力学学报  2019 , 51 (3): 703-713 https://doi.org/10.6052/0459-1879-19-011

无网格粒子类方法专题

自由单元法及其在结构分析中的应用1)

高效伟2), 徐兵兵, 吕军, 彭海峰

(大连理工大学航空航天学院 工业装备结构分析国家重点实验室,大连 116024)

FREE ELEMENT METHOD AND ITS APPLICATION IN STRUCTURAL ANALYSIS1)

Gao Xiaowei2), Xu Bingbing, Lü Jun, Peng Haifeng

State Key Laboratory of Structural Analysis for Industrial Equipment, School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China

中图分类号:  O341

文献标识码:  A

通讯作者:  2)高效伟,教授,主要研究方向:计算力学. E-mail:xwgao@dlut.edu.cn

收稿日期: 2019-01-7

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(11672061, 11772083).

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摘要

通过吸收有限元与无网格法的优点,提出了一种新的数值方法------自由单元法.此方法在离散方面,采用有限元法中的等参单元,表征几何形状和进行物理量的插值;在算法方面,采用单元配点技术,逐点产生系统方程.主要特点是,在每个配置点只需要一个和周围自由选择的节点而形成的一个独立的等参单元,因而不需要考虑物理量在单元之间的相互连接关系与导数连续性问题. 本文介绍强形式与弱形式两种自由单元法,前者直接由控制方程和边界条件直接产生系统方程,后者通过在自由单元上建立控制方程的加权余量式产生弱形式积分式,并通过像传统有限元法中的积分过程建立系统方程组.本文提出的方法是一种单元配点法,对于域内点为了获得较高的导数精度,需要采用至少具有一个内部点的等参单元,为此除了可使用各阶次的拉格朗日四边形单元外, 还 给出了七节点三角形等参单元,用于模拟较为复杂的几何形状问题.

关键词: 自由单元法 ; 强形式自由单元法 ; 弱形式自由单元法 ; 单元配点法 ; 有限单元法 ; 无网格法

Abstract

By absorbing advantages of the finite element and meshless methods, a new numerical method, free element method, is proposed in the paper. In the discretization, the isoparametric elements as used in FEM are employed to represent the geometry and interpolate physical variables; and in the algorithm, the point collocation technique using elements is employed to generate the system of equations point by point. The main feature of the method is that only one independent element formed by freely selecting surrounding points is required for each collocation point, without need to consider the connective relationship between adjacent elements and the continuity of physical variables and their spatial derivatives at interfaces of the connected elements. Two types of free element methods, the strong-form method and weak form method, will be described in the paper. The former directly generates the system of equations from the governing equations and the Neumann boundary conditions, while the latter establishes the weak-form integral expression of the governing equations by the weighted residual technique over the free element first and then generates the system of equations through an integration process similar to that employed in the standard FEM. The method proposed in the paper is an element collocation method. To achieve highly accurate spatial derivatives for internal collocation points of the computational domain, isoparametric elements with at least one internal node are required. For this purpose, apart from the arbitrary order quadrilateral Lagrange elements, a new seven-node triangle element is constructed in the paper, which can be used to model problems with complex geometries.

Keywords: free element method ; strong-form free element method ; weak-form free element method ; element collocation method ; finite element method ; meshless method

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高效伟, 徐兵兵, 吕军, 彭海峰. 自由单元法及其在结构分析中的应用1)[J]. 力学学报, 2019, 51(3): 703-713 https://doi.org/10.6052/0459-1879-19-011

Gao Xiaowei, Xu Bingbing, Jun, Peng Haifeng. FREE ELEMENT METHOD AND ITS APPLICATION IN STRUCTURAL ANALYSIS1)[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 703-713 https://doi.org/10.6052/0459-1879-19-011

