Numerical Differentiation Formulas Research Articles

Приложение биоинспирированных алгоритмов глобальной оптимизации в задаче подбора коэффициентов модели усталостной деградации жесткости композиционного материала

<p>The problem of finding the fatigue characteristics of a material based on the test results of a composite material is considered. The initial data are the properties of the material, load parameters and the tabulated dependence of the elastic modulus on the number of tests performed. A differential mathematical model has been formed that describes the change in the elastic modulus with increasing number of loading cycles. Its parameters are found by approximating the rate of change of the elastic modulus using numerical differentiation formulas of various orders and solving the parametric identification problem. The model coefficients are determined by using a method that simulates the behavior of a swarm of moths, related to bio–inspired global optimization algorithms. A solution to the problem for a specific composite material is given.</p>

Application of the Fourier Modulation Analysis Method to the Problem of Derivatives Recovery

Abstract. In this work, formulas are obtained for finding higher derivatives of a function (dependence), expressed through its Fourier coefficients (harmonic amplitudes). These formulas were found by differentiating both a function with a harmonically modulated argument and its Fourier series. The expressions given make it possible to find higher derivatives of functions numerically with any accuracy by means of different methods. For example, numerical integration of the Euler-Fourier formulas may be used for harmonic amplitudes. Derivatives also can be found experimentally when studying a nonlinear physical process by digitally recording the harmonic amplitudes of the dependence under study under simultaneous static and harmonic influences. The problem of reconstructing derivatives from Fourier coefficients is posed and an analysis of its correctness is performed. Formulas for estimating recovery errors are defined and recommendations are given to reduce these errors. Examples are given with different smoothness of analytical functions and of functions used to explain experiments: 1) an analytical function used to explain the magnetic properties of superconductors, whose Fourier coefficients are determined numerically with the error of the computational algorithm in the Mathcad software environment; 2) current-voltage characteristic (CVC) of 2 back-to-back semiconductor diodes, whose harmonic amplitudes were determined experimentally with a given measurement error of the device. The resulting derivative of the current-voltage characteristic is compared with the derivative obtained using numerical differentiation formulas; 3) an analytical function whose derivative has a discontinuity of the first kind. Errors in measuring Fourier coefficients were added artificially using a random number generator.

Active Kriging-based conjugate first-order reliability method for highly efficient structural reliability analysis using resample strategy

Efficient structural reliability analysis method is crucial to solving reliability analysis of complex structural problems. High-computational cost and low-failure probability problems greatly limit the efficiency in structural reliability analysis problems, causing the safety and reliability of the structure to be questioned. In this work, a highly efficient structural reliability analysis method coupling active Kriging algorithm with conjugate first order reliability method (AK-CFORM) is proposed. Specifically, the resample strategy is considered to reduce the number of samples evaluated in each active learning process; the uniform sampling is used to better balance global and local optimal problems; the conjugate map is used to improve the robustness of analytical first order reliability method; and the approximate numerical differential formula is proposed to solve the problems of non-convergence when solving the gradient of the Kriging surrogate model. Finally, three numerical cases and four engineering cases are used to illustrate the effectiveness and robustness of the proposed method. The results show that the proposed AK-CFORM has greater advantages in the number of calling system response and surrogate model with robust and accurate performance.

The barycentric rational numerical differentiation formulas for stiff ODEs and DAEs

The barycentric rational numerical differentiation formulas for stiff ODEs and DAEs

The construction of an optimal fourth-order fractional-compact-type numerical differential formula of the Riesz derivative and its application

In this paper, a novel optimal fourth-order fractional-compact-type numerical differential formula for the Riesz derivatives is derived by constructing the appropriate generating function. Meanwhile, some interesting and important properties of the proposed formula are also presented and proved. Then, we take the nonlinear space fractional Ginzburg–Landau equations as an example and establish a high-order finite difference scheme by combining the compact integral factor method with Padé approximation. Furthermore, based on the discrete energy analysis method, the proposed difference scheme is proved to be uniquely solvable and convergent with order O(τ2+h4) under different norms, where τ and h denote the temporal and spatial mesh step sizes, respectively. Finally, some numerical results are provided to illustrate the accuracy of numerical differential formula and the effectiveness of numerical method, and further demonstrate the correctness of theoretical analysis.

Formulas for Numerical Differentiation of Functions with Large Gradients1

Formulas for Numerical Differentiation of Functions with Large Gradients1

Numerical analysis of the high-order scheme of the damped nonlinear space fraction Schrödinger equation

Firstly, we construct a fourth-order numerical differential formula for approximating Riesz derivative. Substituting it into the damped nonlinear space fractional Schrödinger equation, the original equation becomes a matrix form of nonlinear ordinary differential equation system. Then, by means of implicit integrating factor method and Padé approximation, we obtain a new numerical scheme whose convergence order is higher than the existing algorithms. Finally, a numerical example is given to verify the effectiveness of the numerical algorithm and the correctness of the theoretical analysis.

