Quintic Spline Research Articles

Modeling and Simulation of Vibrations of Non-Homogeneous Annular Plate of Quadratic Thickness Resting on Elastic Foundation

Mathematical modeling is presented to analyze natural frequencies of vibrations of an isotropic annular plate of quadratic varying thickness resting on Winkler type elastic foundation where numerical simulation is carried out using quintic spline technique for three different combinations of edge conditions. Effect of elastic foundation, together with nonhomogeneity variation, on the natural frequencies of vibration is illustrated for variety of thickness variation for the first three modes. To compare parametric effect on a specific plate, transverse displacements are presented in normalized form. Accuracy of the results and validity of numerical method is demonstrated by comparing the existing results in the literature.

Fast Approximation of Over-Determined Second-Order Linear Boundary Value Problems by Cubic and Quintic Spline Collocation

We present an efficient and generic algorithm for approximating second-order linear boundary value problems through spline collocation. In contrast to the majority of other approaches, our algorithm is designed for over-determined problems. These typically occur in control theory, where a system, e.g., a robot, should be transferred from a certain initial state to a desired target state while respecting characteristic system dynamics. Our method uses polynomials of maximum degree three/five as base functions and generates a cubic/quintic spline, which is C 2 / C 4 continuous and satisfies the underlying ordinary differential equation at user-defined collocation sites. Moreover, the approximation is forced to fulfill an over-determined set of two-point boundary conditions, which are specified by the given control problem. The algorithm is suitable for time-critical applications, where accuracy only plays a secondary role. For consistent boundary conditions, we experimentally validate convergence towards the analytic solution, while for inconsistent boundary conditions our algorithm is still able to find a “reasonable” approximation. However, to avoid divergence, collocation sites have to be appropriately chosen. The proposed scheme is evaluated experimentally through comparison with the analytical solution of a simple test system. Furthermore, a fully documented C++ implementation with unit tests as example applications is provided.

Smoothing and Differentiation of Kinematic Data Using Functional Data Analysis Approach: An Application of Automatic and Subjective Methods

Smoothing is one of the fundamental procedures in functional data analysis (FDA). The smoothing parameter λ influences data smoothness and fitting, which is governed by selecting automatic methods, namely, cross-validation (CV) and generalized cross-validation (GCV) or subjective assessment. However, previous biomechanics research has only applied subjective assessment in choosing optimal λ without using any automatic methods beforehand. None of that research demonstrated how the subjective assessment was made. Thus, the goal of this research was to apply the FDA method to smoothing and differentiating kinematic data, specifically right hip flexion/extension (F/E) angle during the American kettlebell swing (AKS) and determine the optimal λ . CV and GCV were applied prior to the subjective assessment with various values of λ together with cubic and quintic spline (B-spline) bases using the FDA approach. The selection of optimal λ was based on smoothed and well-fitted first and second derivatives. The chosen optimal λ was 1 × 10 − 12 with a quintic spline (B-spline) basis and penalized fourth-order derivative. Quintic spline is a better smoothing and differentiation method compared to cubic spline, as it does not produce zero acceleration at endpoints. CV and GCV did not give optimal λ , forcing subjective assessment to be employed instead.

Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines

In this paper, a hybrid numerical technique is developed to solve the Benjamin-Bona-Mahony-Burgers (BBMB) equation. This technique involves the application of quintic Hermite collocation method (QHCM) and weighted finite difference scheme. The discretization of BBMB equation is done using collocation method with Hermite splines of order five along spatial direction and using weighted finite difference scheme along temporal direction. The nonlinear equation is discretized without using the Hopf-Cole transformation. Stability of the proposed technique using Von-Neumann method is checked. Convergence analysis of the technique is also done in detail. Validity of the technique is shown by solving four test examples with different boundary and initial conditions. The calculated results are found to be in good agreement with the exact solutions and are better than the results from other techniques given in the literature.

