Novel method for exactly evaluating the energy of Catmull-Clark subdivision surfaces

A novel method is produced to evaluate the energy of the Catmull-Clark subdivision surface including extraordinary points in the control mesh. A closed-form analytic formula for thin plate energy of the Catmull-Clark subdivision surface of arbitrary topology is derived through translating the Catmull-Clark subdivision surface into bi-cubic B-spline surface pieces. Using this method, both the membrane energy and the thin plate energy can be evaluated without requiring recursive subdivision. Therefore, it is more efficient and more accurate than the existing methods for calculating the energy of the Catmull-Clark subdivision surface with arbitrary topology. The example of surface fairing demonstrates that this method is efficient and successful for evaluating the energy of subdivision surfaces.

SUBDIVISION建模在室内有机形态的应用研究

通过探索Subdivision建模技术在室内设计中的潜力,提高室内设计有机形态的吸引力和竞争力,并且有效提高设计师在有机形态设计中的效率。通过分析Subdivision建模原理,介绍其在室内运用的可能性、高效性、调节性、仿生性的特点。对其Subdivision建模在室内设计应用中的有机拓扑形态、空间界面软化、泰森多边形应用、渐消面的运用等几点优势展开讨论。结合乐嘉伦敦展厅和BarinSkiResort、Batwings案例,阐述其非线性美学和有机形态,解读其设计特点。进一步分析Subdivision建模的有机形态表达优势与局限性,探讨Subdivision建模在室内设计中的有机结构建模的可行性。探析出Subdivision建模技术能够更好设计具有非线性美学的有机形态,更有助于创造出独特的艺术形式和空间感受。为相关室内空间有机形态的设计提供了创新的设计思路。

Generalized nth-Order Perturbation Method Based on Loop Subdivision Surface Boundary Element Method for Three-Dimensional Broadband Structural Acoustic Uncertainty Analysis

In this paper,a generalized nth-order perturbation method based on the isogeometric boundary element method is proposed for the uncertainty analysis of broadband structural acoustic scattering problems.The Burton-Miller method is employed to solve the problem of non-unique solutions that may be encountered in the external acoustic field,and the nth-order discretization formulation of the boundary integral equation is derived.In addition,the computation of loop subdivision surfaces and the subdivision rules are introduced.In order to confirm the effectiveness of the algorithm,the computed results are contrasted and analyzed with the results under Monte Carlo simulations(MCs)through several numerical examples.

A Subdivision-Based Combined Shape and Topology Optimization in Acoustics

We propose a combined shape and topology optimization approach in this research for 3D acoustics by using the isogeometric boundary element method with subdivision surfaces.The existing structural optimization methods mainly contain shape and topology schemes,with the former changing the surface geometric profile of the structure and the latter changing thematerial distribution topology or hole topology of the structure.In the present acoustic performance optimization,the coordinates of the control points in the subdivision surfaces fine mesh are selected as the shape design parameters of the structure,the artificial density of the sound absorbing material covered on the structure surface is set as the topology design parameter,and the combined topology and shape optimization approach is established through the sound field analysis of the subdivision surfaces boundary element method as a bridge.The topology and shape sensitivities of the approach are calculated using the adjoint variable method,which ensures the efficiency of the optimization.The geometric jaggedness and material distribution discontinuities that appear in the optimization process are overcome to a certain degree by the multiresolution method and solid isotropic material with penalization.Numerical examples are given to validate the effectiveness of the presented optimization approach.

Conjugate-Gradient Progressive-Iterative Approximation for Loop and Catmull-Clark Subdivision Surface Interpolation

Loop and Catmull-Clark are the most famous approximation subdivision schemes,but their limit surfaces do not interpolate the vertices of the given mesh.Progressive-iterative approximation(PIA)is an efficient method for data interpolation and has a wide range of applications in many fields such as subdivision surface fitting,parametric curve and surface fitting among others.However,the convergence rate of classical PIA is slow.In this paper,we present a new and fast PIA format for constructing interpolation subdivision surface that interpolates the vertices of a mesh with arbitrary topology.The proposed method,named Conjugate-Gradient Progressive-Iterative Approximation(CG-PIA),is based on the Conjugate-Gradient Iterative algorithm and the Progressive Iterative Approximation(PIA)algorithm.The method is presented using Loop and Catmull-Clark subdivision surfaces.CG-PIA preserves the features of the classical PIA method,such as the advantages of both the local and global scheme and resemblance with the given mesh.Moreover,CG-PIA has the following features.1)It has a faster convergence rate compared with the classical PIA and W-PIA.2)CG-PIA avoids the selection of weights compared with W-PIA.3)CG-PIA does not need to modify the subdivision schemes compared with other methods with fairness measure.Numerous examples for Loop and Catmull-Clark subdivision surfaces are provided in this paper to demonstrate the efficiency and effectiveness of CG-PIA.

