Mathematics

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18 pages, 1685 KiB

Open AccessArticle

IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method

by Yini Jiang and Hongwei Lin

Mathematics 2023, 11(4), 898; https://doi.org/10.3390/math11040898 - 10 Feb 2023

Cited by 1 | Viewed by 1216

The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation [...] Read more.

The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation (IG-LSPIA), to solve the fitting problem in the collocation method. IG-LSPIA starts with an initial blending function, where the control coefficients are combined with the B-spline basis functions and their derivatives. A new blending function is generated by constructing the differences for collocation points (DCP) and control coefficients (DCC), and then adding the DCC to the corresponding control coefficients. The procedure is performed iteratively until the stop criterion is reached. We prove the convergence of IG-LSPIA and show that the computation complexity in each iteration of IG-LSPIA is related only to the number of collocation points and unrelated to the number of control coefficients. Moreover, an incremental algorithm is designed; it alternates with knot refinement until the desired precision is achieved. After each knot refinement, the result of the last round of IG-LSPIA iterations is used to generate the initial blending function of the new round of iteration, thereby saving great computation. Experiments show that the proposed method is stable and efficient. In the three-dimensional case, the total computation time is saved twice compared to the traditional method. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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16 pages, 2375 KiB

Open AccessArticle

Improved Least-Squares Progressive Iterative Approximation for Tensor Product Surfaces

by Qianqian Hu, Zhifang Wang and Ruyi Liang

Mathematics 2023, 11(3), 670; https://doi.org/10.3390/math11030670 - 28 Jan 2023

Cited by 1 | Viewed by 1197

Abstract

Geometric iterative methods, including progressive iterative approximation and geometric interpolation methods, are efficient for fitting a given data set. With the development of big data technology, the number of fitting data points has become massive, and the progressive iterative approximation for least-squares fitting [...] Read more.

Geometric iterative methods, including progressive iterative approximation and geometric interpolation methods, are efficient for fitting a given data set. With the development of big data technology, the number of fitting data points has become massive, and the progressive iterative approximation for least-squares fitting (LSPIA) is generally applied to fit mass data. Combining the Schulz iterative method for calculating the Moore–Penrose generalized inverse matrix with the traditional LSPIA method, this paper presents an accelerated LSPIA method for tensor product surfaces and shows that the corresponding iterative surface sequence converged to the least-squares fitting surface of the given data set. The iterative format is that of a non-stationary iterative method, and the convergence rate increased rapidly as the iteration number increased. Some numerical examples are provided to illustrate that the proposed method has a faster convergence rate. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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22 pages, 10368 KiB

Open AccessArticle

Polynomial-Based Non-Uniform Ternary Interpolation Surface Subdivision on Quadrilateral Mesh

by Kaijun Peng, Jieqing Tan and Li Zhang

Mathematics 2023, 11(2), 486; https://doi.org/10.3390/math11020486 - 16 Jan 2023

Viewed by 1175

Abstract

For non-uniform control polygons, a parameterized four-point interpolation curve ternary subdivision scheme is proposed, and its convergence and continuity are demonstrated. Following curve subdivision, a non-uniform interpolation surface ternary subdivision on regular quadrilateral meshes is proposed by applying the tensor product method. Analyses [...] Read more.

For non-uniform control polygons, a parameterized four-point interpolation curve ternary subdivision scheme is proposed, and its convergence and continuity are demonstrated. Following curve subdivision, a non-uniform interpolation surface ternary subdivision on regular quadrilateral meshes is proposed by applying the tensor product method. Analyses were conducted on the updating rules of parameters, proving that the limit surface is continuous. In this paper, we present a novel interpolation subdivision method to generate new virtual edge points and new face points of the extraordinary points of quadrilateral mesh. We also provide numerical examples to assess the validity of various interpolation methods. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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15 pages, 405 KiB

Open AccessArticle

On Control Polygons of Planar Sextic Pythagorean Hodograph Curves

by Yujun Li, Lincong Fang, Zhihao Zheng and Juan Cao

Mathematics 2023, 11(2), 383; https://doi.org/10.3390/math11020383 - 11 Jan 2023

Cited by 1 | Viewed by 1163

Abstract

In this paper, we analyze planar parametric sextic curves to determine conditions for Pythagorean hodograph (PH) curves. By expressing the curves to be analyzed in the complex form, the analysis is conducted in algebraic form. Since sextic PH curves can be classified into [...] Read more.

