Analysis of first‐order partial differential equations via shifted Chebyshev polynomials

Abstract

The method of Chebyshev polynomials is introduced to represent approximate solutions of first‐order partial differential equations consisting of two independent variables. A set of linear algebraic equations is obtained by using the properties of Chebyshev polynomials and Kronecker product to analyse first‐order partial differential equations. The coefficient vector of Chebyshev polynomials of the first‐order partial differential equations can be obtained directly from Kronecker product formulas, which are suitable for computer computation. A numerical example for a set of first‐order partial differential equations is solved by a Chebyshev polynomials approximation and the results are satisfactory.     

A pseudospectral tau approximation for time delay systems and its comparison with other weighted‐residual‐type methods

This paper presents a method which applies pseudospectral tau approximation for retarded functional differential equations (RFDEs). The goal is to construct a system of ordinary differential equations, which provides a finite dimensional approximation of the original RFDE. The method can be used to determine approximate stability diagrams for RFDEs. Thorough numerical case studies show that the rightmost characteristic roots of the ordinary differential equation approximation converge to the rightmost characteristic roots of the original RFDE. Application of the method to time‐periodic RFDEs is also demonstrated, and the convergence of the stability boundaries is verified numerically. The method is compared with recently developed highly efficient numerical methods: the pseudospectral collocation (also called Chebyshev spectral continuous‐time approximation), the spectral Legendre tau method, and the spectral element method. The comparison is based on the stability analysis of three linear autonomous RFDEs. The efficiency of the methods is measured by the convergence rate of stability boundaries in the space of system parameters, by the convergence rate of the rightmost characteristic exponent and by the computation time of the stability charts. Copyright © 2016 John Wiley & Sons, Ltd.     

Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

This study proposes a new uncertain analysis method for multibody dynamics of mechanical systems based on Chebyshev inclusion functions The interval model accounts for the uncertainties in multibody mechanical systems comprising uncertain‐but‐bounded parameters, which only requires lower and upper bounds of uncertain parameters, without having to know probability distributions. A Chebyshev inclusion function based on the truncated Chebyshev series, rather than the Taylor inclusion function, is proposed to achieve sharper and tighter bounds for meaningful solutions of interval functions, to effectively handle the overestimation caused by the wrapping effect, intrinsic to interval computations. The Mehler integral is used to evaluate the coefficients of Chebyshev polynomials in the numerical implementation. The multibody dynamics of mechanical systems are governed by index‐3 differential algebraic equations (DAEs), including a combination of differential equations and algebraic equations, responsible for the dynamics of the system subject to certain constraints. The proposed interval method with Chebyshev inclusion functions is applied to solve the DAEs in association with appropriate numerical solvers. This study employs HHT‐I3 as the numerical solver to transform the DAEs into a series of nonlinear algebraic equations at each integration time step, which are solved further by using the Newton–Raphson iterative method at the current time step. Two typical multibody dynamic systems with interval parameters, the slider crank and double pendulum mechanisms, are employed to demonstrate the effectiveness of the proposed methodology. The results show that the proposed methodology can supply sufficient numerical accuracy with a reasonable computational cost and is able to effectively handle the wrapping effect, as cosine functions are incorporated to sharpen the range of non‐monotonic interval functions. Copyright © 2013 John Wiley & Sons, Ltd.     

An essentially non‐oscillatory semi‐Lagrangian method for tidal flow simulations

We develop an essentially non‐oscillatory semi‐Lagrangian method for solving two‐dimensional tidal flows. The governing equations are derived from the incompressible Navier–Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The method employs the modified method of characteristics to discretize the convective term in a finite element framework. Limiters are incorporated in the method to reconstruct an essentially non‐oscillatory algorithm at minor additional cost. The central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized. The resulting semi‐discretized system is then solved by an explicit Runge–Kutta Chebyshev scheme with extended stages. This scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. The proposed method is verified for the recirculation tidal flow in a channel with forward‐facing step. We also apply the method for simulation of tidal flows in the Strait of Gibraltar. In both test problems, the proposed method demonstrates its ability to handle the interaction between water free‐surface and bed frictions. Copyright © 2009 John Wiley & Sons, Ltd.     

A polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non‐smooth boundaries

The aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1D and 2D weakly singular and singular integral equations. The introduction of a smoothing transformation, besides smoothing the solution, allows also the use of a very simple and efficient collocation method based on Chebyshev polynomials of the first kind and their zeros. Further, we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements, which are given by weakly singular, singular or nearly singular integrals. Several numerical tests are given to point out the efficiency of the numerical approach we propose. Copyright © 2003 John Wiley & Sons, Ltd.     

Displacement discontinuity formulation for modeling cracks in orthotropic shear deformable plates

A displacement discontinuity formulation is presented for modeling cracks in orthotropic Reisnner plates. Fundamental solutions for displacement discontinuity are derived for the first time using a Fourier transform method. Boundary integral equations are presented in terms of discontinuity rotations on the crack surfaces for opening mode problems. As the fundamental solutions have singularity of O (1/r ²), Chebyshev polynomials of the second kind are used to evaluate the integral equations. By solving for coefficients of the Chebyshev polynomials, the stress intensity factors at the crack tips are obtained directly. Comparisons are made with solutions using the finite element method to demonstrate that the displacement discontinuity method is an efficient and accurate method for solving crack problems in orthotropic Reissner plates.     

基于Magnus-Gaussian 截断的铣削系统稳定性的半离散分析法

机床铣削系统的动力学可以通过一微分差分方程组来描述。对该动力学模型进行稳定性分析,可以确定稳定的铣削参数域及稳定性图。半离散法作为一种有效的半解析稳定性分析法,在对铣削系统动力学模型进行零阶半离散化时的误差级数较大,往往需要进行多步长的运算才能获得比较精确的稳定性分析结果。本文通过引入Magnus-Gaussian 截断法,推导出了基于Magnus-Gaussian 截断的零阶半离散稳定性分析法。用该方法对铣床切削稳定性进行了分析,并获得了各种工况下的铣床稳定性图。结果表明：该方法比原来的零阶半离散稳定性分析法能够更快的收敛于稳定解,从而节省了稳定性分析的运算时间。     

Three‐dimensional magnetostatic problem

The paper is devoted to an approximation of the solution of Maxwell's equations in three‐dimensional space. We present two methods which couple a finite element method inside the magnetic materials with a boundary integral method which uses Poincaré–Steklov's operator to describe the exterior domain. A computer code has been implemented for each method and a number of numerical experiments have been performed to validate each proposed methodology. Namely, we present numerical results concerning a non‐linear magnetostatic problem in ℝ³. Copyright © 1999 John Wiley & Sons, Ltd.     

Ausgleichsrechnung mit Tschebyscheffpolynomen

A method is presented approximating a data set with additional conditions in an interval using Chebyshev polynomials. An appropriate segmentation of the interval allows in each interval a Chebyshev approximation of low degree and altogether less Chebyshev coefficients may be needed than in the case of a unique approximation of high degree for the whole interval. At the boundaries of the inner segments we demand continuity up to the second derivative as well as the fulfilment of given values at the border intervals. We use the least square method and introduce Lagrange factors to determine the Chebyshev coefficients. The result is a system of linear algebraic equations which has to be solved numerically.     

Bifurcation analysis of nonlinear time‐periodic time‐delay systems via semidiscretization

Bifurcations of the periodic stationary solutions of nonlinear time‐periodic time‐delay dynamical systems are analyzed. The solution operator of the governing nonlinear delay‐differential equation is approximated by a sequence of nonlinear maps via semidiscretization. The subsequent nonlinear maps are combined to a single resultant nonlinear map that describes the evolution over the time period. Fold, flip, and Neimark‐Sacker bifurcations related to the fixed point of this map are analyzed via center manifold reduction and normal form theorems. The analysis unfolds the approximate stability properties and bifurcations of the stationary solution of the delay‐differential equation and, at the same time, allows the approximate computation of the arising period‐1, period‐2, and quasi‐periodic solution branches. The method is demonstrated for the delayed Mathieu‐Duffing equation, and the results are verified by numerical continuation.     

