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A new multivariate approximation scheme to scattered data on arbitrary bounded domains in R d is developed. The approximant is selected from a space spanned (essentially) by corresponding translates of the ‘shifted’ thin-plate spline (‘essentially,’ since the space is augmented by certain functions in order to eliminate boundary effects). This scheme applies to noisy data as well as to noiseless data, but its main advantage seems to be in the former case. We suggest an algorithm for the new approximation scheme with a detailed description (in a MATLAB-like program). Some numerical examples are presented along with comparisons with thin-plate spline interpolation and Wahba's thin-plate smoothing spline approximation.

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 be calculated directly. Several numerical examples are given. The results which are presented to demonstrate the accuracy and efficiency of the present method are quite saisfactory.

An implicit numerical method is introduced that gives piecewise polynomial spline approximations for the solution of an initial value problem for the equationy^'(t)=f(t,y(t)).Here, f is a function having continuous derivatives up to the order r. The method presented in this work, provides spline functions which approximate the solution of the problem in the most regular function space C^r^+^1. Error estimates in C and C^r^+^1 are given and the stability of the method is investigated.

This paper introduces an accurate and efficient engine whose purpose is the dynamic animation of curvilinear objects which are modelled as successions of splines. At each time step the object shape conforms to its spline definitions, thus ensuring that each property implied by the chosen spline models is verified. This is achieved by animating spline control points. However, these control points are not considered as material points but rather as the degrees of freedom of the continuous object. The chosen dynamic equations (Lagrangian formalism) reflect this modelling scheme and yield an exact and very proficient linear system. In the chosen formalism, forces are introduced by either their potential energy or their power ratings in the virtual movements instilled by the degrees of freedom. Both methods are carried out in three cases: gravity, viscosity and generic force. Suitable and classical methods for constraint handling and numerical resolution are briefly discussed. Finally, this animation engine is applied to knitted patterns. Copyright © 2000 John Wiley & Sons, Ltd.

A simple derivation is given for a kernel-type approximation of the weight function for a linear smoothing spline that explicitly includes boundary corrections. As an application, asymptotic approximations are obtained for the estimator's pointwise and integrated mean squared errors.

Controlling the convexity and the strain energy of the interpolating curve can be carried out by controlling the second-order derivative of the interpolating function. In [1], the rational cubic spline with linear denominator has been used to constrain the convexity and the strain energy of the interpolating curves, but it does not work in some case. This paper deals with the weighted rational cubic spline with linear denominator for this kind of constraint, the sufficient and necessary condition for controlling the convexity and strain energy of the interpolating curves are derived, and a numerical example is given.

Numerical simulation of solute transport in heterogeneous porous media is greatly complicated by the large velocity and concentration gradients induced by spatial variations in hydraulic conductivity. Eulerian-Lagrangian methods for solving the transport equation can give accurate solutions to heterogeneous problems if their interpolation algorithms are properly selected. This paper compares the performance of four Eulerian-Lagrangian solvers that rely on linear, quadratic, cubic spline, and taut spline interpolators. In each case a tensor product decomposition is used to reduce the general n -dimensional interpolation problem to a sequence of n one-dimensional problems. Comparisons of a set of test problems indicate that the linear and taut spline interpolators are dispersive while the quadratic and cubic spline interpolators are oscillatory. The cubic and taut spline interpolators give consistently better accuracy than the more conventional linear and quadratic alternatives. Simulation experiments in two- and three-dimensional heterogeneous media indicate that the taut spline interpolator, which is applied here for the first time to a solute transport problem, is able to yield accurate essentially nonoscillatory solutions for high grid Peclet numbers. The cubic spline interpolator requires significantly less computational effort to achieve performance comparable to the other methods.

Bezier spline fitting is a useful tool for generating smooth curves from a set of digitized points. These smooth curves may replace polylines for more accurate representation of natural linear entities. Maintaining its mathematical framework, the Bezier spline fitting can be improved so as to enhance its accuracy in representation. Key for improvement is developing methods to calculate tangent vectors and two coefficients for intermediate Bezier points, which allow higher flexibility of the control polygons than that of the Bezier spline fitting. Validity of the improved new spline fitting was tested by comparing accuracy of spline curves from the new spline fitting with spline curves from the Bezier spline fitting. Test was done using a set of mathematically simulated curves of the natural linear entities.