High-order numerical algorithm and error analysis for the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation

In this paper, we first construct an appropriate new generating function, and then based on this function, we establish a fourth-order numerical differential formula approximating the Riesz derivative with order γ∈(1,2]. Subsequently, we apply the formula to numerically study the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation and obtain a difference scheme with convergence order Oτ2+hx4+hy4, where τ denotes the time step size, hx and hy denote the space step sizes, respectively. Furthermore, with the help of some newly derived discrete fractional Sobolev embedding inequalities, the unique solvability, the unconditional stability, and the convergence of the constructed numerical algorithm under different norms are proved by using the discrete energy method. Finally, some numerical results are presented to confirm the correctness of the theoretical results and verify the effectiveness of the proposed scheme.

Improved stencil selection for meshless finite difference methods in 3D

We introduce a geometric stencil selection algorithm for Laplacian in 3D that significantly improves octant-based selection considered earlier. The goal of the algorithm is to choose a small subset from a set of irregular points surrounding a given point that admits an accurate numerical differentiation formula. The subset serves as an influence set for the numerical approximation of the Laplacian in meshless finite difference methods using either polynomial or kernel-based techniques. Numerical experiments demonstrate a competitive performance of this method in comparison to the finite element method and to other selection methods for solving the Dirichlet problems for the Poisson equation on several STL models. Discretization nodes for these domains are obtained either by 3D triangulations or from Cartesian grids or Halton quasi-random sequences.

Formulas for numerical differentiation of functions with large gradients

Formulas for numerical differentiation of functions with large gradients

Structure preserving fourth-order difference scheme for the nonlinear spatial fractional Schrödinger equation in two dimensions

In this paper, we focus on develop high-order and structure-preserving numerical algorithm for the two-dimensional nonlinear space fractional Schrödinger equations. By constructing a new generating function, we obtain a fourth-order numerical differential formula and use it to approximate the spatial Riesz derivative, while the Crank–Nicolson method is applied for the time derivative. Based on the energy method, the conservation, solvability and convergence of the numerical algorithm are proved. Finally, some numerical examples are used to verify the correctness of the theoretical analysis and the validity of the numerical algorithm.

Nonlenear Integro-differential Equations and Splines of the Fifth Order of Approximation

In this paper, we consider the solution of nonlinear Volterra–Fredholm integro-differential equation, which contains the first derivative of the function. Our method transforms the nonlinear Volterra-Fredholm integro-differential equations into a system of nonlinear algebraic equations. The method based on the application of the local polynomial splines of the fifth order of approximation is proposed. Theorems about the errors of the approximation of a function and its first derivative by these splines are given. With the help of the proposed splines, the function and the derivative are replaced by the corresponding approximation. Note that at the beginning, in the middle and at the end of the interval of the definition of the integro-differential equation, the corresponding types of splines are used: the left, the right or the middle splines of the fifth order of approximation. When using the spline approximations, we also obtain the corresponding formulas for numerical differentiation. which we also apply for the solution of integro-differential equations. The formulas for approximation of the function and its derivative are presented. The results of the numerical solution of several integro-differential equations are presented. The proposed method is shown that it can be applied to solve integro-differential equations containing the second derivative of the solution.

A high-efficient strain-stress method for calculating higher-order elastic constants from first-principles

Though the method for calculating higher-order elastic constants (HOECs) have a long history, there still exist some unsolved issues (e.g. low efficiency), making the development of HOECs much slower than second-order elastic constants. In this paper, we present a general and efficient strain-stress method (SSM) for calculating HOECs from first-principles. In this method, the required number of strain modes is sharply cut down, which ensures a higher efficiency. The time spent on traditional methods is about 3-5 times that of SSM for calculating TOECs of diamond. By taking the HOECs into consideration, the convergence against maximum strain in SSM gets improved significantly, and the results of diamond, gold and magnesium obtained in SSM agree well with previous calculations by other methods. To accelerate the development of the calculation tools for HOECs, we present an algorithm, as well as an open source code, to deduce the strain modes and corresponding coefficients. Specially, we give an explicit expression of strain modes and corresponding coefficients for calculating TOECs in arbitrary symmetry and fifth-order elastic constants in CI (Laue group) symmetry. In addition, we make some extension, e.g. high-accurate numerical differentiation formula, of some existing methods.

Local Interpolation Splines and Solution of Integro-Differential Equations of Mechanic’s Problems

Integro-differential equations are encountered when solving various problems of mechanics. Although Integro-Differential equations are encountered frequently in mathematical analysis of mechanical problems, very few of these equations will ever give us analytic solutions in a closed form. So that construction of numerical methods is the only way to find the approximate solution. This paper discusses the calculation schemes for solving integro-differential equations using local polynomial spline approximations of the Lagrangian type of the fourth and fifth orders of approximation. The features of solving integro-differential equations with the first derivative and the Fredholm and Volterra integrals of the second kind are discussed. Using the proposed spline approximations, formulas for numerical differentiation are obtained. These formulas are used to approximate the first derivative of a function. The numerical experiments are presented.

