具有确定运动姿势的柔性体的动力学分析研究

讨论了具有确定运动姿态的柔性多体系统的非线性动力学控制方程． 将飞行器在空间的运动看作是已知的,分析了飞行器上的挠性构件对飞行器运动和姿态的影响,利用假设模态,将挠性构件的变形,看作是空间直角坐标轴方向的线元振动所构成的,根据动力学中的Kane方法,建立了动力学方程,方程中包含表示弹性变形的结构刚度矩阵及表示变形体非线性变形几何刚度矩阵,方程推导从应力-应变关系入手,使用了有限元法．经简化,得到了带帆板结构的平面挠性体对飞行器运动影响的动力学方程,这种方程可通过计算机实现其数值解．     

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.     

Wavelet Collocation Methods for a First Kind Boundary Integral Equation in Acoustic Scattering

In this paper we consider a wavelet algorithm for the piecewise constant collocation method applied to the boundary element solution of a first kind integral equation arising in acoustic scattering. The conventional stiffness matrix is transformed into the corresponding matrix with respect to wavelet bases, and it is approximated by a compressed matrix. Finally, the stiffness matrix is multiplied by diagonal preconditioners such that the resulting matrix of the system of linear equations is well conditioned and sparse. Using this matrix, the boundary integral equation can be solved effectively.     

A fast collocation method for acoustic scattering in shallow oceans

In this paper, we investigate the numerical solution of the integral equation of the second kind reduced by acoustic scattering in shallow oceans with Dirichlet condition. Based on analyzing the singularity of the truncating kernel with a sum of infinite series, using our trigonometric interpolatory wavelets and collocation method, we obtain the numerical solution which possesses a fast convergence rate like o(2^−j). Moreover, the entries of the stiffness matrix can be obtained by FFT, which lead the computational complexity to decrease obviously.     

Three-Dimensional Green's Function for a Layered Isotropic Half-space

A numerical approach to calculate the Green's function for a layered half space is presented. It is based on the precise integration method (PIM), which is an efficient and accurate numerical method for the solution of one order ordinary differential equations. In the numerical implementation, the layered half space is divided into numerous mini-layers; and the dual vector form of the wave motion equation is introduced to combine two adjacent mini-layers/layers. The advantages of the proposed algorithm are: (a) it overcomes the exponent overflow generally encountered with employing the transfer matrix method; (b) it avoids solving the intractable transcendental functions in the stiffness matrix method and the huge matrix calculation in the thin layer method; (c) it imposes no limit to the thickness of layered strata and ensures convergence at high-frequency range. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dual approaches to finite element model updating

The problem of finding the least change adjustment to a stiffness matrix modeled by finite element method is considered in this paper. Desired stiffness matrix properties such as symmetry, sparsity, positive semidefiniteness, and satisfaction of the characteristic equation are imposed as side constraints of the constructed optimal matrix approximation for updating the stiffness matrix, which matches measured data better. The dual problems of the original constrained minimization are presented and solved by subgradient algorithms with different line search strategies. Some numerical results are included to illustrate the performance and application of the proposed methods.     

A fast numerical method for harmonic equation based on natural boundary integral

Based on the coupling of the natural boundary integral method and the finite elements method, we mainly investigate the numerical solution of Neumann problem of harmonic equation in an exterior elliptic. Using our trigonometric wavelets and Galerkin method, there obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. On the other hand, we prove that the numerical solution possesses exponential convergence rate. Especially, examples state that our method still has good accuracy for small j when the solution u ₀(θ) is almost singular.     

含高次逆幂的矩阵方程对称解的双迭代算法

本文研究了在控制理论和随机滤波等领域中遇到的一类含高次逆幂的矩阵方程的等价矩阵方程对称解的数值计算问题.采用牛顿算法求等价矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立了求这类矩阵方程对称解的双迭代算法,数值算例验证了双迭代算法是有效的.     

Solvability of the fractional Volterra‐Fredholm integro differential equation by hybridizable discontinuous Galerkin method

This article proves the existence and uniqueness of the solution obtained by the hybridizable discontinuous Galerkin (HDG) method of the fractional Volterra‐Fredholm integro differential equation. The method based on local solvers and transmission condition is applied to the equation using two auxiliary variables. The form of the equation is amenable for achieving the solvability criteria of the problem according to the HDG method. We also calculate a numerical solution of the problem whose exact solution is taken as a smooth or fractional function. This results in a tridiagonal, symmetric, and positive definite stiffness matrix.     

Compatible wavelet-Galerkin method to solve stokes interior problem with circular boundary

This article addresses Neumann boundary value interior problem of Stokes equations with circular boundary. By using natural boundary element method, the Stokes interior problem is reduced into an equivalent natural integral equation with a hyper-singular kernel, which is viewed as Hadamard finite part. Based on trigonometric wavelet functions, the compatible wavelet space is constructed so that it can serve as Galerkin trial function space. In proposed compatible wavelet-Galerkin method, the simple and accurate computational formulae of the entries in stiffness matrix are obtained by singularity removal technique. It is also proved that the stiffness matrix is almost a block diagonal matrix, and its diagonal sub-blocks all are both symmetric and circulant submatrices. These good properties indicate that a 2 ^J+3 × 2 ^J+3 stiffness matrix can be determined only by its 2 ^J + 3J + 1 entries. It greatly decreases the computational complexity. Some error estimates for the compatible wavelet-Galerkin projection solutions are established. Finally, several numerical examples are given to demonstrate the validity of the proposed approach.     