引 言

常用的传统数值方法是有限单元法(FEM)[1-4]、有限差分法(FDM)[5-8]、有限体积法(FVM)[9-11]、边界元法(BEM)[12-15]和无网格法(MLM)[16-20]. FEM采用有固定格式节点分布的等参单元,因而计算结果稳定性好,是求解众多工程问题的主流方法. 然而,建立FEM计算格式需要变分原理或能量原理,不同的问题所用的表达形式不同,不便于建立多场耦合问题的统一求解格式. FDM使用简单,但不适合求解不规则几何问题. FVM通过在每个计算点建立一个控制体的方式,由一次分部积分后的控制方程建立代数方程,具有系数矩阵带宽小、容易捕抓激波等间断现象的优点,是流体力学中的主流方法. 然而,FVM计算通量需要用到相邻控制体之间的联络关系,因此不易建立高阶格式的算法,通量的计算精度低. BEM具有只在边界上划分单元的优点,但在求解非线性、非均质问题时优势不明显. MLM不需要划分网格,便于对复杂几何问题的模拟,然而MLM采用计算点周围无规则分布的节点团构建形函数,具有计算效率低、结果稳定性差和不易施加边界条件与不同物性的弱点.

近年来,本文作者提出了一种求解一般热传导[21]和固体力学[22]问题的新型强形式有限单元法------单元微分法(EDM),导出了计算二维(2D)和三维(3D)问题的一阶和二阶空间偏导数的显式表达式. 该表达式是由单元的形函数表示的,因而可用于任何连续量的微分计算. 单元微分法是一种强形式算法,在物理量表征方面,采用了有限元法中的等参单元表征技术,从而可以得到稳定的计算结果;而在算法方面,采用了配点法中的直接配点技术建立系统方程:对于单元内的节点,直接由控制方程建立系统方程,而对于单元之间的节点以及外部节点,则采用节点力平衡关系,由纽曼边界条件建立系统方程. EDM的特点是使用简单,不需要变分原理和积分计算建立系统方程组,其缺点是对于单元间的节点所建立的方程与所有与其相关连的单元的节点有关,因而系统方程的带宽较大.

为了克服上述算法的缺点,在EDM的基础上,作者在2017年提出了在每个节点只使用一个局部单元进行配点的思想[23],并称其为自由单元配点法(FECM)[24],简称自由单元法. 该方法吸收了上述方法的优点,具有不需要变分原理、容易采用高阶格式、计算结果稳定等优点. 其中,最突出的特点是在每个配置点,可以自由选择周围的节点形成一个自由的与周围单元没有连接的等参单元,并对所有的内部点采用控制方程、外部边界点采用纽曼边界条件进行配点建立系统方程组. 虽然自由单元法形成的系统方程是非对称的,但由于每个节点只需要一个单元,因此建立的系统方程矩阵是极度稀疏的,这可以在一定程度上弥补非对称性带来的求解效率降低的损失. 本文将系统介绍自由单元法在求解固体力学问题方面的情况,阐述其基本理论与应用潜力.

1 弹性力学定解问题

弹性力学问题的控制微分方程可表示为[1]

$$\dfrac{\partial { \sigma}_{ij} ({\pmb x})}{\partial x_j } + f_i ({\pmb x}) = 0 (1)$$

其中,${ \sigma}_{ij} $为应力张量,$f_i $为体积力,重复指标表示求和(以下皆同).

应力张量与应变张量之间的关系可由下列广义胡克定律给出

$${ \sigma}_{ij} = { D}_{ijkl} { \varepsilon}_{kl} (2)$$

其中,${ \varepsilon}_{kl} $为应变张量,${ D}_{ijkl} $为弹性本构张量,其表达式为

${ \varepsilon }_{kl} = \dfrac{1}{2}\left( {\dfrac{\partial { u}_k }{\partial x_l }+ \dfrac{\partial { u}_l }{\partial x_k }} \right) (3)$

${ D}_{ijkl} = \lambda \delta _{ij} \delta _{kl} + \mu \left( {\delta _{ik} \delta _{jl} + \delta _{il} \delta _{jk} } \right) (4)$

$\lambda = \dfrac{2\mu \nu }{\left( {1 - 2\nu } \right)} (5)$

式 中,$u_k $为位移矢量,$\mu $为弹性剪切模量,$\nu $为泊松比.

弹性力学的边界条件可以表示为

$$ \left.\!\!\begin{array}{ll} {u}_i ({\pmb x}) = \bar {u}_i ({\pmb x}) \,, & {\pmb x} \in \varGamma _u { t}_i ({\pmb x}) = \sigma _{ij} ({\pmb x})n_j (x) = \bar {t}_i ({\pmb x}) \,, & {\pmb x} \in \varGamma _t \end{array} \!\! \right \} (6) $$

其中,${ t}_i $为边界面力矢量,$\varGamma _u $为已知位移的边界,$\varGamma _t $为已知面力的边界.

