有限元法求解广义热弹耦合一维热冲击问题

为了避免积分变换方法在求解Lord-Shulman(L-S)型广义热弹性耦合问题时由于数值反变换所引起的计算精度降低的问题,该文应用直接有限元方法,求解了基于L-S型广义热弹性理论的窄条薄板受热冲击作用的动态响应问题,结果表明,该方法对求解L-S型广义热弹性耦合的一维问题具有很高的精度。该文给出了L-S型广义热弹性理论下的热弹耦合的控制方程,建立了L-S型的广义热弹性问题的虚位移原理,推导得到了相应的有限元方程。计算得到了窄条薄板中无量纲温度、无量纲位移及无量纲应力的分布规律,从温度分布图上可以清晰地观察到热波波前的特有属性,即热波波前处存在明显的温度梯度的突变。     

薄板屈曲分析的局部Petrov-Galerkin方法

利用薄板控制微分方程的等效积分对称弱形式和对变量(挠度)采用移动最小二乘近似函数进行插值,研究了无网格局部Petrov-Galerkin方法在薄板屈曲问题中的应用。它不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行,并用罚因子法施加本质边界条件。数值算例表明,无网格局部Petrov-Galerkin法不但能够求解弹性静力学问题,而且在求解弹性稳定性问题时仍具有收敛快,稳定性好,精度高的特点。     

基于欧拉插值的最小二乘混合配点法在弹性力学平面问题中的应用

由于常规配点型无网格法存在求解不稳定、精度差和求解高阶导数等问题,提出了基于欧拉插值的最小二乘混合配点法。该方法同时以位移和应变作为未知量,通过欧拉插值将未知变量的导数表达出来,同时在插值中引入高斯权函数,并代入微分方程,从而形成以位移和应变为未知数的超定方程组,然后形成最小二乘意义下的法方程,法方程和相应的位移边界条件、应力边界条件一起形成定解体系。该方法不需要域积分,是一种真正的无网格法。一些典型的弹性力学平面问题表明本文方法具有良好的精度。     

非线性振动分析的重心插值配点法

将计算区间采用第二类Chebyshev点离散,利用数值稳定性好、计算精度高的重心Lagrange插值近似未知函数,建立未知函数各阶导数在计算节点上的微分矩阵.利用重心Lagrange插值公式离散非线性振动微分方程为非线性代数方程,采用Newton法求解非线性代数方程.计算得到振动位移后,采用微分矩阵和重心Lagange插值计算非线性振动的速度、加速度和振动周期.采用重心插值配点法计算了Duffing型非线性振动方程和非线性单摆振动方程.数值算例表明本文方法具有计算公式简单、程序实施方便和计算精度高的优点.     

分阶拟合直接配点无网格法

在子域插值的基础上提出了分阶拟合直接配点无网格方法。该方法通过分阶拟合使近似函数在节点的残差达到最小,边界条件直接引入,然后使用直接配点法求解方程。与其它插值或拟合方法相比,分阶拟合避免了矩阵奇异产生的困难;与最小移动二乘法(MLS)相比,分阶拟合只需用六个点来构造二次基近似函数,减小了计算量;而与其它基于Galerkin法的无网格法相比,分阶拟合直接配点无网格法计算量小。     

非均质材料的扩展无单元Galerkin法模拟

该文基于滑动Kriging插值法,提出了求解含夹杂非均匀材料问题的扩展无单元Galerkin法。该方法利用水平集函数对滑动Kriging插值形函数进行扩展,从而来反映材料交界面的几何形状和不连续位移场。相比传统的移动最小二乘法形函数,滑动Kriging插值形函数由于满足Kronecker delta函数性质,因此能准确施加位移边界条件。在含夹杂非均匀材料问题求解时,阐述了扩展无单元Galerkin法位移模式的构造以及控制方程的建立。最后通过单夹杂和多夹杂算例表明,扩展无单元Galerkin法相比扩展有限元法,计算精度更高、收敛速率更快。     

样条函数线法分析壳体非线性问题

采用样条函数线法计算圆柱壳有具有封闭截面的一类壳体的几何线性问题，用样条函数插值交二维非线性偏微分问题化为一组用径向结线位移增量表示的非线性常微分方程，然后用常微分方程求解器迭代求解，文中导出了用于非线性分析的样条函数线增量方程，最后给出了算例。     

