Article
Mathematics, Interdisciplinary Applications
Jinggai Li, Ye Ji, Chungang Zhu
Summary: De Casteljau algorithm and degree elevation of toric surface patches, including tensor product and triangular rational Bezier surfaces, are important techniques in computer aided geometric design. This paper introduces these algorithms and their applications, demonstrating their effectiveness through representative examples of toric surface patches with common shapes.
JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY
(2021)
Article
Mathematics
Wei Meng, Caiyun Li, Qianqian Liu
Summary: By introducing the parametric and geometric continuity constraints of generalized C-Bezier curves and surfaces with shape parameters, it is possible to easily change curves and surfaces while preserving their characteristics and geometrical configuration.
Article
Mathematics, Applied
Syed Ahmad Aidil Adha Said Mad Zain, Md Yushalify Misro
Summary: Designing complex surfaces is a major problem in various industries, and traditional continuity methods have limitations. This paper proposes fractional continuity for generalized fractional Bezier surfaces, which provides a more convenient and faster solution for generating complex surfaces.
Article
Computer Science, Software Engineering
Benjamin Marussig, Ulrich Reif
Summary: We analyze surface patches with rounded corners, providing sufficient conditions for G(1) smoothness and investigating curvature integrability.
COMPUTER AIDED GEOMETRIC DESIGN
(2022)
Article
Mathematics, Applied
Ales Vavpetic, Emil Zagar
Summary: This paper points out that the results on the optimal approximation of symmetric surfaces in [1] are incorrect and proves it by considering the optimal approximation of spherical squares. The paper provides a detailed analysis and a numerical algorithm, offering the best approximant based on the (simplified) radial error, which differs from the result obtained in [1]. The paper also examines the approximation of the sphere using continuous splines of two and six tensor product quadratic Bezier patches and discusses the problem of approximating spherical rectangles, indicating that multiple optimal approximants might exist in some cases.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics
Haibin Fu, Shaojun Bian, Ouwen Li, Jon Macey, Andres Iglesias, Ehtzaz Chaudhry, Lihua You, Jian Jun Zhang
Summary: In this paper, a new modeling method is introduced to create 3D models by generating characteristic cross section curves. The approach ensures C-2 continuity between adjacent surface patches and saves manual operations, reduces information storage, and generates 3D models quickly.
Article
Mathematics
Samia BiBi, Md Yushalify Misro, Muhammad Abbas, Abdul Majeed, Tahir Nazir
Summary: This article introduces a novel method for constructing generalized hybrid trigonometric (GHT-Bezier) developable surfaces, which allows for the establishment of a class of GHT-Bezier developable surfaces with different shape control parameters and possess many properties of GHT-Bezier surfaces. The article also derives the continuity conditions between GHT-Bezier developable surfaces.
Article
Computer Science, Interdisciplinary Applications
Chengzhi Liu, Zhongyun Liu, Xuli Han
Summary: This paper presents the preconditioned progressive iterative approximation (PPIA) based on diagonally compensated reduction for tensor product Bezier patches, which significantly accelerates the convergence rate of progressive iterative approximation (PIA). An inexact PPIA format is introduced to enhance the robustness and reduce the computational complexity. Several numerical examples are provided to demonstrate the effectiveness of the proposed methods.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Computer Science, Interdisciplinary Applications
S. J. P. Pamela, G. T. A. Huijsmans, M. Hoelzl
Summary: The international tokamak ITER is making progress towards assembly completion and first plasma operation, posing a physics and engineering challenge. Non-linear MHD simulations play a vital role in preparing for ITER experimental scenarios, helping to understand and predict the behavior and stability of tokamak plasmas in future fusion power plants.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Software Engineering
Ying-Ying Yu, Ye Ji, Jing-Gai Li, Chun-Gang Zhu
Summary: Parameterizations are widely used in various fields like computer aided design and computer graphics, and the injective property of toric volumes is discussed in this paper. An algorithm for checking the compatibility of two sets using the mixed product of three vectors is proposed, and the effectiveness of the method is demonstrated through examples. Additionally, the algorithm is improved based on properties of clean and empty tetrahedrons in combinatorics.