In this paper, we analyze planar parametric sextic curves to determine conditions for Pythagorean hodograph (PH) curves. By expressing the curves to be analyzed in the complex form, the analysis is conducted in algebraic form. Since sextic PH curves can be classified into two classes according to the degrees of their derivatives’ factors, we introduce auxiliary control points to reconstruct the internal algebraic structure for both classes. We prove that a sextic curve is completely characterized by the lengths of legs and angles formed by the legs of their Bézier control polygons. As such conditions are invariant under rotations and translations, we call them the geometric characteristics of sextic PH curves. We demonstrate that the geometric characteristics form the basis for an easy and intuitive method for identifying sextic PH curves. Benefiting from our results, the computations of the parameters of cusps and/or inflection points can also be simplified. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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20 pages, 525 KiB

Open AccessArticle

Generalized de Boor–Cox Formulas and Pyramids for Multi-Degree Spline Basis Functions

by Xu Ma and Wanqiang Shen

Mathematics 2023, 11(2), 367; https://doi.org/10.3390/math11020367 - 10 Jan 2023

Cited by 3 | Viewed by 1534

Abstract

The conventional B-splines possess the de Boor–Cox formula, which relates to a pyramid algorithm. However, for multi-degree splines, a de Boor–Cox-type evaluation algorithm only exists in some special cases. This paper considers any multi-degree spline with arbitrary degree and continuity, and provides two [...] Read more.

The conventional B-splines possess the de Boor–Cox formula, which relates to a pyramid algorithm. However, for multi-degree splines, a de Boor–Cox-type evaluation algorithm only exists in some special cases. This paper considers any multi-degree spline with arbitrary degree and continuity, and provides two generalized de Boor–Cox-type relations. One uses several lower degree polynomials to build a combination to evaluate basis functions, whose form is similar to using the de Boor–Cox formula several times. The other is a linear combination of two functions out of the recursive definition, which keeps the combination coefficient polynomials of degree 1, so it is more similar to the de Boor–Cox formula and can be illustrated by several pyramids with different heights. In the process of calculating the recursions, a recursive representation using the Bernstein basis is used and numerically analyzed. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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20 pages, 4341 KiB

Open AccessFeature PaperArticle

G² Hermite Interpolation by Segmented Spirals

by Yuxuan Zhou, Yajuan Li and Chongyang Deng

Mathematics 2022, 10(24), 4757; https://doi.org/10.3390/math10244757 - 14 Dec 2022

Viewed by 1234

Abstract

A curve with single-signed, monotonically increasing or decreasing curvatures is referred to as a planar spiral.

G^{2}

Hermite data are spiral

G^{2}

Hermite data for which only interpolation by a spiral is possible. In this study, we design segmented spirals to [...] Read more.

A curve with single-signed, monotonically increasing or decreasing curvatures is referred to as a planar spiral.

G^{2}

Hermite data are spiral

G^{2}

Hermite data for which only interpolation by a spiral is possible. In this study, we design segmented spirals to geometrically interpolate arbitrary C-shaped

G^{2}

Hermite data. To separate the data into two or three spiral data sets, we add one or two new points, related tangent vectors and curvatures. We provide different approaches in accordance with the various locations of the external homothetic centers of two end-curvature circles. We then match new data by constructing two or three segmented spirals. We generate at most three piecewise spirals for arbitrary C-shaped data. Furthermore, we illustrate the suggested techniques with several examples. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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11 pages, 1667 KiB

Open AccessArticle

Equiareal Parameterization of Triangular Bézier Surfaces

by Jun Chen, Xiang Kong and Huixia Xu

Mathematics 2022, 10(23), 4620; https://doi.org/10.3390/math10234620 - 06 Dec 2022

Viewed by 1063

Abstract

Parameterization is the key property of a parametric surface and significantly affects many kinds of applications. To improve the quality of parameterization, equiareal parameterization minimizes the equiareal energy, which is presented as a measure to describe the uniformity of iso-parametric curves. With the [...] Read more.

Parameterization is the key property of a parametric surface and significantly affects many kinds of applications. To improve the quality of parameterization, equiareal parameterization minimizes the equiareal energy, which is presented as a measure to describe the uniformity of iso-parametric curves. With the help of the binary Möbius transformation, the equiareal parameterization is extended to the triangular Bézier surface on the triangular domain for the first time. The solution of the corresponding nonlinear minimization problem can be equivalently converted into solving a system of bivariate polynomial equations with an order of three. All the exact solutions of the equations can be obtained, and one of them is chosen as the global optimal solution of the minimization problem. Particularly, the coefficients in the system of equations can be explicitly formulated from the control points. Equiareal parameterization keeps the degree, control points, and shape of the triangular Bézier surface unchanged. It improves the distribution of iso-parametric curves only. The iso-parametric curves from the new expression are more uniform than the original one, which is displayed by numerical examples. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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26 pages, 3896 KiB

