An orthogonal basis for non-uniform algebraic-trigonometric spline space

Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonai. We construct an orthogonal basis for the n-order（n ≥ 3） algebraic-trigonometric spline space in order to resolve the theo- retical problem that there is not an explicit orthogonai basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions, which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.

THE DIMENSION OF SPLINE SPACE S_3~1(△) ON A TYPE OF TRIANGULATION

A harmonic condition that can distinguish whether the dimension of spline space S 1 3( △ ) depends on the geometrical character of triangulation is presented, then on a type of general triangulation the dimension is got.

ON EXPLICIT BASIS OF BIVARIATE SPLINE SPACE

For a subdivision △ of a region in d-dimensional Euclidean space. we consider computation of dimension and of basis function in spline space S(△)consisting of all C(?)piecewise Polynomial func- tions over△of degree at most k. A computational scheme is presented for computing the dimension and bases of spline space S(?)(△).This scheme based On the Grobner basis algorithm and the smooth co-factor method for computing multivariate spline. For bivariate splines, explicit basis functions of S(△)age obtained for any integer k and r when△is a cross-cut partition.

Space Mapping of Spline Spaces over Hierarchical T-meshes

In this paper,we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph(CVR graph).We propose a novel structure,by which we offer an effective and easy operative method for constructing the basis functions of the biquadratic spline space.Themapping we construct is an isomorphism.The basis functions of the biquadratic spline space hold the properties such as linear independence,completeness and the property of partition of unity,which are the same as the properties for the basis functions of piecewise constant space over the CVR graph.To demonstrate that the new basis functions are efficient,we apply the basis functions to fit some open surfaces.

NEW PROOF OF DIMENSION FORMULA OF SPLINE SPACES OVER T-MESHES VIA SMOOTHING COFACTORS

A T-mesh is basically a rectangular grid that allows T-junctions. Recently, Deng etal introduced splines over T-meshes, which are generalizations of T-splines invented by Sederberg etal, and proposed a dimension formula based on the B-net method. In this paper, we derive an equivalent dimension formula in a different form with the smoothing cofactor method.

ON THE PROBLEM OF INSTABILITY IN THE DIMENSIONS OF SPLINE SPACES OVER T-MESHES WITH T-CYCLES

The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes.

THE BLOSSOM APPROACH TO THE DIMENSION OF THE BIVARIATE SPLINE SPACE

Presents information on a study which outlined the blossom approach to the dimension count of bivariate spline space. Smoothness conditions in blossoming form; Application of the approach to the case of Morgan-Scott partition.

On Singularity of Spline Space Over Morgan-Scott's Type Partition

Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.

The Instability in the Dimensions of Spline Spaces Over T-Meshes with Nested T-Cycles

This paper studies on the dimensions of spline spaces over some given T-meshes.Using the smoothing cofactor-conformality method,we study the instability in the dimensions of the spline spaces over T-meshes with 2-nested and 3-nested T-cycles.We define a singularity factor of each simple T-cycle,the instability and the structure’s degeneration are associated with the singularity factors.In order to get a stable dimension formula over T-mesh with a N-nested T-cycle,a constraint on the T-mesh is introduced.Finally,a possible degeneration for a case of parallel T-cycles is illustrated.

BASES OF BIQUADRATIC POLYNOMIAL SPLINE SPACES OVER HIERARCHICAL T-MESHES

Basis functions of biquadratic polynomial spline spaces over hierarchical T-meshes are constructed. The basis functions are all tensor-product B-splines, which are linearly independent, nonnegative and complete. To make basis functions more efficient for geometric modeling, we also give out a new basis with the property of unit partition. Two preliminary applications are given to demonstrate that the new basis is efficient.

THE INSTABILITY DEGREE IN THE DIEMNSION OF SPACES OF BIVARIATE SPLINE

In this paper, the dimension of the spaces of bivariate spline with degree less that 2r and smoothness order r on the Morgan-Scott triangulation is considered. The concept of the instability degree in the dimension of spaces of bivariate spline is presented. The results in the paper make us conjecture the instability degree in the dimension of spaces of bivariate spline is infinity.

