B-spline Approximation Research Articles

Discontinuous Galerkin Isogeometric Analysis for the biharmonic equation

We present and analyze an interior penalty discontinuous Galerkin Isogeometric Analysis (dG-IgA) method for the biharmonic equation in computational domain in Rd with d=2,3. The computational domain consists of several non-overlapping sub-domains or patches. We construct B-Spline approximation spaces which are discontinuous across patch interfaces. We present a priori error estimate in a discrete norm and numerical experiments to confirm the theory.

Time-Suboptimal Satellite Formation Maneuvers Using Inverse Dynamics and Differential Evolution

This paper presents an open-loop planner for near-time-optimal maneuvers performed by satellite formations during proximity operations and reconfiguration maneuvers. Using a differential flatness parameterization, which is an inverse-dynamics method with a minimal set of independent functions, a new optimization technique based on the differential evolution is presented. The linearized dynamical model including the perturbation is taken into account. The differential flatness formulation is implemented to obtain the control policy and the full state as a function of the relative displacements in the local reference frame. The differential evolution algorithm with a local neighborhood, in which the risk of stopping at local minima is reduced, is employed. The technique is computationally efficient and evaluates near-optimal solutions characterized by approximated bang–bang control policies. Results are reported to evaluate the performances of the proposed technique, and a Monte Carlo simulation has been run to prove the efficiency of the planner over a variety of different scenarios. Moreover, the Chebyshev and the B-spline approximations are compared to establish that the latter approach guarantees better results than the former, in terms of maneuver time and computational effort.

New cubic B-spline approximation for solving third order Emden–Flower type equations

In this article, the typical cubic B-spline collocation method equipped with new approximations for second and third order derivatives is employed to explore the numerical solution of a class of third order non-linear singular boundary value problems. The singularity is removed by means of L’Hospital’s Rule. The Taylor’s series expansion of the error term reveals that our new scheme is fifth order accurate. The proposed technique is tested on several third order Emden–Flower type equations and the numerical results are compared with those found in the current literature. It is found that our new approximation technique performs superior to the existing methods due to its simple implementation, straight forward interpolation and very less computational cost.

MKSOR iterative method with cubic b-spline approximation for solving two-point boundary value problems

In this study, two-point boundary value problems have been discretized by using cubic B-spline discretization scheme to derive the cubic B-spline approximation equations that corresponds. Then, this approximation equation is used to develop system of cubic B-spline approximation equations. To get the numerical solutions, there are three iterative methods such as Gauss-Seidel (GS), Successive Over Relaxation (SOR) and Modified Kaudd Successive Over Relaxation (MKSOR) used to solve the generated systems of linear equations. For the purpose of comparison, the GS iterative method has been designated as a control method for the SOR and MKSOR iterative methods. Three examples of problems also have been considered to test the effectiveness of these proposed iterative methods. From the numerical results, MKSOR iterative method is superior method in terms of number of iterations and computational time. Keywords: cubic B-spline approximation; two-point boundary value problem; MKSOR iteration

Single-index partially functional linear regression model

In this paper, we propose a flexible single-index partially functional linear regression model, which combines single-index model with functional linear regression model. All the unknown functions are estimated by B-spline approximation. Under some mild conditions, the convergence rates and asymptotic normality of the estimators are obtained. Finally, simulation studies and a real data analysis are conducted to investigate the performance of the proposed methodologies.

DIC deformation analyses of Mg specimens at elevated temperatures

In this contribution the experimental analysis of the deformation behavior of AZ31 (TRC) specimens under bending load at elevated temperatures is demonstrated. The in-plane displacement fields at the surface of the specimens were determined using Digital Image Correlation (DIC) technique. By means of polynomial and B-spline approximation the evaluation of the DIC displacements was performed. Using B-spline approximation of the displacement data shows better results of the determined strain distribution compared to the method using global polynomial functions, particularly at the edges of the specimen surface.