A macro spline finite shell element is presented. This element is capable of analysing flat plates, folded plate structures, and rectangular frames. Two dimensional cubic B-spline interpolation is used for both membrane and bending behaviour. Kirchhoff's theory is used to describe the bending behaviour of the element. Within the macro element C2-continuity for in-plane and out-of-plane behaviour is achieved with only three degrees-of-freedom per node. This reduced number of dof results in an analysis tool which is more efficient and versatile than the spline finite strip method. Several numerical examples are included to demonstrate the versatility, accuracy and numerical efficiency of this approach in the transient analysis of thin-walled structures.

A new group explicit iterative method based on cubic spline approximations is presented for the numerical solution of partial differential equations. The numerical results obtained confirm the viability of the method.

Thin plate spline methods provide an interpolant to values of a real function and are highly useful in many applications. The need for iterative procedures arises, since hardly any sparsity occurs in the linear system of interpolation equations. This paper considers a generalization of the iterative algorithm developed by Beatson, Goodsell and Powell. A proof of convergence of this method is given. It depends mainly on the remark that all the changes to the thin plate spline coefficients reduce a certain semi‐norm of the difference between the required interpolant and the current approximation. The analysis applies also to analogous algorithms for other radial basis functions.

A rational spline based on function values only is constructed, which is of the same order as a cubic spline. The rational spline may be used to control the position and shape of a curve or surface. The approximation properties of this spline are studied.

The accuracy of interpolating semi-Lagrangian (SL) discretization methods depends on the choice of the interpolating function. Results from barotropic transport simulations on the sphere are presented, using either bicubic Lagrangian or bicubic spline SL discretization. The spline-based scheme is shown to generate excessively noisy fields in these simulations. The two methods are then tested in a one-dimensional advection problem. The damping and dispersion relations for the schemes are examined. The analysis and numerical experiments suggest that the excessive noise found in the spline-based simulations is a consequence of insufficient damping of the small scales for small and near-integer values of the Courant number. Inspection of the local Courant number for the two-dimensional spline-based simulation confirms this hypothesis. This noise can be controlled by adding a scale-selective diffusion term to the spline-based scheme, while retaining its excellent dispersion characteristics.

Fuzzy systems built with spline wavelets can approximate any function in a general Hilbert space. Since wavelets are used extensively for the efficient solution of various types of differential equations, it is demonstrated that fuzzy spline wavelets can be used for the solution of the same type of problems. The advantage in using fuzzy spline wavelets for the solution of such problems is that the solution would enjoy the excellent numerical and computational characteristics of the fast wavelet transform, while retaining the explanatory power of fuzzy system. The method is demonstrated with the feedforward control of a flexible robotic arm.

This paper presents an overview of the suitability of the polynomial spline wavelet transform to qualify as an elegant tool for multiresolution signal decomposition. The paper also reviews the effectiveness of the polynomial spline function that has maintained its leadership among its clan having wide applications in different areas ranging from the field of function approximation to computer vision. Construction of different scaling functions and their properties are shown. Polynomial spline wavelets and their characteristics are also shown. We have introduced the special cases of generalised spline wavelets. The distinctive features of these various representations are discussed with respect to the multiresolution signal decomposition. The main thrust is to emphasize a signal and image processing point of view which should make these techniques more attractive for mathematicians and signal processors.

Summary Yee and Wild have presented additive extensions to multivariate regression models including generalized estimating equations. The back-bone of their methods was a vector spline, which was solved by direct solution. A back-fitting solution to the vector spline problem is presented here, with the implication that existing generalized additive model software can, with moderate effort, be modified to implement multivariate models.

