Quintic Hermite Research Articles

Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique.

Stability Analysis of Hermite Collocation Method for Pulp Washing Models

Pulp washing is concerned with detaching cellulose fibres from black liquor with the use of a minimal amount of wash liquor. An efficient numerical technique of hermite collocation method is used for the solution of mathematical models related to pulp washing. The linear and non-linear models are solved using Quintic Hermite collocation method with Dirichlet’s and mixed Robin’s boundary conditions. Numerical solution of the models are derived using MATLAB ode15s. This study deals with the justification of accuracy of the method with stability analysis. The present method is more convenient, simple and elegant for solving the two point boundary value problems and the results found are very much stable from numerical point of view

A Robust Quintic Hermite Collocation Method for One-Dimensional Heat Conduction Equation

A Robust Quintic Hermite Collocation Method for One-Dimensional Heat Conduction Equation

Travelling wave solution of fourth order reaction diffusion equation using hybrid quintic hermite splines collocation technique

Fourth order extended Fisher Kolmogorov reaction diffusion equation has been solved numerically using a hybrid technique. The temporal direction has been discretized using Crank Nicolson technique. The space direction has been split into second order equation using twice continuously differentiable function. The space splitting results into a system of equations with linear heat equation and non linear reaction diffusion equation. Quintic Hermite interpolating polynomials have been implemented to discretize the space direction which gives a system of collocation equations to be solved numerically. The hybrid technique ensures the fourth order convergence in space and second order in time direction. Unconditional stability has been obtained by plotting the eigen values of the matrix of iterations. Travelling wave behaviour of dependent variable has been obtained and the computed numerical values are shown by surfaces and curves for analyzing the behaviour of the numerical solution in both space and time directions.

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

<abstract><p>Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.</p></abstract>

Improvement on the axial force accuracy of the ANCF Euler-Bernoulli beam element with the second-order approximate function of the centerline and precise constraint equations

The conventional Euler-Bernoulli beam element of the absolute nodal coordinate formulation (ANCF) yields incorrect axial force due to the existence of the first-order approximate function of the centerline and over-constraint equations. This work aims to improve the axial force accuracy of the element by using the second-order centerline approximate function and precise constraint equations. The second-order function includes the quintic Hermite shape function and the vector of nodal coordinates containing zeroth to second-order derivatives of position. The elastic force vector of the element and its tangent stiffness were then formulated based on the decoupled elastic line method. The exact kinematic requirements of three types of constraint points were fully clarified. On this basis, the corresponding concise constraint equations, constraint Jacobian and Hessian matrices were rigorously derived in the context of ANCF, where G2 continuity was demonstrated to be satisfied at the node acted by an inclined concentrated force, the relative rotation constraint of the rigid joint was described using the dot product of the gradients on both sides of the joint, and the fixing gradient direction at the partially-clamped end was expressed by using the zero cross product of the end gradients in the initial and deformed configurations. Large deformation problems of typical beams under fixed and follower loads were solved using five types of elements, where the results of the ABAQUS B22 elements were taken as a reference for validation. It is found that when using the first- and second-order beam elements having over-constraint equations of the three types of constraint points, obvious axial force error occurs near the corresponding constraint points. In comparison, a few second-order elements containing the derived constraint equations output correct axial force consistent with ABAQUS. However, a large number of the first-order elements containing the correct constraint equations still yield inaccurate axial force in certain beam regions.

Wave propagation in periodic nano structures through second strain gradient elasticity

The study of wave propagation in the periodic nano-waveguides is of importance to advance the development of energy and wave controlling systems. In this paper, firstly, the element discretization of nano structures is introduced based on the second strain gradient theory. The displacement field is determined through the utilization of six quintic Hermite polynomial interpolating functions. By applying Hamilton’s principle, the weak form, which incorporates the element matrices, is determined and the global dynamic stiffness matrix of a unit cell is assembled. Then, the wave finite element method, based on the inverse form, direct form and contour integral solution, is used to analyze the two-dimensional free wave propagation properties of complex nano structures by solving the eigenvalue problems. The interpretation of the frequency spectrum is confined to the irreducible first Brillouin zone, encompassing both the dispersion relation and band structure. Furthermore, the slowness surfaces and energy flow vector fields are discussed via the second strain gradient theory. The novel work, which utilizes the second strain gradient theory equipped with the wave finite element method to investigate two-dimensional complex nano structures, has revealed significant potential in elucidating the two-dimensional wave propagation characteristics of the complex nano-waveguides.

