Arc Length Parameter Research Articles

A local pseudo arc-length method for hyperbolic conservation laws

A local pseudo arc-length method (LPALM) for solving hyperbolic conservation laws is presented in this paper. The key idea of this method comes from the original arc-length method, through which the critical points are bypassed by transforming the computational space. The method is based on local changes of physical variables to choose the discontinuous stencil and introduce the pseudo arc-length parameter, and then transform the governing equations from physical space to arc-length space. In order to solve these equations in arc-length coordinate, it is necessary to combine the velocity of mesh points in the moving mesh method, and then convert the physical variable in arclength space back to physical space. Numerical examples have proved the effectiveness and generality of the new approach for linear equation, nonlinear equation and system of equations with discontinuous initial values. Non-oscillation solution can be obtained by adjusting the parameter and the mesh refinement number for problems containing both shock and rarefaction waves.

CHARACTERIZATION OF THE SLANT HELIX AS SUCCESSOR CURVE OF THE GENERAL HELIX

In classical curve theory, the geometry of a curve in three dimen- sions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of nding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, including notably the plane curves and general helices. This paper shows constructively how to solve the natural equations explic- itly for an innite series of curve classes. For every Frenet curve, a family of successor curves can be constructed which have the tangent of the original curve as principal normal. Helices are exactly the successor curves of plane curves and applying the successor transformation to helices leads to slant he- lices, a class of curves that has received considerable attention in recent years as a natural extension of the concept of general helices. The present paper gives for the rst time a generic characterization of the slant helix in three-dimensional Euclidian space in terms of its curvature and torsion, and derives an explicit arc-length parametrization of its tangent vector. These results expand on and put into perspective earlier work on Salkowski curves and curves of constant precession, both of which are subclasses of the slant helix.

Affine arc length polylines and curvature continuous uniform B-splines

We study the recently introduced notion of polylines that form a discrete version of planar curves in affine arc length parametrization, showing that they match the control polylines of curvature continuous uniform quadratic B-splines (with analogous results in Rn). It is demonstrated how inflection-free planar curves may be approximated by such affine arc length polylines in a way that the polyline is close to an affinely equidistant discretization of the curve and allows good approximations of the smooth affine curvature.

Curve cuspless reconstructionviasub-Riemannian geometry

We consider the problem of minimizing $\int_{0}^L \sqrt{\xi^2 +K^2(s)}\, ds $ for a planar curve having fixed initial and final positions and directions. The total length $L$ is free. Here $s$ is the variable of arclength parametrization, $K(s)$ is the curvature of the curve and $\xi>0$ a parameter. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.

On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms

We consider semilinear and quasilinear parabolic systems with a nonsmooth rate-independent and a viscous dissipation term in the limit of very slow loading rates, or equivalently with fixed loading and vanishing viscosity > 0. Because for nonconvex energies the solutions will develop jumps, we consider the vanishing-viscosity limit for the graphs of the solutions in the extended state space in arclength parametrization. Here the choice of the viscosity norm for parametrization is crucial to keep the subdifferential structure of the problem. A crucial point in the analysis are new a priori estimates that are rate independent and that allow us to show that the total length of the graph remains bounded in the vanishing-viscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rate-independent systems.

Gamma-Limit of a Model for the Elastic Energy of an Inextensible Ribbon

A Γ-convergence result involving the elastic bending energy of a narrow inextensible ribbon is established. As a consequence of the result, the energy is reduced to a one-dimensional integral, over the centerline of the ribbon, in which the aspect ratio of the ribbon appears as a small parameter. That integral is observed to increase monotonically with the aspect ratio. The Γ-limit of the family of energies is taken in a Sobolev space of centerlines with nonvanishing curvature. In that space, it is shown that the Γ-limit is a functional first proposed by Sadowsky in the context of narrow ribbons that form Möbius bands. The results obtained here do not apply to such ribbons, since the centerline of a Möbius band must have at least one inflection point. As a first step toward dealing with such inflection points, a result concerning the lower semicontinuity of the Sadowsky functional with inflection points comprising a set of measure zero within the domain of an arclength parameterization is presented.

A Pseudo Arc-Length Method for Numerical Simulation of Shock Waves

A pseudo arc-length method is proposed for the numerical simulation of shock wave propagations. This method passes the discontinuities and establishes adaptive moving meshes in the physical space by introducing the arc-length parameter and transforming the computational domain. Numerical experiments of the Sod problem, double Mach reflection problem and explosion problem demonstrate that this approach is more efficient than traditional numerical methods in capturing and tracking discontinuous solutions of singular or nearly singular problems.

