Piecewise Cubic Research Articles

The theoretical analysis for an iterative envelope algorithm

Cubic spline interpolating the local maximal/minimal points is often employed to calculate the envelopes of a signal approximately. However, the undershoots occur frequently in the cubic spline envelopes. To improve them, in our previous paper we proposed a new envelope algorithm, which is an iterative process by using the Monotone Piecewise Cubic Interpolation. Experiments show very satisfying results. But the theoretical analysis on why and how it works well was not given there. This paper establishes the theoretical foundation for the algorithm. We will study the structure of undershoots, prove rigorously that the algorithm converges to an envelope without undershoots with exponential rate of convergence, which can be used to determine the number of iterations needed in the algorithm for a good envelope in applications.

New Relation for the Computation of Settling Velocities and Diameters of Spheres

Single relations that can be used to calculate both the terminal settling velocities of spheres and the equivalent diameter of particles from their settling velocities are developed. The literature going back to Newton is reviewed and the relations developed tabulated. It is shown how the standard drag curve has developed into the dimensionless velocity versus dimensionless diameter curve. No relations that cover the full range that can conveniently be used for both velocity and diameter calculation were found, however a relation by Concha and Almendra covers most of the range. The standard drag curve data are constructed by utilizing 535 data points available in the literature in a Reynolds number range of 2.4 × 10−5 to 2 × 105. The settling velocities are corrected for experiments in finite width columns that do not satisfy the infinite medium dimensions. The data are converted to the dimensionless diameter and dimensionless velocity terms, which is more convenient for calculation purposes. The data are analyzed using piecewise cubic functions. Data from sources with excessive scatter and a few outliers are removed leaving 443 data points. The resulting piecewise cubic can be used to obtain velocity from diameter or diameter from velocity. To give an algebraic expression a hyperbola is fitted to the data giving an expression that can be solved to give explicit relations for both dimensionless velocity and dimensionless diameter. This provides an accuracy that compares well with expressions given in the literature.

Research on Non-Circular Gear Hobbing Simulation Based on Piecewise Cubic Spline Fitting

In order to solve the hobbing machining data calculation and hobbing simulation of non-circular gear with various shapes, the piecewise cubic spline fit method has been adopted to unify the polar coordinate expression of non-circular pitch curve. The method to generate hobbing machining data was deduced based on hobbing linkage mathematical model, and the machining data error caused by piecewise cubic spline fitting method was analyzed. The three-dimension software has been used as simulation tool to make secondary development, and then non-circular gear hobbing simulation was realized based on machining data, which can verify the validity of machining data and the relevant design drawbacks of non-circular gear. The work of this paper lay a good foundation for design parameters optimization and hobbing of non-circular gear.

Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation

Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation

Quantitative evaluation of parameters used in wear testing simulators of total hip arthroplasty components

Biomechanical simulators are extensively used in wear tests of total hip arthroplasty components. Those tests should be in accordance with ISO 14242-1, which specifies three or four abscissæ ordinates of angular displacements and six loads to be attained by the simulators. Nevertheless, the standard does not provide directions on the interpolation method; consequently, wear rates from similar specimens tested in distinct simulators present considerable dispersion which, in turn, prevents the establishment of quality patterns for those components. In this work, continuous gait curves are generated from ISO 14242-1 specified points through Piecewise Cubic Hermite Interpolation Polynomials. A subset of 100 points that best fit ISO specified tolerances, extracted from the continuous curves, is provided for using in wear simulators as a means of decreasing discrepancies in results from different machines. Moreover, a cross-correlation comparison between interpolated and actual gait curves demonstrates that maximisation of the correlation between them (time lags of −24 for flexion/extension, −71 for abduction/adduction, +3 for inward/outward rotation, and −10 for load data) causes loss of synchronism in the heel strike and toe-off instants. Such a result evinces an intrinsic disagreement between actual and standard-prescribed biomechanics of gait.

Piecewise Cubic Approximation for Data

In this paper, we consider features concerning approximation for data by using piecewise interpolation techniques. Numerical examples are given which compare piecewise cubic interpolation methods.

