一种对称非均匀细分曲面算法

为了得到能更好应用于CAD系统的细分曲面造型方法,提出一种基于B-样条的对称非均匀细分算法,其中的思想和均匀Lane-Riesenfeld节点插入算法相似。基于B-样条的节点插入算法,以Blossoming为工具,计算出细分后的新控制顶点。细分后得到的极限曲面由张量积样条曲面组成,在奇异点达到2C连续。与传统的细分曲面算法相比,该细分曲面算法具有良好的局部支撑性,大大降低了算法的复杂度,而且该算法是对称的,不用考虑定向问题。

A prediction method of rail grinding profile using non-uniform rational B-spline curves and Kriging model

Non-uniform rational B-spline(NURBS)curves are combined with the Kriging model to present a prediction method of the rail grinding profile for a grinding train.As a worn rail profile is a free-form curve,the parameterized model of a rail profile is constructed by using the cubic NURBS curve.Taking the removed area of the rail profile cross-section by grinding as the calculation index of the grinding amount,the grinding amount calculation model of a grinding wheel is established based on the area integral formula of the cubic NURBS curve.To predict the grinding amount of a grinding wheel in different modes,a Kriging model of the grinding amount is constructed,taking the travel speed of a grinding train,the grinding angle and grinding pressure of a grinding wheel as the variables,and considering the grinding amount of a grinding wheel as the response.On this basis,the prediction method of the rail grinding profile is presented based on the order-forming mechanisms.The effectiveness of this method is verified based on a practical application.

An Approach to Representing Heterogeneous Non-Uniform Rational B-Spline Objects

The representation method of heterogeneous material information is one of the key technologies of heterogeneous object modeling, but almost all the existing methods cannot represent non-uniform rational B-spline （NURBS） entity. According to the characteristics of NURBS, a novel data structure, named NURBS material data structure, is proposed, in which the geometrical coordinates, weights and material coordinates of NURBS heterogene- ous objects can be represented simultaneously. Based on this data structure, both direct representation method and inverse construction method of heterogeneous NURBS objects are introduced. In the direct representation method, three forms of NURBS heterogeneous objects are introduced by giving the geometry and material information of con- trol points, among which the homogeneous coordinates form is employed for its brevity and easy programming. In the inverse construction method, continuous heterogeneous curves and surfaces can he obtained by interpolating discrete points and curves with specified material information. Some examples are given to show the effectiveness of the pro- posed methods.

Use of Non-uniform Rational B-splines for Discharge Calculation in the Velocity Area Method

The velocity area method belongs to the group of primary methods for discharge measurement in hydropower plants. The measurements require an appropriate application of measuring devices and carrying out correctly the process of data analyzing including integration technique. The authors present their own experiences gathered during many years of utilizing the current meter method for discharge measurement in many hydropower plants. They have developed the special integration techniques using the progressive numerical algorithms. The techniques differ from the recommendations contained in the relevant international standards. The authors＇ own software for calculating the discharge from the measured local velocity distribution （obtained using current meters） adopts advanced spline functions, the so-called NURBS （non-uniform rational B-splines）. Nowadays, this kind of splines is commonly used in modeling of the complex geometrical shapes because of their smoothness. It is assessed that it represents much better quality of interpolation than the classic spline functions （classic cubic spline technique）. Particularly, the better properties of the NURBS splines can be observed for velocity profile area characterized by very strong velocity gradients where boundary layers meet the core regions of the flow （mainstream）. In the developed software the boundary layer thickness and exponent of von Karman function is calculated in accordance with the ISO 3354 standard. The software has been successfully used during many performance tests of the hydraulic turbines in Poland for several years. Paper presents the results of flow rate measurements for two different flow systems of Kaplan turbines. First case concerns the application of the current meters in a long circular penstock whereas the second one in short rectangular turbine intake. A comparative analysis of three flow calculation procedures applied for these two cases is presented in the paper-（1） the integration procedure according to the ISO 3354 standard; （2） the integration procedure based on the NS （natural splines）; and （3） the integration procedure based on the NURBS. The results obtained using these three procedures for the first case （intake via long circular penstock） were compared with the results of discharge measurements conducted using the pressure-time method.

基于曲率参数的NURBS曲线插值

提出一种新的用于NURBS曲线插值的构造边界条件的方法 .该方法不需要已知曲线端点的切矢 ,并且解决了端点曲率为零的问题 ;它使得设计者可以通过简单地调整首末端点曲率参数便可以随意地且很容易地控制曲线首末两段曲线的形状 .最后还讨论了曲率参数的取值对曲线端部形状的影响 .

Weight-based shape modification of NURBS curves by constrained optimization

A new method for shape modification of non-uniform rational B-splines (NURBS) curves was presented, which was based on constrained optimization by means of altering the corresponding weights of their control points. Using this method, the original NURBS curve was modified to satisfy the specified geometric constraints, including single point and multi-point constraints. With the introduction of free parameters, the shapes of modified NURBS curves could be further controlled by users without destroying geometric constraints and seem more naturally. Since explicit formulae were derived to compute new weights of the modified curve, the method was simple and easy to program. Practical examples showed that the method was applicable for computer aided design (CAD) system.