为了方便,将式(3)代入式(2),然后将其结果代入式(1),则弹性力学问题的控制方程式可写为

$${ D}_{ijkl} \dfrac{\partial^2 u_k ({\pmb x})}{\partial x_j \partial x_l } + f_i ({\pmb x}) = 0 (7)$$

而边界面力与应力的关系也可由位移梯度表示为

$${ t}_i ({\pmb x}) = { D}_{ijkl} \dfrac{\partial u_k ({\pmb x})}{\partial x_l }n_j ({\pmb x}) (8)$$

式(8)也称为纽曼边界条件. 从式(7)和式(8)可以看出,控制方程与边界条件涉及到的关键项是物理量对空间总体坐标的一阶与二阶偏导数,本文采用作者在文献[22]导出的单元微分公式计算各阶空间偏导数.

2 单元微分公式

作者在文献[22]中导出了任意等参单元形函数对总体坐标求一阶与二阶偏导数的显式表达式. 本文采用任意阶次的拉格朗日单元与七节点三角形单元.

2.1 拉格朗日等参单元

一维问题的拉格朗日插值函数可以表示为[21]

$$L_I^ (\xi ) = \prod\limits_{i = 1,i \ne I}^m \dfrac{\xi - \xi _i}{\xi _I - \xi _i } \,, \ \ I = 1 \sim m,\;\; - 1 \leqslant \xi \leqslant 1 (9)$$

其中,$m$为等参坐标(也叫自然坐标)$\xi $的插值点数. 通常,插值点$\xi _i$取区间[-1, 1]之间的等分节点,但本文采用不等间隔单元,使其适应性更强,其中各节点的等参坐标值$\xi_i $由单元相邻节点的距离来确定. 图1为四节点线单元节点分布示意图.

图1   四节点一维等参线单元

Fig. 1   Four-node line element

二维问题的拉格朗日形函数可由一维问题的插值函数构成[22]

$$N_\alpha (\xi ,\eta ) = L_I (\xi )L_J (\eta ) (10)$$

其中,下标$\alpha $的值是通过$\xi $方向的指标$I$和$\eta $方向的指标$J$的顺序排列形成的. 图2展示了16节点的二维等参单元节点分布,形函数由式(10)形成,如$N_7 $可以表示为$N_7 \left( {\xi ,\eta } \right) = L_3 \left( \xi \right)L_2 \left( \eta \right)$.

图2   十六节点矩形等参单元

Fig. 2   16-node quadrilateral element

2.2 七节点三角形等参单元

拉格朗日单元具有容易构造高阶单元的特点,但对于一些不规则的几何形状采用三角形单元更具灵活性. 为此,作者构造了一种具有一个内部点的七节点二次三角形等参单元(见图3),其形函数见附录.

图3   七节点三角形等参单元

Fig. 3   7-node triangle element

2.3 单元形函数对总体坐标的偏导数计算式

有了上述等参单元形函数,坐标和物理量在一个单元内的变化关系则可由单元结点值来表示

$x_i (\xi ,\eta ) = N_\alpha (\xi ,\eta )x_i^\alpha (11)$

$u_i (\xi ,\eta ) = N_\alpha (\xi ,\eta )u_i^\alpha (12)$

其中,$x_i^\alpha $和$u_i^\alpha $分别为单元结点$\alpha $的坐标和位移值,重复指标$\alpha$表示遍及所有单元结点求和. 对式(12)中的位移求偏导数可得[22]

$\dfrac{\partial u_i }{\partial x_j } = \dfrac{\partial N_\alpha }{\partial x_j }u_i^\alpha (13)$

$\dfrac{\partial ^2u_i }{\partial x_k \partial x_l } = \dfrac{\partial ^2N_\alpha }{\partial x_k \partial x_l }u_i^\alpha (14)$

从式(9)和式(10)可以看出,形函数是等参坐标$\xi _k $的显式函数,因此有

$\dfrac{\partial N_\alpha }{\partial x_i } = \dfrac{\partial N_\alpha }{\partial \xi _k }\dfrac{\partial \xi _k }{\partial x_i } = [J]_{ik}^{ - 1} \dfrac{\partial N_\alpha }{\partial \xi _k } (15)$

$\dfrac{\partial ^2N_\alpha }{\partial x_i \partial x_j } = \left[ {[J]_{ik}^{ - 1} \dfrac{\partial ^2N_\alpha }{\partial \xi _k \partial \xi _l } + \dfrac{\partial [J]_{ik}^{ - 1} }{\partial \xi _l }\dfrac{\partial N_\alpha }{\partial \xi _k }} \right]\dfrac{\partial \xi _l }{\partial x_j } (16)$

其中,$[J] = [\partial x / \partial \xi ]$为从总体坐标系到等参坐标系转换的雅可比矩阵.文献[22]导出了式(15)和式(16)中的各项显式计算式.