Least Squares Interpolation Methods for LTE SystemChannel Estimation over Extended ITU Channels

Abstract — Performance of pilot-aided channel estimation techniques such as the Least Squares (LS) method depends on not only on the signal-to-noise ratio (SNR), channel conditions and pilot ratio, but also on the choice of interpolation method for deriving channel estimates at non-pilot subcarriers. This paper investigates the bit-error-rate (BER) performance of linear, spline and Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) interpolation methods in LS channel estimation over the three extended ITU channel profiles defined for Long Term Evolution (LTE) testing. Simulation results show that applying the linear interpolation method produces the best BER performance over the fading channel with the smallest multipath delay spread. It is also shown that the choice of best interpolation method actually depends on the SNR in some of the fading channel profiles. Index Terms—ITU, LS, LTE, PCHIP, spline. I. INTRODUCTION Orthogonal frequency division multiplexing (OFDM) technology has been adopted in many of the next generation communication systems such as Long Term Evolution (LTE) and Worldwide Interoperability for Microwave Access (WiMAX) [1]. OFDM is preferred not only because it offers high data rate transmission but also because of its flexibility and robustness against multipath fading [1]. Multipath channel fading exists when multiple delayed copies of a transmitted signal are received at the receiver, thereby resulting in the distortion of the transmitted signal. These distortions are collectively termed „channel fading‟, causing reception errors and hence increased bit-error-rate (BER). Channel fading can be broadly classified as time-selective and/or frequency selective, depending on the relationship between the fading channel and the transmitted signal parameters [2]. To combat multipath fading and improve BER, some form of channel estimation method is required in order to estimate the channel distortion effect for each of the active subcarriers within an OFDM symbol [3].

IRIS: a generic three-dimensional radiative transfer code

We present IRIS, a new generic three-dimensional (3D) spectral radiative transfer code that generates synthetic spectra, or images. It can be used as a diagnostic tool for comparison with astrophysical observations or laboratory astrophysics experiments. We have developed a 3D short-characteristic solver that works with a 3D nonuniform Cartesian grid. We have implemented a piecewise cubic, locally monotonic, interpolation technique that dramatically reduces the numerical diffusion effect. The code takes into account the velocity gradient effect resulting in gradual Doppler shifts of photon frequencies and subsequent alterations of spectral line profiles. It can also handle periodic boundary conditions. This first version of the code assumes Local Thermodynamic Equilibrium (LTE) and no scattering. The opacities and source functions are specified by the user. In the near future, the capabilities of IRIS will be extended to allow for non-LTE and scattering modeling. IRIS has been validated through a number of tests. We provide the results for the most relevant ones, in particular a searchlight beam test, a comparison with a 1D plane-parallel model, and a test of the velocity gradient effect. IRIS is a generic code to address a wide variety of astrophysical issues applied to different objects or structures, such as accretion shocks, jets in young stellar objects, stellar atmospheres, exoplanet atmospheres, accretion disks, rotating stellar winds, cosmological structures. It can also be applied to model laboratory astrophysics experiments, such as radiative shocks produced with high power lasers.

Optimal fin shape in finned double pipe with fully developed laminar flow

Optimal shape of longitudinal fins augmented to the outer surface of the inner pipe enclosed within a concentric outer pipe, is investigated for maximizing the Nusselt number in a steady laminar and fully-developed flow. The wall-fin assembly is made up of highly conductive material and is subjected to the uniform heat flux boundary condition. Taking triangular cross-section as the initial profile, the fin shape is represented by Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) at each step of the optimization process with control points as the design variables. Genetic algorithm and trust region method have been used as the optimizers with FEM as the solver of the governing equations. The results show that the optimal profile is strongly dependent on the number of fins, the ratio of radii, the number of control points and the characteristic length. It gives improvement in the Nusselt number upto 138%, 312% and 263% over the conventional fins of trapezoidal, triangular and parabolic shapes for the case of equivalent diameter and for the case of hydraulic diameter it gives such improvements upto 212%, 59% and 90% respectively. For large number of fins the optimal fin shape based on the equivalent diameter is usually triangular on smaller inner pipes and wavy on larger inner pipes. More and more articulates appear as the larger number of higher fins is employed on bigger inner pipes. For the case of hydraulic diameter, it may either be regarded as parabolic or wavy. Comparison of the present results with the available literature results proves their validity and accuracy.