Efficient parameter estimation and variable selection in partial linear varying coefficient quantile regression model with longitudinal data

Efficient estimation and variable selection in partial linear varying coefficient quantile regression model with longitudinal data is concerned in this paper. To improve estimation efficiency in quantile regression, based on B-spline basis approximation for nonparametric parts, we propose a new estimating function, which can incorporate the correlation structure between repeated measures. In order to reduce computational burdens, the induced smoothing method is used. The new method is empirically shown to be much more efficient and robust than the popular generalized estimating equations based methods. Under mild conditions, the asymptotically normal distribution of the estimators for the parametric components and the optimal convergence rate of the estimators for the nonparametric functions are established. Furthermore, to do variable selection, a smooth-threshold estimating equation is proposed, which can use the correlation structure and select the nonparametric and parametric parts simultaneously. Theoretically, the variable selection procedure works beautifully, including consistency in variable selection and oracle property in estimation. Simulation studies and real data analysis are included to show the finite sample performance.

A point projection approach for improving the accuracy of the multilevel B-spline approximation

Abstract In this study, we present a method for improving the accuracy of the multilevel B-spline approximation (MBA) method. We combine a point projection method with the MBA method for reducing the approximation error by directly adjusting the control points in the local area. An initial surface is generated by the MBA method, and grid points are produced on the surface. These grid points are projected onto the scattered point set, and the distances between the grid points and the projected points are computed. The control points are then modified based on the distances. The proposed method shows better approximations even with the same number of control points and ensures C2-continuity. The experimental results with examples verify the validity of the proposed method. Highlights We propose a method for improving the multilevel B-spline approximation method. We use a point projection method for computing the amount of errors. The computed errors are directly applied to the control points for reducing the approximation error.

One dimensional nonlocal integro-differential model & gradient elasticity model : Approximate solutions and size effects

ABSTRACTIn the present work, the close similarity that exists between Mindlin’s strain gradient elasticity and Eringens nonlocal integro-differential model is explored. A relation between length scales of nonlocal-differential model and gradient elasticity model has been arrived. Further, a relation has also been arrived between the standard and nonstandard boundary conditions in both the cases. C0-based finite element methods (FEMs) are extensively used for the implementation of integro-differential equations. This results in standard diagonally dominant global stiffness matrix with off diagonal elements occupied largely by the kernel values evaluated at various locations. The global stiffness matrix is enriched in this process by nonzero off diagonal terms and helps in incorporation of the nonlocal effect, there by accounting the long-range interactions. In this case, the diagonally dominant stiffness matrix has a band width equal to influence domain of basis function. In such cases, a very fine discretization with larger number of degrees of freedom is required to predict nonlocal effect, thereby making it computationally expensive. In the numerical examples, both nonlocal-differential and gradient elasticity model are considered to predict the size effect of tensile bar example. The solutions to integro-differential equations obtained by using various higher-order approximations are compared. Lagrangian, Bèzier and B-Spline approximations are considered for the analysis. It has been shown that such higher-order approximations have higher inter-element continuity there by increasing the band width and the nonlocal character of the stiffness matrix. The effect of considering the higher-order and higher-continuous approximation on computational effort is made. In conclusion, both the models predict size effect for one-dimensional example. Further, the higher-continuous approximation results in less computational effort for nonlocal-differential model.

TLS-based composite structure deformation analysis validated with laser tracker

This paper focuses on the adjustment of B-spline approximation to terrestrial laser scanning (TLS) data and its contribution to deformation analysis of composite arched structures. The deformation of an arch structure under static loading conditions are investigated with TLS technology which is accurate, time-efficient, and capable to obtain dense 3D coordinates of point clouds for the measured objects.Highly accurate approximation methods for the point clouds data are extremely important for structural deformation analysis. The innovation of this paper is that parameters of B-spline is optimized with laser tracker (LT) to improve the accuracy of deformation monitoring and analysis, where the result is justified with LT data and the uncertainties are analyzed. It is revealed that optimized B-spline approximation agrees better with LT result. The improvement to the B-spline curve with minimum standard deviation reaches 52%, and its improvement to polynomial approximation reaches 77%.

