二维定常不可压缩粘性流动N-S方程的数值流形方法

将流形方法应用于定常不可压缩粘性流动N-S方程的直接数值求解,建立基于Galerkin加权余量法的N-S方程数值流形格式,有限覆盖系统采用混合覆盖形式,即速度分量取1阶和压力取0阶多项式覆盖函数,非线性流形方程组采用直接线性化交替迭代方法和Nowton-Raphson迭代方法进行求解.将混合覆盖的四节点矩形流形单元用于阶梯流和方腔驱动流动的数值算例,以较少单元获得的数值解与经典数值解十分吻合.数值实验证明,流形方法是求解定常不可压缩粘性流动N-S方程有效的高精度数值方法.     

数值流形单元法数学网格自适应

基于数值流形方法和有限覆盖技术,将有限元法的后验误差估计理论及h型网格自适应技术推广应用到数值流形单元法中,提出了数值流形单元法的后验误差估计方法和数学网格自适应技术,并编制了相应的程序。数值算例表明,经过网格自适应,可以在粗糙的初始网格基础上得到质量比较理想的网格,计算结果可达到用户要求的精度。     

Trefftz法中的自锁现象及变量减缩法

用Trefftz型边界解法分析了中厚板弯曲问题,发现了一类新的、因Trefftz函数溢机引起的“自锁现象”,并提出了一种消除这类自锁问题的“变量减缩法”。     

三维数值流形方法及其在复合材料中的应用

数值流形方法在进行接触判断时,传统的直接判断法在三维情况下检索困难,计算量大,对大规模工程问题是不适用的。为此,本文将公共面法引入三维数值流形方法的接触判断,使接触判断的计算量大大减少。目前,数值流形方法主要应用于岩石力学分析,为了拓宽其应用领域,作者比较了复合材料与岩石结构的异同,将其应用于复合材料的数值模拟,数值结果表明,该方法收敛快、精度高,弥补了有限元的不足。     

弹性力学的复变量数值流形方法

数值流形方法通过引入数学和物理双重网格,将插值域和积分域分别定义在两个不同的覆盖上来完成系统能量泛函积分运算. 当采用高阶函数构造位移函数时,广义节点自由度将大大增加. 在求解系统的平衡方程中,运算量是与自由度的三次方成正比的,因此数值流形方法的计算量是较大的. 为此,在复变量理论的基础上,采用一维基函数建立二维问题的逼近试函数,然后将其应用于弹性力学的数值流形方法,提出了复变量数值流形方法,推导了弹性力学的复变量数值流形方法的公式. 与传统的数值流形方法相比,复变量数值流形方法具有计算量小、精度高的优点.      

弹性地基上Timoshenko梁的精确数值解

研究了弹性地基上Timoshenko梁的高精度有限元分析方法,利用控制微分方程的基本解建立了单元形函数,提出了弹性地基上Timoshenko梁分析的Trefftz单元。通过对引入的非节点自由度进行静力凝聚,得到的精确单元与常规单元具有相同的节点自由度。文中还讨论了有效降低计算过程中舍入误差的方法。算例结果表明,采用提出...     

Trefftz有限元法的研究进展

Trefftz有限元法(Trefftz finite element method,TFEM)是一种高效的数值计算方法,兼有传统有限元法和边界元法的诸多优点.基于双独立插值模式,结合杂交泛函和高斯散度定理,推得仅含边界积分的有限元格式.简述了过去10年间(2007—2016)Trefftz有限元法在单元域内插值函数、源项处理、特殊功能单元以及非各向同性材料等方面的研究进展,并对未来的发展趋势给出了几点展望.     

求解非线性模态的相不变流形法与复不变流形法分析比较

本文分析比较了求解非一性模态的相不变流形法和复不变流形法的异同、求解与计算精度。     

基于适合分析T样条的高阶数值流形方法

数值流形方法是一种非常灵活的数值计算方法,连续体的有限单元方法和块体系统的非连续变形分析方法只是这一数值方法的特例.数值流形方法中高阶位移函数的构造可通过提高权函数的阶次来实现,这种方法往往需要沿单元边界配置适当的边内节点,这些结点的出现增加了前处理的复杂性,特别是对于大型复杂的空间问题.另一方面,在数值流形方法中可通过缩小单元尺寸(h加密)来提高求解精度.当模拟裂纹扩展时,这种细化策略可用来克服裂纹尖端的奇异性.一个传统的解决方案是细化整个网格,但这会导致计算效率的显著降低.将适合分析的T样条(analysis-suitable T-spline,AST)引入数值流形方法中来建立高阶数值流形方法的分析格式,有效的避免了该问题的出现.AST样条基函数具有线性无关,单位分解,局部加密等许多重要性质,使得其非常适合用于工程设计及分析.在引入AST样条后,可通过改变数学覆盖的构造形式建立不同阶次的数值流形方法分析格式;AST样条自身的局部加密性质也使得数值流形方法中的数学网格局部加密更容易实现.算例结果表明:随着AST样条基函数阶次的提高,数值流形方法的计算结果有了明显的改善;基于AST样条基函数的数值流形方法在保持计算精度的前提下降低了自由度的数量.     