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

B样条拟插值算子在Orlicz空间中的逼近

为了研究B样条拟插值算子在Orlicz空间的逼近性质,根据m阶B样条算子所构造出的B样条拟插值算子的定义,利用B样条算子的基本性质、Orliz空间的相关性质和光滑模、K泛函及其等价性质等工具,参照B样条拟插值算子在Lp空间逼近的研究方法对B样条拟插值算子进行了研究,得到B样条拟插值算子的有界性,最后得出B样条拟插值算子在Orlicz空间的逼近正定理。

一种三角内花键环规的棒间距测量与判定方法

本文研究了测长机及T型测球对三角内花键环规棒间距的检测方法,通过分析T型测球与三角内花键环规的几何关系,得到不同直径测球与最佳棒径对应棒间距的换算关系,并结合实例进行验证。

DYNAMIC RESPONSE OF ELASTIC RECTANGULAR PLATES BY SPLINE STATE VARIABLE METHOD

Application of spline element and state space method for analysis of dynamic response of elastic rectangular plates is presented. The spline element method is used for space domain and the state space method in control theory of system is used for time domain. A state variable recursive scheme is developed, then the dynamic response of structure can he calculated directly. Several numerical examples are given. The results which are presented to demonstrate the accuracy and efficiency of the present method are quite satisfactory.

An Unconditionally Monotone <i>C</i>²Quartic Spline Method with Nonoscillation Derivatives

A one-dimensional monotone interpolation method based on interface reconstruction with partial volumes in the slope-space utilizing the Hermite cubic-spline, is proposed. The new method is only quartic, however is C2 and unconditionally monotone. A set of control points is employed to constrain the curvature of the interpolation function and to eliminate possible nonphysical oscillations in the slope space. An extension of this method in two-dimensions is also discussed.

Computational Analysis for Solving the Linear Space-Fractional Telegraph Equation

Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.

B样条过渡的二维柱塞泵空间凸轮轨道机构运动特性多目标优化

空间凸轮机构作为二维柱塞泵实现二维运动的关键机构,其从动件运动特性决定了二维柱塞泵的工作特性。以5次B样条曲线作为无停歇运动规律的过渡曲线,建立其数学模型,使用多目标遗传算法以最大无因次速度和加速度作为优化目标对运动规律进行多目标优化,获得Pareto解集。优化实例结果表明,5次B样条曲线作为无停歇运动规律的过渡段,可以构造出高阶连续且可调的无停歇运动规律,多目标遗传算法可实现对B样条过渡段的多目标优化,获得凸轮综合性能良好的运动规律。

Research on a toolpath generation method of NC milling based on space-filling curve

Tool path generated by space-filling curve always turns frequently causing trembling to machine,reducing toollife and affecting workpiece quality. Length and generation time of tool paths are both relatively long. In order to solve these problems,a toolpath generation method of NC milling based on space-filling curve is proposed. First,T-spline surface is regarded as the modeling surface,the grid,which is based on the limited scallop-height,can be got in the parameter space,and the influence value of grid node is determined. Second,a box is defined and planned,and the tool paths are got preliminarily,which is based on minimal spanning tree; Finally,based on an improved chamfering algorithm,the whole tool paths are got. A simulation system is developed for computer simulation,and an experiment is carried out to verify the method. The results of simulation and experiment show that the method is effective and feasible,and length and time of the tool paths are reduced.

一类椭圆型界面问题的数值算法

该文基于再生核空间相关理论提出了一种一维椭圆界面问题的数值算法.该方法通过积分W_(2)^(1)空间的再生核函数,得到了三次样条空间的一组新基,在此基础上建立了一个破裂的三次样条空间,然后利用最小二乘法得出了此类界面问题的近似解,并且讨论了在H2范数、H1范数和L2范数下的收敛阶,最后通过几个数值算例得到了验证.