Numerical simulation of entropy generation due to natural convection heat transfer using Kernel Derivative-Free (KDF) Incompressible Smoothed Particle Hydrodynamics (ISPH) model

This paper develops and applies a Kernel Derivative-Free (KDF) Incompressible Smoothed Particle Hydrodynamics (ISPH) model for analysis of entropy generation and heat transfer in fluid-structure coupling problems. A modified high order Laplacian operator is applied for the treatment of pressure-velocity coupling (Poisson's equation), while an explicit third-order TVD Runge-Kutta scheme is used for time integration of the momentum, energy and displacement equations. To improve the consistency and stability of the model, a new particle regularization technique based on the particle shifting is also introduced for simulating free-surface flows. The developed KDF-ISPH model is validated and evaluated for a series of challenging benchmark cases, including, dam break, stretching water drop, rotating square patch of fluid, and natural convection in square cavity. Accuracy and applicability of the method are further validated by analyzing entropy generation due to the natural convection heat transfer in three well-known geometries including: square cavity with hot obstacle inside, C-shaped enclosure, and square enclosure containing a pair of hot and cold horizontal pipes (heat exchanger). The results are found to be in good agreement with available numerical and experimental data. The accuracy of the developed KDF-ISPH with new Laplacian operator, for use in prediction of fluid flow and heat transfer characteristics is also proven. Finally, by combining the cosine and signal functions, a new high order smoothing kernel is constructed. The evaluation of this new kernel for the propagation of shock wave in 1D tube demonstrates better global stability and consistency properties compared to two frequently used SPH kernels (i.e. cubic and quintic spline functions).

Automatic segment filtering procedure for processing non-stationary signals

Computing time derivatives is a frequent stage in the processing of biomechanical data. Unfortunately, differentiation amplifies the high frequency noise inherent within the signal hampering the accuracy of signal derivatives. A low-pass Butterworth filter is commonly used to reduce the sampled signal noise prior to differentiation. One hurdle lies in selecting an appropriate filter cut-off frequency which retains the signal of interest while reducing deleterious noise. Most biomechanics data processing approaches utilize the same cut-off frequency for the whole sampled signal, but the frequency components of a signal can vary with time. To accommodate such signals, the Automatic Segment Filtering Procedure (ASFP) is proposed which uses different automatically determined Butterworth filter cut-off frequencies for separate segments of a sampled signal. The Teager-Kaiser Energy Operator of the signal is computed and used to determine segments of the signal with different energy content. The Autocorrelation-Based Procedure (ABP) is used on each of these segments to determine filter cut-off frequencies. This new procedure was evaluated by estimating acceleration values from the test data set of Dowling (1985). The ASFP produced a root mean square error (RMSE) of 16.4 rad s−2 (26.6%) whereas a single ABP determined filter cut-off frequency applied to the whole Dowling (1985) signal, representing the common approach, produced a RMSE of 25.5 rad s−2 (41.4%). As a point of comparison, a Generalized Cross-Validated Quintic Spline, a common non-Butterworth filter, produced a RMSE of 23.6 rad s−2 (38.4%). This new automatic approach is advantageous in biomechanics for preserving high frequency content of non-stationary signals.

Simulation of nonlinear Cahn-Hilliard equation based on local refinement pure meshless method

