A Geometric Approach for Multi-Degree Spline

Li, Xin; Huang, Zhangjin; Liu, Zhao

doi:10.1007/s11390-012-1268-2

Cited by 16 publications

(17 citation statements)

References 9 publications

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“…Multi-degree splines were introduced in [34] for degrees (1,2,3) and (1, n). In a more general framework, basis functions for multi-degree splines were constructed in [37,36,38] by means of recursive integral relations involving global quantities that [19] observed as being difficult to compute. Instead, [19] presented a recursive geometric algorithm for computing multi-degree spline curves.…”

Section: Related Literaturementioning

confidence: 99%

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

Toshniwal

Speleers

Hiemstra

et al. 2017

Computer Methods in Applied Mechanics and Engineering

101

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We develop a multi-degree polar spline framework with applications to both geometric modeling and isogeometric analysis. First, multi-degree splines are introduced as piecewise non-uniform rational B-splines (NURBS) of non-uniform or variable polynomial degree, and a simple algorithm for their construction is presented. Then, an extension to two-dimensional polar configurations is provided by means of a tensor-product construction with a collapsed edge. Suitable combinations of these basis functions yield C k smooth polar splines for any k ≥ 0. We show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality. Explicit constructions for k ∈ {0, 1, 2} are presented. Optimal approximation behavior is observed numerically, and examples of free-form design, smooth hole-filling, and high-order partial differential equations demonstrate the applicability of the developed framework.

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Section: Related Literaturementioning

confidence: 99%

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

Toshniwal

Speleers

Hiemstra

et al. 2017

Computer Methods in Applied Mechanics and Engineering

101

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“…where η = 1 2 4 √ Ω(ξ − ξ 0 ) with Ω and k in (33). Note that for p = 1, (33) yields Ω = 1 169 , k = 1, and the cnoidal wave (34) becomes the solitary wave (14).…”

Section: A Cn Solutions For the Fifth-order Kdv Equationmentioning

confidence: 99%

“…Solutions in closed analytical form serve as benchmarks for numerical solvers [30,31] and comparison with experimental data. From a dynamical systems perspective, closed-form solutions play an important role in studies of bifurcations of traveling wave solutions [32,33]. Usually, hump-type solutions correspond to homoclinic orbits, kink-type solutions to heteroclinic orbits (also called connecting orbits), and periodic traveling waves to periodic orbits in phase space.…”

Section: Introductionmentioning

confidence: 99%

Traveling Wave Solutions to Fifth- and Seventh-order Korteweg–de Vries Equations: Sech and Cn Solutions

Mancas

Hereman

2018

J. Phys. Soc. Jpn.

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In this paper we review the physical relevance of a Korteweg-de Vries (KdV) equation with higherorder dispersion terms which is used in the applied sciences and engineering. We also present exact traveling wave solutions to this generalized KdV equation using an elliptic function method which can be readily applied to any scalar evolution or wave equation with polynomial terms involving only odd derivatives. We show that the generalized KdV equation still supports hump-shaped solitary waves as well as cnoidal wave solutions provided that the coefficients satisfy specific algebraic constraints.Analytical solutions in closed form serve as benchmarks for numerical solvers or comparison with experimental data. They often correspond to homoclinic orbits in the phase space and serve as separatrices of stable and unstable regions. Some of the solutions presented in this paper correct, complement, and illustrate results previously reported in the literature, while others are novel.

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“…Most of them do not include the possibility of using multiple knots, and as a consequence, they do not allow to control the degree of continuity as we do with conventional splines. In particular the constructions proposed in [10] and [13] yield splines that are exactly C d−1 -continuous at the join between two segments of same degree d. Between two segments of degrees d i−1 and d i , instead, continuity of order C min(d i−1 ,d i ) is attained by the method in [13], while the splines in [10] can be C 1 -continuous only. However, both constructions do not allow for using multiple knots in order to reduce the continuity.…”

Section: Introductionmentioning

confidence: 99%

On multi-degree splines

Beccari

Casciola

Morigi

2017

Computer Aided Geometric Design

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Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multidegree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to Bézier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines.

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A Geometric Approach for Multi-Degree Spline

Cited by 16 publications

References 9 publications

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

Traveling Wave Solutions to Fifth- and Seventh-order Korteweg–de Vries Equations: Sech and Cn Solutions

On multi-degree splines

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