空间RRRP机械臂的刚柔耦合非线性动力学特性研究

摘 要 机械臂的刚柔耦合非线性动力学特性分析是实现其运动精度控制的基础。针对某航天飞行器装载的RRRP型空间机械臂,采用模态组合函数描述各臂的弹性变形,将转动副角位移、移动幅线位移以及各臂的模态坐标作为广义自由度,利用Lagrange定理建立了空间RRRP机械臂的非线性动力学方程,采用4-5阶变步长 Longgekuta法,对非线性微分方程组进行了数值求解。研究了机械臂刚柔耦合的非线性特性及结构参数变化对末端振动的影响,为进一步实现其结构优化和精度控制奠定基础。     

层合板分析的无网格局部Petrov-Galerkin方法

基于Kirchhoff均匀各向异性板控制方程的等效积分弱形式和对挠度函数采用移动最小二乘近似函数进行插值, 进一步研究无网格局部Petrov-Galerkin方法在纤维增强对称层合板弯曲问题中的应用。该方法不需要任何形式的网格划分, 所有的积分都在规则形状的子域及其边界上进行,其问题的本质边界条件采用罚因子法来施加。通过数值算例和与其他方法的结果比较, 表明无网格局部Petrov-Galerkin法求解层合薄板弯曲问题具有解的精度高、收敛性好等一系列优点。      

压电热弹性材料四边简支层合板的精确解

根据压电热弹性材料的控制方程和热传导关系,重构压电热弹性材料的本构关系,通过新本构关系并结合压电热弹性材料热平衡方程,得出压电热弹性材料机-电-热耦合问题的齐次状态方程。应用精细积分法,状态方程可独立求解。此方法在分析压电热弹性体耦合问题时,避免了求解关于热传导方程和热平衡方程的二阶微分方程,大大减少了数值计算的工作量。     

The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems

In this paper, the moving least-squares (MLS) approximation and the interpolating moving least-squares (IMLS) method proposed by Lancaster are discussed first. A new method for deriving the MLS approximation is presented, and the IMLS method is improved. Compared with the IMLS method proposed by Lancaster, the shape function of the improved IMLS method in this paper is simpler so that the new method has higher computing efficiency. Then combining the shape function of the improved IMLS method with Galerkin weak form of the potential problem, the interpolating element-free Galerkin (IEFG) method for the two- dimensional potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional element-free Galerkin (EFG) method, the boundary conditions can be applied directly in the IEFG method, which makes the computing efficiency higher. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method.     

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.     

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.     

RBF meshless method for large deflection of thin plates with immovable edges

An efficient meshless formulation is presented for large deflection of thin plates with immovable edges. In this method, a fifth-order polynomial radial basis function (RBF) is used to approximate the solution variables. The governing equations are formulated in terms of the three displacement components u, v and w. The solution is obtained by satisfying three coupled partial differential equations and their boundary conditions inside the domain and over the boundary of the plate, respectively. The collocation procedure produces a system of coupled non-linear algebraic equations, which are solved using an incremental-iterative procedure. The numerical efficiency of the proposed method is illustrated through numerical examples.     

The improved element-free Galerkin method for two-dimensional elastodynamics problems

In this paper, we derive an improved element-free Galerkin (IEFG) method for two-dimensional linear elastodynamics by employing the improved moving least-squares (IMLS) approximation. In comparison with the conventional moving least-squares (MLS) approximation function, the algebraic equation system in IMLS approximation is well-conditioned. It can be solved without having to derive the inverse matrix. Thus the IEFG method may result in a higher computing speed. In the IEFG method for two-dimensional linear elastodynamics, we employed the Galerkin weak form to derive the discretized system equations, and the Newmark time integration method for the time history analyses. In the modeling process, the penalty method is used to impose the essential boundary conditions to obtain the corresponding formulae of the IEFG method for two-dimensional elastodynamics. The numerical studies illustrated that the IEFG method is efficient by comparing it with the analytical method and the finite element method.     