B-Spline Curve Approximation with Nearly Arc-Length Parameterization

An algorithm of B-spline curve approximation with the three-dimensional data is presented in this paper. In this algorithm, we will get a smooth curve which is nearly arc-length parameterization. The smoothness and uniform parameterization are key factors of the approximating curve, specifically in skinning surface and surface approximation. Firstly, the data points are fitted using local interpolation, this local fitting algorithm yields n Bezier segments, each segment having speed equal to 1 at their end and midpoints. Then segments are composed of a C1 continuous cubic B-spline curve which named controlling curve. But the controlling curves control points are redundancy, so we find another curve to approximate the controlling curve using least square approximation with smoothness

Adaptive feedrate interpolation with multiconstraints for five-axis parametric toolpath

A good adaptive feedrate will be helpful for improving machining accuracy and efficiency, as well as avoiding the excess of the machine’s physical capabilities and feed fluctuations during machining. Therefore, it is highly desirable to consider the constraints of geometric error, cutting performance, and drive constraints in the feedrate scheduling of the parametric curve interpolator for five-axis computer numerical control machining. In this paper, a novel multiconstraints feedrate scheduling method is proposed for the parametric curve interpolator in five-axis machining. In the method, the feed optimization model is first built with the constraints of geometric error, the maximum feedrate and acceleration of cutter tip, and the maximum feedrate and acceleration of five-drive axes. Then, the relations between each constraint and the cutter tip feedrate are derived by means of near arc length parameterization. After that, a linear programming algorithm is applied to obtain the optimal feed profile on the sampling positions of the given tool path. Finally, illustrated examples are given to validate the feasibility and applicability of the proposed feedrate scheduling method. The comparison results show that the proposed method has an ability of the simultaneous guarantees of geometric accuracy, cutting performance, and drive characters of machine tools.

Strong Coupling Asymptotics for a Singular Schrödinger Operator with an Interaction Supported by an Open Arc

We consider a singular Schrödinger operator in L 2(ℝ2) written formally as − Δ − βδ(x − γ) where γ is a C 4 smooth open arc in ℝ2 of length L with regular ends. It is shown that the jth negative eigenvalue of this operator behaves in the strong-coupling limit, β → + ∞, asymptotically as where is the jth Dirichlet eigenvalue of the operator on L 2(0, L) with κ(s) being the signed curvature of γ at the point s ∈ (0, L) and s is the arc length parameter.

Arclength Parametrized Hamilton's Equations for the Calculation of Instantons

A method is presented to compute minimizers (instantons) of action functionals using arclength parametrization of Hamilton's equations. This method can be interpreted as a local variant of the geometric minimum action method (gMAM) introduced to compute minimizers of the Freidlin-Wentzell action functional that arises in the context of large deviation theory for stochastic differential equations. The method is particularly well-suited to calculate expectations dominated by noise-induced excursions from deterministically stable fixpoints. Its simplicity and computational efficiency are illustrated here using several examples: a finite-dimensional stochastic dynamical system (an Ornstein-Uhlenbeck model) and two models based on stochastic partial differential equations: the $\phi^4$-model and the stochastically driven Burgers equation.

EFFECTIVENESS OF INPUT SHAPING AND REAL-TIME NURBS INTERPOLATION FOR REDUCING FEEDRATE FLUCTUATION

In modern CAD/CAM system, NURBS (Non-Uniform Rational B-Spline) interpolation is adopted for the purpose of achieving high speed and high accuracy performance. The main problem with using NURBS curve is the relationship approximation between arc length and spline parameter. In case the toolpath planning can be assumed with no feed fluctuations, implementing of driving mechanism still can lead CNC system to vibrate. Input shaping is a method of preshaping control input to reduce residual vibration in the flexible systems. Input shaping alters the reference command by convolving it with an impulse sequence. The convolution product is then used to drive the system. In this paper, NURBS curve representation was employed to generate the planning trajectory for a two axis gantry crane which is considered as a CNC system. The effectiveness of input shaping based on genetic algorithm was investigated by following the desired NURBS curve.