This paper develops a new family of convexity-preserving splines of order n , hereby entitled the CP n -spline, that preserves convexity when derivatives at the data points satisfy some reasonable conditions. The spline comprises four components: a constant term, a first order term, and two n th order binomials. A slope-averaging-method is proposed for the general implementation of the new spline. Numerical results that allow for an assessment of the new spline are provided. In particular, a comparative analysis of the CP n -spline, the cubic spline, and of the Carnicer '92 spline is performed. By varying two parameters, the spline shape can be controlled at the local level, while other conventional means can be used to control the shape at the global level. The CP n -spline has no singularities in the case where inflection points are present. Additionally, a less general form of the CP n -spline that applies to most practical cases can be implemented with extreme ease.

In this paper an alternative approach to model - based predictive control of a class of continuous - time linear systems is formulated. The approach is based on a spline approximation of given continuous - time cost function and spline approximation of convolution type system description with structured and unstructured parts. Proposed predictive controller produces the control input signal in the form of a polynomial spline of given order. The various physical constraints on the signal can be easily set.

It is desirable to use approximated cubic splines to obtain reliable processing of experimental data on creep and for identification of creep parameters. The spline approximation makes it possible to allow for such characteristic features of creep as the three-stage nature of the process of deformation and the scattering of Theological properties of materials; it also makes it possible to computer the first and second derivatives for an arbitrary point in time to obtain the corresponding speed and acceleration of the creep process.

Linear, twelfth-order boundary-value problems (special case) are solved, using polynomial splines of degree 12. The spline function values at the midknots of the interpolation interval, and the corresponding values of the even-order derivatives are related through consistency relations. The algorithm developed approximates the solutions, and their higher-order derivatives, of differential equations. Two numerical illustrations are given to show the practical usefulness of the algorithm developed. It is observed that this algorithm is second-order convergent.

This paper investigates a numerical technique using cubic splines for solving the continuous batch grinding equation. The same approach is also used to solve the similarity equation in the case where the breakage function has a more complicated form than that for which analytical solutions are available. In the case of the similarity equation the spline coefficients are obtained from the solution of an algebraic generalized eigenvalue problem. Numerical solutions of the continuous equation for simplified selection and breakage functions are compared with those of analytical solutions in order to assess the validity of the numerical procedures. Experimental results from a limestone ore are used to test the validity of the simplified model. The spline method is also used to solve the continuous equation for a more complicated breakage function evaluated for a gold bearing ore body.

Usually the linear scanner system, especially the airborne system, suffers from the line-to-line geometric variability. About the orientation of the exterior orientation parameters of an airborne system, one of the already proposed methods is to describe the behaviors of the exterior orientation parameters by spline functions [3] . To this method, the selection of the spline function is of great concern to the accuracy of relative orientation of the exterior orientation parameters. In this paper, a method for the selection of the order and range of spline function is proposed. The smoothness conditions defined in the spline functions are considered in this method. The mathematical model of the method is presented, and an evaluation is carried out with real data obtained in a flight test to prove the feasibility of the method.

Authors propose a nonpolynomial spline based Empirical Mode Decomposition (EMD) algorithm to reduce mode mixing, and detrend uncertainty in analysis of time series. This new algorithm first locates original and pseudo extrema and then uses nonpolynomial spline interpolation to determine the upper and lower envelope at each decomposition step. A set of algebraic equations for the nonpolynomial spline interpolation is derived. A numerical simulation has been carried out for the analysis of error in spline interpolations. Various time series analysis have been preformed to show comparison among EMD and ensemble EMD (EEMD) based on polynomial spline, and nonpolynomial spline based EMD. Nonpolynomial spline based EMD algorithm is promising and generating better results.

In this paper the companding quantizer design using the approximative spline functions of the first and the second degree is done. The spline function of the first degree and the second degree is used for the approximation of the optimal compressor function. In order to reduce realization complexity of the designed model, in this paper it is proposed that the optimal compressor function in different segments is approximated by different spline functions, which means that the approximation of the optimal compressor function is done by combining the spline function of the first degree and the second degree. Comprehensive analysis of the results obtained for the proposed companding quantizer for the number of segments 2L = 4 i 2L = 8 is done.