Stationary Subdivision 方法的一个推广及其收敛性

给出了ＳｔａｔｉｏｎａｒｙＳｕｂｄｉｖｉｓｉｏｎ方法的一个推广——矩阵Ｙ－剖分方法。利用泛函分析及样条函数给出了矩阵Ｙ－剖分方法的一个重要的收敛性质。并证明了它的收敛性。

L^p空间中的多变量Subdivision算法(英文)

本文讨论了与一般伸缩矩阵Ｍ相关的Ｓｕｂｄｉｖｓｉｏｎ算法的Ｌｐ－收敛性，多个细分方程的解可由一个给定的细分函数通过迭代算法得到。

Accurate Evaluation of Free-form Surface Profile Error Based on Quasi Particle Swarm Optimization Algorithm and Surface Subdivision

Although significant progress has been made in precision machining of free-form surfaces recently, inspection of such surfaces remains a difficult problem. In order to solve the problem that no specific standards for the verification of free-form surface profile are available, the profile parameters of free-form surface are proposed by referring to ISO standards regarding form tolerances and considering its complexity and non-rotational symmetry. Non-uniform rational basis spline（NURBS） for describing free-form surface is formulated. Crucial issues in surface inspection and profile error verification are localization between the design coordinate system（DCS） and measurement coordinate system（MCS） for searching the closest points on the design model corresponding to measured points. A quasi particle swarm optimization（QPSO） is proposed to search the transformation parameters to implement localization between DCS and MCS. Surface subdivide method which does the searching in a recursively reduced range of the parameters u and v of the NURBS design model is developed to find the closest points. In order to verify the effectiveness of the proposed methods, the design model is generated by NURBS and the measurement data of simulation example are generated by transforming the design model to arbitrary position and orientation, and the parts are machined based on the design model and are measured on CMM. The profile errors of simulation example and actual parts are calculated by the proposed method. The results verify that the evaluation precision of freeform surface profile error by the proposed method is higher 10%-22% than that by CMM software. The proposed method deals with the hard problem that it has a lower precision in profile error evaluation of free-form surface.

Resources calculation of cobalt-rich crusts with the grid subdivision and integral method

On the basis of three geological models and several orebody boundaries, a method of grid subdivision and integral has been proposed to calculate and evaluate the resources of cobalt-rich crusts on the seamounts in the central Pacific Ocean. The formulas of this method are deduced and the interface of program module is designed. The method is carried out in the software ＂Auto mapping system of submarine topography and geomorphology MBChart＂. This method and program will possibly become a potential tool to calculate the resources of seamounts and determine the target diggings for China＇ s next Five-year Plan.

Reverse Loop subdivision with sharp features

In this paper we introduce a new reverse Loop subdivision method. In contrast to current wavelets based Loop subdivision scheme, our method applies the same rules to both regular and extraordinary vertices and reconstructs the sharp features easily. Furthermore, our method runs faster because it does not need analysis and synthesis procedural. Our main goal is the design of a reverse subdivision method that can reconstruct the coarser mesh from a finer subdivision surface with sharp features for multiresolution representation, The proposed method only needs a little memory storage and brings little error, and it is easy to implement.

Subdivision surface modeling system based on arbitrary topological curves network and combined subdivision

Arbitrary topological curve network has no restriction in topology structure,so it has more powerful representing ability in defining complex surfaces.A complex surface modeling system is presented based on arbitrary topological curve network and the improved combined subdivision method,its functions including creating and editing curve network,and generating and modifying curve network＇s interpolated surface.This modeling system can be used to the process of products＇concept design,and its applications is also significant to the development of subdivision method.

The <i>m</i>-Point Quaternary Approximating Subdivision Schemes

In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. The proposed algorithm has been derived from uniform B-spline basis function using the Cox-de Boor recursion formula. In order to determine the convergence and smoothness of the proposed schemes, the Laurent polynomial method has been used.

Improvement of the termination criterion for subdivision of the rational Bézier curves

By using some elementary inequalities, authors in this paper makes further improvement for estimating the heights of Bézier curve and rational Bézier curve. And the termination criterion for subdivision of the rational Bézier curve is also improved. The conclusion of the extreme value problem is thus further confirmed.