Super convergence analysis of fully discrete Hermite splines to simulate wave behaviour of Kuramoto–Sivashinsky equation

An improved collocation technique has been proposed to discretize the fourth-order multi-parameter non-linear Kuramoto -Sivashinsky (K-S) equation. The spatial direction has been discretized with quintic Hermite splines, whereas the temporal direction has been discretized with a weighted finite difference scheme. The fourth-order equation in space direction has been decomposed into second-order using space splitting by introducing a new variable. Space splitting has been proposed to improve the convergence of the approximate solution. The proposed equation has been analyzed on a uniform grid in both space and time directions. Error bounds for general order Hermite splines are established for fully discrete scheme. Stability analysis for the proposed scheme has also been discussed elaborately. Periodic and non periodic problems of K-S equation type have been discussed to study the technique. The error growth has been addressed by computing L2− norm and L∞− norm.

Mathematical modeling and simulation of the inverse kinematic of a redundant robotic manipulator using azimuthal angles and spherical polar piecewise interpolation

In this study, smooth trajectory generation and the mathematical modeling of the inverse kinematics and dynamics of a redundant robotic manipulator is considered. The path of the end-effector of the robotic manipulator is generated – along a particular set of via-points – in a system of spherical coordinates by combining the Hermite polar piecewise interpolants that approximate each intermediate radial distance on the reference plane using the azimuthal angle and each corresponding point height of the path computed in the rotating radial plane through the corresponding polar angle. The smoothness of the end-effector path is guaranteed by the use of quintic Hermite piecewise interpolants having functional continuity, while the inverse kinematics and inverse dynamics used to compute the coordinates of the joints and the equations of motion of the robotic manipulator are considered for optimum trajectory planning. Optimization solutions to minimize the joint velocities, the end-effector positioning error, the traveling time and mechanical energy, or the consumed power, are presented. The direct computation of the joints trajectory and a optimal trajectory with minimization of the end-effector position error while preserving trajectory smoothness of the joints are considered. To assess and verify the approach numerical examples – implemented in Matlab – are introduced, and the outcomes are examined.

Multi-mode propagation and diffusion analysis using the three-dimensional second strain gradient elasticity

The multi-mode propagation and diffusion properties are crucial informations when studying complex waveguides. In this paper, firstly, the three-dimensional modeling of micro-sized structures is introduced by using the second strain gradient theory. The constitutive relation is deduced while the six quintic Hermite polynomial shape functions are employed for the displacement field. The weak formulations including element stiffness and mass matrices and the force vector are calculated through the Hamilton’s principle and the global dynamic stiffness matrix of a unit cell is assembled. Then, free wave propagation characteristics are analyzed by solving eigenvalue problems within the direct wave finite element method framework. The dispersion relations of positive going waves considering the size effects are illustrated. Furthermore, the effects of higher order parameters on the dispersion curves are discussed and the forced responses with two boundary conditions are expounded. Eventually, the wave diffusion including reflection and transmission coefficients are illustrated through simple and complex coupling conditions, respectively. The dynamic analysis of coupled waveguides through the wave finite element method equipped with the second strain gradient is a novel work. The results show that the proposed approach is of significant potential for investigating the wave propagation and diffusion characteristics of micro-sized structures.

