有限元线法求解非线性模型问题——Ⅱ.扭转杆的最优截面

作为有限元线法(FEMOL)求解非线性问题的系列工作之二,本文将该法应用于形状优化问题,对扭转杆的截面优化这一模型问题作了分析求解。文中首先对双连域截面的扭转问题作了FEMOL推导,然后允许结线的长度改变以描述不同的截面形状,再利用若干变换技巧将形状变量及优化条件引入常微分方程(ODE)体系中,从而将问题转换成标准的非线性ODE问题,并由ODE求解器进行求解。文中算例显示了本法对形状优化问题的求解具有方法简洁、实施方便、效率显著等优点。     

Analytical and Numerical Investigations of Unsteady Graphene Oxide Nanofluid Flow Between Two Parallel Plates

In this research, different analytical methods were applied to characterize thermal behavior of unsteady graphene oxide–water nanofluid flow between two parallel moving plates. First of all, partial differential equations (PDEs) were transformed to a system of nonlinear ordinary differential equations (ODEs) using similarity solution. Then, collocation method (CM), least square method (LSM) and Galerkin method (GM) were used to solve the system of ODEs and determine velocity and temperature distribution functions. In addition, effects of moving parameter, concentration, Eckert and Prandtl numbers on nanofluid velocity and temperature profiles were examined. Next, using numerical solution of the obtained system of differential equations, the results obtained from the analytical solutions were validated with that of the numerical solution. The validation results indicated high and appropriate accuracy of the analytical solutions compared to the numerical one.     

双轴受压反对称角铺设复合材料层合板在固支边界下的后屈曲和模态跃迁

为有效分析双轴受压反对称角铺设复合材料层压板在固支边界下的后屈曲性能, 由渐近修正几何非线性理论推导其双耦合四阶偏微分方程(即应变协调方程和稳定性控制方程), 通过双Fourier级数将耦合非线性控制偏微分方程转换为系列非线性常微分方程, 从而获得相对简单的求解方法。使用广义Galerkin方法求解与角交铺设复合层合板相关的边界值问题, 研究了模态跃迁前后不同复杂程度的后屈曲模式。对四层固支边界复合层合板的数值模拟结果表明: 该解析法与有限元方法在主后屈曲区域的线性屈曲荷载计算结果吻合良好; 有限元方法在解靠近二次分岔点时失去收敛性, 而解析方法可深入后屈曲区域, 准确捕捉模态跃迁现象; 对于反对称角铺设层合板, 可仅用纯对称模态来定性预测主后屈曲分支、二次分岔荷载及远程跃迁路径。      

动力刚度法求解平面曲梁面外自由振动问题

该文将动力刚度法应用于平面曲梁面外自由振动的分析。通过建立单元动力刚度所满足的常微分方程边值问题,用具有自适应求解功能的常微分方程求解器COLSYS 进行求解,获得单元动力刚度的数值精确解。以COLSYS 求解单元动力刚度的网格作为单元上固端频率计数求解的子网格,由单元动力刚度的边值问题解答线性组合出该子网格下各子单元的动力刚度,由Wittrick-Williams 算法获得单元固端频率的计数。从而实现整体结构的Wittrick-Williams频率计数。通过建立单元动力刚度对频率的导数所满足的常微分方程边值问题,调用COLSYS求其数值精确解,并将其引入导护型牛顿法,可迅速求得结构精确的频率和振型。数值算例表明,该文方法准确、可靠、有效。     

A precise numerical method for Rayleigh waves in a stratified half space

This paper presents theory and results for Rayleigh waves propagating in a transversely isotropic stratified solid resting on an elastic semi‐infinite space. It uses the precise integration method (PIM) and the extended Wittrick–Williams (W–W) algorithm. This problem can be reduced into the eigenvalue problem of ordinary differential equations (ODEs) in the frequency–wavenumber domain. The PIM used here is a precise method for solving the ODEs with two‐point boundary conditions, and the eigenvalue counting method of the extended W–W algorithm is used to ensure that all eigenvalues are found without the possibility of any being missed. Hence the theory presented in this paper is exact, in the sense that its precision is limited only by the precision of the computer used. Copyright © 2006 John Wiley & Sons, Ltd.     