Modification of the boundary integral equation for the Navier–Lamé equations in 2D elastoplastic problems and its numerical solving

The paper presents the generalization of the modification of classical boundary integral equation and obtaining parametric integral equation system for 2D elastoplastic problems. The modification was made to obtain such equations for which numerical solving does not require application of finite or boundary elements. This was achieved through the use of curves and surfaces for modeling introduced at the stage of analytical modification of the classic boundary integral equation. For approximation of plastic strains the Lagrange polynomials with various number and arrangement of interpolation nodes were used. Reliability of the modification was verified on examples with analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

A new particular solution strategy for hyperbolic boundary value problems using hybrid‐Trefftz displacement elements

A new indirect approach to the problem of approximating the particular solution of non‐homogeneous hyperbolic boundary value problems is presented. Unlike the dual reciprocity method, which constructs approximate particular solutions using radial basis functions, polynomials or trigonometric functions, the method reported here uses the homogeneous solutions of the problem obtained by discarding all time‐derivative terms from the governing equation. Nevertheless, what typifies the present approach from a conceptual standpoint is the option of not using these trial functions exclusively for the approximation of the particular solution but to fully integrate them with the (Trefftz‐compliant) homogeneous solution basis. The particular solution trial basis is capable of significantly improving the Trefftz solution even when the original equation is genuinely homogeneous, an advantage that is lost if the basis is used exclusively for the recovery of the source terms. Similarly, a sufficiently refined Trefftz‐compliant basis is able to compensate for possible weaknesses of the particular solution approximation. The method is implemented using the displacement model of the hybrid‐Trefftz finite element method. The functions used in the particular solution basis reduce most terms of the matrix of coefficients to boundary integral expressions and preserve the Hermitian, sparse and localized structure of the solving system that typifies hybrid‐Trefftz formulations. Even when domain integrals are present, they are generally easy to handle, because the integrand presents no singularity. Copyright © 2015 John Wiley & Sons, Ltd.     

Approximation of multivariate functions and evaluation of particular solutions using Chebyshev polynomial and trigonometric basis functions

A two‐stage numerical procedure using Chebyshev polynomials and trigonometric functions is proposed to approximate the source term of a given partial differential equation. The purpose of such numerical schemes is crucial for the evaluation of particular solutions of a large class of partial differential equations. Our proposed scheme provides a highly efficient and accurate approximation of multivariate functions and particular solution of certain partial differential equations simultaneously. Numerical results on the approximation of eight two‐dimensional test functions and their derivatives are given. To demonstrate that the scheme for the approximation of functions can be easily extended to evaluate the particular solution of certain partial differential equations, we solve a modified Helmholtz equation. Near machine precision can be achieved for all these test problems. Copyright © 2006 John Wiley & Sons, Ltd.     

Two‐dimensional finite element method solution of a class of integro‐differential equations: Application to non‐Fickian transport in disordered media

A finite element method is developed to solve a class of integro‐differential equations and demonstrated for the important specific problem of non‐Fickian contaminant transport in disordered porous media. This transient transport equation, derived from a continuous time random walk approach, includes a memory function. An integral element is the incorporation of the well‐known sum‐of‐exponential approximation of the kernel function, which allows a simple recurrence relation rather than storage of the entire history. A two‐dimensional linear element is implemented, including a streamline upwind Petrov–Galerkin weighting scheme. The developed solver is compared with an analytical solution in the Laplace domain, transformed numerically to the time domain, followed by a concise convergence assessment. The analysis shows the power and potential of the method developed here. Copyright © 2017 John Wiley & Sons, Ltd.     

