空间地基沉降分析半差分半加权余量法

本文针对三维地基沉降分析提出了一种差分伽辽金耦合方法,这是一个半差分半加权余量法,文末给出了一个二层三维弹性层状地基的算例.     

压电结构动力分析的无网格自然单元法

自然单元法是一种新兴的无网格数值计算方法,在本质边界条件的施加上较采用移动最小二乘法的无网格法具有明显的优势。将无网格自然单元法与精细积分法相结合,提出了压电结构动力响应分析的一条新途径。在空间域上采用自然单元法离散,并运用加权余量法推导了压电结构动力分析的离散控制方程。然后,采用精细积分法在时间域上进行求解。最后给出了数值算例,并验证了所提方法的有效性和正确性。     

两端铰支阶梯状柱弹性失稳的加权余量解

本文利用加权余量法推导出两端铰支阶梯状柱，发生弹性失稳时临界荷载Ｐ的计算式。     

高速管流弹性失稳临界流速的加权余量解

本文利用加权余量法推导出管道在高速流体，流经后，管道发生弹性失稳时，流体临界流速Ｖ的计算公式。     

用加权余量法求圆环径向位移的近似方程

本文首先推导出圆环径向变形的微分方程，然后利用加权余量法求出圆环在竖向直径两端Ｂ与Ｄ 承受一对等值以向拉力Ｐ，圆环径向位移的近似方程，并计算出Ｂ与Ｄ相对位移。     

厚截锥壳结构的自由振动

本文对基于Mindlin理论所导出的厚截锥壳的一阶基本微分方程组采用了子结构离散变量法,求解了这类结构的固有频率和相应的振型。文中给出了算例,其结果与试验结果和有限无法计算结果相比是令人满意的。     

分析薄板塑性动力响应问题的一种数值方法

对于结构塑性动力响应问题,一般的是先从静态极限分析的完全解出发,假设其速度场,然后根据各种条件进行求解。本文以圆板塑性动力响应问题为例,给出一种不同的求解方法,即联合运用拉普拉斯变换和加权余量法进行分析和求解。先通过拉普拉斯变换将薄板的动力问题转化为静力问题,然后根据弯矩 M 和挠度 w 的边界条件分别假设试函数,再应用加权余量法进行求解。     

二维有限元线法超收敛解答计算的EEP法

有限元线法(FEMOL)是一种优良的半解析、半离散方法,但其解答存在解析方向和离散方向的精度不相称的弱点。本文提出将二维有限元线法比拟为广义一维问题的概念,遂可将新近提出的一维有限元超收敛计算的单元能量投影(EEP)法推广到二维有限元线法分析中。经有限元线法后处理中EEP超收敛计算而获得的解答,继承和保留了一维有限元中的出色表现,不但使任意一点的位移和应力的解答在两个方向具有相当的精度,而且都具有超收敛性质。文中以二维Poisson方程问题为例,具体给出了有限元线法EEP超收敛的公式,并给出了数值算例,用以表明本法的可行性和有效性。     

超空泡运动体圆柱薄壳动力屈曲及可靠性分析

首先将超空泡运动体模拟成受动态轴向载荷作用的圆柱薄壳,推导了结构的动力稳定性微分方程和动力不稳定区域,然后考虑动态轴向载荷的随机性,采用有限步长迭代法将给出的动力屈曲失稳的多个安全余量方程线性化,并利用逐步搜索法找出有效的安全余量方程,最后结合逐步等效平面法计算了舱段动力屈曲的可靠性指标。通过算例分别分析了载荷频率、速度和载荷比例系数这三个随机参数的变化对动力屈曲可靠性的影响。计算结果为如何选择载荷频率、速度和载荷比例系数的安全范围提供了理论依据。     

热弹性动力学耦合问题的插值型移动最小二乘无网格法研究

该文基于插值型移动最小二乘法,将无网格局部Petrov-Galerkin（MLPG）法用于二维耦合热弹性动力学问题的求解。修正的Fourier热传导方程和弹性动力控制方程通过加权余量法来离散,Heaviside分段函数作为局部弱形式的权函数,从而得到描述热耦合问题的二阶常微分方程组。然后利用微分代数方法,温度和位移作为辅助变量,将上述二阶常微分方程组转换成常微分代数系统,采用Newmark逐步积分法进行求解。该方法无需Laplace变换可直接得到温度场和位移场数值结果,同时插值型移动最小二乘法构造的形函数由于满足Kroneckerdelta特性,因此能直接施加本质边界条件。最后通过两个数值算例来验证该方法的有效性。     

用QR法和样条加权残数法求解高层框剪结构的地震反应

本文将QR法和样条加权残数法结合起来,求解高层框剪结构的地震动力反应。先用QR法建立结构的振动方程,然后,用样条加权残数法来求解。本文所建立的计算格式,其未知量的个数不仅与结构的楼层数无关,而且与划分的单元多少无关。本文方法能方便地在微机上实现。     

分析不同材料界面应力奇异性的一维杂交有限元方法

该文提出了一种计算效率较高的分析不同材料界面应力奇异性的一维杂交有限元方法。为了推导该方法,首先列出了用于求解不同材料界面裂纹奇异应力场特征解的基本方程和边界条件,然后利用加权残量方法(weighted residual method),得到上述基本方程和边界条件的弱形式,该弱形式的基本变量为位移和应力。运用Galerkin有限元方法的思想及上述弱形式,最后得到了一个一维杂交有限元方法,该一维杂交有限元方法只需对扇形区域在角度方向上离散,其总体方程为一个二次特征矩阵方程。数值算例表明:该方法可以准确而高效地计算不同材料界面奇异应力场的特征解。     