ODE求解器求解高层双肢剪力墙结构稳定特征值问题

本文在用连续介质法推导出高层双肢剪力墙结构稳定特征方程的基础上，用常微分方程（简称：ＯＤＥ──ＯｒｄｉｎａｒｙＤｉｆｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎ）求解器研究该结构的稳定特征值问题。首先，将该特征值问题归结为标准的非线性ＯＤＥ边值问题；然后用ＯＤＥ求解器求解这一等价的非线性问题。得到的结果与加权余量法和有限差分法结果进行比较，吻合得很好，表明稳定特征值问题能够凭借ＯＤＥ求解器的功效得以精确、可靠、方便地求解。     

Improving the accuracy of DRBEM for convective partial differential equations

The dual reciprocity boundary element method (DRBEM) has been successfully applied to the solution of many examples of linear partial differential equations (PDEs). Our experience with the method shows that it encounters some convergence problems when dealing with convective equations. The main problem lies in the differentiation of a radial basis function in a bounded domain. Here following Zhu and Zhang [Zhu SP, Zhang YL. Improvement on dual reciprocity boundary element method for equations with convective terms. Comm Appl Numer Meth 1994;10:361–371], we have proposed and analysed a transformation technique for removing convective terms. We have shown that our transformation improves and leads to better convergence results with DRBEM than solving the original PDEs directly. Five numerical examples are illustrated.     

A Fictitious Time Integration Method for Solving Delay Ordinary Differential Equations

A new numerical method is proposed for solving the delay ordinary differential equations (DODEs) under multiple time-varying delays or state-dependent delays. The finite difference scheme is used to approximate the ODEs, which together with the initial conditions constitute a system of nonlinear algebraic equations (NAEs). Then, a Fictitious Time Integration Method (FTIM) is used to solve these NAEs. Numerical examples confirm that the present approach is highly accurate and efficient with a fast convergence.     

The Scalar Homotopy Method for Solving Non-Linear Obstacle Problem

In this study, the nonlinear obstacle problems, which are also known as the nonlinear free boundary problems, are analyzed by the scalar homotopy method (SHM) and the finite difference method. The one- and two-dimensional nonlinear obstacle problems, formulated as the nonlinear complementarity problems (NCPs), are discretized by the finite difference method and form a system of nonlinear algebraic equations (NAEs) with the aid of Fischer-Burmeister NCP-function. Additionally, the system of NAEs is solved by the SHM, which is globally convergent and can get rid of calculating the inverse of Jacobian matrix. In SHM, by introducing a scalar homotopy function and a fictitious time, the NAEs are transformed to the ordinary differential equations (ODEs), which can be integrated numerically to obtain the solutions of NAEs. Owing to the characteristic of global convergence in SHM, the restart algorithm is adopted to fasten the convergence of numerical integration for ODEs. Several numerical examples are provided to validate the efficiency and consistency of the proposed scheme. Besides, some factors, which might influence on the accuracy of the numerical results, are examined by a series of numerical experiments.     

Guided particle swarm optimization method to solve general nonlinear optimization problems

The development of hybrid algorithms is becoming an important topic in the global optimization research area. This article proposes a new technique in hybridizing the particle swarm optimization (PSO) algorithm and the Nelder–Mead (NM) simplex search algorithm to solve general nonlinear unconstrained optimization problems. Unlike traditional hybrid methods, the proposed method hybridizes the NM algorithm inside the PSO to improve the velocities and positions of the particles iteratively. The new hybridization considers the PSO algorithm and NM algorithm as one heuristic, not in a sequential or hierarchical manner. The NM algorithm is applied to improve the initial random solution of the PSO algorithm and iteratively in every step to improve the overall performance of the method. The performance of the proposed method was tested over 20 optimization test functions with varying dimensions. Comprehensive comparisons with other methods in the literature indicate that the proposed solution method is promising and competitive.     

