钢筋砼圆形贮液池的优化设计

本文研究了钢筋砼圆形贮液池池壁的优化设计问题,文中用插值矩阵法作内力分析,以池壁造价为目标函数,依照设计规范要求,用复合形法求出使目标函数最小时的池壁厚度变化和配筋量。并编制了专用程序。与已有实例比较,本文结果的效益显著。     

求解具有弹性接头桩基动力学大变形的微分—代数方法

本文采用弧坐标首先建立了求解具有弹性接头的桩基大变形分析的非线性动力学微分方程,其中, 广义Winkler模型用来模拟土对桩基的抗力.其次,在空间域内应用微分求积单元法来离散非线性微分方程组,并给出了处理弹性接头处连接条件的微分求积单元公式,得到了时间域内的一组微分-代数方程,采用二阶向后差分来代替二阶时间导数离散微分-代数方程组,得到一组离散化的非线性代数方程,应用Newton-Raphson方法求解了该非线性代数方程组.最后给出了数值算例,得到了桩基在顶部处受到组合动载荷作用时的响应,考察了弹性接头的刚度、位置对桩基动力学行为的影响.     

对称三对角矩阵的两个FORTRAN程序

在工程力学等许多学科领域里,常会遇到如下两个线代数计算问题: 问题1 解线代数方程组 Cx=d (1) 其中C为对称三对角矩阵     

阶梯式Timoshenko梁自由振动的DCE解

本文基于微分容积法和区域叠加技术提出了微分容积单元法（Differential Cubature Element method，以下简称DCE方法），并用之求解阶梯式变截面Timoshenko梁的自由振动问题。根据梁的变截面情况将其划分为几个单元，在每个单元内应用微分容积法将梁的控制微分方程和边界约束方程离散成为一组关于该单元内配点位移的线性代数方程组，将这些方程组写在一起并在各单元之间应用连续性条件和平衡条件得到一组关于整个域内各点位移的齐次线性代数方程组，这是一广义特征值问题，由子空间迭代法求解该特征问题便可求得系统的自振动频率。数值算例表明，本方法能稳定收敛、并有较高的数值精度和计算效率。     

高层建筑结构分层模型弹塑性动力分析

1.概述输入地震波对高层建筑结构进行弹塑性动力分析能反映出地震过程中结构的动力特性.多质点体系(图1)在动力荷载作用下的振动方程为:[M]{(?)}+[C]{(?)}+[K]{(?)}=-[M]{(?)} (1)式中{x}、{(?)}、{(?)}——质点的位移、速度、加速度向量;[M]、[C]、[K]——质量矩阵、阻尼矩阵和刚度矩阵;{(?)}——地面加速度向量.方程(1)是二阶微分方程组,按已知的地震加速度记录(?)(t)对时间t 求解这方程组,便可求得地震过程中各质点在每一时刻的位移、速度和加速度,从而计算结构的内力.     

考虑弯扭形变的薄壁箱形梁的非线性后屈曲

在大位移和扭转的前提下,通过一中等弯曲扭转的位移场描述了薄壁箱形梁在偏心载荷作用下的静稳定性问题.该非线性公式可用于分析简支薄壁箱形梁在不同载荷作用下的屈曲和后屈曲行为.采用伽辽金方法将非线性微分系统离散,并通过牛顿-拉普森增量迭代法求解得代数方程组.数值计算结果表明,当前屈曲位移不可忽略时,经典的横向屈曲预测是保守的...     

基础-饱和土地基耦合系统分析的微分求积单元法

研究了基础-饱和土地基耦合系统的动力学特性.首先根据多孔介质理论,在小变形的假设下,分别建立了耦合系统中不可压流体饱和土地基和弹性基础的运动微分方程以及相应的边界条件,连接条件和初始条件;然后在空间域采用微分求积单元法对基础-饱和土地基的控制方程进行离散,并提供了正确处理耦合系统界面之间连接条件和间断性条件的方法,从而得到时间域内的一组代数-微分方程;接着运用隐式二阶向后差分格式处理了代数-微分方程组;最后利用牛顿迭代法数值求解了该方程组,得到了耦合系统的数值解,考察了所布节点数和参数对数值结果的影响.     

