Local and singularity-free G 1 triangular spline surfaces using a minimum degree scheme

We develop a scheme for constructing G ¹ triangular spline surfaces of arbitrary topological type. To assure that the scheme is local and singularity-free, we analyze the selection of scalar weight functions and the construction of the boundary curve network in detail. With the further requirements of interpolating positions, normals, and surface curvatures, we show that the minimum degree of such a triangular spline surface is 6. And we present a method for constructing boundary curves network, which consists of cubic Bézier curves. To deal with certain singular cases, the base mesh must be locally subdivided and we proposed an adaptive subdivision strategy for it. An application of our G ¹ triangular spline surfaces to the approximation of implicit surfaces is described. The visual quality of this scheme is demonstrated by some examples.     

On the evaluation of one-dimensional Cauchy principal value integrals by rules based on cubic spline interpolation

In this note we consider the numerical evaluation of one dimensional Cauchy principal value integrals of the form $$\rlap{--} \smallint _a^b \frac{{k(x)f(x)}}{{x - \lambda }}dx, a< \lambda< b,$$ by rules obtained by “subtracting out” the singularity and then applying product quadratures based on cubic spline interpolation at equally spaced nodes. Convergence results are established for Hölder continuous functions of order, μ, 0<μ≤1, and asymptotic rates are obtained for functionsf≠C ^k [a, b],k=1, 2, 3 or 4. Some comparisons with other methods and numerical examples are also given.     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.     

Accurate finite element method for atomic calculations based on density functional theory and Hartree–Fock method

An accurate finite element method is developed for atomic calculations based on density functional theory (DFT) within local density approximation (LDA) and Hartree–Fock (HF) method. The radial wave functions are expanded by cubic Hermite spline functions on a uniform mesh for , and all the associated integrals are analytically evaluated in conjunction with fitting procedures of the Hartree and the exchange–correlation potentials to the same cubic Hermite spline functions using a set of recurrence formulas. The total energy of atoms systematically converges from above, and the error algebraically decays as the mesh spacing decreases. When the mesh spacing d is taken to be , the total energy for an atom of atomic number Z can be calculated within error of ¹⁰−7 hartree for both the LDA and HF methods. The equal applicability of the method to DFT and the HF method with a similar degree of high accuracy enables the method to be a reliable platform for development of new functionals in DFT such as hybrid functionals.     

多层积分值三次样条拟插值

目的 在实际问题中,某些插值问题结点处的函数值往往是未知的,而仅仅知道一些连续等距区间上的积分值。为此提出了一种基于未知函数在连续等距区间上的积分值和多层样条拟插值技术来解决函数重构。该方法称之为多层积分值三次样条拟插值方法。方法 首先,利用积分值的线性组合来逼近结点处的函数值;然后,利用传统的三次B-样条拟插值和相应的误差函数来实现多层三次样条拟插值;最后,给出两层积分值三次样条拟插值算子的多项式再生性和误差估计。结果 选取无穷次可微函数对多层积分值三次样条拟插值方法和已有的积分值三次样条拟插值方法进行对比分析。数值实验印证了本文方法在逼近误差和数值收敛阶均稍占优。结论本文多层三次样条拟插值函数能够在整体上很好的逼近原始函数,一阶和二阶导函数。本文方法较之于已有的积分值三次样条拟插值方法具有更好的逼近误差和数值收敛阶。该方法对连续等距区间上积分值的函数重构具有普适性。     

A class of C 2piecewise quintic polynomials

A new class of C ² piecewise quintic interpolatory polynomials is defined. It is shown that this new class contains a number of interpolatory functions which present practical advantages, when compared with the conventional cubic spline.     

Shape preservation of scientific data through rational fractal splines

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $\mathcal C ^1$ - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $\frac{p_i(x)}{q_i(x)},$ where $p_i(x)$ and $q_i(x)$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $\mathcal C ^2$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.     

Erzeugende Funktionen bei Spline-Interpolation mit äquidistanten Knoten

Using generating functions we obtain in the case ofn+1 equidistant data points a method for the calculation of the interpolating spline functions(x) of degree 2k+1 with boundary conditionss ^(κ) (x₀)=y ₀ ^(κ) ,s ^(κ) (x _n )=y _n ^(κ) , κ=1(1)k, which only needs the inversion of a matrix of orderk. The applicability of our method in the case of general boundary conditions is also mentioned.     

High-accuracy cubic spline alternating group explicit methods for 1D quasi-linear parabolic equations†

In this article, we study the application of the alternating group explicit (AGE) and Newton-AGE iterative methods to a two-level implicit cubic spline formula of O(k ²+kh ²+h ⁴) for the solution of 1D quasi-linear parabolic equation u _xx =φ (x, t, u, u _x , u _t ), 0<x<1, t>0 subject to appropriate initial and natural boundary conditions prescribed, where k>0 and h>0 are mesh sizes in t- and x-directions, respectively. The proposed cubic spline methods require 3-spatial grid points and are applicable to problems in both rectangular and polar coordinates. The convergence analysis at advanced time level is briefly discussed. The proposed methods are then compared with the corresponding successive over relaxation (SOR) and Newton-SOR iterative methods both in terms of accuracy and performance.     

Recent advances in the construction of polygonal finite element interpolants

Summary This paper is an overview of recent developments in the construction of finite element interpolants, which areC ⁰-conforming on polygonal domains. In 1975, Wachspress proposed a general method for constructing finite element shape functions on convex polygons. Only recently has renewed interest in such interpolants surfaced in various disciplines including: geometric modeling, computer graphics, and finite element computations. This survey focuses specifically on polygonal shape functions that satisfy the properties of barycentric coordinates: (a) form a partition of unity, and are non-negative; (b) interpolate nodal data (Kronecker-delta property), (c) are linearly complete or satisfy linear precision, and (d) are smooth within the domain. We compare and contrast the construction and properties of various polygonal interpolants—Wachspress basis functions, mean value coordinates, metric coordinate method, natural neighbor-based coordinates, and maximum entropy shape functions. Numerical integration of the Galerkin weak form on polygonal domains is discussed, and the performance of these polygonal interpolants on the patch test is studied.     

A characterization of residual implications derived from left-continuous uninorms

In this paper, a set of axioms is given that characterizes those functions I:[0, 1]² → [0, 1] for which a left-continuous uninorm U exists in such a way that I is the residual implication derived from U. A characterization for the particular cases when U is representable, when U is continuous in ]0,1[² and when U is idempotent are also given.     

Bending of skew plates by spline-finite-strip method

Spline finite strip was devised by Cheung et al. in 1982 [1]. Unlike the standard finite element method, this method employs B-3spline function for interpolation in one direction and local Hermite cubic polynomial in the other direction. The general form of displacement function is given as their product. Extensive numerical examples on right plates and shells were well documented by Cheung et al. [1]. but the applicability of this method in the analysis of skew plates remains unexplored. The main theme of the present paper is to generalize the technique to include parallelogram plates. As this extension still retains the banded nature, only a small amount of extra computing effort is required. The convergence of the method is established and it is supported by numerous examples of different loading and support conditions. It has shown that spline finite strip method, which requires less variables for interpolation, can achieve the same order of accuracy as the conforming finite element.     

The search for stable finite element methods for simple relativistic systems

Three finite element methods are applied to the describing static, spherically symmetric fluid bodies in general relativity. The usual Ritz method with linear splines is unconditionally unstable, although the instability is mild. As most Ritz and Galerkin methods will produce similar unstable methods a weighted residual method with cubic Lagrange splines is tried. This again proves unstable, and it seems that this approximation is inherently inappropriate for hyperbolic equations. However, the cubic Hermite spline approximation, with some modification, proves to be both A-stable and S-Stable, and is accurate to order h³.     

A Hermite cubic collocation scheme for plane strain hydraulic fractures

We describe a novel cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain. Special blended cubic Hermite-power–law basis functions, with arbitrary index 0 < α < 1, are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. The implementation of blended infinite elements to model semi-infinite crack problems is also described. Explicit formulae for the integrated kernels associated with the cubic Hermite and blended basis functions are provided. The cubic Hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions. The convergence rate of the cubic Hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the O(h⁴) error due to the Hermite cubic interpolation. The errors due to these two approximations need to be matched in order to achieve optimal convergence. Backward Euler time-stepping yields a robust algorithm that, along with geometric increments in the time-step, can be used to explore the transition between propagation regimes over many orders of magnitude in time.     

Efficient algorithms for volterra integral equations of the second kind

Spline function of degreem, deficiencyJ?1, i. e. inC ^m?J , are used in conjunction with (Gaussian) quadrature rules to construct algorthms for the numerical solution of a general Volterra integral equation of the second kind. For a givenm, the method is of order (m+1) and, in general, requires 0(N) evaluations of the kernel. This is in sharp contrast to the 0(N ²) evaluations required by hitherto known methods. It is shown that the method for spline functions with full continuity (J=1) is numerically unstable for allm>2. However, stability is established forJ=m, m?1, for allm. Furthermore, form=3,J=1, it is demonstrated that by appropriately modifying the original method, a whole family of stable methods is obtained.     

形状可调的5次组合样条及其参数选择

目的 为了克服3次参数B样条在形状调整与局部性方面的不足,提出带参数的5次多项式组合样条。方法 首先构造一组带参数的5次多项式基函数;然后采用与3次B样条曲线相同的组合方式定义带参数的5次多项式组合样条曲线,并讨论基于能量优化法的5次组合样条曲线参数最佳取值问题;最后定义相应的组合样条曲面,并研究利用粒子群算法求解曲面的最佳参数取值。结果 5次组合样条不仅继承了3次B样条的诸多性质,而且还比3次B样条具有更强的局部性及形状可调性。由于5次组合样条仍为多项式模型,因此方程结构相对较为简单,符合实际工程的需要。利用能量优化法可获得光顺的5次组合样条曲线与曲面。结论 所提出5次多项式组合样条克服了3次参数B样条在形状调整与局部性方面的不足,是一种实用的自由曲线曲面造型方法。     

Runge-Kutta interpolants based on values from two successive integration steps

New interpolants of the explicit Runge-Kutta method for the initial value problem are proposed. These interpolants are based on values of the solution and its derivative from two successive integration steps. In this paper, three interpolants withO(h ⁶) local error (l.e.), for the fifth order solution, of the methods Fehlberg 4(5) (RKF 4(5)), Dormand and Prince 5(4) (RKDP 5(4)) and Verner 5(6) (RKV 5(6)) without extra cost are derived. An interpolant withO(h ⁷) (l.e.) for the sixth order solution of the Verner's method with only one extra function evaluation per integration step is also constructed. The above advantages are obtained without any cost in the magnitude of the error.     

A New Local Control Spline with Shape Parameters for CAD/CAM

Anew local control spline based on shape parameterw with G^3 continuity,called BLC-spline,is pro* posed.Not only is BLC-spline very smoot,but also the spline curve‘s characteristic polygon has only three control vertices,and the characteristic polyhedron has only nine control vertices.The behavior of Iocal control of BLC-spline is better than that of the other splines such as cubic Bezier,B and Beta-spline.The three shape parameters β0，β1and β2 of BLC-spline,which are independent of the control vertices,may be altered to change the shape of the curve or surface.It is shown that BLC-spline may be used to construcet a space are spline for DNC machining directly.That is a powerful tool for the design and manufacture of curves and surfaces in integrated CAD/CAM systems.     

Smooth spline surface generation over meshes of irregular topology

An efficient method for generating a smooth spline surface over an irregular mesh is presented in this paper. Similar to the methods proposed by [1, 2, 3, 4], this method generates a generalised bi-quadratic B-spline surface and achieves C ¹ smoothness. However, the rules to construct the control points for the proposed spline surfaces are much simpler and easier to follow. The construction process consists of two steps: subdividing the initial mesh once using the Catmull–Clark [5] subdivision rules and generating a collection of smoothly connected surface patches using the resultant mesh. As most of the final mesh is quadrilateral apart from the neighbourhood of the extraordinary points, most of the surface patches are regular quadratic B-splines. The neighbourhood of the extraordinary points is covered by quadratic Zheng–Ball patches [6].     

Curve fitting and fairing using conic splines

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. For parts of the G¹ conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bézier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G² continuous at convex or concave parts and G¹ continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method.