Boundary element‐free method (BEFM) and its application to two‐dimensional elasticity problems

In this study, we first discuss the moving least‐square approximation (MLS) method. In some cases, the MLS may form an ill‐conditioned system of equations so that the solution cannot be correctly obtained. Hence, in this paper, we propose an improved moving least‐square approximation (IMLS) method. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill‐conditioned system of equations. Combining the boundary integral equation (BIE) method and the IMLS approximation method, a direct meshless BIE method, the boundary element‐free method (BEFM), for two‐dimensional elasticity is presented. Compared to other meshless BIE methods, BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied easily; hence, it has higher computational precision. For demonstration purpose, selected numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd.     

Compression of time‐generated matrices in two‐dimensional time‐domain elastodynamic BEM analysis

This paper describes a new scheme to improve the efficiency of time‐domain BEM algorithms. The discussion is focused on the two‐dimensional elastodynamic formulation, however, the ideas presented apply equally to any step‐by‐step convolution based algorithm whose kernels decay with time increase. The algorithm presented interpolates the time‐domain matrices generated along the time‐stepping process, for time‐steps sufficiently far from the current time. Two interpolation procedures are considered here (a large number of alternative approaches is possible): Chebyshev–Lagrange polynomials and linear. A criterion to indicate the discrete time at which interpolation should start is proposed. Two numerical examples and conclusions are presented at the end of the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

A fast boundary cloud method for 3D exterior electrostatic analysis

An accelerated boundary cloud method (BCM) for boundary‐only analysis of 3D electrostatic problems is presented here. BCM uses scattered points unlike the classical boundary element method (BEM) which uses boundary elements to discretize the surface of the conductors. BCM combines the weighted least‐squares approach for the construction of approximation functions with a boundary integral formulation for the governing equations. A linear base interpolating polynomial that can vary from cloud to cloud is employed. The boundary integrals are computed by using a cell structure and different schemes have been used to evaluate the weakly singular and non‐singular integrals. A singular value decomposition (SVD) based acceleration technique is employed to solve the dense linear system of equations arising in BCM. The performance of BCM is compared with BEM for several 3D examples. Copyright © 2004 John Wiley & Sons, Ltd.     

Precise integration methods based on Lagrange piecewise interpolation polynomials

This paper introduces two new types of precise integration methods for dynamic response analysis of structures, namely, the integral formula method and the homogenized initial system method. The applied loading vectors in the two algorithms are simulated by the Lagrange piecewise interpolation polynomials based on the zeros of the first Chebyshev polynomial. Developed on the basis of the integral formula and the Lagrange piecewise interpolation polynomial and combined with the recurrence relationship of some key parameters in the integral computation suggested in this paper with the solving process of linear algebraic equations, the integral formula method has been set up. On the basis of the Lagrange piecewise interpolation polynomial, and transforming the non‐homogenous initial system into the homogeneous dynamic system, the homogenized initial system method without dimensional expanding is presented; this homogenized initial system method avoids the matrix inversion operation and is a general homogenized high‐precision direct integration scheme. The accuracy of the presented time integration schemes is studied and is compared with those of other commonly used schemes; the presented time integration schemes have arbitrary order of accuracy, wider application and are less time consuming. Two numerical examples are also presented to demonstrate the applicability of these new methods. Copyright © 2008 John Wiley & Sons, Ltd.     

A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials

The accuracy of metamodelling is determined by both the sampling and approximation. This article proposes a new sampling method based on the zeros of Chebyshev polynomials to capture the sampling information effectively. First, the zeros of one-dimensional Chebyshev polynomials are applied to construct Chebyshev tensor product (CTP) sampling, and the CTP is then used to construct high-order multi-dimensional metamodels using the ‘hypercube’ polynomials. Secondly, the CTP sampling is further enhanced to develop Chebyshev collocation method (CCM) sampling, to construct the ‘simplex’ polynomials. The samples of CCM are randomly and directly chosen from the CTP samples. Two widely studied sampling methods, namely the Smolyak sparse grid and Hammersley, are used to demonstrate the effectiveness of the proposed sampling method. Several numerical examples are utilized to validate the approximation accuracy of the proposed metamodel under different dimensions.