Traveling waves from the arclength parameterization: Vortex sheets with surface tension

We study traveling waves for the vortex sheet with surface tension. We use the angle-arclength description of the interface rather than Cartesian coordinates, and we utilize an arclength parameterization as well. In this setting, we make a new formulation of the traveling wave ansatz. For this problem, it should be possible for traveling waves to overturn, and notably, our formulation does allow for waves with multi-valued height. We prove that there exist traveling vortex sheets with surface tension bifurcating from equilibrium. We compute these waves by means of a quasi-Newton iteration in Fourier space; we find continua of traveling waves bifurcating from equilibrium and extending to include overturning waves, for a variety of values of the mean vortex sheet strength.

On Reduction of Stress Concentration around the Pin Hole in Turbine Discs

The stress concentration phenomenon around holes in rotated components is discussed. Firstly, finite element analysis of a simplified model of turbine disc flange is carried out, and then a method is developed by designing an intended-cut notch on the model to reduce the stress around the pin hole. Furthermore, two characteristic parameters which determine the configuration of the notch, namely notch depth and arc length parameter, are investigated by changing the notch dimensions. The results show that this method can reduce the stress around the hole significantly, notch depth and arc length have an important impact on the stress concentration factor (SCF), the SCF decreases with the increase of notch depth, and increasing the arc length within a critical scope results in the decrease of the SCF.

Segmentation algorithm of intervertebral disc magnetic resonance images based on two-dimensional automatic active shape model

In response to the issue that the intervertebral disk manual modeling was time-consuming and subjective, and the existing segmentation method was not accurate enough, a new method named two-diememsional Automatic Active Shape Model( 2D-AASM) was proposed. It included three parts: automatic statistical shape modeling of intervertebral disk based on minimum description length, 2D local gradient modeling and segmentation. Adopting the manual segmentation results of 25sets of spinal MR images as the training set, the study used minimum description length method to determine the point correspondence, built statistical shape model and 2D local gradient model for intervertebral disk T4-5. The generated shape model had lower variance and the objective function value than the manual and arc length parameter method. After the model was built, three sets of Magnetic Resonance Image( MRI) images were used to test the proposed method. Compared with the traditional ASM and 1D-ASM, the segmentation result of the proposed method had a higher Dice coefficient and lower oversegmentation and under-segmentation rate. The experiment results indicate that the proposed method generates a better model and more accurate segmentation result.

RECOGNITION OF OCCLUSIONS IN CT IMAGES USING A CURVE-BASED PARAMETERIZATION METHOD

It is an important way for segmentations of CT images to extract contours of objects slice-by-slice. For such a way, an important idea is analogy. That is to say, correct the contour in current slice (current contour) according to the contour in previous slice (previous contour). The key to properly correct the current contour is the ability to recognize occlusions (or say leaking parts) in the current contour. We present a curve-based curve parameterization method to recognize occlusions. The previous contour is evolved to the current contour using line projections. In the process of evolution, the parameterization is realized, which includes two types of information for every point in the evolved contour: the arc length parameter on the previous contour, and distance moved from the initial position to the present position. Using these two parameters, we are able to recognize occlusions in the current contour. Many experiments indicate that the method can recognize all of the occlusions in a given contour. Consequently, the method is robust and can be used as a part of an algorithm to automatically extract contours for CT images.

A New Piecewise Rational Reparameterization Method

A new piecewise rational re-parameterization of a polynomial curve is presented. The method relies on a parametric polygonal decomposition method. With the proposed piecewise rational re-parameterization method, the optimal parameterization can be reached, and the optimal parameterizations asymptotic convergence to the arc-length parameterization is analyzed. Computing instances demonstrates the efficiency of the proposed method.

A Re-Parameterization Transformation of Bézier Curve

A re-parameterization transformation is discussed. With this re-parameterization transformation, a polynomial curve can be reformulated as a rational curve, with a parameterization that is optimal in the sense that no other rational representation of the curve approximates more closely arc-length parameterization. Computing instances are included.

Surface-diffusion-driven decay of high-aspect-ratio gratings: Existence of morphologically related classes

We present numerical and theoretical results concerning the technologically important process of evolution of high-aspect-ratio profiles due to surface diffusion under thermal treatment. We show how a broad class of initial gratings adopt, after a short transient stage, a typical shape that can be accurately described as a curve whose curvature has only two single Fourier modes as a function of the arc-length parameter. Moreover, we introduce a set of evolution equations for the relevant parameters that accounts very accurately for both morphological and kinetic aspects of the transformation processes for these curves in a wide region in parameter space. Regarding the decay of rectangular gratings, our numerical results show the existence of geometrically related classes that asymptotically approach to the same trajectory in parameter space. Gratings belonging to the same class pass through the same sequence of morphologies before reaching the final equilibrium state.

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.