Fault tolerant shape control for particulate process systems under simultaneous actuator and sensor faults

The problem of fault tolerant shape control for particle size distribution (PSD) process subject to simultaneous time-varying actuator and sensor faults is studied. In this work, sensor and actuator faults are taken into consideration in a unified framework, and estimated by using adaptive observer technique. For the PSD process, the output distribution, rather than the system output signal itself, is adopted for shape control. Square root rational B-spline approximation is used to fit the distribution shape. An innovative on-line fault estimation scheme is proposed for simultaneous actuator and sensor faults estimation. Then an augmented controller based on the virtual actuator and virtual sensor techniques is designed to compensate for the faults and realise PSD shape tracking. A simulation example of emulsion polymerisation is used to validate the proposed scheme and satisfactory performance is obtained.

Cubic B-spline solution for two-point boundary value problem with AOR iterative method

In this study, the cubic B-spline approximation equation has been derived by using the cubic B-spline discretization scheme to solve two-point boundary value problems. In addition to that, system of cubic B-spline approximation equations is generated from this spline approximation equation in order to get the numerical solutions. To do this, the Accelerated Over Relaxation (AOR) iterative method has been used to solve the generated linear system. For the purpose of comparison, the GS iterative method is designated as a control method to compare between SOR and AOR iterative methods. There are two examples of proposed problems that have been considered to examine the efficiency of these proposed iterative methods via three parameters such as their number of iterations, computational time and maximum absolute error. The numerical results are obtained from these iterative methods, it can be concluded that the AOR iterative method is slightly efficient as compared with SOR iterative method.

Shape retrieval through normalized B-splines curves

This paper proposes a new technique for 2D shape modeling and retrieval based only on curves defined from shape boundary. Firstly, a shape representation system is build based on the decomposition of the outline into its constituent parts and their geometric description. The process decomposition is done using high curvature points located along the boundary. These obtained parts are then described by parametric curves using the B-spline approximation and normalized in order to eliminate scaling transformation. Finally, the resulting curves allow matching of shapes and retrieving that is robust to rotation, scale change and deformation. Experiments conducted on a variety of shape databases including Kimia-99, Kimia-216, MPEG-7 and our database created from a selection of ETH-80 shape database, illustrate the performance of the proposed approach when compared with existing algorithms in literature. Obtained results are presented and discussed.

B-Spline Approximations of the Gaussian, their Gabor Frame Properties, and Approximately Dual Frames

We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters $$a,b>0$$ with $$ab<1$$ , the scaled version of $$B_N$$ generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to “almost perfect reconstruction” within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the $$L^p$$ -spaces, as well in the time-domain as in the frequency domain.

Trajectory Planning and Tracking Control of a Differential-Drive Mobile Robot in a Picture Drawing Application

This paper proposes a method for trajectory planning and control of a mobile robot for application in picture drawing from images. The robot is an accurate differential drive mobile robot platform controlled by a field-programmable-gate-array (FPGA) controller. By not locating the tip of the pen at the middle between two wheels, we are able to construct an omnidirectional mobile platform, thus implementing a simple and effective trajectory control method. The reference trajectories are generated based on line simplification and B-spline approximation of digitized input curves obtained from Canny’s edge-detection algorithm on a gray image. Experimental results for image picture drawing show the advantage of this proposed method.

B-spline curve fitting with invasive weed optimization

• The performance of IWO has been observed on the discrete optimization problem. • IWO algorithm is adapted to the problem of B-spline curve fitting. • Seven different case study have been handled in order to deal with the problem. • It has been proposed to prevent IWO algorithm from tackling to the local minima. • The number of offspring is determined according to the error rates of their parents. B-spline curves and surfaces are generally used in computer aided design (CAD), data visualization, virtual reality, surface modeling and many other fields. Especially, data fitting with B-splines is a challenging problem in reverse engineering. In addition to this, B-splines are the most preferred approximating curve because they are very flexible and have powerful mathematical properties and, can represent a large variety of shapes efficiently [1] . The selection of the knots in B-spline approximation has an important and considerable effect on the behavior of the final approximation. Recently, in literature, there has been a considerable attention paid to employing algorithms inspired by natural processes or events to solve optimization problems such as genetic algorithms, simulated annealing, ant colony optimization and particle swarm optimization. Invasive weed optimization (IWO) is a novel optimization method inspired from ecological events and is a phenomenon used in agriculture. In this paper, optimal knots are selected for B-spline curve fitting through invasive weed optimization method. Test functions which are selected from the literature are used to measure performance. Results are compared with other approaches used in B-spline curve fitting such as Lasso, particle swarm optimization, the improved clustering algorithm, genetic algorithms and artificial immune system. The experimental results illustrate that results from IWO are generally better than results from other methods.