DESCRIPTIONS OF THE CYLINDRICAL HELIX WITH NURBS

The representation of a cylindrical helix by Non-Uniform Rational B-Spline (NURBS) curves is presented in this paper. A method is proposed to assess the influences produced by different ways to determine the control ver-texes positions of helix. The error distribution cases between the helix approximated by NURBS curves and the original theoretical one are also analyzed. Meanwhile a computational method that guarantees the precision requirements is presented.

Alternative 3D Modeling Approaches Based on Complex Multi-Source Geological Data Interpretation

Due to the complex nature of multi-source geological data, it is difficult to rebuild every geological structure through a single 3D modeling method. The multi-source data interpretation method put forward in this analysis is based on a database-driven pattern and focuses on the discrete and irregular features of geological data. The geological data from a variety of sources covering a range of accuracy, resolution, quantity and quality are classified and integrated according to their reliability and consistency for 3D modeling. The new interpolation-approximation fitting construction algorithm of geological surfaces with the non-uniform rational B-spline(NURBS) technique is then presented. The NURBS technique can retain the balance among the requirements for accuracy, surface continuity and data storage of geological structures. Finally, four alternative 3D modeling approaches are demonstrated with reference to some examples, which are selected according to the data quantity and accuracy specification. The proposed approaches offer flexible modeling patterns for different practical engineering demands.

A Coupled Analysis of Nonlinear Sloshing and Ship Motion

Nonlinear interactions among incident wave, tank-sloshing and floating body coupling motion are investigated. The fully nonlinear sloshing and body-surface nonlinear free surface hydrodynamics is simulated using a Non-Uniform Rational B-Spline （NURBS） higher-order panel method in time domain based on the potential theory. A robust and stable improved iterative procedure （Yan and Ma, 2007） for floating bodies is used for calculating the time derivative of velocity potential and floating body motion. An energy dissipation condition based on linear theory adopted by Huang （2011） is developed to consider flow viscosity effects of sloshing flow in nonlinear model. A two-dimensional tank model test was performed to identify its validity. The present nonlinear coupling sway motion results are subsequently compared with the corresponding Rognebakke and Faltinsen （2003）＇s experimental results, showing fair agreement. Thus, the numerical approach presented in this paper is expected to be very efficient and realistic in evaluating the coupling effects of nonlinear sloshing and body motion.

3D Simulation of Two-bar Warp-knitted Structures

In this study, 3D computer modeling of simple warp-knitted structures is achieved based on 3D model of warp-knitted loops. Firstly, according to the studying on the geometric structure of warp-knitted loops, Goktepe＇s 3D solid yarn model is developed, and the dimensions of the warp-knitted loops are obtained; then 3D models of stitch defined by eleven given points and in-lay defined by five given points are built with the method of Non-Uniform Rational B-Spline （NURBS） curves and surfaces. Secondly, according to the chain notations of warp-knitted structures, the loop＇s shape can be decided, and with the 3D models the loops can be connected freely and smoothly in some special warp-knitted structures, such as miss-lapping and two- course in-lay. At last, with the tools of Visual C ＋＋ and OpenGL, computer 3D simulation of two-bar warp-knitted fabrics is successfully realized, taking the factor of fabric light and materials into 3D structural model.

Numerical Calculation of Channel Dredging Volume Using 3D Digital Stratum Model

Prediction of channel dredging volume is critical for project cost estimation. However, many proposed approximate methods are not accurate. This paper presents a novel numerical method to accurately calculate the dredg- ing volume using a 3D stratum model (DSM) and a channel surface model. First, the 3D DSM is constructed rapidly yet accurately from non-uniform rational B-splines (NURBS) surfaces through Boolean operation between a physical terrain model and a stratum surfaces model. Then, a parametric channel surface model is built from cross-section data and a channel center line using code implemented in the VC++ programming language. Finally, the volumes of different types of physical stratums can be calculated automatically and hierarchically to determine the dredging volume. Practical application shows that the DSM method is more precise and faster compared to the section method, and that the implementation of the developed software provides an interactive graphical user interface and visual presentation.

Automatic Generation Method of Geological Cross-Sections in Dredging Engineering Based on 3D Solid NURBS Models

An automatic generation method of geological cross-sections in dredging engineering based on 3D geological solid models is presented.The 3D geological models are built applying the non-uniform rational B-splines(NURBS) technique,and a 2D profile can be calculated and generated automatically through Boolean operation to meet the demands of dredging projects.Moreover,an automatic marking method for geological attributes is put forward based on database technology,and the geological attributes include the profile name,scale,horizontal and vertical relative coordinates,geological lithology,and 2D standard lithology legend.At the same time,the automatic marking method can also provide an interactive mode for geological engineers to edit and modify the profile in the modeling system.Practical engineering applications show that the automatic generation method is a simple,flexible,fast and precise visual graphics rendering process that can create 2D standard profiles automatically and efficiently.This method also provides a convenient support tool for geological engineering digital analysis.