In the present work a nonpolynomial spline function is used to approximate the solution of the second order two point boundary value problems. The classes of numerical methods of second order, for a specific choice of parameters involved in nonpolynomial spline, have been developed. Numerical examples are presented to illustrate the applications of this method. The solutions of these examples are found at the nodal points with various step sizes and with various parameters ( α, β). The absolute errors in each example are estimated, and the comparison of approximate values, exact values, and absolute errors of at the nodal points are shown graphically. Further, shown that nonpolynomial spline produces accurate results in comparison with the results obtained by the B-spline method and finite difference method.

International audience; In this paper we generate and study new cubature formulas based on spline quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations of a polygonal domain in $ℝ^2$. By using a specific refinement of a generic triangulation, optimal convergence orders are obtained for some of these rules. Numerical tests are presented for illustrating the theoretical results.

In this paper we propose using free-knot splines with a common knot vector to approximate multiresponse model functions and their associated error variance-covariance matrix. A “loose coupling” type of algorithm for minimizing the determinant parameter estimation criterion is constructed for multiresponse spline models. The algorithm evaluates the increment on the parameters of the common knot positions first, then evaluates the increment on spline coefficients individually. One constraint on using the determinant criterion for multiresponse estimation is that the number of design points has to be greater than the number of parameters. This makes spline models infeasible when a large number of responses are measured. We loosen the constraint on the number of parameters allowed and propose a more appropriate residual degrees of freedom for the spline models.

In this paper we have obtained a precise error estimate concerning deficient discrete cubic spline interpolant matching with the given function at the intermediate points between successive mesh points.

Linear, eighth-order boundary-value problems (special case) are solved, using polynomial splines of degree six. The spline function values at the midknots of the interpolation interval, and the corresponding values of the even-order derivatives are related through consistency relations. The algorithm developed approximates the solution, and their higher-order derivatives, of differential equations. Four numerical illustrations are given to show the practical usefulness of the algorithm developed. It is observed that this algorithm is second-order convergent.

A generalized multiresolution of multiplicityr, generated byr linearly independent spline functions with multiple knots, is introduced. With the help of the autocorrelation symbol and the two-scale symbol of the scaling functions, spline wavelets with multiple knots can be completely characterized. New decomposition and reconstruction algorithms, based on the Fourier technique, are presented.

An algorithm is presented for smoothing arbitrarily distributed noisy measurement data with a Powell-Sabin spline surface that satisfies necessary and sufficient monotonicity conditions. The Powell-Sabin spline is expressed as a linear combination of locally supported basis functions used in their Bernstein-Bezier representation. Numerical examples are given to illustrate the performance of the algorithm.

Linear, sixth-order boundary-value problems (special case) are solved, using polynomial splines of degree six. The spline function values at the midknots of the interpolation interval, and the corresponding values of the even-order derivatives are related through consistency relations. The algorithm developed approximates the solutions, and their higher-order derivatives, of differential equations. Two numerical illustrations are given to show the practical usefulness of the algorithm developed. It is observed that this algorithm is second-order convergent.

The spline method of Loscalzo [1] is adapted to the neutral delay-differential equations (NDDES). The resulting method enables us to construct an approximate spline solution for the (NDDES). The existence and uniqueness of the approximate spline solution are investigated. Numerical experiments and comparisons with other methods are given.

We use quadratic spline functions to develop a numerical method for computing approximate solution of a system of second order boundary value problems associated with obstacle, unilateral and contact problems. The present method outperforms other available techniques and it has extra added advantage of approximating the first derivative of the solution. Numerical example is presented to illustrate the applicability of the new method.

The aim of this paper is to present a modification to the Chui and Quak (1992) spline multiresolution analysis for the finite interval. Boundary scaling functions with multipole nodes at interval endpoints are rejected, in favor of the classical B-spline scaling function restricted to the interval. This necessitates derivation of revised boundary wavelets. In addition, a direct method of decomposition results in bandwidth reduction on solving some associated linear systems, and image distortion is reduced by employing natural spline projection. Finally, a hybrid projection scheme is proposed, which particularly for large systems further lowers operation count. Numerical experiments proving the algorithm are indicated.

The paper shows how a finite dimensional representation of a cubic smoothing spline can be put in the framework of a dynamic linear model. The formulation provides an updating scheme when observations do not occur sequentially in time or space.