Fault detection and diagnosis for delay-range-dependent stochastic systems using output PDFs

This paper investigates a new fault detection and diagnosis(FDD) scheme for delay-range-dependent stochastic systems. Compared with classical FDD problem, the measurable information in this paper is supposed to be the output probability density function(PDF), rather than the output itself. By using the square root B-spline approximation technique, the dynamic weight model of the output PDFs is established and the considered problem is converted into a nonlinear FDD problem for stochastic systems with delays. The main objective of this paper is to construct a filter based residual generator such that the fault can be detected and estimated. The FDD criteria is provided on the basis of linear matrix inequalities(LMIs). Besides, to improve the FDD performance, the tuning parameters, slack variables as well as the free-weighting matrices are applied to optimize the FDD criteria. Finally, the simulations are given to demonstrate the effectiveness of the proposed method.

Path smoothing for five-axis machine tools using dual quaternion approximation with dominant points

Nowadays, G01 blocks are still the most widely utilized representation form for the tool paths in the five-axis machine tools. Tangency and curvature discontinuities of the linear tool paths may lead to frequent feedrate fluctuation and acceleration variation, which inevitably deteriorate the machining efficiency and quality. To solve this problem, this paper proposed a dual quaternion B-spline approximation method of dominant points to generate smooth tool paths for five-axis CNC machine tools. First, the dominant points, which can characterize the shape of the linear path, are selected for the tool tip location and the tool orientation simultaneously. Such scheme can significantly compress the number of the control points and release the computational load. Then, the linear tool paths are converted to dual quaternions and approximated to dual quaternion B-spline, which can realize the parameter synchronization for the tool tip and tool orientation, and avoid the fitting oscillatory as well. To verify the proposed method, a trajectory generation method for the dual quaternion B-spline is developed. Simulations and experiments on a five-axis machine tool are conducted, and the results demonstrate the feasibility and efficiency of the proposed method.

Meshless Local B-Spline Collocation Method for Two-Dimensional Heat Conduction Problems With Nonhomogenous and Time-Dependent Heat Sources

This paper presents a new approach of meshless local B-spline based finite difference (FD) method for transient 2D heat conduction problems with nonhomogenous and time-dependent heat sources. In this method, any governing equations are discretized by B-spline approximation which is implemented as a generalized FD technique using local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighboring nodal values based on B-spline interpolants. The set of neighboring nodes is allowed to be randomly distributed. This allows enhanced flexibility to be obtained in the simulation. The method is truly meshless as no mesh connectivity is required for field variable approximation or integration. Galerkin implicit scheme is employed for time integration. Several transient 2D heat conduction problems with nonuniform heat sources in arbitrary complex geometries are examined to show the efficacy of the method. Comparison of the simulation results with solutions from other numerical methods in the literature is given. Good agreement with reference numerical methods is obtained. The method is shown to be simple and accurate for the time-dependent problems.

A time varying filter approach for empirical mode decomposition

A modified version of empirical mode decomposition (EMD) is presented to solve the mode mixing problem. The sifting process is completed using a time varying filter technique. In this paper, the local cut-off frequency is adaptively designed by fully facilitating the instantaneous amplitude and frequency information. Then nonuniform B-spline approximation is adopted as a time varying filter. In order to solve the intermittence problem, a cut-off frequency realignment algorithm is also introduced. Aimed at improving the performance under low sampling rates, a bandwidth criterion for intrinsic mode function (IMF) is proposed. The proposed method is fully adaptive and suitable for the analysis of linear and non-stationary signals. Compared with EMD, the proposed method is able to improve the frequency separation performance, as well as the stability under low sampling rates. Besides, the proposed method is robust against noise interference.