Minimal quasi-Bézier surface

Hao, Yong-Xia; Wang, Ren-Hong; Li, Chong–Jun

doi:10.1016/j.apm.2012.01.040

Cited by 17 publications

(10 citation statements)

References 12 publications

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“…The Willmore energy has been demonstrated to have possible applications in many different aspects of molecular, biological, and nanotechnology science. On the other hand, catenoids, as absolute minimizers of the Willmore energy, have numerous uses in nanomagnetism [ 5 – 7 , 15 , 26 ]. This paper expands the method of using catenoids to establish the conformation of the connection of BN nanostructures, namely, a BN nanotube to a BN fullerene and two BN fullerenes to each other.…”

Section: Introductionmentioning

confidence: 99%

Mathematical energy minimization model for joining boron nitride fullerene with several BN nanostructures

Alshammari

2021

J Mol Model

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Nanoscale materials have gained considerable interest because of their special properties and wide range of applications. Many types of boron nitride at the nanoscale have been realized, including nanotubes, nanocones, fullerenes, tori, and graphene sheets. The connection of these structures at the nanoscale leads to merged structures that have enhanced features and applications. Modeling the joining between nanostructures has been adopted by different methods. Namely, carbon nanostructures have been joined by minimizing the elastic energy in symmetric configurations. In other words, the only considerable curvature in the elastic energy is the axial curvature. Accordingly, because it has nanoscale structures similar to those in carbon, BN can also be joined and connected by using this method. On the other hand, different methods have been proposed to consider the rotational curvature because it has a similar size. Based on that argument, the Willmore energy, which depends on both curvatures, has been minimized to join carbon nanostructures. This energy is used to identify the joining region, especially for a three-dimensional structure. In this paper, we expand the use of Willmore energy to cover the joining of boron nitride nanostructures. Therefore, because catenoids are absolute minimizers of this energy, pieces of catenoids can be used to connect nanostructures. In particular, we joined boron nitride fullerene to three other BN nanostructures: nanotube, fullerene, and torus. For now, there are no experimental or simulation data for comparison with the theoretical connecting structures predicted by this study, which is some justification for the suggested simple model shown in this research. Ultimately, various nanoscale BN structures might be connected by considering the same method, which may be considered in future work.

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Section: Introductionmentioning

confidence: 99%

Mathematical energy minimization model for joining boron nitride fullerene with several BN nanostructures

Alshammari

2021

J Mol Model

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“…Based on this functional, a multiresolution analysis method with B-splines is proposed to obtain the parametric surface of minimal area in [7]. Moreover, the minimal quasi-Bézier surfaces in non-polynomial space are also investigated by the Dirichlet method and harmonic method in [8]. A new energy functional called quasi-harmonic energy functional is proposed in [21] as the objective functional to obtain the quasi-harmonic Bézier surface from given boundaries.…”

Section: Introductionmentioning

confidence: 99%

The Plateau-Bézier Problem with Weak-Area Functional

Hao¹

2020

JCM

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In this paper, we present a new method to solve the Plateau-Bézier problem. A new energy functional called weak-area functional is proposed as the objective functional to obtain the approximate minimal Bézier surface from given boundaries. This functional is constructed based on Dirichlet energy and weak isothermal parameterization condition. Experimental comparisons of the weak-area functional method with existing Dirichlet, quasi-harmonic, the strain energy-minimizing, harmonic and biharmonic masks are performed which show that the weak-area functional method are among the best by choosing appropriate parameters.

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“…A more general case of borders is taken in ref. [25] that study the Plateau-quasi-Bézier problem which includes the case when the boundary curves are catenaries and circular arcs. The Plateau-quasi-Bézier problem is related to the quasi-Bézier surface with minimal area among all the quasi-Bézier surfaces with prescribed border.…”

Section: Introductionmentioning

confidence: 99%

“…(11) to write the Coons patch spanned by N > 4-number of curves which may then be used to find the associated minimal surface by the ansatz eq. (25). Using that formalism our technique can be applied to any number of curves, we have implemented it in full though numerical implementation has been limited to five straight lines.…”

Section: Introductionmentioning

confidence: 99%

Variational minimization on string-rearrangement surfaces, illustrated by an analysis of the bilinear interpolation

Ahmad

Masud

2014

Applied Mathematics and Computation

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In this paper we present an algorithm to reduce the area of a surface spanned by a finite number of boundary curves by initiating a variational improvement in the surface. The ansatz we suggest consists of original surface plus a variational parameter t multiplying the numerator H0 of mean curvature function defined over the surface. We point out that the integral of the square of the mean curvature with respect to the surface parameter becomes a polynomial in this variational parameter. Finding a zero, if there is any, of this polynomial would end up at the same (minimal) surface as obtained by minimizing more complicated area functional itself. We have instead minimized this polynomial. Moreover, our minimization is restricted to a search in the class of all surfaces allowed by our ansatz. All in all, we have not yet obtained the exact minimal but we do reduce the area for the same fixed boundary. This reduction is significant for a surface (hemiellipsoid) for which we know the exact minimal surface. But for the bilinear interpolation spanned by four bounding straight lines, which can model the initial and final configurations of re-arranging strings, the decrease remains less than 0.8 percent of the original area. This may suggest that bilinear interpolation is already a near minimal surface.

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Minimal quasi-Bézier surface

Cited by 17 publications

References 12 publications

Mathematical energy minimization model for joining boron nitride fullerene with several BN nanostructures

Mathematical energy minimization model for joining boron nitride fullerene with several BN nanostructures

The Plateau-Bézier Problem with Weak-Area Functional

Variational minimization on string-rearrangement surfaces, illustrated by an analysis of the bilinear interpolation

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