10 research outputs found

    A C r Trivariate Macro-Element Based on the Alfeld Split of Tetrahedra

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    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 C1C^1 cubic Clough-Tocher macro-triangles

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    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

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    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

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    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

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    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

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    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

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    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 Techniques for the Modeling, Processing and Visualization of Surfaces and Volumes

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    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
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