10 research outputs found
A C r Trivariate Macro-Element Based on the Alfeld Split of Tetrahedra
Abstract We construct trivariate macro-elements of class C r for any r ≥ 1 over the Alfeld refinement of any tetrahedral partition in R 3 . In our construction, the degree of polynomials used for these macro-elements is the lowest possible. We also give the dimension formula for the subspace of consisting of these macro-elements
Gaussian quadrature for cubic Clough-Tocher macro-triangles
A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed
by Hammer and Stroud [14]. The quadrature rule requires n + 2 quadrature points: the barycentre of the
simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the
simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that
admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule
is exact for a larger space, namely the C1 cubic Clough-Tocher spline space over macro-triangles if and only
if the split-point is the barycentre. This results into a factor of three reduction in the number of quadrature
points needed to integrate the Clough-Tocher spline space exactly
Generalized Finite Element Systems for smooth differential forms and Stokes problem
We provide both a general framework for discretizing de Rham sequences of
differential forms of high regularity, and some examples of finite element
spaces that fit in the framework. The general framework is an extension of the
previously introduced notion of Finite Element Systems, and the examples
include conforming mixed finite elements for Stokes' equation. In dimension 2
we detail four low order finite element complexes and one infinite family of
highorder finite element complexes. In dimension 3 we define one low order
complex, which may be branched into Whitney forms at a chosen index. Stokes
pairs with continuous or discontinuous pressure are provided in arbitrary
dimension. The finite element spaces all consist of composite polynomials. The
framework guarantees some nice properties of the spaces, in particular the
existence of commuting interpolators. It also shows that some of the examples
are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8
figures and several comments adde
Exact conversion from BĂ©zier tetrahedra to BĂ©zier hexahedra
International audienceModeling and computing of trivariate parametric volumes is an important research topic in the field of three-dimensional isogeo-metric analysis. In this paper, we propose two kinds of exact conversion approaches from BĂ©zier tetrahedra to BĂ©zier hexahedra with the same degree by reparametrization technique. In the first method, a BĂ©zier tetrahedron is converted into a degenerate BĂ©zier hexahedron, and in the second approach, a non-degenerate BĂ©zier tetrahedron is converted into four non-degenerate BĂ©zier hexahedra. For the proposed methods, explicit formulas are given to compute the control points of the resulting tensor-product BĂ©zier hexahedra. Furthermore, in the second method, we prove that tetrahedral spline solids with C k-continuity can be converted into a set of tensor-product BĂ©zier volumes with G k-continuity. The proposed methods can be used for the volumetric data exchange problems between different trivariate spline representations in CAD/CAE. Several experimental results are presented to show the effectiveness of the proposed methods
Lagrange interpolation and quasi-interpolation using trivariate splines on a uniform partition
We develop quasi-interpolation methods and a Lagrange interpolation method for trivariate splines on a regular tetrahedral partition, based on the Bernstein-BĂ©zier representation of polynomials. The partition is based on the bodycentered cubic grid.
Our quasi-interpolation operators use quintic C2 splines and are defined by giving explicit formulae for each coefficient. One operator satisfies a certain convexity condition, but has sub-optimal approximation order. A second
operator has optimal approximation order, while a third operator interpolates the provided data values. The first two operators are defined by a small set of computation rules which can be applied independently to all tetrahedra of the underlying partition. The interpolating operator is more complex while maintaining the best-possible approximation order for the spline space. It relies on a decomposition of the partition into four classes, for each of which a set of computation rules is provided.
Moreover, we develop algorithms that construct blending operators which are based on two quasi-interpolation operators defined for the same spline space, one of which is convex. The resulting blending operator satisfies the
convexity condition for a given data set.
The local Lagrange interpolation method is based on cubic C1 splines and focuses on low locality. Our method is 2-local, while comparable methods are at least 4-local.
We provide numerical tests which confirm the results, and high-quality visualizations of both artificial and real-world data sets
New Models for High-Quality Surface Reconstruction and Rendering
The efficient reconstruction and artifact-free visualization of surfaces from measured real-world data is an important issue in various applications, such as medical and scientific visualization, quality control, and the media-related industry. The main contribution of this thesis is the development of the first efficient GPU-based reconstruction and visualization methods using trivariate splines, i.e., splines defined on tetrahedral partitions. Our methods show that these models are very well-suited for real-time reconstruction and high-quality visualizations of surfaces from volume data. We create a new quasi-interpolating operator which for the first time solves the problem of finding a globally C1-smooth quadratic spline approximating data and where no tetrahedra need to be further subdivided. In addition, we devise a new projection method for point sets arising from a sufficiently dense sampling of objects. Compared with existing approaches, high-quality surface triangulations can be generated with guaranteed numerical stability. Keywords. Piecewise polynomials; trivariate splines; quasi-interpolation; volume data; GPU ray casting; surface reconstruction; point set surface
New Models for High-Quality Surface Reconstruction and Rendering
The efficient reconstruction and artifact-free visualization of surfaces from measured real-world data is an important issue in various applications, such as medical and scientific visualization, quality control, and the media-related industry. The main contribution of this thesis is the development of the first efficient GPU-based reconstruction and visualization methods using trivariate splines, i.e., splines defined on tetrahedral partitions. Our methods show that these models are very well-suited for real-time reconstruction and high-quality visualizations of surfaces from volume data. We create a new quasi-interpolating operator which for the first time solves the problem of finding a globally C1-smooth quadratic spline approximating data and where no tetrahedra need to be further subdivided. In addition, we devise a new projection method for point sets arising from a sufficiently dense sampling of objects. Compared with existing approaches, high-quality surface triangulations can be generated with guaranteed numerical stability. Keywords. Piecewise polynomials; trivariate splines; quasi-interpolation; volume data; GPU ray casting; surface reconstruction; point set surface
Recommended from our members
Multivariate Splines and Algebraic Geometry
Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems
New Techniques for the Modeling, Processing and Visualization of Surfaces and Volumes
With the advent of powerful 3D acquisition technology, there is a growing demand
for the modeling, processing, and visualization of surfaces and volumes. The
proposed methods must be efficient and robust, and they must be able to extract the essential structure of the data and to easily and quickly convey the most significant information to a human observer. Independent of the specific nature of the data, the following fundamental problems can be identified: shape reconstruction from discrete samples, data analysis, and data compression.
This thesis presents several novel solutions to these problems for surfaces
(Part I) and volumes (Part II). For surfaces, we adopt the well-known triangle
mesh representation and develop new algorithms for discrete curvature estimation,detection of feature lines, and line-art rendering (Chapter 3), for connectivity encoding (Chapter 4), and for topology preserving compression of 2D vector fields (Chapter 5). For volumes, that are often given as discrete samples, we base our approach for reconstruction and visualization on the use of new trivariate spline spaces on a certain tetrahedral partition. We study the properties of the new spline spaces (Chapter 7) and present efficient algorithms for reconstruction and visualization by iso-surface rendering for both, regularly (Chapter 8) and irregularly (Chapter 9) distributed data samples