The study of the stress-strain state of plates and shells subjected to various types of loads with different support conditions entails the formulation of boundary-value problems that generally involve systems of partial differential equations with variable coefficients. The complexity of solving these problems stems not only from the high order of the system and the variability of the coefficients, but also from the need to exactly satisfy prescribed boundary conditions. The use of a given method to obtains a solution with a satisfactorily high degree of accuracy depends to a significant extent on the geometric and mechanical parameters characterizing certain aspects of the problem and the type of boundary conditions. These factors sometimes limit the possibilities of solving problems in the important (in a practical sense) cases in which the stiffness of the shell or plate supports is also variable. In addition, problems of the shell theory entail local and edge effects, which imposes certain stiffness conditions on boundary-value problems related to the phenomenon of instability in the computation. Spline functions have recently come into wide use to solve such problems in the areas of computational mathematics, mathematical physics, and mechanics. The popularity of this approach stems from the advantagesmore » offered by spline functions compared to other methods. Among these advantages: the stability of splines in relation to local perturbations, i.e. the behavior of the spline in the neighborhood of a point has no effect on the behavior of the spline as a whole (which is not the case in a polynomial approximation); good convergence of a spline-interpolation, in contrast to a polynomial interpolation; simplicity and ease of use of algorithms that construct and calculate splines on computers. 65 refs., 5 figs., 7 tabs.« less

An approximate spline is constructed for the solution of initial-value problems in delay differential equations. The existence and uniqueness of the approximate spline solution are investigated. Samples of our computational experiments are given and compared with other methods.

A spline wavelets element method that combines the versatility of the finite element method with the accuracy of spline functions approximation and the multiresolution strategy of wavelets is proposed for frame structures vibration analysis. Instead of exploring orthogonal wavelets for specific differential operators, the spline wavelets are applied directly in finite element implementation for general differential operators. Although lacking orthogonality, the “two-scale relations” of spline functions and its corresponding wavelets from multiresolution analysis are employed to facilitate the elemental matrices manipulation by constructing two transform matrices under the constraint of finite domain of elements. In the actual formulation, the segmental approach for spline functions is provided to simplify the computation, much as conventional finite element procedure does. The assembled system matrices at any resolution level are reusable for the furthur finer resolution improvement. The local approximation and hiararchy merits make the approach competitive especially for higher mode vibration analysis. Some examples are studied as verification and demonstration of the approach.

Spline functions have come into increasingly wide use recently in the solution of boundary-value problems of the theory of elasticity of plates and shells. This development stems from the advantages offered by spline approximations compared to other methods. Among the most important advantages are the following: (1) the behavior of the spline in the neighborhood of a point has no effect on the behavior of the spline as a whole; (2) spline interpolation converges well compared to polynomial interpolation; (3) algorithms for spline construction are simple and convenient to use. The use of spline functions to solve linear two-dimensional problems on the stress-strain state of shallow shells and plates that are rectangular in plan has proven their efficiency and made it possible to expand the range of problems that can be solved. The approach proposed in these investigations is based on reducing a linear two-dimensional problem to a unidimensional problem by the spline unidimensional problem by the method of discrete orthogonalization in the other coordinate direction. Such an approach makes it possible to account for local and edge effects in the stress state of plates and shells and obtain reliable solutions with complex boundary conditions. In the present study, wemore » take the above approach, employing spline functions to solve linear problems, and use it to also solve geometrically nonlinear problems of the statics of shallow shells and plates with variable parameters.« less

The isoparametric spline finite strip method for degenerate shells is presented. In the formulation, both the geometry and the displacement field are represented by uniform cubic B-spline curves. In this paper, the general theory of isoparametric spline finite strip for analysis of shell structures is outlined. The method, when applied to most problems, yields a relatively narrow band matrix and requires little computational effort. Solutions of a number of problems using this method are compared with other available analytical and numerical solutions, and in all cases very good agreement is observed.

A method of constructing a mathematical model of a surface using a disordered set of measured points is described. The results are used to construct a shoe model on a machine tool with numerical program control.