Application of Three-Dimensional Visual Simulation in Tidal Defense Engineering

Any tidal defense engineering involves the collection and analysis of massive information about engineering structures and their surrounding environment. Traditional method, which is carried out mainly by means of twodimensional drawings and textures, is not efficient and intuitive enough to analyze the whole project and reflect its spatial relationship. Three-dimensional visual simulation provides an advanced technical means of solving this problem. In this paper, triangular irregular network (TIN) model simplified by non-uniform rational B-splines (NURBS) technique was used to establish the digital terrain model (DTM) of a super large region. Simulation of dynamic water surface was realized by combining noise function with sine wave superposition method. Models of different objects were established with different modeling techniques according to their characteristics. Application of texture mapping technology remarkably improved the authenticity of the models. Taking the tidal defense engineering in the new coastal region of Tianjin as a case study, three-dimensional visual simulation and dynamic roaming of the study area were realized, providing visual analysis and visible demonstration method for the management and emergency decision-making associated with construction.

A Geometric Proof of the Convexity Preserving Property of NURBS Curves

NURBS curves are convexity preserving, i.e. once the control polygon is convex, the associated NURBS curve will also be convex. In this paper this property is proved geometrically.

Combined back-analysis method of ground stress based on refined geological modeling

A new back-analysis method of ground stress is proposed with comprehensive consideration of influence of topography, geology and nonlinear physical mechanical properties of rock on ground stress. This method based on non-uniform rational B-spline (NURBS) technology provides the means to build a refined three-dimensional finite element model with more accurate meshing under complex terrain and geological conditions. Meanwhile, this method is a back-analysis of ground stress with combination of multivariable linear regression model and neural network (ANN) model. Firstly, the regression model is used to fit approximately boundary loads. Regarding the regressed loads as mean value, some sets of boundary loads with the same interval are constructed according to the principle of orthogonal design, to calculate the corresponding ground stress at the observation positions using finite element method. The results (boundary loads and the corresponding ground stress) are added to the samples for ANN training. And on this basis, an ANN model is established to implement higher precise back-analysis of initial ground stress. A practical application case shows that the relative error between the inversed ground stress and observed value is mostly less than 10 %, which can meet the need of engineering design and construction requirements.

NURBS curve blending using extension

Curve and surface blending is an important operation in CAD systems, in which a non-uniform rational B-spline （NURBS） has been used as the de facto standard. In local comer blending, two curves intersecting at that comer are first made disjoint, and then the third blending curve is added-in to smoothly join the two curves with G^1- or G^2-continuity. In this paper we present a study to solve the joint problem based on curve extension. The following nice properties of this extension algorithm are exploited in depth： （1） The parameterization of the original shapes does not change; （2） No additional fragments are created. Various examples are presented to demonstrate that our solution is simple and efficient.

Numerical manifold method based on isogeometric analysis

In this study, numerical manifold method(NMM) coupled with non-uniform rational B-splines(NURBS) and T-splines in the context of isogeometric analysis is proposed to allow for the treatments of complex geometries and local refinement. Computational formula for a 9-node NMM based on quadratic B-splines is derived. In order to exactly represent some common free-form shapes such as circles, arcs, and ellipsoids, quadratic non-uniform rational B-splines(NURBS) are introduced into NMM. The coordinate transformation based on the basis function of NURBS is established to enable exact integration for the manifold elements containing those shapes. For the case of crack propagation problems where singular fields around crack tips exist, local refinement technique by the application of T-spline discretizations is incorporated into NMM, which facilitates a truly local refinement without extending the entire row of control points. A local refinement strategy for the 4-node mathematical cover mesh based on T-splines and Lagrange interpolation polynomial is proposed. Results from numerical examples show that the 9-node NMM based on NURBS has higher accuracies. The coordinate transformation based on the NURBS basis function improves the accuracy of NMM by exact integration. The local mesh refinement using T-splines reduces the number of degrees of freedom while maintaining calculation accuracy at the same time.

A strain smoothing formulation for NURBS-based isogeometric finite element analysis

A strain smoothing formulation for NURBS (non-uniform rational B-splines) based isogeometric finite element analysis is presented. This approach is formulated within the framework of assumed strain methods and strain smoothing operations. The strain smoothing is defined through strain averaging in the element sub-domains which are subsequently used for numerical integration of the Galerkin weak form. This formulation satisfies the orthogonality condition of the assumed strain methods. Meanwhile the present formulation totally avoids the gradient computation of the rational NURBS basis functions in the formulation of stiffness matrix. A transformation method is employed to accurately enforce the displacement boundary conditions. Numerical results demonstrate that the present formation gives very satisfactory solution accuracy simultaneously with improved computational efficiency.