Wavelet Solution of Plane Elasticity Problem in the Upper Half-plane

The plane elasticity problem includes plane strain problem and plane stress problem which are widely applied in mechanics and engineering. In this article, we first reduce the plane elasticity problem in the upper half-plane into natural boundary integral equation and then apply wavelet-Galerkin method to deal with the numerical solution of the natural boundary integral equation. The test and trial functions used are the scaling basis functions of Shannon wavelet. In our fast algorithm, the computational formulae of entries of the stiffness matrix yield simple close-form and only 3 K entries need to be computed for one 4 K ‐ 4 K stiffness matrix.     

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong‐form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second‐order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well‐known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher‐order derivatives using the approximation of lower‐order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second‐ and higher‐orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031–1053, 2015     

A wavelets method for plane elasticity problem

A Neumann boundary value problem of plane elasticity problem in the exterior circular domain is reduced into an equivalent natural boundary integral equation and a Poisson integral formula with the DtN method. Using the trigonometric wavelets and Galerkin method, we obtain a fast numerical method for the natural boundary integral equation which has an unique solution in the quotient space. We decompose the stiffness matrix in our numerical method into four circulant and symmetrical or antisymmetrical submatrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform (FFT) and the inverse fast Fourier transform (IFFT) instead of the inverse matrix. Examples are given for demonstrating our method has good accuracy of our method even though the exact solution is almost singular.     

Numerical approach for solving fractional relaxation–oscillation equation

In this study, we will obtain the approximate solutions of relaxation–oscillation equation by developing the Taylor matrix method. A relaxation oscillator is a kind of oscillator based on a behavior of physical system’s return to equilibrium after being disturbed. The relaxation–oscillation equation is the primary equation of relaxation and oscillation processes. The relaxation–oscillation equation is a fractional differential equation with initial conditions. For this propose, generalized Taylor matrix method is introduced. This method is based on first taking the truncated fractional Taylor expansions of the functions in the relaxation–oscillation equation and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown fractional Taylor coefficients can be found approximately. The reliability and efficiency of the proposed approach are demonstrated in the numerical examples with aid of symbolic algebra program, Maple.     

矩阵方程AXB+CYD=E对称最小范数最小二乘解的极小残差法

<正>1引言本文用R~(n×m)表示全体n×m实矩阵集合,用SR~(n×n)表示全体n×n实对称矩阵集合,OR~(n×n)表示全体n×n实正交矩阵集合.用I_n表示n阶单位矩阵,用A*B表示矩阵A与B的Hadamard乘积.对任意矩阵A,B∈R~(n×m),定义内积〈A,B〉=tr(B~T A),其中     

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

非线性动力系统的自适应显式Magnus数值方法

基于最近发展的矩阵李群上非线性微分方程的显式Magnus展式,给出了非线性动力系统的有效的数值算法,并且在数值求解过程中具有自适应的步长控制特点,可以显著地提高计算效率．最后,通过非线性动力系统典型问题Duffing方程和强刚性的Van derPol方程以及非线性振子的Hamilton方程的数值实验来说明方法的有效性．     

结构刚度函数识别的一个途径

为了计算结构的刚度函数,将结构振动微分方程分解为关于已知的原始刚度函数的微分方程和关于未知待求的刚度函数的第一类Fredholm积分方程,利用p个光滑因子进行外插值的求解方法,数值计算当光滑因子为零时的积分方程的稳定解．从而可得到结构的刚度函数．通过数值模拟说明方法是可行的．     

A MPI PARALLEL PRECONDITIONED SPECTRAL ELEMENT METHOD FOR THE HELMHOLTZ EQUATION

Spectral element method is well known as high-order method, and has potential better parallel feature as compared with low order methods. In this paper, a parallel preconditioned conjugate gradient iterative method is proposed to solving the spectral element approximation of the Helmholtz equation. The parallel algorithm is shown to have good performance as compared to non parallel cases, especially when the stiffness matrix is not memorized. A series of numerical experiments in one dimensional case is carried out to demonstrate the efficiency of the proposed method.     

Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW

Fractional calculus is an extension of derivatives and integrals to non-integer orders and has been widely used to model scientific and engineering problems. In this paper, we describe the fractional derivative in the Caputo sense and give the second kind Chebyshev wavelet (SCW) operational matrix of fractional integration. Then based on above results we propose the SCW operational matrix method to solve a kind of nonlinear fractional-order Volterra integro-differential equations. The main characteristic of this approach is that it reduces the integro-differential equations into a nonlinear system of algebraic equations. Thus, it can simplify the problem of fractional order equation solving. The obtained numerical results indicate that the proposed method is efficient and accurate for this kind equations.