A Galerkin-reproducing kernel method: Application to the 2D nonlinear coupled Burgers' equations

The paper introduces a Galerkin method in the reproducing kernel Hilbert space. It is implemented as a meshless method based on spatial trial spaces spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. For the time-dependent PDEs it leads to a system of ordinary differential equations. The method is used for solving the 2D nonlinear coupled Burgers' equations having Dirichlet and mixed boundary conditions. The numerical solutions for different values of Reynolds number (Re) are compared with analytical solutions as well as other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Re in the case of Dirichlet boundary conditions.     

悬索承重梁索耦合结构的垂向运动动力学模型及主共振分析

该文建立了描述结构大变形和主缆初始曲率产生的几何非线性对系统动力学影响的悬索承重梁索耦合结构垂向运动动力学偏微分方程组。通过Galerkin方法一次截断把偏微分方程组化为时域上的两自由度常微分方程组。使用多尺度法得到简谐激励下常微分方程组主共振时的一次近似解。结果显示,当外激励仅激发低频或高频主共振时,系统的振幅随激励的幅值或激励频率的变化出现突然的跳跃。当激励同时激发低频和高频主共振时则有两种情况:1) 若固定高频激励幅值和频率,则系统的低频和高频振动成分的振幅随低频激励参数变化同时增加或减小;2) 若固定低频激励的幅值和频率,则系统的低频和高频振动成分的振幅随高频激励参数变化以相反的趋势变化。即高频振动幅值增大时,低频振幅减小,反之亦然。     

Bending analysis of thick laminated plates using extended Kantorovich method

This study is concerned with bending of moderately thick rectangular laminated plates with clamped edges. The governing equations, based on Reissner first-order shear deformation plate theory; in terms of deflection and rotations of the plate include a system of three second-order, partial differential equations (PDEs). Application of extended Kantorovich method (EKM) to the system of partial differential equations reduces the governing equations to a double set of three second-order ordinary differential equations in the variables x and y. These sets of equations were then solved in an iterative manner until convergence was achieved. Normally three to four iterations are enough to get the final results with desired accuracy. It is demonstrated that, unlike other weighted residual methods, in the extended Kantorovich method initial guesses to start iterations are arbitrary and not even necessary to satisfy the boundary conditions. Results of this study also reveal that the convergence of the EKM is rapid and the method is an efficient way to solve system of PDEs of the same type. To compare the results of this study, the problem was also analyzed using commercial finite element software, ANSYS. Results show reasonably good agreement with the finite element analysis.     

Method of particular solutions for solving PDEs of the second and fourth orders with variable coefficients

The paper presents a new meshless numerical method for solving partial differential equations of the second and fourth orders with variable coefficients. The key idea of the method is the use of modified particular solutions which satisfy the homogeneous boundary conditions of the problem. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem. As a result we separate the approximation of the boundary conditions and the PDE inside the solution domain. Numerical experiments are carried out for accuracy and convergence investigations. A comparison of the numerical results obtained in the paper with the exact solutions or other numerical methods indicates that the proposed method is accurate in dealing with PDEs with variable coefficients.     

Homotopy method of fundamental solutions for solving certain nonlinear partial differential equations

In this study, the homotopy analysis method (HAM) is combined with the method of fundamental solutions (MFS) and the augmented polyharmonic spline (APS) to solve certain nonlinear partial differential equations (PDE). The method of fundamental solutions with high-order augmented polyharmonic spline (MFS–APS) is a very accurate meshless numerical method which is capable of solving inhomogeneous PDEs if the fundamental solution and the analytical particular solutions of the APS associated with the considered operator are known. In the solution procedure, the HAM is applied to convert the considered nonlinear PDEs into a hierarchy of linear inhomogeneous PDEs, which can be sequentially solved by the MFS–APS. In order to solve strongly nonlinear problems, two auxiliary parameters are introduced to ensure the convergence of the HAM. Therefore, the homotopy method of fundamental solutions can be applied to solve problems of strongly nonlinear PDEs, including even those whose governing equation and boundary conditions do not contain any linear terms. Therefore, it can greatly enlarge the application areas of the MFS. Several numerical experiments were carried out to validate the proposed method.     

