Generating bicubic B-spline surfaces by a sixth order PDE

Yan Wu; Chun-Gang Zhu; Yan Wu; Chun-Gang Zhu

doi:10.3934/math.2021099

AIMS Mathematics

2021, Volume 6, Issue 2: 1677-1694. doi: 10.3934/math.2021099

Previous Article Next Article

Research article

Generating bicubic B-spline surfaces by a sixth order PDE

Yan Wu ^1,2,
Chun-Gang Zhu ^{1,2
,
,}

1.
School of Mathematics, Dalian University of Technology, Dalian 116024, China
2.
Key Laboratory for Computational Mathematics and Data Intelligence of Liaoning Province, Dalian University of Technology, Dalian 116024, China

Received: 17 September 2020 Accepted: 16 November 2020 Published: 26 November 2020
MSC : 65D07, 65D17

As the solutions of partial differential equations (PDEs), PDE surfaces provide an effective way for physical-based surface design in surface modeling. The bicubic B-spline surface is a useful tool for surface modeling in computer aided geometric design (CAGD). In this paper, we present a method for generating bicubic B-spline surfaces with the uniform knots and the quasi-uniform knots from the sixth order PDEs. From the given boundary condition, based on the cubic B-spline basis representation and the PDE mask, the resulting bicubic B-spline surface can be generated uniquely. The boundary condition is more flexible and can be applied for curvature-continuous surface design, surface blending and hole filling. Some representative examples show the effectiveness of the presented method.
- bicubic B-spline surfaces,
- sixth order PDE,
- PDE surfaces
Citation: Yan Wu, Chun-Gang Zhu. Generating bicubic B-spline surfaces by a sixth order PDE[J]. AIMS Mathematics, 2021, 6(2): 1677-1694. doi: 10.3934/math.2021099

Related Papers:

Abstract

As the solutions of partial differential equations (PDEs), PDE surfaces provide an effective way for physical-based surface design in surface modeling. The bicubic B-spline surface is a useful tool for surface modeling in computer aided geometric design (CAGD). In this paper, we present a method for generating bicubic B-spline surfaces with the uniform knots and the quasi-uniform knots from the sixth order PDEs. From the given boundary condition, based on the cubic B-spline basis representation and the PDE mask, the resulting bicubic B-spline surface can be generated uniquely. The boundary condition is more flexible and can be applied for curvature-continuous surface design, surface blending and hole filling. Some representative examples show the effectiveness of the presented method.