3 强形式的自由单元法(SFECM)

自由单元法的基本思想是像无网格法一样,将计算域离散成一系列节点,系统方程逐点配置而成.具体思路是,在每个配置点周围自由选择一些节点与其形成一个独立的等参单元(叫做配点单元),此单元是自由的(不与其他单元连接),并由式(13)$\sim$式(16)表达位移对坐标的一阶和二阶偏导数,然后将其直接代入控制方程与纽曼边界条件产生代数方程组.图4给出了采用九节点拉格朗日单元(黑线表示)和五节点十字单元[23](蓝线表示)作为配点单元的示意图.其他类型的等参单元[25]也可以作为配点单元,但对于内部点,为了获得较高的计算精度,需要使用至少有一个内部点的等参单元,以便将配置点放在单元的内部节点上;对于边界节点,可以使用任意形式的等参单元,因为配置点只能放在单元的边界节点上.

图 4   配置点及其等参单元示意图

Fig. 4   Collocation points and isoparametric element diagrams

这样,对于内部点,将微分关系式(14)直接代入控制方程(7)可得到下列方程

$${ D}_{ijkl} \dfrac{\partial ^2N_\alpha ({\pmb \xi} )}{\partial x_j \partial x_l }u_k^\alpha + f_i ({\pmb x})= 0 (17)$$

其中,${\pmb\xi }$表示对应于配置点的单元结点的等参坐标值,${\pmb x}$表示该配置点的总体坐标.

对于边界节点,将式(13)代入面力$\!$-$\!$-$\!$位移关系式(8)可得到下列方程

$$D_{ijkl} \dfrac{\partial N_\alpha ({\pmb \xi })}{\partial x_l }n_j u_k^\alpha ={ t}_i (18)$$

由式(17)和式(18)可见,内部点和边界点的代数方程为由所在单元各节点位移表示的线性方程组. 对所有的节点进行配点,并施加已知的边界条件后,则可建立起形如$AX=b$或$KU=F$的系统方程组(其处理与文献[22]和一般有限元法中的一样). 当所有节点的位移通过求解该系统方程获得后,则可由式(2) $\sim$式(4)和式(13)、式(15)计算各节点的应力值.

4 弱形式的自由单元法(WFECM)

与上述强形式一样,弱形式的自由单元法也是在每个节点采用一个与周围节点自由形成的等参单元. 不同之处是,为了使用高斯积分公式计算积分,所用的自由单元需要是等参变量变化范围为$[-1,+1]$之间的等参单元.这样,对每个配置点(用符号*标记),对控制方程(1)建立如下形式的加权余量积分式

$$\int_\varOmega {w\dfrac{\partial \sigma _{ij} }{\partial x_j }\varOmega } + \int_\varOmega {w\,f_i \varOmega } = 0 (19)$$

式中,积分域$\varOmega $为当前自由单元的面积(二维)或体积(三维),$w$为在$\varOmega $内连续的权函数.

对式(19)进行分部积分,并使用高斯散度公式和考虑式(2) $\sim$式(6)则可得到下列积分式

$$ \int_\varGamma {w\,t_j \varGamma } - \int_\varOmega {\dfrac{\partial w}{\partial x_j }D_{ijkl} \dfrac{\partial u_k }{\partial x_l } \varOmega } +\int_\varOmega {w\,f_i \varOmega } = 0 (20) $$

式中,$\varGamma $为积分域$\varOmega $的边界.

