 4 quadrilaterals meet) reduces the requirements on quad-meshing and increases the flexibility for polyhedral design with associated smooth surfaces. This paper introduces a family of piecewise polynomial, geometrically continuous surface constructions that yield good highlight line distributions also in the presence of irregular nodes next to a T-gon. Such tight juxtaposition can further reduce the quad-meshing requirements and increase the space of polyhedral design control structures. The surfaces can be chosen to cover T-gons with G 1 caps of degree bi-4 - or with caps of degree bi-3 that are almost G 1 and preserve the good highlight line distribution of the bi-4 G 1 surfaces.

Refinable bi-quartics for design and analysis.

To be directly useful both for shape design and a thin shell analysis, a surface representation has to satisfy three properties: (1) be compatible with CAD surface representations, (2) yield generically a good highlight distribution, and (3) offer a refinable space of functions on the surface. Here we propose a new construction, based on a number of recently-developed techniques, that satisfies all three criteria. The construction converts quad meshes with irregularities, where more or fewer than four quads meet, to C 1 (or, at the cost of more pieces, C 2) bi-4 surface consisting of subdivision rings for the main body completed by a tiny G 1 cap.

Improved shape for refinable surfaces with singularly parameterized irregularities.

To date, singularly-parameterized surface constructions suffer from poor highlight line distributions, ruling them out as a surface representation of choice for primary design surfaces. This paper explores graded, many-piece, everywhere C1 singularly-parameterized surface caps that mimic the shape of a high-quality guide surface. The approach illustrates the trade-off between polynomial degree and surface quality. For bi-degree 5, minor flaws in the highlight line distribution are still visible when zooming in on the singularity, but the distribution is good at the macroscopic level. Constructions of degree bi-4 or bi-3 may require one or more steps of guided subdivision to reach the same macroscopic quality. Akin to subdivision surfaces, singularly-parameterized functions on the surfaces are straightforward to refine.

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

Geometrically continuous (Gk ) constructions naturally yield families of finite elements for isogeometric analysis (IGA) that are Ck also for non-tensor-product layout. This paper describes and analyzes one such concrete C1 geometrically generalized IGA element (short: gIGA element) that generalizes bi-quadratic splines to quad meshes with irregularities. The new gIGA element is based on a recently-developed G1 surface construction that recommends itself by its a B-spline-like control net, low (least) polynomial degree, good shape properties and reproduction of quadratics at irregular (extraordinary) points. Remarkably, for Poisson's equation on the disk using interior vertices of valence 3 and symmetric layout, we observe O(h3) convergence in the L∞ norm for this family of elements. Numerical experiments confirm the elements to be effective for solving the trivariate Poisson equation on the solid cylinder, deformations thereof (a turbine blade), modeling and computing geodesics on smooth free-form surfaces via the heat equation, for solving the biharmonic equation on the disk and for Koiter-type thin-shell analysis.