A Review of Collocation Approximations to Solutions of Differential Equations

This review considers piecewise polynomial functions, that have long been known to be a useful and versatile tool in numerical analysis, for solving problems which have solutions with irregular features, such as steep gradients and oscillatory behaviour. Examples of piecewise polynomial functions used include splines, in particular B-splines, and Hermite functions. Spline functions are useful for obtaining global approximations whilst Hermite functions are useful for approximation over finite elements. Our aim in this review is to study quintic Hermite functions and develop a numerical collocation scheme for solving ODEs and PDEs. This choice of basis functions is further motivated by the fact that we are interested in solving problems having solutions with steep gradients and oscillatory properties, for which this approximation basis seems to be a suitable choice. We derive the quintic Hermite basis and use it to formulate the orthogonal collocation on finite element (OCFE) method. We present the error analysis for third order ODEs and derive both global and nodal error bounds to illustrate the super-convergence property at the nodes. Numerical simulations using the Julia programming language are performed for both ODEs and PDEs and enhance the theoretical results.

Numerical simulation of time variable fractional order mobile–immobile advection–dispersion model based on an efficient hybrid numerical method with stability and convergence analysis

In this paper, we propose a hybrid numerical method based on the weighted finite difference and the quintic Hermite collocation methods. The proposed method is used for solving the variable-order time fractional mobile–immobile advection–dispersion(VOMIM-AD) model, such that the discretization is done by applying collocation method with Hermite splines in the spatial direction and weighted finite difference method in the temporal direction. Provided examples confirm the studied stability and convergence properties of the proposed method. The obtained results from the graphical illustration and numerical simulations, in comparison with other methods in the literature, demonstrate that the reported method is very robust and accurate.

Flutter analysis of sandwich panel with the constrained viscoelastic layer considering the effects of the imperfection and the core thickness deformation

The supersonic flutter analysis of panels treated with the viscoelastic constrained layer damping (CLD) is performed with a special focus on the effects of the initial geometric imperfection and the core through-the-thickness deformation on the flutter boundaries. For kinematic modeling of the CLD core, a higher-order shear deformation theory is used where the through-the-thickness deformation is also taken into account. The core viscoelastic property is also described using the frequency-dependent complex-modulus approach and the first-order Piston theory is employed to determine the aerodynamic pressure. The discretized equations of motion are then derived using the finite element method (FEM). The integrations required in the FEM for obtaining each element stiffness matrix are also facilitated by approximating the imperfection function using the quintic Hermite interpolation function. Moreover, due to the large dimension of the final global matrices, the model is reduced using the oscillatory damped modes of the panel, which greatly reduces the computation time needed for flutter analysis. Numerical studies are performed for two different viscoelastic materials (VEMs), which shows that an efficient flutter suppression could not always be achieved with any type of VEMs. The thickness deformation effect is also found to be most prominent when the CLD length is about 80% of the panel length. Moreover, the mode crossing phenomenon that occurs by increasing the imperfection amplitude is found to significantly reduce the flutter boundaries.

A computationally efficient technique for the solution of pulp washing models

An efficient numerical technique for the solution of the pulp washing model is proposed in this study. Two linear and one nonlinear model are explained with quintic Hermite collocation method. In this technique, quintic Hermite polynomials (C2 continuous) are used as a basis function and orthogonal collocation method is applied within each element of the partitioned domain. For accuracy and applicability of the method, a comparison of the numerical results with analytic ones is made. The method is found to be stable using stability analysis and convergence criteria. The effect of Peclet number on exit solute concentration and other parameters is presented in the form of breakthrough curves. The results are derived for a broad range of parameters and the present method is found to be more useful and refined for solving the two-point boundary value problems.