Subdivision Surface-Based Isogeometric Boundary Element Method for Steady Heat Conduction Problems with Variable Coefficient

The present work couples isogeometric analysis(IGA)and boundary element methods(BEM)for three dimensional steady heat conduction problems with variable coefficients.The Computer-Aided Design(CAD)geometries are built by subdivision surfaces,and meantime the basis functions of subdivision surfaces are employed to discretize the boundary integral equations for heat conduction analysis.Moreover,the radial integration method is adopted to transform the additional domain integrals caused by variable coefficients to the boundary integrals.Several numerical examples are provided to demonstrate the correctness and advantages of the proposed algorithm in the integration of CAD and numerical analysis.

Unification and Application of 3-point Approximating Subdivision Schemes of Varying Arity

In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds.? The usefulness of the scheme is illustrated by considering different examples along with its comparison with the established subdivision schemes. Moreover, B-splines of degree 4and well known 3-point schemes [1, 2, 3, 4, 6, 11, 12, 14, 15] are special cases of our proposed scheme.

GPU-based composite subdivision

This paper proposes a novel strategy based on fragment meshes of Shiue et al. for GPU rendering of compos- ite subdivision surfaces. Two enumeration systems are established to label the primitives of each fragment mesh. A sector-layer-index enumeration system is responsible for retrieving proximities for subdivision masks while a sec- tor-index enumeration system designates a 2D texture buffer in GPU. Recurring to the free conversion between them, our approach may get rid of lookup tables that are designed to record subdivision masks. In addition, relatively small composite subdivision masks make it easy to develop automatically retrieving techniques. Finally, as center vertices are often irregular, their computation is related to an average with alterable number of items. Considering that variable loop is not efficient in GPU, we evaluate the center vertex of each fragment mesh using the linear combination of its level 0, level 1 and limit positions instead of averaging schemes. Experiments demonstrate that our approach generally outper- forms that of Shiue et al. in FPS by a long way.

Reducing Uncertainty in Subdivision Optimization

Design of watertight subdivision inherently involves its optimization with the objective to increase the index ＂A＂ above its minimum required value. In view of a big popularity of probabilistic search methods such as genetic algorithms, this task is intrinsically time consuming. Thus, even when an optimal subdivision layout （i.e. topology） is determined, it can be found that the optimal bulkhead positions can be a great challenge time-wise, often forcing designers to satisfy with suboptimal solutions. The fundamental reason why this happens is that the nature of the optimized function （e.g., index ＂A＂ as a function of bulkhead positions） is unknown and hence it has no effect upon the choice of optimization strategy, which therefore reflects subjective but not factual preferences. In this paper we study the nature of functional dependency between the subdivision index and bulkhead positions, as a simplest case, and indicate pertinent optimization strategies that consequently reduce the optimization time. In our study we use a cruise ship model to demonstrate the application results of our findings.

Improved Butterfly Subdivision Scheme for Meshes with Arbitrary Topology

Based on the butterfly subdivision scheme and the modified butterfly subdivision scheme, an improved butterfly subdivision scheme is proposed. The scheme uses a small stencil of six points to calculate new inserting vertex, 2n new vertices are inserted in the 2n triangle faces in each recursion, and the n old vertices are kept, special treatment is given to the boundary, achieving higher smoothness while using small stencils is realized. With the proposed scheme, the number of triangle faces increases only by a factor of 3 in each refinement step. Compared with the butterfly subdivision scheme and the modified butterfly subdivision scheme, the size of triangle faces changes more gradually, which allows one to have greater control over the resolution of a refined mesh.

Improved non-uniform subdivision scheme with modified Eigen-polyhedron

In this study,a systematic refinement method was developed for non-uniform Catmull-Clark subdivision surfaces to improve the quality of the surface at extraordinary points(EPs).The developed method modifies the eigenpolyhedron by designing the angles between two adjacent edges that contain an EP.Refinement rules are then formulated with the help of the modified eigenpolyhedron.Numerical experiments show that the method significantly improves the performance of the subdivision surface for non-uniform parameterization.

Subdivision建模在汽车造型数字化设计中的应用研究

Subdivision建模具有快速便捷,易于修改的特点,在汽车造型设计开发中具有巨大的应用价值。通过分析Subdivision建模原理,总结出在汽车造型数字化设计应用的可行性、优势与局限性,结合汽车造型数字化设计开发的一般流程,提出具体的应用策略与方法,对提高汽车造型数字化设计效率具有重要的意义。