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.

FCA 필릿 파이프 용접에서 다중 비선형 회귀 모형과 구간적 3차 에르미트 보간법을 통한 비드 형상 예측

Pipe welding is used in various ranges such as civil engineering and ship building engineering. Until now, many technicians work for pipe welding manually under harmful, dangerous and difficult conditions. So it is necessary to install automation process. For automation pipe welding, relation between welding parameters & bead shape should be considered. Using this relation, bead shape could be expected from welding parameters. FCAW was used in this study. Instead of pipe workpiece, fillet joint plate is used, which were inclined 0,45,90,135,180 degree. By analyzing between welding parameters (current, welding speed, voltage) and bead shape parameters with non-linear multiple regression, bead shape parameters could be expected. Piecewise Cubic Hermite Interpolation was used to expect smooth curved bead shape with bead shape parameters. From these processes, bead shape could be expected from welding parameters.

Formula omitted] piecewise cubic quasi-interpolants on a 6-direction mesh

We study two kinds of quasi-interpolants (abbr. QI) in the space of C 2 piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type. As those QIs are exact on the space of cubic polynomials, their approximation order is 4 for sufficiently smooth functions. In addition, they exhibit nice superconvergent properties at some specific points. Moreover, the infinity norms of the discrete QIs being small, they give excellent approximations of a smooth function and of its first order partial derivatives. The approximation properties of the QIs are illustrated by numerical examples.

Animating oscillatory motion with overlap

Oscillatory motion is ubiquitous in computer graphics, yet existing animation techniques are ill-suited to its authoring. We introduce a new type of spline for this purpose, known as a "Wiggly Spline." The spline generalizes traditional piecewise cubics when its resonance and damping are set to zero, but creates oscillatory animation when its resonance and damping are changed. The spline provides a combination of direct manipulation and physical realism. To create overlapped and propagating motion, we generate phase shifts of the Wiggly Spline, and use these to control appropriate degrees of freedom in a model. The phase shifts can be created directly by procedural techniques or through a paint-like interface. A further option is to derive the phase shifts statistically by analyzing a time-series of a simulation. In this case, the Wiggly Spline makes it possible to canonicalize a simulation, generalize it by providing frequency and damping controls and control it through direct manipulation.

Image signal‐to‐noise ratio estimation using Shape‐Preserving Piecewise Cubic Hermite Autoregressive Moving Average model

We propose to cascade the Shape-Preserving Piecewise Cubic Hermite model with the Autoregressive Moving Average (ARMA) interpolator; we call this technique the Shape-Preserving Piecewise Cubic Hermite Autoregressive Moving Average (SP2CHARMA) model. In a few test cases involving different images, this model is found to deliver an optimum solution for signal to noise ratio (SNR) estimation problems under different noise environments. The performance of the proposed estimator is compared with two existing methods: the autoregressive-based and autoregressive moving average estimators. Being more robust with noise, the SP2CHARMA estimator has efficiency that is significantly greater than those of the two methods.

Accurate numerical estimation of interphase momentum transfer in Lagrangian–Eulerian simulations of dispersed two-phase flows

The Lagrangian–Eulerian (LE) approach is used in many computational methods to simulate two-way coupled dispersed two-phase flows. These include averaged equation solvers, as well as direct numerical simulations (DNS) and large-eddy simulations (LES) that approximate the dispersed-phase particles (or droplets or bubbles) as point sources. Accurate calculation of the interphase momentum transfer term in LE simulations is crucial for predicting qualitatively correct physical behavior, as well as for quantitative comparison with experiments. Numerical error in the interphase momentum transfer calculation arises from both forward interpolation/approximation of fluid velocity at grid nodes to particle locations, and from backward estimation of the interphase momentum transfer term at particle locations to grid nodes. A novel test that admits an analytical form for the interphase momentum transfer term is devised to test the accuracy of the following numerical schemes: (1) fourth-order Lagrange Polynomial Interpolation (LPI-4), (3) Piecewise Cubic Approximation (PCA), (3) second-order Lagrange Polynomial Interpolation (LPI-2) which is basically linear interpolation, and (4) a Two-Stage Estimation algorithm (TSE). A number of tests are performed to systematically characterize the effects of varying the particle velocity variance, the distribution of particle positions, and fluid velocity field spectrum on estimation of the mean interphase momentum transfer term. Numerical error resulting from backward estimation is decomposed into statistical and deterministic (bias and discretization) components, and their convergence with number of particles and grid resolution is characterized. It is found that when the interphase momentum transfer is computed using values for these numerical parameters typically encountered in the literature, it can incur errors as high as 80% for the LPI-4 scheme, whereas TSE incurs a maximum error of 20%. The tests reveal that using multiple independent simulations and higher number of particles per cell are required for accurate estimation using current algorithms. The study motivates further testing of LE numerical methods, and the development of better algorithms for computing interphase transfer terms.