在式(20)中,如果取权函数$w$为1,则可得到像有限体积法[9]中那样的算法,但在本文中我们取$w$为式(12)中表示位移变化的形函数$N_\alpha $中对应于配置点的形函数$N_{\ast }$. 这样,使用式(13),由式(20)可得到如下伽辽金形式的积分公式

$$ \int_{\varGamma _{\rm s} } {N_{\ast } N_\beta \varGamma t_j^\beta } - \int_\varOmega {\dfrac{\partial N_{\ast } }{\partial x_j }D_{ijkl} \dfrac{\partial N_\alpha }{\partial x_l } \varOmega } \,u_k^\alpha + \int_\varOmega {N_{\ast } N_\alpha \varOmega } f_i^\alpha = 0 (21) $$

式中,边界积分域$\varGamma _s $为积分单元暴露在整个计算域外边的边界,当配置点为内部点时,该积分为零;指标$\beta $只取积分单元在外边界$\varGamma _{\rm s}$上的节点,$f_i^\alpha $表示体积力$f_i $在第$\alpha$个节点上的值.

需要说明的是,式(21)已考虑了配置点的形函数$N_{\ast }$在不包括其的其他边上的值为零的特性,因此该公式不适用于其他不完备的退化单元,如十字单元[23]和最简单元[25],但拉格朗日单元以及有限元法中的Serendipity单元都适用于该公式.此外,式(21)虽然与传统有限元法的公式在形式上相似,但在本文方法中,一个单元只用于一个节点,不需要与其他单元的结果进行组集,这是主要的不同.

5 数值算例

利用本文所推导的公式,我们编制了FORTRAN程序,并对一些弹性力学问题做了计算分析,用以验证强、弱两种形式方法的正确性.

5.1 端部受剪悬臂梁

为了验证算法的精度,这里给出了一个简单的二维平面问题算例. 现考虑一悬臂梁,在其自由端施加一抛物线型面力,模型如图5所示. 按照平面应力求解.

图5   自由端受剪力的悬臂梁模型

Fig. 5   Cantilever beam with shear free force at the end

假设右端抛物线型切向载荷满足公式

$$p = \dfrac{Py}{I}(y - D) (22)$$

则可以得到应力解析解为[16]

$$ \left.\begin{array}{l} \sigma _{11} = \dfrac{P}{I}(L - x)\left( {y - \dfrac{D}{2}} \right) \\ \sigma _{22} = 0 \sigma _{12} = \dfrac{Py}{I}(y - D) \end{array} \right \} (23) $$

上两式中,$L$为悬臂梁的长度,$D$为宽度,$I$为惯性矩,对于单位厚度的矩形截面有$I= D^3 / 12$.

在本问题中,梁的弹性模量取为200,GPa,泊松比取为0.3,几何尺寸参数$L = 10$,mm,$D = 4$,mm,右端载荷参数$P =1$,GPa.

在本算例中,分别采用高阶强形式自由单元法SFECM和低阶弱形式自由单元法WFECM进行计算.对于SFECM模型,整个悬臂梁被离 散成1071个节点,其分布如图6所示. 每个节点与周围的24个节点形成$5\times 5 =25$节点的高阶拉格朗日单元用于计算. 对于弱形式WFECM,这里采用比强形式更密一倍的网格,配点与周围的8个节点形成$3\times 3 = 9$节点的低阶拉格朗日单元.同时采用ANSYS计算并与解析解比较误差. ANSYS所采用单元类型为八节点单元,总单元数1000,总节点数为3141.为了和有限元对比,WFECM采用均匀分布的节点. 经过计算,上的节点比较剪应力分布,得到如图7的误差曲线.

图6   悬臂梁计算模型不等间隔节点分布图

Fig. 6   Nodes distribution over the cantilever beam

图7   $x=L/2$线上剪应力的误差曲线

Fig. 7   Error of shear stress on line of $x=L/2 $

图7中可以看出,相比较ANSYS,采用弱形式的WFECM误差更小. 对于高阶强形式SFECM其计算误差最小.由此可见SFECM计算的 应力最精确,并且在整条路径上应力误差在较小的范围内变化,说明SFECM计算精度高,稳定性好.

同时,绘制了WFECM,SFECM和ANSYS三种方法所得的底面上的Mises应力和$y$方向位移曲线(如图8图9所示),并在图10中绘制出了ANSYS和SFECM所得的整体悬臂梁的Mises应力云图.