The phase separation phenomenon between different matters plays an important role in many science fields. And the high order nonlinear Cahn-Hilliard (C-H) equation is often used to describe the phase separation phenomenon between different matters. However, it is difficult to solve the high-order nonlinear C-H equations by the theorical methods and the grid-based methods. Therefore, in this work the meshless methods are addressed, and a local refinement finite pointset method (LR-FPM) is proposed to numerically investigate the high-order nonlinear C-H equations with different boundary conditions. Its constructive process is as follows. 1) The fourth derivative is decomposed into two second derivatives, and then the spatial derivative is discretized by FPM based on the Taylor series expansion and weighted least square method. 2) The local refinement and quintic spline kernel function are employed to improve the numerical accuracy. 3) The Neumann boundary condition with high-order derivatives is accurately imposed when solving the local linear equation sets. The 1D/2D C-H equations with different boundary conditions are first solved to show the ability of the LR-FPM, and the analytical solutions are available for comparison. Meanwhile, we also investigate the numerical error and convergence order of LR-FPM with uniform/non-uniform particle distribution and local refinement. Finally, 1D/2D C-H equation without analytical solution is predicted by using LR-FPM, and compared with the FDM result. The numerical results show that the implement of the boundary value condition is accurate, the LR-FPM indeed has a higher numerical accuracy and convergence order, is more flexible and applicable than the grid-based FDM, and can accurately predict the time evolution of nonlinear diffusive phase separation phenomenon between different materials time.

A fourth order non-polynomial quintic spline collocation technique for solving time fractional superdiffusion equations

The purpose of this article is to present a technique for the numerical solution of Caputo time fractional superdiffusion equation. The central difference approximation is used to discretize the time derivative, while non-polynomial quintic spline is employed as an interpolating function in the spatial direction. The proposed method is shown to be unconditionally stable and O(h^{4}+Delta t^{2}) accurate. In order to check the feasibility of the proposed technique, some test examples have been considered and the simulation results are compared with those available in the existing literature.

Fractal Quintic Spline Solutions for Fourth-Order Boundary-Value Problems

A new numerical method is developed for approximating the solution of the fourth-order boundary-value problems with the help fractal quintic spline. The proposed method has second-order convergence. Numerical examples are experimented for the numerical illustration of the proposed method. It is shown that the method developed in this paper is more efficient than the method developed by the quintic spline.

A Special Quintic Spline for (0,1,4) Lacunary Interpolation and Cauchy Initial Value Problem

A Special Quintic Spline for (0,1,4) Lacunary Interpolation and Cauchy Initial Value Problem

Composite non-polynomial spline solution of boundary value problems in plate deflection theory

This article presents the numerical solution of a system of linear fourth-order boundary value problems using a different amalgamation of non-polynomial splines. A novel approach was developed using quintic spline function together with exponential and trigonometric functions. Our method is convergent and second-order accurate. Numerical examples show that the method congregates with sufficient accuracy to the exact solutions. Our methodology has the advantages over some existing quintic spline method, direct method, and finite difference method.

Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations

This paper presents a novel approach to numerical solution of a class of fourth-order time fractional partial differential equations (PDEs). The finite difference formulation has been used for temporal discretization, whereas the space discretization is achieved by means of non-polynomial quintic spline method. The proposed algorithm is proved to be stable and convergent. In order to corroborate this work, some test problems have been considered, and the computational outcomes are compared with those found in the exiting literature. It is revealed that the presented scheme is more accurate as compared to current variants on the topic.

Quintic Non-polynomial Spline Method for Solving the Time Fractional Biharmonic Equation

Quintic Non-polynomial Spline Method for Solving the Time Fractional Biharmonic Equation

Numerical solutions of nonlinear fractional Wu–Zhang system for water surface versus three approximate schemes

This paper examines the effects of three distinct numerical schemes (Adomian Decomposition, quintic & septic Spline methods) to investigate semi-analytical and approximate solutions on Wu–Zhang (ZW) system. It describes the (1+1)-dimensional dispersive long wave in two horizontal directions on shallow waters. The ZW model is one of the fractional nonlinear partial differential equations. Conformable derivatives properties are employed to convert the nonlinear fractional partial differential equation into an ordinary differential equation with integer order so as to obtain the approximate solutions for this model. The solutions obtained for each technique were compared to reveal their relationship to their characteristics illustrated under the suitable choice of the parameters values. The obtained solutions showed the power, easiness, and effectiveness of these methods on nonlinear partial differential equations.