Simple modifications for stabilization of the finite point method

A stabilized version of the finite point method (FPM) is presented. A source of instability due to the evaluation of the base function using a least square procedure is discussed. A suitable mapping is proposed and employed to eliminate the ill‐conditioning effect due to directional arrangement of the points. A step by step algorithm is given for finding the local rotated axes and the dimensions of the cloud using local average spacing and inertia moments of the points distribution. It is shown that the conventional version of FPM may lead to wrong results when the proposed mapping algorithm is not used. It is shown that another source for instability and non‐monotonic convergence rate in collocation methods lies in the treatment of Neumann boundary conditions. Unlike the conventional FPM, in this work the Neumann boundary conditions and the equilibrium equations appear simultaneously in a weight equation similar to that of weighted residual methods. The stabilization procedure may be considered as an interpretation of the finite calculus (FIC) method. The main difference between the two stabilization procedures lies in choosing the characteristic length in FIC and the weight of the boundary residual in the proposed method. The new approach also provides a unique definition for the sign of the stabilization terms. The reasons for using stabilization terms only at the boundaries is discussed and the two methods are compared. Several numerical examples are presented to demonstrate the performance and convergence of the proposed methods. Copyright © 2005 John Wiley & Sons, Ltd.     

用分域加权残值法求解部分区域有任意几何缺陷的圆柱壳

本文以实际工程中壳体的工况为背景，以缺陷产生区域为依据，将具有部分任意几何缺陷的圆柱壳划分成若干个计算区域（子域）的组合体。建立各子域的微分方程及各子域间的交界条件，进而运用加权残值的方法，求解组合体。文中所给算例，与已有文献吻合较好。     

Least squares finite element analysis of laminar boundary layer flows

Based on the least squares error criterion, a class of finite element is formulated for the numerical analysis of steady state viscous boundary layer flow problems. The method is essentially a discrete element-wise minimization of square and weighted residuals which arise from the attempts in approximately satisfying boundary layer equations. An iterative linearization scheme is developed to circumvent the mathematical difficulties posed by the non-linear boundary layer equations. It results in a process of successive least squares minimizations of residual errors arising from satisfying a set of linear differential equations. A mathematical justification for the method is presented. A major feature of the method lies in the linearization approach which renders non-linear differential equations amenable to linear least squares finite element analysis. Another important feature rests on the proposed finite element formulation which preserves the symmetric nature of finite element matrix equations through the use of the least squares error criterion. Numerical examples of viscous flow along a flat plate are presented to demonstrate the applicability of the method as well as to illuminate discussions on the theoretical aspects of the method.     

A comparison of Galerkin and streamline techniques for integrating strains from an Eulerian flow field

Two methods are compared for integrating the strains that can arise in finite element solutions for Eulerian velocity fields associated with large strain material forming processes. With the Galerkin formulation, partial differential equations for the deformation gradient are solved over the entire domain based on a weighted residual; with the streamline integrated technique, the corresponding ordinary differential equations are integrated along characteristic lines. Both methods have yielded accurate integrations for the radial flow and planar rolling problems studied. A finite element technique is also presented for ensuring that the free surfaces of the fluid flow are streamlines. This technique has been used for ensuring proper boundary conditions in the rolling analysis.     

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。     

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.     

力-变位关系全过程模拟的有限元位移控制新方法

材料、构件及结构的力-变位关系非线性全过程曲线有限元数值模拟中,极限强度和其后的软化下降段模拟一直是未得到较好解决的难题。位移增量控制法可以方便跨越力-变位关系中的极值点,因此它常被用于求解材料、构件及结构包含极限强度后软化下降段的力-变位关系非线性全过程曲线。但是,传统的位移增量控制法需要重新排列有限元方程的刚度矩阵,并存在求解非对称和非带状系数方程的问题,因而限制了其推广应用。该文提出了求解材料、构件及结构的力-变位关系非线性全过程曲线的一种新的位移增量控制方法,该方法通过修改刚度矩阵中相关对角系数的方式,将边界和力作用点的控制位移条件隐含到有限元方程中,从而可以采用有限元荷载控制方法同样的方式进行求解,保证了有限元方程系数矩阵在求解过程中的对称性和带状性,并可方便地在现有通用商业有限元软件中实现。算例分析表明了该文方法的有效性。     

A new least-squares finite element model for combined Navier–Stokes and Darcy flows in geometrically complicated domains with solid and porous boundaries

In this paper we present a novel method for linking Navier–Stokes and Darcy equations along a porous inner boundary in a flow regime which is governed by both types of these equations. The method is based on a least-squares finite element technique and uses isoparametric C¹ continuous Hermite elements for domain discretization. We show that our technique is superior to previously developed models for the combined Navier–Stokes/Darcy flows. The previous works use weighted residual finite element procedures in conjunction with C⁰ elements which are inherently incapable of linking Navier–Stokes and Darcy equations. The paper includes the application of our model to a geometrically complicated axisymmetric slurry filtration system.