COMPUTERS & GRAPHICS-UK
(2021)
Article
Mathematics, Applied
Ivan Cheltsov, Adrien Dubouloz, Takashi Kishimoto
Summary: This study focuses on toric G-solid Fano threefolds with at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups. All varieties are assumed to be projective and defined over the field of complex numbers.
SELECTA MATHEMATICA-NEW SERIES
(2023)
Article
Mathematics, Applied
Xuanyi Zhao, Ying Wang, Jinggai Li, Chungang Zhu
Summary: This paper investigates algorithms for constructing offset curves of toric curves, including algorithms based on control polygons or on degree elevation, as well as approximate offsetting algorithms. These algorithms are able to deal with curves exhibiting self-intersections and cusps. Examples of constructing non-self-intersecting offset curves and comparison with other methods are proposed.
COMPUTATIONAL & APPLIED MATHEMATICS
(2022)
Article
Computer Science, Software Engineering
Peter Salvi, Marton Vaitkus, Tamas Varady
Summary: We investigate multi-sided patches that interpolate ribbon surfaces along their boundaries. Algorithms are proposed to set the cross-derivatives of these patches, and a constrained editing technique based on control vectors is suggested. Continuity constraints and smooth connections are ensured in defining a nice surface.
COMPUTERS & GRAPHICS-UK
(2023)
Article
Mathematics
Arman Sarikyan
Summary: We classify toric Fano threefolds with at most terminal singularities, where G is a finite group acting on the variety by automorphisms and the rank of the G-invariant part of the class group equals one.
INTERNATIONAL JOURNAL OF MATHEMATICS
(2022)
Article
Computer Science, Software Engineering
Jiong Yang, Tao Ning, YunChao Shen
Summary: This paper investigates planar G(3) Hermite interpolation using a quintic Bdzier curve, constructing first and second derivatives satisfying G(2) condition and deriving two parameters satisfying G(3) condition. By ensuring equality of the first derivative of curvature with respect to arc length, the two parameters can be computed as solutions of linear systems, obtaining the control points of the quintic Bdzier curve.
Article
Mathematics, Interdisciplinary Applications
Jinggai Li, Ye Ji, Chungang Zhu
Summary: De Casteljau algorithm and degree elevation of toric surface patches, including tensor product and triangular rational Bezier surfaces, are important techniques in computer aided geometric design. This paper introduces these algorithms and their applications, demonstrating their effectiveness through representative examples of toric surface patches with common shapes.
JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY
(2021)
Article
Computer Science, Software Engineering
Ying-Ying Yu, Ye Ji, Jing-Gai Li, Chun-Gang Zhu
Summary: Parameterizations are widely used in various fields like computer aided design and computer graphics, and the injective property of toric volumes is discussed in this paper. An algorithm for checking the compatibility of two sets using the mixed product of three vectors is proposed, and the effectiveness of the method is demonstrated through examples. Additionally, the algorithm is improved based on properties of clean and empty tetrahedrons in combinatorics.
COMPUTERS & GRAPHICS-UK
(2021)
Article
Mathematics, Applied
Shenggang Zhang, Chungang Zhu, Qinjiao Gao
Summary: In this paper, we propose a modification to the classical cubic B-spline quasi-interpolant Q(3) by using multi-node higher-order expansions and the P-t-exact A-discretization approach. The proposed quartic spline quasi-interpolant Q(3)(1) achieves higher accuracy with relatively small computational complexity, without requiring any additional information of the target function. We also develop high accuracy quadrature formulas and differentiation matrices using the proposed quasi-interpolant, providing explicit coefficient error estimates. Numerical experiments demonstrate the effectiveness of the proposed quasi-interpolants.
APPLICABLE ANALYSIS
(2023)
Article
Computer Science, Software Engineering
Ye Ji, Jing-Gai Li, Ying-Ying Yu, Chun-Gang Zhu
Summary: This paper proposes an improved subdivision algorithm for toric surface patches that can divide an N-sided toric surface patch into rational tensor product Bézier surface patches while maintaining continuity. By combining it with the knot insertion algorithm of nonuniform rational B-splines, a novel h-refinement scheme for isogeometric analysis with planar toric parameterizations is developed.