2-D Modeling of Dual-Gate MOSFET Devices Using Quintic Splines

This paper is the first known composition to demonstrate the application of quintic splines in the development of a dual-gate MOSFET model based on the well-known drift-diffusion system of differential equations used in semiconductor analysis. The techniques and methodologies presented advance the current state of the art in this particular area by implementing improved 2-D numerical techniques and avoiding the use of common accuracy reducing assumptions to decrease semiconductor equation complexity and execution time. This results in a model that is valid in all regions of operation, includes Non-Quasi static transient behavior, accounts for short channel effects, and allows researchers to predict the real world manufacturing effects of gate oxide thickness imperfections and applied gate voltage disparities. Model validation was accomplished by comparing simulation results to previously published experimental data for long and short channel devices as well as data generated using an established commercial product. In addition, a subsequent paper submission is planned to demonstrate the application of this modeling method in determining frequency response for simple CMOS circuits.

Quintic trigonometric spline based numerical scheme for nonlinear modified Burgers' equation

This paper presents a numerical method based on quintic trigonometric B‐splines for solving modified Burgers' equation (MBE). Here, the MBE is first discretized in time by Crank–Nicolson scheme and the resulting scheme is solved by quintic trigonometric B‐splines. The proposed method tackles nonlinearity by using a linearization process known as quasilinearization. A rigorous analysis of the stability and convergence of the proposed method are carried out, which proves that the method is unconditionally stable and has order of convergence O(h4 + k2). Numerical results presented are very much in accordance with the exact solution, which is established by the negligible values of L2 and L∞ errors. Computational efficiency of the scheme is proved by small values of CPU time. The method furnishes results better than those obtained by using most of the existing methods for solving MBE.

An exploration of quintic Hermite splines to solve Burgers\u2019 equation

An innovative scheme of collocation having quintic Hermite splines as base functions has been followed to solve Burgers’ equation. The scheme relies on approximation of Burgers’ equation directly in non-linear form without using Hopf–Cole transformation (Hopf in Commun Pure Appl Math 3:201–216, 1950; Cole in Q Appl Math 9:225–236, 1951). The significance of the numerical technique is demonstrated by comparing the numerical results to the exact solution and published results (Asaithambi in Appl Math Comput 216:2700–2708, 2010; Mittal and Jain in Appl Math Comput 218:7839–7855, 2012). Five problems with different initial conditions have been examined to validate the efficiency and accuracy of the scheme. Euclidean and supremum norms have been reckoned to scrutinize the stability of the numerical scheme. Results have been demonstrated in plane and surface plots to indicate the effectiveness of the scheme.

Non-polynomial quintic spline for solving fourth-order fractional boundary value problems involving product terms

In this article, we have explored the numerical solution of fourth order fractional boundary value problems, involving product terms, by means of quintic spline collocation method. The proposed numerical approach is based on non-polynomial quintic spline functions comprised of a trigonometric part and polynomial part. The second and fourth order convergence of the presented algorithm has been discussed rigorously. Some test examples have been considered and the approximate results are found to be more accurate as compared to the other variants on the topic.

A spline-based computational technique applicable for solution of boundary value problem arising in human physiology

Non-polynomial quintic spline functions based algorithms are used for computing an approximation to the nonlinear two point second order singular boundary value problems arising in human physiology. After removing the singularity by L' hospital rule, the resulting boundary value problem is then efficiently treated by employing non-polynomial quintic spline for finding the numerical solution. Two examples have been included and comparison of the numerical results made with cubic extended B-spline method and finite difference method.

A spline-based computational technique applicable for solution of boundary value problem arising in human physiology

Non-polynomial quintic spline functions based algorithms are used for computing an approximation to the nonlinear two point second order singular boundary value problems arising in human physiology. After removing the singularity by L' hospital rule, the resulting boundary value problem is then efficiently treated by employing non-polynomial quintic spline for finding the numerical solution. Two examples have been included and comparison of the numerical results made with cubic extended B-spline method and finite difference method.