特征函数展开解法的研究

以常微分方程的理论为基础,利用新的对偶变量、对偶微分矩阵和正交关系,以单连续坐标弹性体系为例,建立了与弹性力学求解新体系平行的特征函数展开解法．并将正交关系应用于可对角化边界条件的处理,实现了求解待定系数方程组的解耦,求得问题的显式封闭解．     

结构动力响应的时间有限元全域算法

利用状态方程将二阶结构动力学方程组变成一阶线性微分方程组,并采用加权残值伽辽金法将一阶线性微分方程组离散成线性方程组.该方法是一种全域算法,是真正意义上的时间有限元算法.数值算例表明:该方法有很好的计算精度,与解析解吻合较好.     

双模量面板泡沫铝芯圆形层合板的非线性弯曲

双模量材料是典型的拉压弹性模量不同的材料,在均匀外载荷作用下,双模量面板泡沫铝芯圆形层合板相当于三种不同材料组成的层合板。采用弹性理论建立了双模量面板泡沫铝芯圆形层合板在均布载荷作用下的静力平衡方程,利用该静力平衡方程确定了层合板的中性面位置。在此基础上建立了双模量面板泡沫铝芯圆形层合板的大挠度弯曲微分方程组,求得了层合板中心挠度与均布载荷的关系式。该方法计算结果与有限元计算结果的最大误差仅为3.8%,这说明该方法是可靠的。算例分析表明不考虑面板拉压弹性模量相异时其计算结果与实际情况相差较大,超过了工程上所允许的计算误差5%。所以,在计算双模量面板泡沫铝芯圆形层合板的非线性弯曲时,不宜采用相同弹性模量弹性理论,而应该采用拉压弹性模量不同的弹性理论。     

不规则区域平面弹性问题的正则区域重心插值配点法

将不规则区域嵌入到规则的矩形区域,在矩形区域上将弹性平面问题的控制方程采用重心Lagrange插值离散,得到控制方程矩阵形式的离散表达式。在边界节点上利用重心插值离散边界条件,规则区域采用置换法施加边界条件,不规则区域采用附加法施加边界条件,得到求解平面弹性问题的过约束线性代数方程组,采用最小二乘法进行求解,得到整个规则区域上的位移数值解。利用重心插值计算得到不规则区域内任意节点的位移值,计算精度可到10^-14以上。数值算例验证了所建立方法的有效性和计算精度。     

Chebyshev spectral variational integrator and applications

The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.     

Optimal control of attitude for coupled-rigid-body spacecraft via Chebyshev-Gauss pseudospectral method

The attitude optimal control problem(OCP) of a two-rigid-body spacecraft with two rigid bodies coupled by a ball-in-socket joint is considered. Based on conservation of angular momentum of the system without the external torque, a dynamic equation of three-dimensional attitude motion of the system is formulated. The attitude motion planning problem of the coupled-rigid-body spacecraft can be converted to a discrete nonlinear programming(NLP) problem using the Chebyshev-Gauss pseudospectral method(CGPM). Solutions of the NLP problem can be obtained using the sequential quadratic programming(SQP) algorithm. Since the collocation points of the CGPM are Chebyshev-Gauss(CG) points, the integration of cost function can be approximated by the Clenshaw-Curtis quadrature, and the corresponding quadrature weights can be calculated efficiently using the fast Fourier transform(FFT). To improve computational efficiency and numerical stability, the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of state and control variables. Furthermore, numerical float errors of the state differential matrix and barycentric weights can be alleviated using trigonometric identity especially when the number of CG points is large. A simple yet efficient method is used to avoid sensitivity to the initial values for the SQP algorithm using a layered optimization strategy from a feasible solution to an optimal solution. Effectiveness of the proposed algorithm is perfect for attitude motion planning of a two-rigid-body spacecraft coupled by a ball-in-socket joint through numerical simulation.     

非线性MEMS微梁的重心有理插值迭代配点法分析

文章利用重心有理插值迭代配点法分析计算非线性MEMS微梁问题。通过处理MEMS微梁的几何通过假设初始函数,将微梁非线性控制方程转换为线性化微分方程,建立逼近非线性微分方程的线性化迭代格式。采用重心有理插值配点法求解线性化微分方程,提出了数值分析MEMS微梁非线性弯曲问题的重心插值迭代配点法。给出了非线性微分方程的直接线性化和Newton线性化计算公式,详细讨论了非线性积分项的计算方法和公式。利用重心有理插值微分矩阵,建立了矩阵-向量化的重心插值迭代配点法的计算公式。数值算例结果表明,重心插值迭代配点法求解微梁非线性弯曲问题,具有计算公式简单、程序实施方便和计算精度高的特点。     