Post-buckling analysis of viscoelastic plates with fractional derivative models

The post-buckling response of thin plates made of linear viscoelastic materials is investigated. The employed viscoelastic material is described with fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plate element, are three coupled nonlinear fractional partial evolution type differential equations in terms of three displacements. The nonlinearity is due to nonlinear kinematic relations based on the von Kármán assumption. The solution is achieved using the analog equation method (AEM), which transforms the original equations into three uncoupled linear equations, namely a linear plate (biharmonic) equation for the transverse deflection and two linear membrane (Poisson’s) equations for the inplane deformation under fictitious loads. The resulting initial value problem for the fictitious sources is a system of nonlinear fractional ordinary differential equations, which is solved using the numerical method developed recently by Katsikadelis for multi-term nonlinear fractional differential equations. The numerical examples not only demonstrate the efficiency and validate the accuracy of the solution procedure, but also give a better insight into this complicated but very interesting engineering plate problem     

A comparison study of the LMAPS method and the LDQ method for time-dependent problems

This paper compares numerical solutions of spatial-temporal partial differential equations based on two RBF-based meshless methods: the local method of approximate particular solutions (LMAPS) and the local RBFs-based DQ method (LDQ). To avoid the ill-conditioned problems of the global version, the weighting coefficients at the supporting points are determined by solving low-order linear systems instead of large dense linear systems. The Runge–Kutta method is adopted for time stepping schemes. The numerical experiments have shown that the LMPAS method and the LDQ method are capable of solving the initial boundary value problem for spatial-temporal partial differential equations with high accuracy and efficiency.     

Heat transfer during fluidized-bed coating

A fully implicit numerical method for linear parabolic free boundary problems with coupled and integral boundary conditions is described. The partial differential equation and the boundary conditions are time discretized with the method of lines. An auxiliary function is introduced to remove the coupled and integral boundary conditions from the resulting free boundary problem for ordinary differential equations. Once separated boundary conditions are obtained, invariant imbedding is used to solve the free boundary problem numerically. The method is illustrated by solving the heat transfer equations for the fluidized-bed coating of a thin-walled cylinder.     

一种新型磁跟踪模式及其L-M算法研究

为了提高磁跟踪系统的定位精度,设计了分时交流励磁跟踪方法,通过接收线圈检测目标的空间磁场,利用相关电磁场理论求解出目标方位.针对高次非线性定位方程组变量多、不单调、具有多个局部极值点的特点,将方程组求解问题转化成非线性无约束最小二乘问题,运用改进后的列文伯格(Levenberg-Mar-quardt)算法对其求解.仿真结果表明,改进后的定位算法具有不依赖于初值、精度高、收敛速度快的特点,成功解决了定位中多局部极值的高维优化问题,从而为跟踪方案的实现提供了理论依据.     

Development of a new meshless — point weighted least-squares (PWLS) method for computational mechanics

A truly meshless approach, point weighted least-squares (PWLS) method, is developed in this paper. In the present PWLS method, two sets of distributed points are adopted, i.e. fields node and collocation point. The field nodes are used to construct the trial functions. In the construction of the trial functions, the radial point interpolation based on local supported radial base function are employed. The collocation points are independent of the field nodes and adopted to form the total residuals of the problem. The weighted least-squares technique is used to obtain the solution of the problem by minimizing the functional of the summation of residuals. The present PWLS method possesses more advantages compared with the conventional collocation methods, e.g. it is very stable; the boundary conditions can be easily enforced; and the final coefficient matrix is symmetric. Several numerical examples of one- and two-dimensional ordinary and partial differential equations (ODEs and PDEs) are presented to illustrate the performance of the present PWLS method. They show that the developed PWLS method is accurate and efficient for the implementation.