High order compact difference scheme for solving the time multi-term fractional sub-diffusion equations

<abstract><p>In this paper, a high order compact finite difference is established for the time multi-term fractional sub-diffusion equation. The derived numerical differential formula can achieve second order accuracy in time and four order accuracy in space. A unconditionally stable and convergent difference scheme is presented, and a rigorous proof for the stability and convergence is given. Numerical results demonstrate the efficiency of the proposed difference schemes.</p></abstract>

Numerical differentiation on scattered data through multivariate polynomial interpolation

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

Non-Polynomial Interpolation of Functions with Large Gradients and Its Application

Interpolation of a function of one variable with large gradients in the boundary layer region is studied. The problem is that the use of classical polynomial interpolation formulas on a uniform mesh to functions with large gradients can lead to errors of the order of $$O(1)$$ , despite a small mesh size. An interpolation formula based on fitting to the component that defines the boundary-layer growth of the function is investigated. An error estimate, which depends on the number of interpolation nodes and is uniform over the boundary layer component and its derivatives, is obtained. It is shown how the interpolation formula derived can be used to construct formulas for numerical differentiation and integration and in the two-dimensional case. The corresponding error estimates are obtained.

A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability

In the present work, first of all, a new numerical fractional differentiation formula (called the CF2 formula) to approximate the Caputo-Fabrizio fractional derivative of order $alpha,$ $(0<alpha<1)$ is developed. It is established by means of the quadratic interpolation approximation using three points $ (t_{j-2},y(t_{j-2}))$, $(t_{j-1},y(t_{j-1})) $ and $ (t_{j},y(t_{j})) $ on each interval $[t_{j-1},t_{j}]$ for $ ( j geq 2 )$, while the linear interpolation approximation is applied on the first interval $[t_{0},t_{1}]$. As a result, the new formula can be formally viewed as a modification of the classical CF1 formula, which is obtained by the piecewise linear approximation for $y(t)$. Both the computational efficiency and numerical accuracy of the new formula is superior to that of the CF1 formula. The coefficients and truncation errors of this formula are discussed in detail. {Two test example show} the numerical accuracy of the CF2 formula. The CF1 formula demonstrates that the new CF2 is much more effective and more accurate than the CF1 when solving fractional differential equations. Detailed stability analysis and region stability of the CF2 are also carefully investigated.

Optimizing the Coefficients of Numerical Differentiation Formulae Using Neural Networks

The use of a numerical differentiation formula (NDF) is an excellent method for solving stiff ordinary differential equations. However, the NDF method cannot fully adapt to all stiff systems. An optimal general method for optimizing NDF coefficients using a back-propagation neural network is proposed in this work that can be used for different systems of stiff equations. The ranges of stability of the first- to fourth-order coefficients are obtained by analyzing the definition of stability. In order to solve the different stiff systems by changing the NDF coefficients, the relationship between the eigenvalues of the Jacobian matrix and NDF coefficients is analyzed. The back-propagation neural network is used to describe the relationship between them and predict the optimal parameters of different stiff systems. Compared with the NDF coefficients, the absolute and mean square errors of the numerical solutions are smaller. The simulation results show that the numerical solution’s accuracy is higher. Because optimizing the NDF coefficients improves the accuracy of the simulation results, the new approach is more suitable for solving stiff ordinary differential equations.

A New Projected Active Set Conjugate Gradient Approach for Taylor-Type Model Predictive Control: Application to Lower Limb Rehabilitation Robots With Passive and Active Rehabilitation.

In this paper, a three-order Taylor-type numerical differentiation formula is firstly utilized to linearize and discretize constrained conditions of model predictive control (MPC), which can be generalized from lower limb rehabilitation robots. Meanwhile, a new numerical approach that projected an active set conjugate gradient approach is proposed, analyzed, and investigated to solve MPC. This numerical approach not only incorporates both the active set and conjugate gradient approach but also utilizes a projective operator, which can guarantee that the equality constraints are always satisfied. Furthermore, rigorous proof of feasibility and global convergence also shows that the proposed approach can effectively solve MPC with equality and bound constraints. Finally, an echo state network (ESN) is established in simulations to realize intention recognition for human–machine interactive control and active rehabilitation training of lower-limb rehabilitation robots; simulation results are also reported and analyzed to substantiate that ESN can accurately identify motion intention, and the projected active set conjugate gradient approach is feasible and effective for lower-limb rehabilitation robot of MPC with passive and active rehabilitation training. This approach also ensures computational when disturbed by uncertainties in system.