Open AccessArticle

Algorithms for Space Mapping Method on Spline Spaces over Modified Hierarchical T-Meshes

by Jingjing Liu, Li Zhang and Weihong Zhang

Mathematics 2022, 10(20), 3864; https://doi.org/10.3390/math10203864 - 18 Oct 2022

Viewed by 1144

Abstract

The space-mapping method provides a novel method for dimension formulae explanation and basis construction for the spline space over hierarchical T-meshes. By the space-mapping method, we provide a unique basis construction framework that incorporates basis modification of the spline space over modified hierarchical [...] Read more.

The space-mapping method provides a novel method for dimension formulae explanation and basis construction for the spline space over hierarchical T-meshes. By the space-mapping method, we provide a unique basis construction framework that incorporates basis modification of the spline space over modified hierarchical T-meshes. The subdivision rules on the modified hierarchical T-meshes are given to prevent the redundant edges that exist on hierarchical T-meshes. In the basis construction framework, we describe the spline-modification mechanism over the modified hierarchical T-mesh when the cells of the corresponding crossing vertex relationship graph (CVR graph) are adjusted. We provide the framework’s algorithms for basis construction and modification. Moreover, we discuss the application of the splines that are constructed by the framework to surface reconstruction with adaptive refinement. In comparison to splines over hierarchical T-meshes, the modified hierarchical T-meshes have fewer cells subdivided when achieving similar accuracy. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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17 pages, 21407 KiB

Open AccessArticle

Progressive Iterative Approximation of Non-Uniform Cubic B-Spline Curves and Surfaces via Successive Over-Relaxation Iteration

by Huahao Shou, Liangchen Hu and Shiaofen Fang

Mathematics 2022, 10(20), 3766; https://doi.org/10.3390/math10203766 - 12 Oct 2022

Cited by 3 | Viewed by 1205

Abstract

Geometric iteration (GI) is one of the most efficient curve- or surface-fitting techniques in recent years, which is famous for its remarkable geometric significance. In essence, GI can be thought of as the sum of iterative methods for solving systems of linear equations, [...] Read more.

Geometric iteration (GI) is one of the most efficient curve- or surface-fitting techniques in recent years, which is famous for its remarkable geometric significance. In essence, GI can be thought of as the sum of iterative methods for solving systems of linear equations, such as progressive iterative approximation (PIA) which relies on the theory of Richardson iteration. Thus, when the curve- or surface-fitting error is at a desired level, we want to have as few iterations as possible to improve efficiency when dealing with large data sets. Based on the idea of successive over-relaxation (SOR) iteration, we formulate a faster PIA curve and surface interpolation scheme using classical non-uniform cubic B-splines, named SOR-PIA. The genetic algorithm is utilized to estimate the best approximate relaxation factor of SOR-PIA. Similar to standard PIA, SOR-PIA can also be regarded as a process in which the control points move in one direction, but it can greatly reduce the number of iterations in the iterative process with the same fitting accuracy. By comparing with the standard PIA and WPIA algorithms, the effectiveness of the SOR-PIA iterative interpolation algorithm can be verified. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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15 pages, 47473 KiB

Open AccessArticle

HOME: 3D Human–Object Mesh Topology-Enhanced Interaction Recognition in Images

by Weilong Peng, Cong Li, Keke Tang, Xianyong Liu and Meie Fang

Mathematics 2022, 10(16), 2841; https://doi.org/10.3390/math10162841 - 10 Aug 2022

Viewed by 1534

Abstract

Human–object interaction (HOI) recognition is a very challenging task due to the ambiguity brought by occlusions, viewpoints, and poses. Because of the limited interaction information in the image domain, extracting 3D features of a point cloud has been an important means to improve [...] Read more.

Human–object interaction (HOI) recognition is a very challenging task due to the ambiguity brought by occlusions, viewpoints, and poses. Because of the limited interaction information in the image domain, extracting 3D features of a point cloud has been an important means to improve the recognition performance of HOI. However, the features neglect topological features of adjacent points at low level, and the deep topology relation between a human and an object at high level. In this paper, we present a 3D human–object mesh topology enhanced method (HOME) for HOI recognition in images. In the method, human–object mesh (HOM) is built by integrating the reconstructed human and object mesh from images firstly. Therefore, under the assumption that the interaction comes from the macroscopic pattern constructed by spatial position and microscopic topology of human–object, HOM is inputted into MeshCNN to extract the effective edge features by edge-based convolution from bottom to up, as the topological features that encode the invariance of the interaction relationship. At last, topological cues are fused with visual cues to enhance the recognition performance greatly. In the experiment, HOI recognition results have achieved an improvement of about 4.3% mean average precision (mAP) in the Rare cases of the HICO-DET dataset, which verifies the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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20 pages, 3194 KiB