基于单位分解法的无网格数值流形方法

在数值流形方法和单位分解法的基础上，提出了无网格数值流形方法. 无网格数值流形 方法在分析时采用了双重覆盖系统，即数学覆盖和物理覆盖. 数学覆盖提供的节点形成求解 域的有限覆盖和单位分解函数；而物理覆盖描述问题的几何区域及其域内不连续性. 与原有 的数值流形方法相比，无网格数值流形方法的数学覆盖形状更加灵活，可以用一系列节点的 影响域来建立数学覆盖和单位分解函数，具有无网格方法的特性，从而摆脱了传统的数值流 形方法中网格所带来的困难. 与无网格方法相比，由于采用了有限覆盖技术，试函数的构造 不受域内不连续的影响，克服了原有的无网格方法在处理不连续问题时所遇到的困难. 详细推导了无网格数值流形方法的试函数和求解方程，最后给出了算例，验证了该方法的正 确性.     

Trefftz method for hydrodynamic pressure on rigid dams with non-vertical upstream face

Based on a two-dimensional potential flow theory, earthquake-induced hydrodynamic pressures on a rigid dam with a non-vertical upstream face are examined by the Trefftz method. The effect of surface waves on the hydrodynamic pressure distribution is discussed in detail. Numerical values are given for different wave effect parameters and different geometries of the dam–water interface.     

Mode III fracture analysis by Trefftz boundary element method

This paper presents a hybrid Trefftz (HT) boundary element method (BEM) by using two indirect techniques for mode III fracture problems. Two Trefftz complete functions of Laplace equation for normal elements and a special purpose Trefftz function for crack elements are proposed in deriving the Galerkin and the collocation techniques of HT BEM. Then two auxiliary functions are introduced to improve the accuracy of the displacement field near the crack tips, and stress intensity factor (SIF) is evaluated by local crack elements as well. Furthermore, numerical examples are given, including comparisons of the present results with the analytical solution and the other numerical methods, to demonstrate the efficiency for different boundary conditions and to illustrate the convergence influenced by several parameters. It shows that HT BEM by using the Galerkin and the collocation techniques is effective for mode III fracture problems. The project supported by the National Natural Science Foundation of China(10472082). The English text was polished by Keren Wang.     

Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.     

Trefftz间接法解自由面渗流问题

渗流问题是岩体水力学研究的重点之一，较常规的有限元法和传统的边界元法而言，Trefftz型边界元法具有程序简单、计算量小及无须奇异积分等特点。本文给出定常和非定常自由面渗流问题的Trefftz解法，计算结果表明，该法收敛性好，结果比较精确。     

Effects of surface waves on hydrodynamic pressures on rigid dams with arbitrary upstream face

The effects of surface gravity waves on earthquake‐induced hydrodynamic pressures on rigid dams with nonvertical upstream face are examined, taking the compressibility and viscosity of water into account. A simple closed‐form solution is obtained by using the Trefftz numerical method. The boundary under study is reduced to the upstream face only and the degrees of freedom of the problem are restricted to the number of trial functions used for analysis. For harmonic base excitation, a number of numerical examples for various geometries of the upstream face are presented. The hydrodynamic pressure distribution, the resultant lateral force and its effective height of application are evaluated. The results are strongly influenced by the upstream shape and, in a lesser degree, by the consideration of surface waves. Copyright © 2009 John Wiley & Sons, Ltd.     

LOCKING PROBLEM AND VARIABLE REDUCING PROCEDURE IN TREFFTZ METHOD

The Trefftz-type boundary solution methods^[1] are applied in analysing moderately thick plate bending problems. A new type of locking problem caused by the overflow of Trefftz functions has been found and a so-called variable-reducing procedure for eliminating such a phenomenon is also proposed. Supported by National Natural Science Foundation of China(No. 19872019) and Solid Mechanics Open Research Laboratory of Tongji University     

APPLICATION OF TREFFTZ BEM TO ANTI-PLANE PIEZOELECTRIC PROBLEM

Anti-plane electroelastic problems are studied by the Trefftz boundary elementmethod (BEM) in this paper. The Trefftz BEM is based on a weighted residual formulation andindirect boundary approach. In particular the point-collocation and Galerkin techniques, in whichthe basic unknowns are the retained expansion coefficients in the system of complete equations,are considered, Furthermore, special Trefftz functions and auxiliary functions which satisfy ex-actly the specified boundary conditions along the slit boundaries are also used to derive a specialpurpose element with local defects. The path-independent integral is evaluated at the tip of acrack to determine the energy release rate for a mode Ⅲ fracture problem. In final, the accuracyand efficiency of the Trefftz boundary element method are illustrated by an example and thecomparison is made with other methods.     

Nonlocal elasticity defined by Eringen’s integral model: Introduction of a boundary layer method

In this paper we consider a nonlocal elasticity theory defined by Eringen’s integral model and introduce, for the first time, a boundary layer method by presenting the exponential basis functions (EBFs) for such a class of problems. The EBFs, playing the role of the fundamental solutions, are found so that they satisfy the governing equations on an unbounded domain. Some insight to the theory is given by showing that the EBFs satisfying the Navier equations in the classical elasticity theory also satisfy the governing equations in the nonlocal theory. Some additional EBFs are particularly obtained for the nonlocal theory. In order to use the EBFs on bounded domains, the effects of the boundary conditions are taken into account by truncating the kernel/attenuation function in the constitutive equations. This leads to some residuals in the governing equations which appear near the boundaries. A weighted residual approach is employed to minimize the residuals near the boundaries. The method presented in this paper has much in common with Trefftz methods especially when the influence area of the kernel function is much smaller than the main computational domain. Several one/two dimensional problems are solved to demonstrate the way in which the EBFs can be used through the proposed boundary layer method.