力学学报  2019 , 51 (3): 690-702 https://doi.org/10.6052/0459-1879-19-004

无网格粒子类方法专题

薄板分析的线性基梯度光滑伽辽金无网格法1)

邓立克, 王东东2), 王家睿, 吴俊超

(厦门大学土木工程系,厦门 361005);(厦门市交通基础设施智能管养工程技术研究中心,厦门 361005)

A GRADIENT SMOOTHING GALERKIN MESHFREE METHOD FOR THIN PLATE ANALYSIS WITH LINEAR BASIS FUNCTION1)

Deng Like, Wang Dongdong2), Wang Jiarui, Wu Junchao

Department of Civil Engineering, Xiamen University, Xiamen 361005, China;Xiamen Engineering Technology Center for Intelligent Maintenance of Infrastructures, Xiamen 361005, China

中图分类号:  O242.2

文献标识码:  A

通讯作者:  2) 王东东, 教授, 主要研究方向: 计算力学与结构工程.E-mail: ddwang@xmu.edu.cn

收稿日期: 2019-01-2

网络出版日期:  2019-05-18

版权声明:  2019 力学学报期刊社 所有

基金资助:  1) 国家自然科学基金资助项目(11772280, 11472233).

展开

摘要

薄板问题的控制方程为四阶微分方程,因而当采用伽辽金法进行分析时,形函数需要满足C$^{1}$连续性要求,且至少使用二次基函数才能保证方法的收敛性.无网格形函数虽然易于满足C$^{1}$连续性要求,但由于不是多项式,其二阶导数的计算较为复杂耗时,同时也对刚度矩阵的数值积分提出了更高的要求.本文提出了一种薄板分析的线性基梯度光滑伽辽金无网格法,该方法的基础是线性基无网格形函数的光滑梯度.在梯度光滑构造的理论框架内,无网格形函数的二阶光滑梯度可以表示为形函数一阶梯度的线性组合,因而可以提高形函数二阶梯度的计算效率.分析表明,线性基无网格形函数的光滑梯度不仅满足其固有的线性梯度一致性条件,还满足本属于二次基函数对应的额外高阶一致性条件,因此能够恰当地运用到薄板结构的伽辽金分析.此外,插值误差分析也很好地验证了线性基无网格光滑梯度的收敛特性.算例结果进一步表明,线性基梯度光滑伽辽金无网格法的收敛率与传统二次基伽辽金无网格法相当,但精度更高,同时刚度矩阵所需的高斯积分点数明显减少.

关键词: 伽辽金无网格法 ; 线性基函数 ; 薄板问题 ; 光滑梯度 ; 一致性条件

Abstract

The fourth order governing equation of thin plate necessitates the employment of C$^{1}$ continuous shape functions with a minimum degree of two in a Galerkin formulation. Thus at least a quadratic basis function should be utilized in meshfree approximation to enable the Galerkin meshfree thin plate analysis. However, due to the rational nature of reproducing kernel meshfree shape functions, the computation of the second order derivatives of meshfree shape functions is quite complex and costly, which also requires expensive high order Gauss quadrature rules to properly integrate the stiffness matrix. In this work, a gradient smoothing Galerkin meshfree method with particular reference to the linear basis function is proposed for thin plate analysis. The foundation of the present development is the construction of smoothed meshfree gradients with linear basis function, where the second order smoothed gradients are expressed as combinations of standard first order gradients and the computational burden is remarkably reduced. Furthermore, it is shown that the smoothed meshfree gradients with linear basis function satisfy both the linear and quadratic gradient consistency conditions and consequently they are adequate for thin plate analysis in the context of Galerkin formulation. An interpolation error study is given as well to validate the higher order consistency conditions and applicability of smoothed meshfree gradients for Galerkin analysis of thin plates. It turns out that efficient lower order Gauss integration rules now work well for the proposed method. Numerical results demonstrate that compared with the conventional Galerkin meshfree method with quadratic basis function, the proposed gradient smoothing Galerkin meshfree method with linear basis function yields similar convergence rates, but with better accuracy and less integration points for stiffness computation.

Keywords: Galerkin meshfree method ; linear basis function ; thin plate ; gradient smoothing ; consistency condition

0

PDF (12603KB) 元数据 多维度评价 相关文章 收藏文章

本文引用格式 导出 EndNote Ris Bibtex

邓立克, 王东东, 王家睿, 吴俊超. 薄板分析的线性基梯度光滑伽辽金无网格法1)[J]. 力学学报, 2019, 51(3): 690-702 https://doi.org/10.6052/0459-1879-19-004

Deng Like, Wang Dongdong, Wang Jiarui, Wu Junchao. A GRADIENT SMOOTHING GALERKIN MESHFREE METHOD FOR THIN PLATE ANALYSIS WITH LINEAR BASIS FUNCTION1)[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 690-702 https://doi.org/10.6052/0459-1879-19-004

引 言

薄板结构具有广泛的工程应用,但其四阶控制方程给计算分析带来了相当的困难,即使采用基于弱形式的伽辽金法进行分析,也需要具有C$^{1}$连续性的二次基函数构造的形函数才能保证数值解的收敛性[1].受限于单元离散,有限元法仍然缺乏构造C$^{1}$形函数的简单有效的方法[1-2].无网格法基于节点离散,可以不依赖单元建立任意高阶连续的形函数,为薄板分析提供了一种行之有效的方法[3-7].例如,Krysl和Belytschko[8]首先采用伽辽金无网格法分析了薄板结构,Liu和Chen[9]基于伽辽金无网格法进行了薄板振动分析,龙述尧和Atluri[10]将局部彼得罗夫伽辽金无网格法推广至薄板分析,Lu等[11]提出了形函数具有插值特性的再生核有限元薄板分析方法,刘岩等[12]提出了一种薄板分析的Hermite径向点插值无网格法,马丽红等[13]结合区间分析法和无网格法对Winkler地基薄板进行了分析.为了使无网格形函数具有插值特性,Bui和Nguyen[14]应用移动Kriging无网格法研究了薄板振动问题,Cui等[15]提出了一种薄板分析的光滑点插值方法.Millan等[16]研究了形函数具有凸近似特性的薄板壳结构最大熵无网格分析方法,Zhang等[17]基于准凸的等几何无网格耦合方法分析了含裂缝薄板振动问题.Wang等[18-22]系统地发展了薄板分析的Hermite再生核无网格法和配套的稳定子域高效数值积分方法,Tanaka等[23]采用Hermite再生核无网格法进行了薄板的几何非线性分析.然而值得注意的是,无网格形函数通常不是多项式,例如常用的移动最小二乘和再生核无网格形函数[3,5],因此其导数计算,尤其是高阶导数的计算较为复杂耗时,同时也给刚度矩阵的数值积分带来了很大困难[24-26].已有工作虽然针对薄板问题提出了一些高效数值积分方法[18],但发展更为简洁高效的无网格法仍然是一个值得关注的重要问题.

另一方面,文献[27]提出了一种简洁的无网格形函数光滑梯度构造方法,并将其用于构造二阶问题的配点型无网格法,相应计算结果表明该方法可以有效解决奇数次基函数无网格配点法的收敛率下降问题,实现超收敛计算.在此基础上,本文着重讨论了线性基无网格形函数的一阶和二阶光滑梯度,分析了其对应的标准线性一致性条件和额外的二阶一致性条件.因此,仅采用线性基无网格形函数便可实现通常需要采用二次基函数才能满足的二阶一致性条件,所以相关的二阶光滑梯度能够直接用于薄板问题的伽辽金无网格分析,构造一种薄板分析的线性基伽辽金无网格法.文中通过对比分析表明,线性基无网格形函数的光滑梯度有效降低了二次基无网格形函数的标准梯度的震荡性,因此采用较少的积分点便可保证计算精度.例如,二次基伽辽金无网格法通常需要采用6$\times$6的高斯积分才能达到理论收敛率,而本文的线性基梯度光滑无网格法仅用2$\times$2的高斯积分便可以得到与之相当的结果.文中通过系列梁和板算例验证了线性基梯度光滑无网格法的精度和收敛性.

1 薄板控制方程

考虑图1所示的薄板结构,其中面为$\varOmega $,边界为$\varGamma$,厚度为$t$. 中面$\varOmega $内一点${ { x}}$处的挠度为$w({ {x}})$,其相应的转角向量${ { \theta }}({ {x}})$和曲率向量$\kappa ({ { x}})$为

图1   薄板问题示意图

Fig. 1   Sign convention of thin plate

$${\theta} = \left\{ {\begin{matrix} {\theta _x } \\ {\theta _y } \\ \end{matrix} } \right\} = \left\{ {{\begin{matrix} {w_{,x} } \\ {w_{,y} } \\ \end{matrix} }} \right\}\tag{1} $$

$$ \kappa = \left\{ {\begin{matrix} {\kappa _{xx} } \\ {\kappa _{yy} } \\ {2\kappa _{xy} } \\ \end{matrix} } \right\} = \left\{ {\begin{matrix} {w_{,xx} } \\ {w_{,yy} } \\ {2w_{,xy}} \\ \end{matrix} } \right\}\tag{2}$$

对于线弹性材料,应力矩向量${ { m}}$和曲率向量$ \kappa$之间的本构关系为

\begin{equation}\label{eq3} { { m}} = - { { D}} \kappa\tag{3}\end{equation}

其中

$${ { m}} = \mbox{\{}m_{xx} \mbox{ }m_{yy} \mbox{ }m_{xy} \mbox{\}}^{\rm T}\tag{4}$$

$$ { { D}} = \bar {D}\left[ {{\begin{matrix}{*{20}c} 1& \nu& 0\\ \nu& 1& 0\\ 0& 0& {(1 - \nu ) / 2}\\ \end{matrix} }} \right],\mbox{ }\bar {D} = \frac{Et^3}{12(1 - \nu ^2)}\tag{5}$$

式(5)中,$E$和$\nu $分别为材料的杨氏模量和泊松比,$\bar{D}$为薄板的抗弯刚度.

由于无网格形函数一般不具有插值特性,这里采用拉格朗日乘子的方法施加挠度和转角边界条件,相应的薄板问题等效积分弱形式为

$$\int_\varOmega {\delta \kappa ^{\rm T}{ { m}}{\rm d}\varOmega } + \int_\varOmega {\delta wq{\rm d}\varOmega } + \\ \qquad \int_{\varGamma ^w} {\delta \lambda ^w(w - \bar {w}){\rm d}\varGamma } + \int_{\varGamma ^w} {\lambda ^w\delta w{\rm d}\varGamma } + \\ \qquad \int_{\varGamma ^\theta } {\delta \lambda ^\theta (\theta _n-\bar {\theta }_n )} {\rm d}\varGamma + \int_{\varGamma ^\theta } {\lambda ^\theta \delta \theta _n } {\rm d}\varGamma = 0 \tag{6} $$

其中,$q$是作用在薄板上的竖向载荷,$\varGamma ^w$和$\varGamma^\theta $是强制挠度和转角边界,$\theta _n = w_{,n}$是沿着边界外法向${ {n}}$的转角,$\bar {w}$和$\bar {\theta }_n$是给定的挠度和转角,$\lambda ^w$和$\lambda ^\theta$是对应于挠度和转角的拉格朗日乘子. 由于式(6)中的$ \kappa$项包含挠度的二阶导数,因此至少需要采用二次基函数才能保证伽辽金方法的收敛性[1].

2 无网格形函数

本文采用移动最小二乘或再生核无网格形函数,当采用多项式基函数时,这两种无网格形函数其实是等同的.因此,不失一般性,这里以再生核形函数为例说明无网格形函数的构造特点.对于薄板问题,一般将薄板中面$\varOmega $及其边界$\varGamma$离散为一组节点$\left\{ {{ { x}}_I } \right\}_{I = 1}^{NP}$,每个节点${ { x}}_I $对应一个形函数$\varPsi _I ({ { x}})$.根据再生核无网格近似理论[5],薄板挠度$w({ {x}})$对应的无网格近似解$w^h({ { x}})$可表示为\begin{equation}\label{eq7} w^h({ { x}}) = \sum\limits_{I = 1}^{NP} {\varPsi_I ({ { x}})d_I }\tag{7}\end{equation}其中,$d_I $为与节点${ { x}}_I $对应的节点系数.无网格形函数$\varPsi _I ({ { x}})$具有如下形式\begin{equation}\label{eq8} \varPsi _I ({ { x}}) = { { p}}^{\rm T}({ { x}}_I- { {x}}){ { c}}({ { x}})\phi _s ({ { x}}_I - { {x}})\tag{8}\end{equation}式中,${ { c}}({ { x}})$是依赖于位置${ {x}}$的一个待定系数向量. ${ { p}}({ {x}})$为$p$阶单项式基向量,即\begin{equation}\label{eq9} { { p}}({ { x}}) = \{1,\mbox{ }x,\mbox{ }y,\mbox{}x^2,\mbox{ }xy,\mbox{ }y^2,\mbox{ }\cdots,x^p,\mbox{}\cdots,\mbox{ }y^p\}^{\rm T}\mbox{ }\tag{9}\end{equation}式中$p$为基函数的阶次. $\phi _s ({ { x}}_I - { {x}})$为附属于节点${ { x}}_I$的核函数,具有紧支性,下标$s$表示其影响域:$supp({ { x}}_I )$.本文采用如下的五次B样条核函数

$$\varphi _s (r) = \\ \dfrac{1}{5!}\left\{\!\! {{\begin{array}{*{20}l} {(3 \!-\! 3r)^5 \!-\! 6(2 \!-\! 3r)^5 \!+\! 15(1 \!-\! 3r)^5}, {r \le \dfrac{\mbox{1}}{3}} \\ {(3 - 3r)^5 - 6(2 - 3r)^5}, \ \ \ {\dfrac{\mbox{1}}{3}< r \le \dfrac{2}{3}} \\ {(3 - 3r)^5} ,\ \ \ \ {\dfrac{2}{3} < r \le 1} \\ 0, \ \ \ \ {r > 1} \\ \end{array} }} \right.\\ \tag{10} $$

其中,$r$表示相邻节点之间的相对距离.二维核函数可以方便地通过两个一维核函数的张量积形式进行构造\begin{equation}\label{eq11} \phi _s ({ { x}} - { { x}}_I ) = \phi _{s_x }(r_x )\phi _{s_y } (r_y )\tag{11}\end{equation}其中,$r_x = \left\| {x - x_I } \right\| / s_x $,$r_y = \left\|{y - y_I } \right\| / s_y $,$s_x $和$s_y$分别代表了$x$和$y$方向的影响域大小.

为了保证无网格近似的一致性或完备性,形函数$\varPsi _I ({ {x}})$需要满足下列的再生条件或一致性条件

\begin{equation} \label{eq12} \sum\limits_{I = 1}^{NP} {\varPsi _I ({ { x}}){ {p}}\mbox{(}{ { x}}_I )} = { { p}}({ { x}})\tag{12} \end{equation}

\begin{equation} \label{eq13} \sum\limits_{I = 1}^{NP} {\varPsi _I ({ { x}}){ {p}}\mbox{(}{ { x}}_I - { { x}})} = { { p}}({\bf{ 0}})\tag{13} \end{equation}

将式(8)代入式(13)中有

\begin{equation} \label{eq14} { { A}}({ { x}}){ { c}}({ { x}}) = { { p}}({ { x}})\tag{14} \end{equation}

式中${ { A}}({ { x}})$称作矩量矩阵

\begin{equation} \label{eq15} { { A}}({ { x}}) = \sum\limits_{I = 1}^{NP} {{ {p}}({ { x}}_I - { { x}}){ { p}}^{\rm T}({ { x}}_I-{ {x}})\phi _s ({ { x}}_I - { { x}})}\tag{15} \end{equation}

由式(14)有${ { c}}({ { x}}) = { { A}}^{ - 1}({ {x}}){ {p}}({ { x}})$,将其代入式(8)可得无网格形函数

\begin{equation} \label{eq16} \varPsi _I ({ { x}}) = { { p}}^{\rm T}( 0){ { A}}^{ - 1}({ { x}}){ { p}}({ { x}}_I - { { x}})\varphi _s ({ { x}}_I - { { x}})\tag{16} \end{equation}

对式(16)直接微分,无网格形函数的一阶和二阶导数可以表示为

$$\varPsi _{I,i} ({ { x}}) = { { p}}^{\rm T}( 0)\{{ { A}}_{,i}^{ - 1} { { p}}_I \phi _{sI} + { { A}}^{ - 1}{ { p}}_{I,i} \phi _{sI} + \\ { { A}}^{ - 1}{ { p}}_I \phi _{sI,i} \}\tag{17}$$

$$\varPsi _{I,ij} ({ { x}}) = { { p}}^{\rm T}( 0)\{{ { A}}_{,ij}^{ - 1} { { p}}_I \phi _{sI} + { { A}}_{,i}^{ - 1} { { p}}_{I,j} \phi _{sI} + \\ { { A}}_{,i}^{ - 1} { { p}}_I \phi _{sI,j} \mbox{ + }{ { A}}_{,j}^{ - 1} { { p}}_{I,i} \phi _{sI} + { { A}}^{ - 1}{ { p}}_{I,ij} \phi _{sI}+ \\ { { A}}^{ - 1}{ { p}}_{I,i} \phi _{sI,j} + { { A}}_{,j}^{ - 1} { { p}}_I \phi _{sI,i} \mbox{ + }{ { A}}^{ - 1}{ { p}}_{I,j} \phi _{sI,i}+ \\ { { A}}^{ - 1}{ { p}}_I \phi _{sI,ij} \} \tag{18} $$

其中${{ {p}}_I} = { {p}}({{ {x}}_I} - { {x}}), {\phi _{sI}} = {\phi _s}({{ {x}}_I}-{ {x}})$,下标中逗号为微分算子,$\{i,j\} = \{x,y\}$. ${ { A}}_{,i}^{ - 1} $和${ { A}}_{,ij}^{ - 1} $为

$$\left. \begin{array}{l} { {A}}_{,i}^{ - 1} = - {{ {A}}^{-1}}{{ {A}}_{,i}}{{ {A}}^{ - 1}}\\[3mm] { {A}}_{,ij}^{ - 1} = - {{ {A}}^{ - 1}}({{ {A}}_{,ij}}{{ {A}}^{ - 1}} + {{ {A}}_{,i}}{ {A}}_{,j}^{ - 1} + {{ {A}}_{,j}}{ {A}}_{,i}^{ - 1}) \end{array} \right\}\tag{19} $$

由式(16)$\sim$式(19)可以看出,由于无网格形函数为有理式,其导数计算相当繁琐复杂,且计算效率较低.此外,无网格形函数导数的复杂特点也直接导致了通常需要采用高阶的高斯积分才能得到收敛的结果,进一步降低了计算效率.

注意到式(12)定义的无网格形函数的一致性或完备条件是无网格法收敛性的保证,对其直接微分可得到无网格形函数标准梯度应满足的一致性条件.具体到二维二次基函数,相应的一阶和二阶梯度的一致性条件为

\begin{equation} \label{eq20} \left. {\begin{array}{l} \sum\limits_{I = 1}^{NP} {\varPsi _{I,x} ({ { x}})y_I } = \sum\limits_{I = 1}^{NP} {\varPsi _{I,y} ({ { x}})x_I } = 0 \\[5mm] \sum\limits_{I = 1}^{NP} {\varPsi _{I,x} ({ { x}})x_I } = \sum\limits_{I = 1}^{NP} {\varPsi _{I,y} ({ { x}})y_I } = 1 \\ \end{array}} \right\}\tag{20} \end{equation} \begin{equation} \label{eq21} \left. {\begin{array}{l} \sum\limits_{I = 1}^{NP} {\varPsi _{I,xx} ({ { x}})x_I^2 } = \sum\limits_{I = 1}^{NP} {\varPsi _{I,yy} ({ { x}})y_I^2 } = 2 \\[5mm] \sum\limits_{I = 1}^{NP} {\varPsi _{I,xy} ({ { x}})x_I y_I } = 1 \\ \end{array}} \right\}\tag{21} \end{equation}

值得注意的是,要使得式(21)成立,无网格形函数至少要采用二次基函数.

3 无网格梯度光滑方法及一致性条件

3.1 无网格形函数的光滑梯度

根据梯度光滑理论[24],无网格形函数的光滑梯度,记作$\tilde {\varPsi }_{I,i} ({ { x}})$,可以表示成

\begin{equation} \label{eq22} \tilde {\varPsi }_{I,i} ({ { x}}) = \int_\varOmega \vartheta ({ { x}},{ { y}})\varPsi _{I,i} ({ { y}}){\rm d}\varOmega\tag{22} \end{equation}

其中$\vartheta ({ { x}},{ { y}})$是光滑函数.方便起见,计算中可以选取无网格形函数作为光滑函数[27-29],即$\vartheta ({ { x}},{ { x}}_J ) = \varPsi _J ({ {x}})$. 因而,在离散情况下,无网格形函数的一阶光滑导数,记作$\bar{\varPsi }_{I,i} ({ { x}})$,可以表示为如下形式

$$\bar {\varPsi }_{I,i} ({ { x}}) = \sum\limits_{J = 1}^{NP} {\vartheta ({ { x}},{ { x}}_J )\varPsi _{I,i} ({ { x}}_J )} = \\ \sum\limits_{J = 1}^{NP} \varPsi _J ({ { x}})\varPsi _{I,i} ({ { x}}_J )\tag{23} $$

对式(23)直接微分可得无网格形函数的二阶光滑梯度

$$ \label{eq24} \bar {\varPsi }_{I,ij}({ { x}}) = \sum\limits_{J = 1}^{NP} \varPsi _{J,j} ({ { x}})\varPsi _{I,i} ({ { x}}_J )\tag{24} $$

图2图3分别对比了一维和二维无网格形函数的标准梯度与光滑梯度之间的区别,其中标准梯度采用的是二次基函数$(p= 2)$,光滑梯度采用的是线性基函数$(p = 1)$.从图中可以看出,线性基无网格形函数的光滑梯度甚至比采用二次基函数的无网格形函数的标准梯度更加光滑,震荡性更小.同时,经过梯度光滑构造,形函数光滑梯度的影响域出现叠加效应,其影响域大于标准梯度的影响域.因此,基于梯度光滑方法,采用线性基函数仍然可以得到光滑的二阶梯度.

图2   一维无网格形函数标准梯度与光滑梯度对比

Fig. 2   Comparison of 1D standard and smoothed meshfree gradients

图3   二维无网格形函数标准梯度与光滑梯度对比

Fig. 3   Comparison of 2D standard and smoothed meshfree gradients

3.2 线性基形函数光滑梯度的线性一致性条件

如前所述,形函数的一致性条件是收敛性的保证.因此这里首先证明线性基无网格形函数的光滑梯度满足类似于式(20)给出的无网格形函数标准梯度对应的常规一致性条件.对于一阶光滑梯度,有

$$\sum\limits_{I = 1}^{NP} {\bar {\varPsi }_{I,y} ({ { x}})x_I } = \sum\limits_{I = 1}^{NP} {\sum\limits_{J = 1}^{NP} {\varPsi _J ({ {x}})\varPsi _{I,y} ({ { x}}_J )x_I } } =\\ \sum\limits_{J = 1}^{NP} {\varPsi _J ({ { x}})\underbrace {\sum\limits_{I = 1}^{NP} {\varPsi _{I,y} ({ { x}}_J )x_I } }_{ = 0}} = 0 \tag{25} $$

$$\sum\limits_{I = 1}^{NP} {\bar {\varPsi }_{I,x} ({ { x}})x_I } = \sum\limits_{I = 1}^{NP} {\sum\limits_{J = 1}^{NP} {\varPsi _J ({ {x}})\varPsi _{I,x} ({ { x}}_J )x_I } } = \\ \sum\limits_{J = 1}^{NP} {\varPsi _J ({ { x}})\underbrace {\sum\limits_{I = 1}^{NP} {\varPsi _{I,x} ({ { x}}_J )x_I } }_{ = 1}} =\\ \sum\limits_{J = 1}^{NP} {\varPsi _J ({ { x}})} = 1\tag{26}$$

式(25)和式(26)的推导使用了式(12). 同理可得

\begin{equation} \label{eq27} \sum\limits_{I = 1}^{NP} {\bar {\varPsi }_{I,x} ({ {x}})y_I } = 0,\mbox{ }\sum\limits_{I = 1}^{NP} {\bar {\varPsi }_{I,y} ({ { x}})y_I } = 1\tag{27} \end{equation}

图4(a)验证了在一维情况下,线性基无网格形函数光滑梯度的常规一致性条件.可以看出,线性基无网格形函数的一阶光滑梯度是满足式(26)和式(27)给出的一致性条件.与一维情况类似,图4(b)验证了二维线性基无网格形函数的一阶光滑梯度满足一致性条件.

图4   线性基无网格形函数一阶光滑梯度的常规一致性条件

Fig. 4   Standard consistency conditions for the first order smoothed meshfree gradients with linear basis function

3.3 线性基形函数光滑梯度的二阶一致性条件

除了常规的线性一致性条件,这里进一步证明,在均布离散条件下,线性基函数无网格形函数的二阶光滑梯度还满足二次基函数无网格形函数标准二阶梯度对应的式(21)给出的一致性条件.

首先考虑式(21)的第一个条件,即

$$\sum\limits_{I = 1}^{NP} {\bar {\varPsi }_{I,xx} ({ { x}})x_I^2 } = \sum\limits_{J = 1}^{NP} {\varPsi _{J,x} ({ { x}})} \sum\limits_{I = 1}^{NP} {\varPsi _{I,x} ({ { x}}_J )(x_I - x_J + x_J )^2}= \\ \sum\limits_{J = 1}^{NP} {\varPsi _{J,x} ({ { x}})x_J^2 } \underbrace{\sum\limits_{I = 1}^{NP} \varPsi _{I,x} ({ { x}}_J )} _{ = 0} +\\ 2\sum\limits_{J = 1}^{NP} \varPsi _{J,x} ({ { x}})x_J \underbrace {\sum\limits_{I = 1}^{NP} \varPsi _{I,x} ({ { x}}_J )x_{IJ}} _{ = 1}+ \\ \sum\limits_{J = 1}^{NP} {\varPsi _{J,x} ({ { x}})} \sum\limits_{I = 1}^{NP} {\varPsi _{I,x} ({ { x}}_J )} x_{IJ}^2 =\\ 2\underbrace{\sum\limits_{J = 1}^{NP} {\varPsi _{J,x} ({ { x}})x_J } }_{ = 1} +\\ \sum\limits_{J = 1}^{NP} {\varPsi _{J,x} ({ { x}})} \sum\limits_{I = 1}^{NP} {\varPsi _{I,x} ({ { x}}_J )} x_{IJ}^2= \\ 2 + \sum\limits_{J = 1}^{NP} {\varPsi _{J,x} ({ { x}})} \sum\limits_{I = 1}^{NP} {\varPsi _{I,x} ({ { x}}_J )} x_{IJ}^2 \tag{28} $$

其中,$x_{IJ} = x_I - x_J $.此外,注意到对于均布无网格离散,形函数周期性分布的特点使得如下关系成立

\begin{equation} \label{eq29} \sum\limits_{I = 1}^{NP} {\varPsi _{I,x} ({ { x}}_J )} x_{IJ}^2 = \sum\limits_{K = 1}^{NP} {\varPsi _{K,x} ({ { x}}_L )} x_{KL}^2\tag{29} \end{equation}

因而,式(28)可以进一步简化为

\begin{equation} \label{eq30} \sum\limits_{I = 1}^{NP} {\bar {\varPsi }_{I,xx} ({ {x}})x_I^2 } = 2 + \sum\limits_{K = 1}^{NP} {\varPsi _{K,x} ({ {x}}_L )} x_{KL}^2 \underbrace {\sum\limits_{J = 1}^{NP} {\varPsi _{J,x} ({ { x}})} }_{ = 0} = 2\tag{30} \end{equation}

于是有

\begin{equation} \label{eq31} \sum\limits_{I = 1}^{NP} {\bar {\varPsi }_{I,xx} ({ {x}})x_I^2 } = 2\tag{31} \end{equation}

按照类似的方法,可以进一步证明

$$\mathop{\sum}\limits_{I = 1}^{NP} \bar{\varPsi }_{I,yy} ({ {x}})y_I^2 = 2\tag{32} $$

$$ \mathop{\sum}\limits_{I = 1}^{NP} \bar{\varPsi }_{I,xy} ({ {x}})x_I y_I = 1\tag{33}$$

值得注意的是,式(31)$\sim$式(33)的证明过程并不依赖于式(21)所给出的二阶一致性条件.因此,在仅采用线性基函数构造无网格形函数的情况下,通过梯度光滑的方法可以构造出满足二阶一致性条件的二阶光滑梯度.而在标准无网格形函数的范畴,只有采用二次基函数的无网格形函数才能满足二阶一致性条件,所以这里将式(31)$\sim$式(33)称之为额外高阶一致性条件.

图5(a)和图5(b)分别验证了一维和二维线性基无网格形函数二阶光滑梯度满足的额外高阶一致性条件.

图5   线性基无网格形函数二阶光滑梯度的二阶一致性条件

Fig. 5   Extra quadratic consistency conditions for the second order smoothed meshfree gradients with linear basis function

为了便于对比,图中也给出了线性基无网格形函数标准二阶梯度对应的二阶一致性条件.由图可知,线性基无网格形函数的二阶光滑梯度可以满足额外高阶一致性条件,而线性基无网格形函数的标准二阶梯度则不满足.这为构造薄板分析的线性基无网格梯度光滑伽辽金方法提供了必要条件.

3.4 插值误差分析

为了更深入研究线性基无网格形函数光滑梯度,本小节根据下面的误差形式对其进行插值精度分析

$$L_2 \mbox{error = }\bigg(\int_\varOmega {(w - w^h)^2{\rm d}\varOmega \bigg)^{1 / 2}} \bigg/ \bigg(\int_\varOmega {w^2{\rm d}\varOmega } \bigg)^{1 / 2}\tag{34}$$

$$H_{s_1 }\mbox{error} = \bigg(\int_\varOmega {(w_{,i} - w_{,i}^h )^2{\rm d}\varOmega } \bigg)^{1 / 2} \bigg/ \bigg(\int_\varOmega {w_{,i}^2 {\rm d}\varOmega } \bigg)^{1 / 2}\tag{35}$$

$$ H_{s_2 } \mbox{error} = \bigg(\int_\varOmega {(w_{,ij} - w_{,ij}^h )^2{\rm d}\varOmega } \bigg)^{1 / 2} \bigg/ \bigg(\int_\varOmega {w_{,ij}^2 {\rm d}\varOmega } \bigg)^{1 / 2}\tag{36}$$

其中,$w$,$w_{,i} $和$w_{,ij} $及$w^h$,$w_{,i}^h $和$w_{,ij}^h$分别表示精确解及无网格近似解,这些变量分别对应于薄板结构的挠度、转角和曲率.

在插值精度分析中,考虑如下的一维和二维挠度函数

$$w(x) = {\rm e}^x + \frac{1}{6}(1 - {\rm e})x^3 - \frac{1}{2}x^2 + \left(\frac{4}{3} - \frac{5}{6}{\rm e}\right)x - 1\tag{37}$$

$$ w({ { x}}) = \sin (\pi x)\sin (\pi y)\tag{38} $$

另一方面,$w^h$由式(7)给出,$w_{,i}^h $和$w_{,ij}^h $分别为

\begin{equation} \label{eq39} w_{,i}^h ({ { x}}) = \sum\limits_{I\mbox{ = }1}^{NP} {\bar {\varPsi }_{I,i} ({ { x}})d_I }\tag{39} \end{equation} \begin{equation} \label{eq40} w_{,ij}^h ({ { x}}) = \sum\limits_{I\mbox{ = }1}^{NP} {\bar {\varPsi }_{I,ij} ({ { x}})d_I }\tag{40} \end{equation}

式中$d_I = w({ { x}}_I )$.

图6给出了一维和二维线性基无网格形函数及其光滑梯度的插值误差.从图中可见,一维和二维线性基无网格形函数光滑梯度的插值误差有类似的收敛特性,其中对应$L_2$,$H_{s1 } $和$H_{s2 }$三种不同度量形式的收敛率分别为2,2,1,具有与二次基无网格形函数标准梯度相同的收敛特性,但无网格形函数所采用的基函数阶次降低了一次.

图6   线性基无网格形函数及光滑梯度的插值误差

Fig. 6   Comparison of interpolation errors of smoothed meshfree gradients with linear basis function

4 薄板分析的线性基梯度光滑伽辽金无网格法离散方程

将式(7)$\sim$式(39)及式(40)的无网格离散代入薄板问题的等效积分弱形式(6),可得线性基梯度光滑伽辽金无网格法的离散方程

\begin{equation}\label{eq41} {{\bar{ K}\bar {d}}} = {{\bar{ f}}}\tag{41} \end{equation}

其中

$${{ \bar { K}}}\mbox{ = }\left[ {{\begin{array}{*{20}c} { {K}}& {{ { G}}^{w{\rm T}} }& {{ { G}}^{\theta {\rm T}} }\\ {{ { G}}^w }& {\bf{ 0}}& {\bf{ 0}}\\ {{ { G}}^\theta }& {\bf{ 0}}& {\bf{ 0}} \\ \end{array} }} \right]\tag{42}$$

$${{ \bar { d}}} = \left\{ {{\begin{array}{*{20}c} { { d}}\\ { { \lambda }}^w \\ { { \lambda }}^\theta \\ \end{array} }} \right\}\mbox{, }{{ \bar { f}}} = \left\{ {{\begin{array}{*{20}c} { { f}}\\ { { q}}^w \\ { { q}}^\theta \\ \end{array} }} \right\}\tag{43}$$

$${ {K}} = \mathop {\rm A}\limits_{I,J = 1}^{NP} [{K_{IJ}}{\rm{], }}~{ {G}}^w = \mathop {\rm A}\limits_{I,K = 1}^{NP,NW} [G_{IK}^w{\rm{]}},{\rm{ }}{ {G}}_{}^\theta = \mathop {\rm A}\limits_{I,L = 1}^{NP,NR} [G_{IL}^\theta {\rm{]}}\qquad \tag{44}$$

$$ { {d}} = \mathop {\rm A}\limits_{I = 1}^{NP} [{d_I}{\rm{]}},~{\rm{ }}{{ {\lambda }}^w} = \mathop {\rm A}\limits_{K = 1}^{NW} [\lambda _K^w{\rm{], }}~{{ {\lambda }}^\theta } = \mathop {\rm A}\limits_{L = 1}^{NR} [\lambda _L^\theta {\rm{]}}\tag{45}$$

$${ {f}} = \mathop {\rm A}\limits_{I = 1}^{NP} [{f_I}{\rm{]}},~{\rm{ }}{{ {q}}^w} = \mathop {\rm A}\limits_{K = 1}^{NW} [q_K^w{\rm{], }}~{{ {q}}^\theta } = \mathop {\rm A}\limits_{L = 1}^{NR} [q_L^\theta {\rm{]}}\tag{46}$$

$$K_{IJ} \mbox{ = }\int_\varOmega {{{ \bar { B}}}_I^{\rm T} {{ D\bar { B}}}_J {\rm d}\varOmega } ,\mbox{ }{{ \bar { B}}}_I = \left\{ {{\begin{array}{*{20}c} {\bar {\varPsi }_{I,xx} }\\ {\bar {\varPsi }_{I,yy} }\\ {\bar {\varPsi }_{I,xy} + \bar {\varPsi }_{I,yx} }\\ \end{array} }} \right\}\tag{47}$$

$$G_{IK}^w = - \int_{\varGamma ^w} {\varPsi _I N_K } {\rm d}\varGamma ,\mbox{ }G_{IL}^\theta = - \int_{\varGamma ^\theta } {\bar {\varPsi }_{I,n} N_L } {\rm d}\varGamma\tag{48}$$

$$f_I = \int_\varOmega {\varPsi _I q{\rm d}\varOmega }\tag{49}$$

$$q_K^w = - \int_{\varGamma ^w} {N_K \bar {w}{\rm d}\varGamma } \mbox{, }q_L^\theta = - \int_{\varGamma ^\theta } {N_L \bar {\theta }_n {\rm d}\varGamma }\tag{50}$$

式中,A为组装算子[2],$NW$和$NR$分别表示边界$\varGamma^w$和$\varGamma ^\theta $的无网格离散点个数,$N_K$为对应于拉格朗日乘子离散的一维拉格朗日插值函数.对于刚度矩阵,本文采用高斯积分进行计算.例如,二维情况下,对于线性基梯度光滑伽辽金方法,采用2$\times$2的高斯积分方案,而对于标准的二次基伽辽金方法则使用6$\times$6的高斯积分方案.

5 算例

本小节通过3个典型算例,对线性基梯度光滑伽辽金无网格法的精度进行了系统的分析.方便起见,分别用"QMF"和"LGSMF"代表标准二次基伽辽金无网格法和线性基梯度光滑伽辽金无网格法.算例中的影响域大小均为相对影响域尺度,即节点之间距离的倍数.

5.1 欧拉梁问题

首先,考虑与薄板问题具有相同性质的一维梁问题.

本算例简支梁的材料与几何参数为:长度$L = 10$,截面宽度$b =0.1$,高度$t = 0.2$,杨氏模量$E = 2\times 10^6$,受到均布载荷$q(x)= - 1$的作用.

计算中取21,41,81,161个节点进行收敛率分析,QMF的核函数影响域取2.3,LGSMF的核函数影响域取1.3.

图7为相应的挠度,转角和曲率的收敛率结果,即2,2和1,QMF和LGSMF两种方法的收敛率基本相同,与前述的插值误差分析结果一致,但本文所提的基于线性基函数的LGSMF方法的精度高于采用二次基函数的QMF方法.

图7   简支梁问题的收敛率对比

Fig. 7   Convergence comparison for the simply supported beam problem

图8图9给出了21个节点无网格离散模型对应的挠度、转角、曲率的计算结果及其误差分布图,进一步表明LGSMF方法的计算误差小于传统的QMF方法.此外,图10为线性分布载荷$q(x) =-x$作用下悬臂梁问题的收敛特性分析,其结果与简支梁问题的结果完全一致,仍然说明LGSMF方法的计算精度优于QMF方法.

图8   简支梁问题的结果对比

Fig. 8   Comparison of results for the simply supported beam problem

图9   简支梁问题的误差对比

Fig. 9   Error comparison for the simply supported beam problem

图10   悬臂梁问题的收敛率对比

Fig. 10   Convergence comparison for the cantilever beam problem

5.2 简支方板和矩形板问题

考虑图11所示的简支板,其长度为$L_x $,宽度为$L_y $,厚度$t =0.1$,杨氏模量$E = 2.0\times 10^6$,泊松比$\nu = 0.3$.该问题的解析解为

图11   简支方板问题示意图

Fig. 11   Description of the simply supported square plate problem

\begin{equation} \label{eq51} w({ { x}}) = - \sin \left(\frac{\pi x}{L_x }\right)\sin \left(\frac{\pi y}{L_y }\right)\tag{51} \end{equation}

其对应的分布载荷为$q({ { x}}) = \bar {D}\nabla ^2\nabla ^2w({ {x}})$.

$$ q({ x})=-\bar{D}\left(\dfrac{\pi^2}{L^2_x}+\dfrac{\pi^2}{L^2_y}\right)^2\sin \left(\dfrac{\pi x}{L_x}\right)\sin\left(\dfrac{\pi y}{L_y}\right)$$

首先考虑简支方板问题,其几何尺寸为$L_x = L_y =10$,无网格离散模型见图12,其中4个模型的无网格节点数分别11$\times$11,21$\times $21,31$\times $31和41$\times $41.计算中QMF方法的核函数影响域取2.4,而LGSMF方法的核函数影响域取1.1.图13给出了该板问题的挠度、转角和曲率收敛率分析,相应的收敛率为2,2,1,QMF\!和\!LGSMF\!两种方法的收敛率大致相同.

图12   简支方板的无网格离散模型

Fig. 12   Meshfree discretizations for the simply supported square plate problem

图13   简支方板问题的收敛率对比

Fig. 13   Convergence comparison for the simply supported square plate problem

图14给出了41$\times$41个节点无网格离散模型对应的挠度、转角、曲率误差分布图,图13图14的结果均表明LGSMF方法的精度更高.对于$L_x = 10$,$L_y = 5$的矩形板问题,采用11$\times$11,21$\times $21,31$\times $31和41$\times$41的无网格离散模型对应收敛率的结果见图15,同样表明LGSMF方法具有更高的计算精度.

图14   简支方板问题的误差对比

Fig. 14   Error comparison for the simply supported square plate problem

图15   简支矩形板问题的收敛率对比

Fig. 15   Convergence comparison for the simply supported rectangular plate problem

5.3 固支圆板问题

考虑图16所示的受均布载荷$q =-10$作用周边固支圆板,其几何和材料参数为:半径$R = 5$,厚度$t = 0.1$,杨氏模量$E = 2.0\times 10^6$,泊松比$\nu = 0.3$. 该问题的精确解[30]

图16   固支圆板问题示意图

Fig. 16   Description of the clamped circular plate problem

\begin{equation} \label{eq52} w({ { x}}) = \frac{q}{64\bar {D}}(R^2 - x^2-y^2)^2\tag{52} \end{equation}

图17列出了圆板问题的无网格离散模型,分别对应121,361,729,1089个无网格离散节点.由于圆板本身的几何特性,无网格离散模型中节点呈现非均匀分布特性.伽辽金无网格分析中,采用二次基函数的QMF方法的核函数影响域为2.2,采用线性基函数的LGSMF方法的核函数影响域为1.5.注意到由于基函数阶次不同,QMF方法的影响域至少要大于2,而LGSMF方法的最小影响域为1.图18给出了圆板问题的挠度、转角和曲率的收敛率计算结果.可见,对于非均匀节点离散的圆板问题,虽然线性基形函数光滑梯度不严格满足二阶一致性条件,但采用线性基函数的LGSMF方法的精度仍然高于采用二次基函数的传统QMF方法,并能达到与方板问题相当的收敛率.

图17   固支圆板无网格离散模型

Fig. 17   Meshfree discretizations for the clamped circular plate problem

图18   固支圆板问题的收敛率对比

Fig. 18   Convergence comparison for the clamped circular plate problem

6 结论

薄板分析的四阶控制方程要求采用至少二次基无网格形函数来保证伽辽金方法的收敛性.本文通过引入梯度光滑理论,构建了线性基无网格形函数的光滑梯度,进而建立了一种薄板分析的线性基梯度光滑伽辽金无网格法.该方法采用的线性基无网格形函数的光滑梯度不仅满足常规的线性梯度一致性条件,而且在一定条件下还满足二次基无网格形函数对应的标准二阶梯度一致性条件,使得采用线性基无网格形函数就能够进行薄板问题分析,降低了计算复杂度,为采用低次基函数快速求解高阶问题提供了一种有效的数值工具.

与二次基伽辽金无网格法相比,线性基梯度光滑伽辽金无网格法避免了繁琐的无网格形函数二阶梯度计算,而且光滑梯度有效降低了无网格形函数二阶梯度的震荡性,因而采用较少的积分点就可以保证计算精度.例如,文中线性基梯度光滑伽辽金无网格法仅需采用2点高斯积分方案,而传统的二次基伽辽金无网格法需要采用6点高斯积分方案.文中通过插值误差分析和数值算例,系统地验证了线性基梯度光滑伽辽金无网格法的精度.结果表明,线性基梯度光滑伽辽金无网格法和二次基伽辽金无网格法的收敛率相当,但前者精度更优,且所需基函数阶次降低了一次.

The authors have declared that no competing interests exist.


参考文献

[1] Zienkiewicz OC, Taylor RL, Zhu JZ.

The Finite Element Method: Its Basis and Fundamentals (7th Edition)

. Singapore: Elsevier, 2015

URL      [本文引用: 3]      摘要

Описание: Provides an in-depth background to understanding of finite element results and techniques for improving accuracy of finite element methods. This book helps the reader to identify and eliminate errors contained in finite element models. It talks about the three different error analysis techniques developed from a common theoretical foundation.
[2] Hughes TJR.

The Finite Element Method: Linear Static and Dynamic Finite Element Analysis

. New York: Dover Publications, 2000

[本文引用: 2]     

[3] Nayroles B, Touzot G, Villon P.

Generalizing the finite element method: diffuse approximation and diffuse elements

. Computational Mechanics, 1992, 10(5): 307-318

DOI      URL      [本文引用: 2]      摘要

This paper describes the new “diffuse approximation” method, which may be presented as a generalization of the widely used “finite element approximation” method. It removes some of the limitations of the finite element approximation related to the regularity of approximated functions, and to mesh generation requirements. The diffuse approximation method may be used for generating smooth approximations of functions known at given sets of points and for accurately estimating their derivatives. It is useful as well for solving partial differential equations, leading to the so called “diffuse element method” (DEM), which presents several advantages compared to the “finite element method” (FEM), specially for evaluating the derivatives of the unknown functions.
[4] Belytschko T, Lu YY, Gu L.

Element-free Galerkin methods

. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229-256

DOI      URL     

[5] Liu WK, Jun S, Li S, et al.

Reproducing kernel particle methods for structural dynamics

. International Journal for Numerical Methods in Engineering, 2010, 38(10): 1655-1679

[本文引用: 2]     

[6] 张雄, 刘岩, 马上.

无网格法的理论及应用

. 力学进展, 2009, 39(1): 1-36

DOI      URL      Magsci      摘要

详细论述了近年来迅速发展的无网格法的理论基础及其在各个领域内的应 用. 无网格法网格依赖性弱, 避免了传统的有限元、边界元等基于网格的数值方法 中可能出现的网格畸变和扭曲, 在一些有限元、边界元等方法难以较好处理的领域体现 出独特的优势. 以加权余量法为主线归纳了已有的30多种无网格法, 各类 无网格法的主要区别在于使用了不同的加权余量法和近似函数. 详尽介绍 了各种无网格近似方案(包括移动最小二乘近似、核近似和重构核近似、单位分 解近似、径向基函数近似、点插值近似、自然邻接点插值近似等)和无网格法 中常用的各类加权余量法(伽辽金格式、配点格式、局部弱形式、加权最小二乘 格式和边界积分格式等), 并讨论了数值积分方法和边界条件的处理等问题. 在 此基础上较系统地总结了无网格法在冲击爆炸、裂纹传播、超大变形、结 构优化、流固耦合、生物力学和微纳米力学等领域的应用, 展示了无网格法相 对于传统数值方法的优势.

(Zhang Xiong, Liu Yan, Ma Shang.

Meshfree methods and their applications

. Advances in Mechanics, 2009, 39(1): 1-36(in Chinese))

DOI      URL      Magsci      摘要

详细论述了近年来迅速发展的无网格法的理论基础及其在各个领域内的应 用. 无网格法网格依赖性弱, 避免了传统的有限元、边界元等基于网格的数值方法 中可能出现的网格畸变和扭曲, 在一些有限元、边界元等方法难以较好处理的领域体现 出独特的优势. 以加权余量法为主线归纳了已有的30多种无网格法, 各类 无网格法的主要区别在于使用了不同的加权余量法和近似函数. 详尽介绍 了各种无网格近似方案(包括移动最小二乘近似、核近似和重构核近似、单位分 解近似、径向基函数近似、点插值近似、自然邻接点插值近似等)和无网格法 中常用的各类加权余量法(伽辽金格式、配点格式、局部弱形式、加权最小二乘 格式和边界积分格式等), 并讨论了数值积分方法和边界条件的处理等问题. 在 此基础上较系统地总结了无网格法在冲击爆炸、裂纹传播、超大变形、结 构优化、流固耦合、生物力学和微纳米力学等领域的应用, 展示了无网格法相 对于传统数值方法的优势.
[7] Chen JS, Hillman M, Chi SW.

Meshfree methods: progress made after 20 years. Journal of

Engineering Mechanics-ASCE, 2017, 143(4): 04017001

DOI      URL      [本文引用: 1]     

[8] Krysl P, Belytschko T.

Analysis of thin plates by the element-free Galerkin method

. Computational Mechanics, 1995, 17(1-2): 26-35

DOI      URL      [本文引用: 1]      摘要

A meshless approach to the analysis of arbitrary Kirchhoff plates by the Element-Free Galerkin (EFG) method is presented. The method is based on moving least squares approximant. The method is meshless, which means that the discretization is independent of the geometric subdivision into “finite elements”. The satisfaction of the C 1 continuity requirements are easily met by EFG since it requires only C 1 weights; therefore, it is not necessary to resort to Mindlin-Reissner theory or to devices such as discrete Kirchhoff theory. The requirements of consistency are met by the use of a quadratic polynomial basis. A subdivision similar to finite elements is used to provide a background mesh for numerical integration. The essential boundary conditions are enforced by Lagrange multipliers. It is shown, that high accuracy can be achieved for arbitrary grid geometries, for clamped and simply-supported edge conditions, and for regular and irregular grids. Numerical studies are presented which show that the optimal support is about 3.9 node spacings, and that high-order quadrature is required.
[9] Liu GR, Chen XL.

A mesh-free method for static and free vibration analyses of thin plates of complicated shape

. Journal of Sound and Vibration, 2001, 241(5): 839-855

DOI      URL      [本文引用: 1]      摘要

A mesh-free method is presented to analyze the static deflection and the natural frequencies of thin plates of complicated shape. The present method uses moving least-squares (MLS) interpolation to construct shape functions based on a set of nodes arbitrarily distributed in the analysis domain. Discrete system equations are derived from the variational form of system equation. For static analysis, a penalty method is presented to enforce the essential boundary conditions. For frequency analysis of free vibration, the essential boundary conditions are represented through a weak form and imposed using orthogonal transformation techniques. The present EFG method together with techniques for imposing boundary conditions is coded in Fortran. Numerical examples are presented for rectangular, elliptical, polygonal and complicated plates to demonstrate the convergence and efficiency of the present method.
[10] Long SY, Atluri SN.

A meshless local Petrov--Galerkin method for solving the bending problem of a thin plate

. CMES: Computer Modeling in Engineering and Sciences, 2002, 3(1): 53-63

[本文引用: 1]     

[11] Lu H, Li S, Simkins DC, et al.

Reproducing kernel element method Part III: Generalized enrichment and applications

. Computer Methods in Applied Mechanics and Engineering, 2004, 193: 989-1011

DOI      URL      [本文引用: 1]      摘要

In this part of the work, a notion of generalized enrichment is proposed to construct the global partition polynomials or to enrich global partition polynomial basis with extra terms corresponding to the higher order derivatives of primary variable. This is accomplished by either multiplying enrichment functions with the original global partition polynomials, or increasing the order of global partition polynomials in the same mesh. Without refining mesh, high order consistency in interpolation hierarchy with generalized Kronecker delta property can be straightforwardly achieved in quadrilateral and triangular mesh in 2D by the proposed scheme. Comparing with the traditional finite element methods, the construction proposed here has more flexibility and only needs minimal degrees of freedom. The optimal element with high reproducing capacity and overall minimal degrees of freedom can be constructed by the generalized enrichment procedure. Two optimal elements in two dimensional space have been constructed: T10P3I 4 3 triangular element satisfies third order consistency condition with only 10 degrees of freedom, and Q15P4I 4 3 quadrilateral element satisfies fourth order consistency condition with 15 degrees of freedom. The performance of interpolation hierarchy is evaluated through solving some bench-mark problems for thin (Kirchhoff) plates.
[12] Liu Y, Hon YC, Liew KM.

A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems

. International Journal for Numerical Methods in Engineering, 2006, 66(7): 1153-1178

DOI      URL      [本文引用: 1]      摘要

A meshfree computational method is proposed in this paper to solve Kirchhoff plate problems of various geometries. The deflection of the thin plate is approximated by using a Hermite-type radial basis function approximation technique. The standard Galerkin method is adopted to discretize the governing partial differential equations which were derived from using the Kirchhoff's plate theory. The degrees of freedom for the slopes are included in the approximation to make the proposed method effective in enforcing essential boundary conditions. Numerical examples with different geometric shapes and various boundary conditions are given to verify the efficiency, accuracy, and robustness of the method. Copyright 漏 2005 John Wiley & Sons, Ltd.
[13] 马丽红, 邱志平, 王晓军.

Winkler地基板的区间无网格Galerkin方法

. 岩土工程学报, 2008, 30(3): 384-389

DOI      URL      Magsci      [本文引用: 1]      摘要

研究了具有有界不确定结构参数Winkler地基板的弯曲问题。将无网格Galerkin方法与区间数学相结合,将不确定参数用区间数来描述,给出了估计Winkler地基板弯曲挠度范围的区间无网格Galerkin方法。通过数值算例,表明了本文方法的有效性和可行性。

(Ma Lihong, Qiu Zhiping, Wang Xiaojun, et al.

Interval element-free Galerkin method for plates on Winkler foundation. Chinese Journal of

Geotechnical Engineering. 2008, 30(3): 384-389 (in Chinese))

DOI      URL      Magsci      [本文引用: 1]      摘要

研究了具有有界不确定结构参数Winkler地基板的弯曲问题。将无网格Galerkin方法与区间数学相结合,将不确定参数用区间数来描述,给出了估计Winkler地基板弯曲挠度范围的区间无网格Galerkin方法。通过数值算例,表明了本文方法的有效性和可行性。
[14] Bui TQ, Nguyen MN.

A moving Kriging interpolation-based meshfree method for free vibration analysis of Kirchhoff plates

. Computers & Structures, 2011, 89(3-4): 380-394

DOI      URL      [本文引用: 1]      摘要

The present work aims to make a further development of a novel meshfree method for free vibration analysis of classical Kirchhoff’s plates. The deflection of plates is approximated by the moving Kriging interpolation method which possesses the Kronecker’s delta property. This thus makes the proposed method efficient and straightforward in imposing the essential boundary conditions, and no special treatment techniques are required. A standard weak form is adapted to discrete the governing partial differential equations of plates. Numerical examples with different geometric shapes are considered to demonstrate the applicability and the accuracy of the proposed method.
[15] Cui XY, Liu GR, Li GY, et al.

A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells

. International Journal for Numerical Methods in Engineering, 2011, 85(8): 958-986

DOI      URL      [本文引用: 1]      摘要

A formulation for thin plates with only the deflection as nodal variables has been proposed using the generalized gradient smoothing technique and the radial point interpolation method (RPIM). The deflection fields are approximated using the RPIM shape functions which possess the Kronecker Delta property for easy impositions of essential boundary conditions. Three types of smoothing domains, which are also serving as the numerical integrations domains, are constructed based on the background three-node triangular cells and the generalized gradient smoothing operation is performed over each of them to obtain the smoothed curvatures. The generalized smoothed Galerkin weak form is then used to create the discretized system equations. The essential boundary conditions of rotations are imposed in the process of constructing the curvature field, and the translation boundary conditions are imposed as in the standard FEM. A number of numerical examples, including both static and free vibration analysis, are studied using the present methods and the numerical results are compared with the analytical ones and those in the open literatures. The results show that the present formulation can obtain very stable and accurate solutions, even for the extremely irregular background cells. Copyright 漏 2010 John Wiley & Sons, Ltd.
[16] Millan D, Rosolen A, Arroyo M.

Thin shell analysis from scattered points with maximum-entropy approximants

. International Journal for Numerical Methods in Engineering, 2011, 85(6): 723-751

DOI      URL      [本文引用: 1]      摘要

We present a method to process embedded smooth manifolds using sets of points alone. This method avoids any global parameterization and hence is applicable to surfaces of any genus. It combines three ingredients: (1) the automatic detection of the local geometric structure of the manifold by statistical learning methods; (2) the local parameterization of the surface using smooth meshfree (here maximum-entropy) approximants; and (3) patching together the local representations by means of a partition of unity. Mesh-based methods can deal with surfaces of complex topology, since they rely on the element-level parameterizations, but cannot handle high-dimensional manifolds, whereas previous meshfree methods for thin shells consider a global parametric domain, which seriously limits the kinds of surfaces that can be treated. We present the implementation of the method in the context of Kirchhoff鈥揕ove shells, but it is applicable to other calculations on manifolds in any dimension. With the smooth approximants, this fourth-order partial differential equation is treated directly. We show the good performance of the method on the basis of the classical obstacle course. Additional calculations exemplify the flexibility of the proposed approach in treating surfaces of complex topology and geometry. Copyright 漏 2010 John Wiley & Sons, Ltd.
[17] Zhang HJ, Wu JC, Wang DD.

Free vibration analysis of cracked thin plates by quasi-convex coupled isogeometric-meshfree method

. Frontiers of Structural and Civil Engineering, 2015, 9(4): 405-419

DOI      URL      [本文引用: 1]      摘要

The free vibration analysis of cracked thin plates via a quasi-convex coupled isogeometric-meshfree method is presented. This formulation employs the consistently coupled isogeometric-meshfree strategy where a mixed basis vector of the convex B-splines is used to impose the consistency conditions throughout the whole problem domain. Meanwhile, the rigid body modes related to the mixed basis vector and reproducing conditions are also discussed. The mixed basis vector simultaneously offers the consistent isogeometric-meshfree coupling in the coupled region and the quasi-convex property for the meshfree shape functions in the meshfree region, which is particularly attractive for the vibration analysis. The quasi-convex meshfree shape functions mimic the isogeometric basis function as well as offer the meshfree nodal arrangement flexibility. Subsequently, this approach is exploited to study the free vibration analysis of cracked plates, in which the plate geometry is exactly represented by the isogeometric basis functions, while the cracks are discretized by meshfree nodes and highly smoothing approximation is invoked in the rest of the problem domain. The efficacy of the present method is illustrated through several numerical examples.
[18] Wang DD, Chen JS.

A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration

. International Journal for Numerical Methods in Engineering, 2008, 74(3): 368-390

DOI      URL      [本文引用: 2]      摘要

A Hermite reproducing kernel (RK) approximation and a sub-domain stabilized conforming integration (SSCI) are proposed for solving thin-plate problems in which second-order differentiation is involved in the weak form. Although the standard RK approximation can be constructed with an arbitrary order of continuity, the proposed approximation based on both deflection and rotation variables is shown to be more effective in solving plate problems. By imposing the Kirchhoff mode reproducing conditions on deflectional and rotational degrees of freedom simultaneously, it is demonstrated that the minimum normalized support size (coverage) of kernel functions can be significantly reduced. With this proposed approximation, the Galerkin meshfree framework for thin plates is then formulated and the integration constraint for bending exactness is also derived. Subsequently, an SSCI method is developed to achieve the exact pure bending solution as well as to maintain spatial stability. Numerical examples demonstrate that the proposed formulation offers superior convergence rates, accuracy and efficiency, compared with those based on higher-order Gauss quadrature rule. Copyright 漏 2007 John Wiley & Sons, Ltd.
[19] Wang DD, Lin ZT.

Free vibration analysis of thin plates using Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration

. Computational Mechanics, 2010, 46(5): 703-719

DOI      URL      摘要

A Hermite reproducing kernel (HRK) Galerkin meshfree formulation is presented for free vibration analysis of thin plates. In the HRK approximation the plate deflection is approximated by the deflection as well as slope nodal variables. The n th order reproducing conditions are imposed simultaneously on both the deflectional and rotational degrees of freedom. The resulting meshfree shape function turns out to have a much smaller necessary support size than its standard reproducing kernel counterpart. Obviously this reduction of minimum support size will accelerate the computation of meshfree shape function. To meet the bending exactness in the static sense and to remain the spatial stability the domain integration for stiffness as well as mass matrix is consistently carried out by using the sub-domain stabilized conforming integration (SSCI). Subsequently the proposed formulation is applied to study the free vibration of various benchmark thin plate problems. Numerical results uniformly reveal that the present method produces favorable solutions compared to those given by the high order Gauss integration (GI)-based Galerkin meshfree formulation. Moreover the effect of sub-domain refinement for the domain integration is also investigated.
[20] Wang DD, Lin ZT.

Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures

. Computational Mechanics, 2011, 48(1): 47-63

DOI      URL      摘要

A dispersion analysis is carried out to study the dynamic behavior of the Hermite reproducing kernel (HRK) Galerkin meshfree formulation for thin beam and plate problems. The HRK approximation utilizes both the nodal deflectional and rotational variables to construct the meshfree approximation of the deflection field within the reproducing kernel framework. The discrete Galerkin formulation is fulfilled with the method of sub-domain stabilized conforming integration. In the dispersion analysis following the HRK Galerkin meshfree semi-discretization, both the deflectional and rotational nodal variables are expressed by harmonic functions and then substituted into the semi-discretized equation to yield the characteristic equation. Subsequently the numerical frequency and phase speed can be obtained. The transient analysis with full-discretization is performed by using the central difference time integration scheme. The results of dispersion analysis of thin beams and plates show that compared to the conventional Gauss integration-based meshfree formulation, the proposed method has more favorable dispersion performance. Thereafter the superior performance of the present method is also further demonstrated by several transient analysis examples.
[21] Wang DD, Peng HK.

A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates

. Computational Mechanics, 2013, 51(6): 1013-1029

DOI      URL      摘要

http://link.springer.com/article/10.1007%2Fs00466-012-0784-9
[22] Wang DD, Song C, Peng HK.

A circumferentially enhanced Hermite reproducing kernel meshfree method for buckling analysis of Kirchhoff--Love cylindrical shells

. International Journal of Structural Stability and Dynamics, 2015, 15(6): 1450090

DOI      URL      [本文引用: 1]      摘要

A circumferentially enhanced Hermite reproducing kernel (HRK) meshfree method is developed for the buckling analysis of Kirchhoffu2013Love cylindrical shells. In this method, in order to accurately represent the circumferential periodicity of cylindrical shells, the shell mid-surface is first discretized by a set of physical nodes in the two-dimensional parametric space, thereafter another set of dummy nodes are added by a straightforward periodic translation of the physical nodes. Subsequently the meshfree shape functions are constructed using both the physical nodes and the dummy nodes through a periodically linked relationship. The resulting meshfree shape functions exhibit the desired circumferential periodicity. The meshfree shape functions are formulated in the HRK framework which can be degenerated to the standard reproducing kernel (RK) shape functions just by removing the rotational terms. Meanwhile, the cylindrical shell buckling equations are rationally derived from the consistent linearization of the internal virtual work. During the meshfree discretization, the in-plane shell displacements are represented by the conventional RK shape functions, while the out-of-plane shell deflection is approximated by the Hermite meshfree shape functions with both directional and rotational degrees of freedom. The numerical integration of the material as well as the geometric stiffness matrices are carried out by the strain smoothing sub-domain stabilized conforming integration (SSCI) method. Numerical examples show that the proposed approach yields very favorable results for the buckling analysis of cylindrical shells.
[23] Tanaka S, Sadamoto S, Okazawa S.

Nonlinear thin-plate bending analyses using the Hermite reproducing kernel approximation

. International Journal of Computational Methods, 2012, 9(1): 1240012

DOI      URL      [本文引用: 1]      摘要

This study analyzed thin-plate bending problems with a geometrical nonlinearity using the Hermite reproducing kernel approximation and sub-domain-stabilized conforming integration. In thin-plate bending analyses, the deflections and rotations satisfy so-called Kirchhoff mode reproducing conditions. It is then possible to solve large deflection analyses of thin plates, such as elastic bucking problems, with high accuracy and efficiency. Total Lagrangian method is applied to solve the geometrical nonlinearity of the thin plates' deflections and rotations. The Greenu2013Lagrange strain and second Piolau2013Kirchhoff stress forms are adopted to represent the strains and stresses in the thin plates. Mathematical formulation and some numerical examples are also demonstrated.
[24] Chen JS, Wu CT, Yoon S, et al.

A stabilized conforming nodal integration for Galerkin mesh-free methods

. International Journal for Numerical Methods in Engineering, 2015, 50(2): 435-466

URL      [本文引用: 2]     

[25] 吴俊超, 邓俊俊, 王家睿.

伽辽金型无网格法的数值积分方法

. 固体力学学报, 2016, 37(3):208-233

URL      摘要

无网格法直接通过节点信息构造形函数,不依赖于节点之间的有序单元连接,能够建立任意高阶连续的整体协调形函数.与传统的有限元法相比,无网格法对大变形问题、移动边界问题和高阶问题的求解有比较明显的优势.伽辽金型无网格法是目前应用最为广泛的一类无网格法.虽然无网格形函数本身不依赖于单元,但伽辽金型无网格法需要采取合适的方法进行弱形式的数值积分.由于无网格形函数一般不是多项式,具有非插值性且影响域与背景积分网格通常不重合,伽辽金型无网格法通常需要采用高阶的高斯积分进行数值积分,导致了计算效率低下,难于求解大型实际问题.因此,如何通过建立高效积分方法提高无网格法的计算效率成为无网格法研究领域的一个核心问题.论文总结了伽辽金型无网格法中若干常用的数值积分方法,并对伽辽金型无网格法的数值积分方法领域存在的一些问题进行了探讨.

(Wu Junchao, Deng Junjun, Wang Jiarui, et al.

A review of numerical integration approaches for Galerkin meshfree methods. Chinese Journal of

Solid Mechanics. 2016, 37(3):208-233 (in Chinese))

URL      摘要

无网格法直接通过节点信息构造形函数,不依赖于节点之间的有序单元连接,能够建立任意高阶连续的整体协调形函数.与传统的有限元法相比,无网格法对大变形问题、移动边界问题和高阶问题的求解有比较明显的优势.伽辽金型无网格法是目前应用最为广泛的一类无网格法.虽然无网格形函数本身不依赖于单元,但伽辽金型无网格法需要采取合适的方法进行弱形式的数值积分.由于无网格形函数一般不是多项式,具有非插值性且影响域与背景积分网格通常不重合,伽辽金型无网格法通常需要采用高阶的高斯积分进行数值积分,导致了计算效率低下,难于求解大型实际问题.因此,如何通过建立高效积分方法提高无网格法的计算效率成为无网格法研究领域的一个核心问题.论文总结了伽辽金型无网格法中若干常用的数值积分方法,并对伽辽金型无网格法的数值积分方法领域存在的一些问题进行了探讨.
[26] 王冰冰, 段庆林, 李锡夔.

薄板弯曲分析的高阶高效无网格法

. 固体力学学报, 2018, 39(2): 152-161

URL      [本文引用: 1]      摘要

与传统有限元法相比,无网格法具有节点形函数高度光滑、易于形成高阶近似等优势,更适合于以薄板弯曲问题为代表的高阶偏微分方程的数值求解.然而,高阶无网格法的形函数是非多项式的有理函数,导致弱形式的区域积分难以得到精确计算,通常采用的高阶高斯积分方法需使用大量积分点,计算效率低且精度不高.论文针对薄板弯曲问题的高阶(三阶)无网格法分析,首次发展了与该高阶近似相一致的曲率光顺方案,并基于背景三角形积分单元建立了相应的数值积分格式,大幅度减少了所需的积分点数目.所发展方法的关键在于计算刚度阵所需的形函数的二阶导数由形函数及其一阶导数通过散度定理确定,而非对形函数直接求导获得.数值结果表明,基于标准的高斯积分方案的高阶无网格法精度不高,不能精确再现纯弯曲和线性弯曲模式,且得到的弯矩场分布存在严重的虚假数值振荡.而论文所建议的基于曲率光顺方案的高阶无网格法能够方便高效地求解薄板弯曲问题,尤其是它能精确反映纯弯曲和线性弯曲模式.与标准的高斯积分方法和目前主流的常曲率光顺方法相比,论文方法在计算效率、精度、弯矩分布等方面均展现出显著优势,因而具有较好的应用价值.

(Wang Bingbing, Duan Qinglin, Li Xikui, et al.

An efficient higher-order meshfree method for thin plate analysis. Chinese Journal of

Solid Mechanics. 2018, 39(2): 152-161(in Chinese))

URL      [本文引用: 1]      摘要

与传统有限元法相比,无网格法具有节点形函数高度光滑、易于形成高阶近似等优势,更适合于以薄板弯曲问题为代表的高阶偏微分方程的数值求解.然而,高阶无网格法的形函数是非多项式的有理函数,导致弱形式的区域积分难以得到精确计算,通常采用的高阶高斯积分方法需使用大量积分点,计算效率低且精度不高.论文针对薄板弯曲问题的高阶(三阶)无网格法分析,首次发展了与该高阶近似相一致的曲率光顺方案,并基于背景三角形积分单元建立了相应的数值积分格式,大幅度减少了所需的积分点数目.所发展方法的关键在于计算刚度阵所需的形函数的二阶导数由形函数及其一阶导数通过散度定理确定,而非对形函数直接求导获得.数值结果表明,基于标准的高斯积分方案的高阶无网格法精度不高,不能精确再现纯弯曲和线性弯曲模式,且得到的弯矩场分布存在严重的虚假数值振荡.而论文所建议的基于曲率光顺方案的高阶无网格法能够方便高效地求解薄板弯曲问题,尤其是它能精确反映纯弯曲和线性弯曲模式.与标准的高斯积分方法和目前主流的常曲率光顺方法相比,论文方法在计算效率、精度、弯矩分布等方面均展现出显著优势,因而具有较好的应用价值.
[27] Wang DD, Wang JR, Wu JC.

Superconvergent gradient smoothing meshfree collocation method

. Computer Methods in Applied Mechanics and Engineering, 2018, 340: 728-766

DOI      URL      [本文引用: 2]      摘要

A superconvergent meshfree collocation method with smoothed nodal gradients is presented. In this method, the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function, while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients. It is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function, which facilitates a trivial numerical implementation. Subsequently, an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points. Based upon a local truncation error analysis, it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions. A key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation. Another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation, which is non-feasible in the conventional collocation formulation. The effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems. Numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method.
[28] Wang DD, Li ZY.

A two-level strain smoothing regularized meshfree approach with stabilized conforming nodal integration for elastic damage analysis

. International Journal of Damage Mechanics, 2013, 22(3): 440-459

DOI      URL     

[29] Wu YC, Wang DD, Wu CT.

Three dimensional fragmentation simulation of concrete structures with a nodally regularized meshfree method

. Theoretical and Applied Fracture Mechanics, 2014, 72: 89-99

DOI      URL      [本文引用: 1]      摘要

A three dimensional large deformation meshfree simulation of concrete fragmentation is presented by using a nodally regularized Galerkin meshfree method. This nodally regularized meshfree method is established with the two-level Lagrangian nodal gradient smoothing technique to relieve the material instability in failure modeling. The rate formulation is employed for the treatment of large deformation and therefore the two-level gradient smoothing is performed for the rate of deformation tensor and the deformation gradient. The essential characteristic of the present approach is that all the variables are conveniently computed at the meshfree nodes, which allows an efficient evaluation of the Galerkin weak form. The concrete failure is described by the KCC concrete model with three independent strength surfaces. This model has a pressure dependent evolving failure surface that is built with an internal damage variable. The computational implementation of the given concrete model within the context of meshfree formulation is discussed in detail. The effectiveness of the present method is demonstrated through several numerical examples of concrete structures.
[30] Timoshenko S, Krieger W.

Theory of Plates and Shells.

New York: McGraw-Hill, 1959

DOI      URL      [本文引用: 1]      摘要

Timoshenko, S.; Woinowsky-Krieger, S.

/