References

[1]	G. Farin, Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann, 2002.
[2]	M. I. G. Bloor, M. J. Wilson, Generating blend surfaces using partial differential equations, Comput. Aided Design, 21 (1989), 165-171. doi: 10.1016/0010-4485(89)90071-7
[3]	M. I. G. Bloor, M. J. Wilson, Generating n-sided patches with partial differential equations, In: New advances in computer graphics, Springer-Verlag, Tokyo, 1989,129-145.
[4]	M. I. G. Bloor, M. J. Wilson, Blend design as a boundary-value problem, In: Theory and practice of geometric modeling, Springer-Verlag, Berlin, Heidelberg, 1989,221-234.
[5]	M. I. G. Bloor, M. J. Wilson, Using partial differential equations to generate free-form surfaces, Comput. Aided Design, 22 (1990), 202-212. doi: 10.1016/0010-4485(90)90049-I
[6]	M. I. G. Bloor, M. J. Wilson, Spectral approximations to PDE surfaces, Comput. Aided Design, 28 (1996), 145-152. doi: 10.1016/0010-4485(95)00060-7
[7]	H. Ugail, M. I. G. Bloor, M. J. Wilson, Techniques for interactive design using the PDE method, ACM T. Graphic., 18 (1999), 195-212. doi: 10.1145/318009.318078
[8]	C. Cosin, J. Monterde, Bézier surfaces of minimal area, In: Computational science-ICCS 2002, Springer-Verlag, Berlin, Heidelberg, 2002, 72-81.
[9]	J. Monterde, The Plateau-Bézier problem, In: Mathematics of surfaces, Springer-Verlag, Berlin, Heidelberg, 2003,262-273.
[10]	J. Monterde, Bézier surfaces of minimal area: The Dirichlet approach, Comput. Aided Geom. D., 21 (2004), 117-136. doi: 10.1016/j.cagd.2003.07.009
[11]	G. Xu, G. Z. Wang, Parametric polynomial minimal surfaces of degree six with isothermal parameter, In: Advances in geometric modeling and processing, Springer-Verlag, Berlin, Heidelberg, 2008,329-343.
[12]	G. Xu, G. Z. Wang, Quintic parametric polynomial minimal surfaces and their properties, Diff. Geom. Appl., 28 (2010), 697-704. doi: 10.1016/j.difgeo.2010.07.003
[13]	G. Xu, T. Rabczuk, E. Güler, Q. Wu, K. C. Hui, G. Wang, Quasi-harmonic Bézier approximation of minimal surfaces for finding forms of structural membranes, Comput. Struct., 161 (2015), 55-63. doi: 10.1016/j.compstruc.2015.09.002
[14]	J. Monterde, H. Ugail, A general 4th-order PDE method to generate Bézier surfaces from the boundary, Comput. Aided Geom. D., 23 (2006), 208-255. doi: 10.1016/j.cagd.2005.09.001
[15]	B. Jüttler, M. Oberneder, A. Sinwel, On the existence of biharmonic tensor-product Bézier surface patches, Comput. Aided Geom. D., 23 (2006), 612-615. doi: 10.1016/j.cagd.2006.05.003
[16]	M. I. G. Bloor, M. J. Wilson, Representing PDE surfaces in terms of B-splines, Comput. Aided Design, 22 (1990), 324-331. doi: 10.1016/0010-4485(90)90083-O
[17]	P. Kiciak, Bicubic B-spline blending patches with optimized shape, Comput. Aided Design, 43 (2011), 133-144. doi: 10.1016/j.cad.2010.10.003
[18]	G. Arora, B. K. Singh, Numerical solution of Burgers' equation with modified cubic B-spline differential quadrature method, Appl. Math. Comput., 224 (2013), 166-177.
[19]	M. Abbas, A. A. Majid, A. I. M. Ismail, A. Rashid, The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems, Appl. Math. Comput., 239 (2014), 74-88.
[20]	C. G. Zhu, R. H. Wang, Numerical solution of Burgers' equation by cubic B-spline quasiinterpolation, Appl. Math. Comput., 208 (2009), 260-272.
[21]	A. Kouibia, M. Pasadas, Optimization of the parameters of surfaces by interpolating variational bicubic splines, Math. Comput. Simulat., 102 (2014), 76-89. doi: 10.1016/j.matcom.2013.09.003
[22]	G. Farin, D. Hansford, Discrete Coons patches, Comput. Aided Geom. D., 16 (1999), 691-700. doi: 10.1016/S0167-8396(99)00031-X
[23]	X. Han, J. Han, Representation of piecewise biharmonic surfaces using biquadratic B-splines, J. Comput. Appl. Math., 290 (2015), 403-411. doi: 10.1016/j.cam.2015.05.025
[24]	X. Han, J. Han, Bicubic B-spline surfaces constrained by the Biharmonic PDE, Appl. Math. Comput., 361 (2019), 766-776.
[25]	G. G. Castro, H. Ugail, P. Willis, I. Palmer, A survey of partial differential equations in geometric design, Visual Comput., 24 (2008), 213-225. doi: 10.1007/s00371-007-0190-z
[26]	D. Lesnic, On the boundary integral equations for a two-dimensional slowly rotating highly viscous fluid flow, Adv. Appl. Math. Mech., 1 (2009), 140-150.
[27]	M. Kapl, V. Vitrih, Solving the triharmonic equation over multi-patch planar domains using isogeometric analysis, J. Comput. Appl. Math., 358 (2019), 385-404. doi: 10.1016/j.cam.2019.03.020
[28]	Y. Wu, C. G. Zhu, Construction of triharmonic Bézier surfaces from boundary conditions, J. Comput. Appl. Math., 377 (2020), 112906. doi: 10.1016/j.cam.2020.112906
[29]	L. H. You, P. Comninos, J. J. Zhang, PDE blending surfaces with C² continuity, Comput. Graph., 28 (2004), 895-906. doi: 10.1016/j.cag.2004.08.003
[30]	J. J. Zhang, L. H. You, Fast surface modelling using a 6th order PDE, Comput. Graph. Forum, 23 (2004), 311-320. doi: 10.1111/j.1467-8659.2004.00762.x
[31]	M. Botsch, L. Kobbelt, An intuitive framework for real-time freeform modeling, ACM T. Graphic., 23 (2004), 630-634. doi: 10.1145/1015706.1015772
[32]	D. Liu, G. L. Xu, A general sixth order geometric partial differential equation and its application in surface modeling, Journal of Information and Computational Science, 4 (2007), 129-140.
[33]	Q. Zhang, G. L. Xu, J. Sun, A general sixth order geometric flow and its applications in surface processing, In: Proceedings of 2007 International Conference on Cyberworlds (CW'07), Hannover, 2007,447-456.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)