Flexible G1 Interpolation of Quad Meshes

International audienceTransforming an arbitrary mesh into a smooth G1 surface has been the subject of intensive research works. To get a visual pleasing shape without any imperfection even in the presence of extraordinary mesh vertices is still a challenging problem in particular when interpolation of the mesh vertices is required. We present a new local method, which produces visually smooth shapes while solving the interpolation problem. It consists of combining low degree biquartic Bézier patches with minimum number of pieces per mesh face, assembled together with G1-continuity. All surface control points are given explicitly. The construction is local and free of zero-twists. We further show that within this economical class of surfaces it is however possible to derive a sufficient number of meaningful degrees of freedom so that standard optimization techniques result in high quality surfaces

G1-smooth Biquintic Approximation of Catmull-Clark Subdivision Surfaces

International audienceIn this paper a construction of a globally G1 family of Bézier surfaces, defined by smoothing masks approximating the well-known Catmull-Clark (CC) subdivision surface is presented. The resulting surface is a collection of Bézier patches, which are bicubic C2 around regular vertices and biquintic G1 around extraordinary vertices (and C1 on their one-rings vertices). Each Bézier point is computed using a locally defined mask around the neighboring mesh vertices. To define G1 conditions, we assign quadratic gluing data around extraordinary vertices that depend solely on their valence and we use degree five patches to satisfy these G1 constraints. We explore the space of possible solutions, considering several projections on the solution space leading to different explicit formulas for the masks. Certain control points are computed by means of degree elevation of the C0 scheme of Loop and Schaefer [22], while for others, explicit masks are deduced by providing closed-form solutions of the G1 conditions, expressed in terms of the masks. We come up with four different schemes and conduct curvature analysis on an extensive benchmark in order to assert the quality of the resulting surfaces and identify the ones that lead to the best result, both visually and numerically. We demonstrate that the resulting surfaces converge quadratically to the CC limit when the mesh is subdivided

Geometrically smooth spline bases for data fitting and simulation

International audienceGiven a topological complex $M$ with glueing data along edges shared by adjacent faces, we study the associated space of geometrically smooth spline functions that satisfy differentiability properties across shared edges. We present new and efficient constructions of basis functions of the space of $G^{1}$-spline functions on quadrangular meshes, which are tensor product b-spline functions on each quadrangle and with b-spline transition maps across the shared edges. This new strategy for constructing basis functions is based on a local analysis of the edge functions, and does not depend on the global topology of $M$. We show that the separability of the space of $G^{1}$ splines across an edge allows to determine the dimension and a basis of the space of $G^{1}$ splines on $M$.This leads to explicit and effective constructions of basis functions attached to the vertices, edges and faces of $M$.This basis construction has important applications in geometric modeling and simulation. We illustrate it by the fitting of point clouds by $G^{1}$ splines on quadrangular meshes of complex topology and in Isogeometric Analysis methods for the solution of diffusion equations. The ingredients are detailed and experimentation results showing the behavior of the method are presented

G 1 -smooth splines on quad meshes with 4-split macro-patch elements

We analyze the space of smooth spline functions on quad-meshes. These functions are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges so that the space of smooth splines is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of G1 spline functions on each quadrangular face of the mesh. We determine the dimension of this space of G1 spline functions for degree big enough and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. This construction requires the analysis of the module of syzygies of univariate b-spline functions with b-spline function coefficients. We provide new results on their generators and the dimension of the graded pieces.Examples of bases of G1 splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions

Mini-Workshop: Mathematical Foundations of Isogeometric Analysis

Isogeometric Analysis (IgA) is a new paradigm which is designed to merge two so far disjoint disciplines, namely, numerical simulations for partial differential equations (PDEs) and applied geometry. Initiated by the pioneering 2005 paper of one of us organizers (Hughes), this new concept bridges the gap between classical finite element methods and computer aided design concepts. Traditional approaches are based on modeling complex geometries by computer aided design tools which then need to be converted to a computational mesh to allow for simulations of PDEs. This process has for decades presented a severe bottleneck in performing efficient simulations. For example, for complex fluid dynamics applications, the modeling of the surface and the mesh generation may take several weeks while the PDE simulations require only a few hours. On the other hand, simulation methods which exactly represent geometric shapes in terms of the basis functions employed for the numerical simulations bridge the gap and allow from the beginning to eliminate geometry errors. This is accomplished by leaving traditional finite element approaches behind and employing instead more general basis functions such as B-Splines and Non-Uniform Rational B-Splines (NURBS) for the PDE simulations as well. The combined concept of Isogeometric Analysis (IgA) allows for improved convergence and smoothness properties of the PDE solutions and dramatically faster overall simulations. In the last few years, this new paradigm has revolutionized the engineering communities and triggered an enormous amount of simulations and publications mainly in this field. However, there are several profound theoretical issues which have not been well understood and which are currently investigated by researchers in Numerical Analysis, Approximation Theory and Applied Geometry

Locally refinable gradient meshes supporting branching and sharp colour transitions:Towards a more versatile vector graphics primitive

We present a local refinement approach for gradient meshes, a primitive commonly used in the design of vector illustrations with complex colour propagation. Local refinement allows the artist to add more detail only in the regions where it is needed, as opposed to global refinement which often clutters the workspace with undesired detail and potentially slows down the workflow. Moreover, in contrast to existing implementations of gradient mesh refinement, our approach ensures mathematically exact refinement. Additionally, we introduce a branching feature that allows for a wider range of mesh topologies, as well as a feature that enables sharp colour transitions similar to diffusion curves, which turn the gradient mesh into a more versatile and expressive vector graphics primitive

A family of C1C^1 quadrilateral finite elements

We present a novel family of $C^1$ quadrilateral finite elements, which define global $C^1$ spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by (Brenner and Sung, J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product degree $p\geq 6$, to all degrees $p \geq 3$. Thus, we call the family of $C^1$ finite elements Brenner-Sung quadrilaterals. The proposed $C^1$ quadrilateral can be seen as a special case of the Argyris isogeometric element of (Kapl, Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles. Just as for the Argyris triangle, we additionally impose $C^2$ continuity at the vertices. In this paper we focus on the lower degree cases, that may be desirable for their lower computational cost and better conditioning of the basis: We consider indeed the polynomial quadrilateral of (bi-)degree~$5$, and the polynomial degrees $p=3$ and $p=4$ by employing a splitting into $3\times3$ or $2\times2$ polynomial pieces, respectively. The proposed elements reproduce polynomials of total degree $p$. We show that the space provides optimal approximation order. Due to the interpolation properties, the error bounds are local on each element. In addition, we describe the construction of a simple, local basis and give for $p\in\{3,4,5\}$ explicit formulas for the B\'{e}zier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed $C^1$ quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for $p=5$) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom