参数曲线近似弧长参数化的三角函数方法

本描述了一种局部的近似弧长参数化插值方法，用三角函数对曲线的弧长函数进行分段逼近，段与段之间是相互独立的，且插值曲线在插值点处的弧长与原参数曲线的真实弧长相等。     

Existence and uniqueness for Legendre curves

We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.     

A planar cubic Bézier spiral

A planar cubic Bézier curve segment that is a spiral, i.e., its curvature varies monotonically with arc-length, is discussed. Since this curve segment does not have cusps, loops, and inflection points (except for a single inflection point at its beginning), it is suitable for applications such as highway design, in which the clothoid has been traditionally used. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS (nonuniform rational B-splines) and is thus suitable for general curve design applications in which fair curves are important.     

The geometrical construction of pointwise distributions on curves

A method is developed to generate desirable pointwise distributions along curves. This is accomplished with a simple geometrical construction which provides a global parameter for curvature clustering together with other parameters for arbitrary local clustering specifications. The level of available precision is considerable in that exact numbers of points can be assigned to both local clusters and to curvature simultaneously with specified spacing from the endpoints. The basic construction simply involves the generation of an auxiliary curve along outward normal directions from the given one. The distribution results when uniform arc-length increments are taken along the auxiliary curve and are projected back along the normals to our given curve. This construction can be applied either directly or in the form of equivalent weight functions. Moreover, it is valid regardless of whether the curve lies in Euclidian space or in surfaces and regardless of the dimensionality of the space in which the curve lies.     

Instability analysis of space trusses using exact tangent-stiffness matrices

A simple (exact) expression for the tangent-stiffness matrix of a space truss undergoing arbitrarily large deformation, as well as member buckling, is given. An arc-length method is used to solve the tangent-stiffness equations in the post-buckling range of the structural deformation. Several examples to illustrate the viability of the present approaches in analyzing large space structures, simply, efficiently, and accurately, are given.     

Chordal cubic spline interpolation is fourth-order accurate

^** Email: michaelf{at}ifi.uio.no It is well known that complete cubic spline interpolation offunctions with four continuous derivatives is fourth-order accurate.In this paper we show that this kind of interpolation, whenused to construct parametric spline curves through sequencesof points in any space dimension, is again fourth-order accurateif the parameter intervals are chosen by chord length. We alsoshow how such chordal spline interpolants can be used to approximatethe arc-length derivatives of a curve and its length.     

Spiral transitions

Spiral curves are free from singularities and curvature extrema. These are used in path smoothing applications to overcome the abrupt change in curvature and super-elevation of moving object that occurs between tangent and circular curve. Line to circle spiral transition is made of straight line segment and curvature continuous spiral curve. It is extendible to other important types of transitions like line to line and circle to circle. Although the problem of line to circle transition has been addressed by many researchers, there is no comprehensive literature review available. This paper presents state-of-the-art of line to circle spiral transition,applications in different fields, limitations of existing approaches, and recommendations to meet the challenges of compatibility with today's CAD/CAM soft wares, satisfaction of Hermite end conditions, approximation of discrete data for image processing, 3 D path smoothness for an unmanned aerial vehicle(UAV), and arc-length parametrization. Whole discussion is concluded with future research directions in various areas of applications.     

Arc Length Estimation and the Convergence of Polynomial Curve Interpolation

When fitting parametric polynomial curves to sequences of points or derivatives we have to choose suitable parameter values at the interpolation points. This paper investigates the effect of the parameterization on the approximation order of the interpolation. We show that chord length parameter values yield full approximation order when the polynomial degree is at most three. We obtain full approximation order for arbitrary degree by developing an algorithm which generates more and more accurate approximations to arc length: the lengths of the segments of an interpolant of one degree provide parameter intervals for interpolants of degree two higher. The algorithm can also be used to estimate the length of a curve and its arc-length derivatives. AMS subject classification (2000) 65D05, 65D10     

Geometrical nonlinear analysis based on optimization technique

In this paper, two triangular shapes are used to formulate iteration procedures required for nonlinear analysis. These triangles are formed by the structural load–displacement curve. The areas of these shapes are considered as two variables objective functions. To find the optimal solutions, these functions are minimized with respect to each variable. As a result, two new constraint equations are obtained, which can be utilized as the nonlinear solver. To demonstrate the merit of authors’ scheme, some geometric nonlinear analyses of shells, frames and trusses are performed. Furthermore, present formulations are compared with the cylindrical arc-length method.     

The Qualocation Method for Symm's Integral Equation on a Polygon

This paper discusses the convergence of the qualocation method for Symm's integral equation on closed polygonal boundaries in IR². Qualocation is a Petrov-Galerkin method in which the outer integrals are performed numerically by special quadrature rules. Before discretisation a nonlinear parametrisation of the polygon is introduced which varies more slowly than arc-length near each corner and leads to a transformed integral equation with a regular solution. We prove that the qualocation method using smoothest splines of any order k on a uniform mesh (with respect to the new parameter) converges with optimal order O (h^k). Furthermore, the method is shown to produce superconvergent approximations to linear functionals, retaining the same high convergence rates as in the case of a smooth curve.     

Tangent segments and orthogonality types in normed planes

The following characterizations of the Euclidean plane are obtained: the two tangent segments of the unit circle of a normed plane from each point of a disc centered at the origin with sufficiently large diameter have equal lengths; the lengths of the tangent segments from each point of a fixed circle centered at the origin are determined only by the radius of this circle. Three further characterizations of the Euclidean plane are obtained by considering properties of certain points related to an exterior point of the unit disc and the two tangent segments corresponding to it. To obtain one of these characterizations, the notion of arc-length orthogonality is introduced, and the Euclidean plane is also characterized via a relation between arc-length orthogonality and Birkhoff orthogonality.     

分析结构跳跃问题的自适应参数增量迭代方法

通过对Riks切线弧长法和Crisfield圆弧长法的综合及改进,本文提出分析结构跳跃问题的自适应参数增量迭代方法,并建立相应理论基础。最后文中给出中心分布压力作用下圆底扁球壳跳跃问题的数值结果,以验证它的有效性。     

Nonlinear structural finite element analysis using the preconditioned Lanczos method on serial and parallel computers

The application of the Lanczos algorithm in Newton-like methods for solving non-linear systems of equations arising in nonlinear structural finite element analysis is presented. It is shown that with appropriate preconditioners iterative methods can be developed which are robust and efficient even for ill conditioned problems. Though the real advantage of iterative solvers seems to exist on distributed memory machines, even on serial machines the performance can be improved compared with direct solvers while saving memory capacity. With a specific modification of the Lanczos algorithm in combination with arc-length procedures a further speed-up of the nonlinear analysis can be achieved. For parallel implementations domain decomposition methods are used. A parallel preconditioning strategy based on an incomplete factorisation method is presented. An example is taken and the quality and efficiency of two different domain decomposition methods are discussed for a large shell structure. This work was supported by the BMBF (Bundesministerium für Bildung und Forschung) of Germany.     

Curvature functionals for curves in the equi-affine plane

After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail.     

The possibility of a theoretical confirmation of the experimental values of the external critical pressure of thin-walled cylindrical shells

The problem of the buckling of elastic, isotropic, thin-walled cylindrical shells with small initial shape defects that are under the action of an external pressure is solved in a geometrically non-linear formulation. Equations that are identical to Marguerre's equations for a shallow cylindrical shell are used in formulating the problem. The solution is constructed by the Rayleigh–Ritz method with the points of the middle surface of the shell approximated by double functional sums over trigonometric and beam functions. The system of non-linear equations obtained is solved by arc-length methods. Cases of the clamped and supported shells when loading with a lateral and uniform hydrostatic pressure are considered. Its deflections from the limit points of the postbuckling branches of its loading trajectory are used as the initial imperfections. An inspection of the different forms of the initial imperfections when they have maximum values of up to 30% of the shell thickness made it possible to obtain practically the whole range of experimentally found critical pressures.     

Uniform Convergence Analysis of Finite Difference Scheme for Singularly Perturbed Delay Differential Equation on an Adaptively Generated Grid

<正>Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.     

Circle Numbers of Regular Convex Polygons

The circle number function is extended here to regular convex polygons. To this end, the radius of the polygonal circle is defined as the Minkowski functional of the circumscribed polygonal disc, and the arc-length of the polygonal circle is measured in a generalized Minkowski space having the rotated polar body as the unit disc.     

Trace and Density Results on Regular Trees

Potential Analysis - The boundary of a regular tree can be viewed as a Cantor-type set. We equip our tree with a weighted distance and a weighted measure via the Euclidean arc-length and consider...     

Parameter-uniform numerical methods for singularly perturbed mixed boundary value problems using grid equidistribution

In this paper, we present the analysis of an upwind scheme for obtaining the solution of a convection-diffusion two-point boundary value problem with Robin boundary conditions. The solution is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc-length monitor function. It is shown that the discrete solution obtained by the upwind scheme converges uniformly with respect to the perturbation parameter. Numerical results are presented that demonstrate the sharpness of the theoretical estimates.     

On an Extremal Problem About the Arc-Length of Algebraic Polynomials

In this work we investigate polynomials of maximal (minimal) arc-length in the interval [−1, 1] amongst all monic polynomials of fixed degree n with n real zeros in [−1, 1].