Open AccessArticle

On a Nonlinear Three-Point Subdivision Scheme Reproducing Piecewise Constant Functions

by Sofiane Zouaoui, Sergio Amat, Sonia Busquier and Juan Ruiz

Mathematics 2022, 10(15), 2790; https://doi.org/10.3390/math10152790 - 05 Aug 2022

Viewed by 936

Abstract

In this article, a nonlinear binary three-point non-interpolatory subdivision scheme is presented. It is based on a nonlinear perturbation of the three-point subdivision scheme: A new three point approximating

C^{2}

subdivision scheme, where the convergence and the stability of this linear subdivision [...] Read more.

In this article, a nonlinear binary three-point non-interpolatory subdivision scheme is presented. It is based on a nonlinear perturbation of the three-point subdivision scheme: A new three point approximating

C^{2}

subdivision scheme, where the convergence and the stability of this linear subdivision scheme are analyzed. It is possible to prove that this scheme does not present Gibbs oscillations in the limit functions obtained. The numerical experiments show that the linear scheme is stable even in the presence of jump discontinuities. Even though, close to jump discontinuities, the accuracy is loosed. This order reduction is equivalent to the introduction of some diffusion. Diffusion is a good property for subdivision schemes when the discontinuities are numerical, i.e., they appear when discretizing a continuous function close to high gradients. On the other hand, if the initial control points come from the discretization of a piecewise continuous function, it can be interesting that the subdivision scheme produces a piecewise continuous limit function. For instance, in the approximation of conservation laws, real discontinuities appear as shocks in the solution. The nonlinear modification introduced in this work allows to attain this objective. As far as we know, this is the first subdivision scheme that appears in the literature with these properties. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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22 pages, 10798 KiB

Open AccessArticle

Some New n-Point Ternary Subdivision Schemes without the Gibbs Phenomenon

by Sofiane Zouaoui, Sergio Amat, Sonia Busquier and Mª José Legaz

Mathematics 2022, 10(15), 2674; https://doi.org/10.3390/math10152674 - 29 Jul 2022

Cited by 1 | Viewed by 909

Abstract

This paper is devoted to the construction and analysis of some new families of n-point ternary subdivision schemes. Some members of the families were adapted to the presence of discontinuities converging to limit functions without Gibbs oscillations. We present a numerical comparison [...] Read more.

This paper is devoted to the construction and analysis of some new families of n-point ternary subdivision schemes. Some members of the families were adapted to the presence of discontinuities converging to limit functions without Gibbs oscillations. We present a numerical comparison where we check the theoretical properties. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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19 pages, 7602 KiB

Open AccessArticle

Topology Optimization and Fatigue Life Estimation of Sustainable Medical Waste Shredder Blade

by Muhammad Muzammil Azad, Dohoon Kim, Salman Khalid and Heung Soo Kim

Mathematics 2022, 10(11), 1863; https://doi.org/10.3390/math10111863 - 30 May 2022

Cited by 3 | Viewed by 2987

Abstract

There is an increased interest in designing cost-effective lightweight components to meet modern design requirements of improving cost and performance efficiency. This paper describes a significant effort to optimize the medical waste shredder blade through weight reduction by increasing material efficiency. The blade [...] Read more.

There is an increased interest in designing cost-effective lightweight components to meet modern design requirements of improving cost and performance efficiency. This paper describes a significant effort to optimize the medical waste shredder blade through weight reduction by increasing material efficiency. The blade computer-aided design (CAD) model was produced through reverse engineering and converted to the finite element (FE) model to characterize von Mises stress and displacement. The obtained stress characteristics were introduced into the FE-SAFE for fatigue analysis. Furthermore, the FE model was analyzed through topological optimization using strain energy as the objective function while implementing the volume constraint. To obtain the optimal volume constraint for the blade model, several 3D numerical test cases were performed at various volume constraints. A significant weight reduction of 24.7% was observed for the 80% volume constraint (VC80). The FE analysis of optimal geometry indicated a 6 MPa decrease in the von Mises and a 14.5% increase in the fatigue life. Therefore, the proposed optimal design method demonstrated to be effective and easy to apply for the topology optimization of the shredder blade and has significantly decreased the structural weight without compromising the structural integrity and robustness. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design)

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