Approximate parameterization by planar rational curves

Jüttler, Bert; Chalmovianský, Pavel

doi:10.1145/1037210.1037215

Cited by 4 publications

(5 citation statements)

References 13 publications

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“…To select optimal values for m 0 and m 1 , we choose

{bold-italicm}^{*} = \underset{bold-italicm \in {double-struckR}^{2}}{argmin} F false(bold-italicm false),

where

F false(bold-italicm false) = \int_{0}^{1} φ^{2} false(bold-italicc false(t, bold-italicm false) false) d t

is the objective function and m ={ m 0 , m 1 } is now an unknown in the expression for c given in . A similar method to approximate the implicit curve φ ( x )=0 is used by Dokken and Jüttler and Chalmovianský . Jüttler and Chalmovianský adopt rational Bézier curves to represent the implicit curve approximation.…”

Section: Level Set Methodsmentioning

confidence: 99%

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Modeling curved interfaces without element‐partitioning in the extended finite element method

Chin

Sukumar

2019

Numerical Meth Engineering

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In this paper, we model holes and material interfaces (weak discontinuities) in two-dimensional linear elastic continua using the extended finite element method on higher-order (spectral) finite element meshes. Arbitrary parametric curves such as rational Bézier curves and cubic Hermite curves are adopted in conjunction with the level set method to represent curved interfaces. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme-a method that uses Euler's homogeneous function theorem and Stokes' theorem to reduce integration to the boundary of the domain. Numerical integration on cut elements requires the evaluation of a one-dimensional integral over a parametric curve, and hence, the need to partition curved elements is eliminated. To improve stiffness matrix conditioning, ghost penalty stabilization and the Jacobi preconditioner are used.For material interface problems, we develop an enrichment function that captures weak discontinuities on spectral meshes. Taken together, we show through numerical experiments that these advances deliver optimal algebraic rates of convergence with h-refinement (p = 1, 2, … , 5) and exponential rates of convergence with p-refinement (p = 1, 2, … , 7) for elastostatic problems with holes and material inclusions on Cartesian pth-order spectral finite element meshes. KEYWORDSBézier curves, homogeneous numerical integration method, interface stabilization, level set methods, material interface enrichment, spectral X-FEM Int J Numer Methods Eng. 2019;120:607-649.wileyonlinelibrary.com/journal/nme

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“…To select optimal values for m 0 and m 1 , we choose

{bold-italicm}^{*} = \underset{bold-italicm \in {double-struckR}^{2}}{argmin} F false(bold-italicm false),

where

F false(bold-italicm false) = \int_{0}^{1} φ^{2} false(bold-italicc false(t, bold-italicm false) false) d t

Section: Level Set Methodsmentioning

confidence: 99%

“…A similar method to approximate the implicit curve φ ( x )=0 is used by Dokken and Jüttler and Chalmovianský . Jüttler and Chalmovianský adopt rational Bézier curves to represent the implicit curve approximation. Rational Bézier curves are able to reproduce more shapes, as they can exactly represent genus zero algebraic curves.…”

Section: Level Set Methodsmentioning

confidence: 99%

Modeling curved interfaces without element‐partitioning in the extended finite element method

Chin

Sukumar

2019

Numerical Meth Engineering

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“…A method for parameterizing planar curves (i.e., d = 2) has been described by [Jüttler and Chalmovianský 2004]. It is based on minimizing the nonlinear objective function…”

Section: Planar Curvesmentioning

confidence: 99%

Approximate algebraic methods for curves and surfaces and their applications

Jüttler

Chalmovianský

Shalaby

et al. 2005

Proceedings of the 21st Spring Conference on Computer Graphics

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We report on approximate techniques for conversion between the implicit and the parametric representation of curves and surfaces, i.e., implicitization and parameterization. It is shown that these techniques are able to handle general free-form surfaces, and they can therefore be used to exploit the duality of implicit and parametric representations. In addition, we discuss several applications of these techniques, such as detection of self-intersections, raytracing, footpoint computation and parameterization of scattered data for parametric curve or surface fitting.

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“…The definition for surface reconstruction maybe related in terms of parametric form. [2] • Bert Juttler (2004) present a method for approximate parameterization of a planer algebric curve by a relational Bezier (spline) curve. [3]…”

mentioning

confidence: 99%

“…[2] • Bert Juttler (2004) present a method for approximate parameterization of a planer algebric curve by a relational Bezier (spline) curve. [3]…”

mentioning

confidence: 99%

3D Surface Generation Algorithm Using Lagrange Basis Functions in CAD/CAM Application

S. Bedan,

F. Alani

2008

ETJ

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The objective of this paper is to create an efficient and accurate 3D surface interior data depending on primary initial data based on Lagrangian interpolation concept. The presented algorithm of 3D surface generation is an extended of the conventional Lagrangian interpolation (1D). The interior data of the designed surface have been transformed automatically to a vertical CNC milling machine (Bridge-port) through the serial port (RS232) to machine the designed 3D surfaces, where the toolpath have been generated based on linear interpolation techniques. The result of the proposed algorithm have been compared with the well know "Hermit, Bezier and B-Spline" 3D surface generation methods.

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Approximate parameterization by planar rational curves

Cited by 4 publications

References 13 publications

Modeling curved interfaces without element‐partitioning in the extended finite element method

Modeling curved interfaces without element‐partitioning in the extended finite element method

Approximate algebraic methods for curves and surfaces and their applications

3D Surface Generation Algorithm Using Lagrange Basis Functions in CAD/CAM Application

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