一种高阶精度人工边界条件:出平面外域波动问题

针对无限外域中的出平面波动问题,提出一种用于近场波动有限元分析的高阶精度人工边界条件。首先,采用变量分离法求解远场初边值问题,建立了时空全局的精确动力刚度人工边界条件;然后,发展了一种由有理函数近似和辅助变量实现构成的时间局部化方法,并将其应用于动力刚度人工边界条件,得到时间局部的高阶精度人工边界条件;最后,沿人工边界离散高阶精度人工边界条件,并将其与近场集中质量有限元方程耦合,形成对称的时间二阶常微分方程组,采用一种新的显式时间积分方法进行求解。数值算例表明:提出的高阶精度人工边界条件精确、高效、稳定并且容易在现有的有限元代码中实现。     

Numerical Solution of a Problem of Thermal Stresses of a Magnetothermoelastic Cylinder with Rotation by Finite-Difference Method

The present article deals with the investigation thermal stress of a magnetothermoelastic cylinder subjected to rotation, open or closed circuit, thermal and mechanical boundary conditions. The outer and inner surfaces of the cylinder are subjected to both mechanical and thermal boundary conditions. A The transient coupled thermoelasticity in an infinite cylinder with its base abruptly exposed to a heat flux of a decaying exponential function of time is devised solve by the finite-difference method. The fundamental equations’ system is solved by utilizing an implicit finite-difference method. This current method is a second-order accurate in time and space; it is also unconditionally stable. To illustrate the present model’s efficiency, we consider a suitable material and acquire the numerical solution of temperature, displacement components, and the components of stresses with time t and through the radial of an infinite cylinder. The results indicate that the effect of coupled thermoelasticity, magnetic field, and rotation on the temperature, stresses, and displacement is quite pronounced. In order to illustrate and verify the analytical developments, the numerical solution of partial differential equations, stress components, displacement components and temperature is carried out and computer simulated results are presented graphically. This study is helpful in the development of piezoelectric devices.     

Solutions of population balance equations by double Legendre polynomial transformation

Abstract

First‐order partial differential equations of population balance are solved by employing the Legendre polynomials. The key of the method is that the dependent variable of the population density function is assumed to be expressed by a double series of Legendre polynomials with respect to time and space variables. The approach algorithm is that a series of ordinary differential equations are obtained by making the Legendre transformation with respect to the space coordinate. The series of time‐function ordinary differential equations are further transformed into algebraic equations of expansion coefficients with respect to time. The expansion coefficients of the Legendre polynomials are obtained by solving matrix equations which represent the series of algebraic equations. Illustrative examples are given, and the computational results are compared with those of other numerical values given in the literature. Satisfactory agreements are obtained.     

DEM: A new computational approach to sheet metal forming problems

Many current approaches to finite element modelling of large deformation elastic—plastic forming problems use a rate form of the virtual work (equilibrium) equations, and a finite element representation of the displacement components. Called the incremental method, this approach produces a three-field formulation in which displacements, stresses and effective strain are dependent variables. Next, the formulation is converted to a one-field displacement formulation by an algebraic time discretization which uses a low order explicit time-stepping procedure to integrate the equations. This approach does not produce approximations which satisfy the discrete equilibrium equations at all times and, moreover, the advantage of the single-field algebraic formulation is realized at the expense of very small time steps needed to produce stability and accuracy in the numerical calculations. This paper describes a variant of the mixed method in which all three field variables (displacements, stresses and effective strain) are given finite element representations. The discrete equilibrium equations then generate a nonlinear system of algebraic equations whose solutions represent a manifold, while the constitutive equations form a system of ordinary differential equations. A commercially available, variable time step/variable order code is then used to integrate this differential/algebraic system. When applied to the problem of hydrostatic bulging of a membrane, the new approach requires far less computer time than the incremental method.     

Inverse heat conduction problems in three-dimensional anisotropic functionally graded solids

A meshless method based on the local Petrov–Galerkin approach is applied to inverse transient heat conduction problems in three-dimensional solids with continuously inhomogeneous and anisotropic material properties. The Heaviside step function is used as a test function in the local weak form, leading to the derivation of local integral equations. Nodal points are randomly distributed in the domain analyzed, and each node is surrounded by a spherical subdomain in which a local integral equation is applied. A meshless approximation based on the moving least-squares method is employed in the implementation. After performing spatial integrations, we obtain a system of ordinary differential equations for certain nodal unknowns. A backward finite-difference method is used for the approximation of the diffusive term in the heat conduction equation. A truncated singular-value decomposition is used to solve the ill-conditioned linear system of algebraic equations at each time step. The effectiveness of the meshless local Petrov–Galerkin (MLPG) method for this inverse problem is demonstrated by numerical examples.