COMPUTER AIDED GEOMETRIC DESIGN
(2022)
Article
Mathematics, Applied
Ye Ji, Ying-Ying Yu, Meng-Yun Wang, Chun-Gang Zhu
Summary: In this paper, an algorithm for injective NURBS parameterization of computational domains is proposed, using both internal control points and weights as optimization variables to improve parameterization quality and robustness. Numerical examples demonstrate the effectiveness and superiority of this method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Computer Science, Software Engineering
Ye Ji, Meng-Yun Wang, Mao-Dong Pan, Yi Zhang, Chun-Gang Zhu
Summary: In this paper, a robust and efficient volumetric parameterization method based on penalty functions and the Jacobian regularization technique is proposed. The method does not require bijective initialization and makes innovative contributions in the objective function, penalty function, and numerical integration strategy.
COMPUTER AIDED GEOMETRIC DESIGN
(2022)
Article
Mathematics, Applied
Xuanyi Zhao, Ying Wang, Jinggai Li, Chungang Zhu
Summary: This paper investigates algorithms for constructing offset curves of toric curves, including algorithms based on control polygons or on degree elevation, as well as approximate offsetting algorithms. These algorithms are able to deal with curves exhibiting self-intersections and cusps. Examples of constructing non-self-intersecting offset curves and comparison with other methods are proposed.
COMPUTATIONAL & APPLIED MATHEMATICS
(2022)
Article
Computer Science, Software Engineering
Ye Ji, Meng-Yun Wang, Yu Wang, Chun-Gang Zhu
Summary: This study focuses on an r-adaptive parameterization technique for isogeometric analysis, which characterizes numerical errors using the principal curvature of the solution surfaces. The study also develops algorithms for sensitivity propagation and computation for optimization. Several examples demonstrate the effectiveness of the proposed method.
COMPUTER-AIDED DESIGN
(2022)
Article
Mathematics, Interdisciplinary Applications
Ji Ye, Wang Mengyun, Yu Yingying, Zhu Chungang
Summary: Inspired by the r-refinement method in isogeometric analysis, the authors propose a curvature-based r-adaptive isogeometric method for planar multi-sided computational domains parameterized by toric surface patches. The method uses three absolute curvature metrics to characterize gradient information and optimizes internal weights to achieve an analysis-suitable and injectivity-preserving parameterization. The effectiveness and efficiency of the proposed method are demonstrated by solving several PDEs over multi-sided computational domains parameterized by toric surface patches.
JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY
(2023)
Article
Mathematics, Applied
Meng Sun, Lin Lan, Chun -Gang Zhu, Fengchun Lei
Summary: Traditional end conditions for cubic spline interpolation involve values and derivatives of interpolated functions at boundary knots. The not-a-knot end condition proposed by de Boor (1985) is a practical approach that doesn't require derivatives at the end knots. However, it suffers from a significant disadvantage of reduced accuracy at boundary intervals. In this paper, we propose an optimal arrangement of shifted spline knots that improves the interpolation accuracy by approximately 3.4 times compared to de Boor's not-a-knot end condition. We also present optimal end conditions for approximating the first and second derivatives. Representative examples demonstrate the effectiveness and superiority of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Pei Zhou, Chun -Gang Zhu
Summary: In this paper, we propose a new residual parameterization method for planar physical domains in isogeometric collocation (IGC), aiming to improve the numerical accuracy of solving partial differential equations (PDEs). The method minimizes objective functions consisting of geometry-related functionals and analysis-related residual norms in an unconstrained optimization problem. Reduced quadrature rules are applied to simplify the computation of residual norms. Numerical examples show that the proposed method achieves a significantly higher numerical accuracy compared to the standard IGC method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Xuanyi Zhao, Jinggai Li, Shiqi He, Chungang Zhu
Summary: This paper discusses the concept of injectivity of 3D Bezier volumes, and proves that a Bezier volume is injective if and only if its control points set is compatible. An algorithm for checking injectivity is proposed, along with several explicit examples.
Article
Mathematics, Applied
Yan Wu, Chun-Gang Zhu
Summary: This paper presents a method for generating bicubic B-spline surfaces from sixth order PDEs, allowing for flexible and effective surface design based on different boundary conditions. Representative examples demonstrate the method's effectiveness in surface modeling.
Article
Thermodynamics
Lan-Yin Sun, Chun-Gang Zhu
ADVANCES IN MECHANICAL ENGINEERING
(2020)
Article
Mathematics, Applied
Cai-Yun Li, Chun-Gang Zhu