IMPROVEMENTS OF GEOMETRICALLY NONLINEAR ANALYSIS ALGORITHMS FOR SPATIAL FRAME STRUCTURES

以几何精确梁理论为基础,分别采用高阶拉格朗日插值和埃米特插值构造高精度空间梁单元。提出基于单元层次平衡迭代的自由度凝聚方法,以保证单元的通用性。实现了基于载荷控制或柱面弧长控制的结构几何非线性分析算法。算例研究结果表明,提出的改进方法不但提高了计算效率,而且还具有较高的数值稳定性;特别是基于三次埃米特插值构造的单元表现出较好的性态,适用于结构屈曲后分析。     

拟谱-微分求积混合方法求解一类双曲电报方程

拟谱方法和微分求积法是两类重要的无网格法,二者都已在科学和工程计算中获得了广泛应用。采用拉格朗日插值多项式作为二者的试函数,且采用同一种网格点分布,指出了在空间域上,微分求积法是拟谱方法的一种特殊形式。在此基础上,结合二者各自的特点,提出了拟谱-微分求积混合方法用于求解一类双曲电报方程。理论分析和数值测试表明,新方法在空间域上具有谱精度收敛性,在时间域上是A-稳定的,比较适合于求解多维电报方程。     

High-precision stress determination in photoelasticity

Stress separation is usually achieved by solving differential equations of equilibrium after parameter determination from isochromatics and isoclinics. The numerical error resulting from the stress determination is a main concern as it is always a function of parameters in discretization. To improve the accuracy of stress calculation,a novel meshless barycentric rational interpolation collocation method(BRICM) is proposed. The derivatives of the shear stress on the calculation path are determine...     

应变梯度理论自然邻近混合伽辽金法

应变梯度理论考虑了位移二阶梯度对应变能密度函数的贡献,在本构关系中引入了与材料微结构特征尺寸相关的参数,可以唯象地解释尺度效应现象。基于约束变分原理,把位移与位移一阶梯度作为独立场变量,使用拉格朗日乘子法引入二者的协调关系,放松对试探函数连续性与完备性的要求,建立了二维及三维问题的应变梯度理论自然邻近混合伽辽金法。通过算例对方法的计算性能进行了考查,结果表明,该方法具有良好的数值精度,能够模拟材料力学性能的尺度效应。     

多边形有限元研究进展

有限元法是数值求解偏微分方程边值问题的重要方法,采用不规则多边形单元网格, 可以方便有效地模拟材料的力学性能, 又使得区域网格剖分变得灵活方便. 特别是对于复杂的几何形状, 多边形单元网格具有更大的优势. 本文对国内外有关多边形有限元法的最新进展作了初步的总结和评述, 主要以基于位移法的多边形有限元为主.论述了多边形有限元的发展历史, 给出了多边形单元上的Wachspress插值、Laplace插值和重心坐标的一些最新研究成果. 与经典有限元法形函数为多项式形式不同, 多边形单元的形函数为有理函数或者无理函数形式. 多边形单元插值形函数满足线性完备性, 可以再现线性位移场, 像经典有限元法一样直接施加本质边界条件; 插值函数在多边形的边界上是线性的,确保不同单元间的自动协调. 不同单元的插值形函数表达公式形式统一, 方便混合单元网格计算的程序编写. 提出了多边形有限元法今后需要研究的问题.      

A modified finite element method for determining equilibrium capillary surfaces of liquids with specified volumes

This paper describes a modified finite element method (MFEM) for determining the static equilibrium shape of the capillary surface of a liquid with a prescribed volume constrained by rigid boundaries with arbitrary shapes. It is assumed that the liquid is in static equilibrium under the influence of surface tension, adhesion, and gravity forces. This problem can be solved by employing the conventional FEM; however, a major difficulty arises due to the presence of the volume (integral) constraint and usually requires the use of the Lagrange multiplier method, the sequential unconstrained minimization technique, or the augmented Lagrange multiplier method. With the MFEM, the space variables defining the equilibrium surfaces (or curves) are expanded in terms of parametric interpolation functions, which are designed such that the boundary conditions and the integral constraint equation are automatically satisfied during each iteration of a direct numerical search process. Hence, there is no need to include Lagrange multipliers and/or penalty factors and the problem can be treated more simply as one involving unconstrained optimization. This investigation indicates that the MFEM is more efficient and reliable than the other methods. Results are presented for several case study problems involving liquid solder drops. Copyright © 2000 John Wiley & Sons, Ltd.