Numerical investigation of variable‐order fractional Benjamin–Bona–Mahony–Burgers equation using a pseudo‐spectral method

This article introduces a new local variable‐order (VO) fractional differentiation called the VO conformable fractional derivative. This derivative is employed to define the VO time fractional Benjamin–Bona–Mahony–Burgers (BBMB) equation. A pseudo‐spectral algorithm using the Chebyshev cardinal functions (CCFs) is adopted to find a numerical solution for this equation. The presented method with the help of the CCFs derivative matrices (which are obtained in this research) turns problem solving into solving an algebraic system of equations. The proposed approach is applied on some test problems, and its accuracy is examined in terms ofL∞andL2error norms. A numerical comparison is performed between the achieved results of the method with the results obtained from the cubic B‐spline method and the hybrid method generated using the quintic Hermite approach and the finite difference technique.

Analysis of a linear and non-linear model for diffusion–dispersion phenomena of pulp washing by using quintic Hermite interpolation polynomials

Analysis of a linear and non-linear model for diffusion–dispersion phenomena of pulp washing by using quintic Hermite interpolation polynomials

Optimal Control of Robotic Systems Using Finite Elements for Time Integration of Covariant Control Equations

We used an optimal control method involving covariant control equations as optimality conditions, to command the actuators of robot manipulators. These form a coupled system of second order nonlinear ordinary differential equations when associated with the robot motion equations. By solving this system, the control action required to take the robot from an initial to a final state is optimized in a prescribed time. However, the target set of equations exhibited stiffness. Therefore, an adequate solution could only be found for short trajectory durations with readily available numerical methods. We examined a time discretization procedure based on cubic and quintic Hermite finite elements which exhibited superconvergence properties for interpolation. This motivated us to develop a time integration algorithm based on Hermite's technique, where motion and control equations were perturbed to solve the optimal control problem. The optimal motion of a robotic manipulator was simulated using this algorithm. Our method was compared with a commercial differential equations solver on the basis of specific indicators. It outperformed the commercial solver by effectively solving the stiff set of equations for longer trajectory durations, with the cubic elements performing better than the quintic ones in this sense. The convergence analysis of our method confirmed that the quintic elements are more precise at the cost of increased computational burden, but converge at a lower rate than expected. Controlled motion experiments on a robotic manipulator validated our methodology. Trajectories were smoothly tracked and results exposed further methodology improvements.

Real-time quintic Hermite interpolation for robot trajectory execution.

This paper presents a real-time joint trajectory interpolation system for the purpose of frequency scaling the low cycle time of a robot controller, allowing a Python application to real-time control the robot at a moderate cycle time. Interpolation is based on quintic Hermite piece-wise splines. The splines are calculated in real-time, in a piecewise manner between the high-level, long cycle time trajectory points, while sampling of these splines at an appropriate, shorter cycle time for the real-time requirement of the lower-level system. The principle is usable in general, and the specific implementation presented is for control of the Panda robot from Franka Emika. Tracking delay analysis is presented based on a cosine trajectory. A simple test application has been implemented, demonstrating real-time feeding of a pre-calculated trajectory for cutting with a knife. Estimated forces on the robot wrist are recorded during cutting and presented in the paper.

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.

A new selection scheme for spatial Pythagorean hodograph quintic Hermite interpolants

For given C1 Hermite data, there exists a two-parameter family of Pythagorean hodograph (PH) quintic curves which interpolate the data (two end-points and end-derivatives) as observed by Farouki et al. (2002b). As “good” candidate curves for a selection problem, we propose a special type of PH quintic interpolating curves called extremal interpolants and prove that the extremal interpolants preserve planarity, i.e., they are planar curves if the data are planar. Since there are only four distinct extremal interpolants, the selection problem, when only considering extremal interpolants as possible candidates, is reduced to picking one curve from finite candidates.Due to the preservation of planarity, extremal interpolants coincide with one of p0,0(t),p0,π(t),pπ,0(t),pπ,π(t) if the data are planar, where pϕ0,ϕ2(t) denotes the parametrization proposed by Šír and Jüttler (2005). However, any of the four extremal interpolants is generically not identical to the interpolants for non-planar data, and empirical results suggest that being compared with the unique cubic interpolant, the best curve is one of extremal interpolants among all extremal interpolants and p0,0(t),p0,π(t),pπ,0(t),pπ,π(t).