Application of the time discontinuous Galerkin finite element method to heat wave simulation

This paper deals with the numerical simulation of heat wave propagation in the medium subjected to different kinds of heat source, particularly heat impulse. The discontinuous Galerkin finite element method (DGFEM) proposed for the stress wave propagation in solids [X.K. Li, D.M. Yao, R.W. Lewis, A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non-linear solids and saturated porous media. Int. J. Numer. Meth. Eng. 57 (2003) 1775–1800] is extended to numerically solve for the non-Fourier heat transport equation constructed according to the CV model [C. Cattaneo, A form of heat-conduction equation which eliminates the paradox of instantaneous propagation, Compute Rendus 247 (1958) 431–433; P. Vernotte, Les paradoxes de la theorie continue de l’equation de la chaleur, Compute Rendus 246 (1958) 3154–3155]. Temperature and its time-derivative are chosen as primitive variables defined at each FE node. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite’s polynomial) and linear interpolations for both temperature and its time-derivative, respectively, in the time domain is particularly proposed. As a consequence the continuity of temperature at each discrete time instant is exactly ensured, whereas discontinuity of the time-derivative of temperature at discrete time levels remains. Numerical results illustrate good performance of the present method in the numerical simulation of heat wave propagation in eliminating spurious numerical oscillations and in providing more accurate solutions in the time domain.

External approximation of [formula omitted] in a bounded domain of [formula omitted] with piecewise cubics of weak C2-class

A nonconforming finite element method is introduced to approximate triharmonic boundary value problems in R 3 , among other applications. It is constructed upon tetrahedra and piecewise cubic representations. The finite element can be viewed as the primitive of a quadratic one proposed by the first author to solve biharmonic problems, which can be considered in turn as the three-dimensional analogue of the well-known Morley triangle. The new method is proven to be first order convergent in the natural discrete H 3-norm for the problem under consideration. To cite this article: V. Ruas et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

A new subdivision method for bivariate splines on the four-directional mesh

We present a new bivariate subdivision scheme based on two generators of a four-directional spline space. In particular, we are dealing with piecewise linear, continuous splines, and with piecewise cubic, continuously differentiable splines. The subdivision schemes rely on a matrix mask of small support. In addition, we present some results on related quasi-interpolating operators, and on approximation order of the underlying shift-invariant spaces.

A numerical study of the vibrations of a damaged beam

We developed a finite element method (FEM) for examining small transverse vibrations of a damaged beam. The damage is modelled as an elastic joint and standard Hermite piecewise cubics were adapted to use as basis functions in the FEM. This method can be used in situations where the characteristic equation approach is not viable.

An approximation method for the hypersingular heat operator equation

The original model problem is the two-dimensional heat conduction problem with vanishing initial data and a given Neumann-type boundary condition. In particular, certain choices of the representation formula for the heat potential yield the hypersingular heat operator equation of the first kind. In this paper we concentrate on the problem of solving this hypersingular integral equation. Our approximation method is a Petrov–Galerkin method, where we use collocation with respect to the space variable and Galerkin method with respect to the time variable. The trial functions are tensor products of piecewise cubic (space) and piecewise linear (time) smoothest splines. Stability and convergence of the resulting scheme is proved when the spatial domain of the original heat conduction problem is a disc.