图8   底边上Mises应力曲线图

Fig. 8   Mises stress curve on the bottom

图9   底面上$y$方向位移曲线图

Fig. 9   Displacement in $y$-direction on the bottom

图10   两种FECM方法计算所得悬臂梁整体Mises应力云图

Fig. 10   The Mises stress contour of the whole model calculated by two method of FECM respectively

图8可以看出3种算法所得Misis应力解非常接近,但在固定端,SFECM的解更能体现应力集中现象.这里需要指出,文献中提到的应力解 析解是在$L \gg D$的前提下得到的,其只适用于远离固定端的区域,在固定端的应力解析形式并不满足式(23),因此这里并不给出解析形式的Mises应力云图.

5.2 厚壁圆筒受压分析

本算例分析如图11所示的承受均布内压的厚壁圆筒,内半径$d_1 = 50$\,mm,外半径$d_2 = 100$\,mm,内壁面压强$p =0.4$\,MPa,材料的弹性模量$E$为$2.1$\,TPa,泊松比$\nu $为0.3.因为对称,可取1/4部分计算,FECM所采用的节点分布如图12所示.

图11   受内压的厚壁圆筒

Fig. 11   Thick-walled cylinder under inner pressure

图12   圆筒节点分布

Fig. 12   Distribution of nodes over the thick-walled cylinder

此问题的位移和应力解析解可求得为[27]

$u_r = \dfrac{1}{E}\dfrac{d_1^2 p}{d_2^2 - d_1^2 }\left[ {(1 + \nu )\dfrac{d_2^2 }{r} + (1 - \nu )r} \right] (24)$

$ \sigma _r = - \dfrac{\dfrac{d_2^2 }{r^2} - 1}{\dfrac{d_2^2 }{d_1^2 } - 1}p\,,\quad \sigma _\theta = \dfrac{\dfrac{d_2^2 }{r^2} + 1}{\dfrac{d_2^2 }{d_1^2 } - 1}p (25)$

对于本问题,FECM采用3种方式进行计算,即分别采用九节点、二十五节点单元的强形式(SFECM)以及九节点单元的弱形式(WFECM).3种方式采用如图12所示的同一套节点网格,其总节点数为1365.为了比较,同时使用有限元软件ANSYS进行计算,其采用八节点等参单元,总节点数为1045.

我们取1/4模型的底部,即路径$y=0$上的节点进行比较,位移变化如图13所示.从图13可以看出,使用高阶强形式和低阶弱形式所得 的结果和解析解十分相近.

图13   圆筒径向位移曲线图

Fig. 13   Displacement distribution along radial direction

图14给出了ANSYS和3种FECM模型计算所得的径向应力$\sigma _r$的误差曲线,从图中可以看出,对于此密度的节点分布模型,九节点低阶单元强形式模型SFECM-9并不能得到较高的精度,但使用二十五节点高阶单元强形式模型SFECM-25可以得到十分精确的计算结果,同样九节点弱形式算法WFECM可以得到较高的精度.从图14也可以看出,高阶强形式和低阶弱形式的自由单元法计算精度都高于ANSYS的计算精度.图15给出了解析形式的及二十五节点强形式SFECM-25模型计算所得的Mises应力云图.对于弱形式WFECM和九节点强形式SFECM-9的计算结果和其类似.

图14   沿径向的应力$\sigma _r $误差曲线图

Fig. 14   Error of $\sigma _r $ along radial direction

图15   圆筒的Mises应力云图

Fig. 15   Mises stress contour of the cylinder

为了验证本文算法的网格收敛性,针对本算例我们又对另外3种不同节点密度的模型进行了计算,其节点数分别为112,363和3007. 图16给出了$L_2 $误差范数随单元平均尺寸的变化曲线. $L_2$误差范数的变化能体现计算方法的整体收敛性,其定义为$L_2 = \sqrt {(u - \mathop{u}\limits^{\_}) \cdot (u -\mathop{u}\limits^{\_}) / n} $,其中$u$和$\bar{u}$分别为计算所得的各节点位移矢量和解析解位移矢量,$n$为模型的总节点数.从图16可以看出,计算结果的整体收敛性很好,表明本文算法具有良好的稳定性.

图16   $L_2$ 误差范数随单元尺寸的变化曲线

Fig. 16   The curve of L2 norm versus element size

5.3 对称边界的带圆孔复合圆弧

以上两算例模型中均是采用结构化(拉格朗日)单元,本算例将讨论具有非结构化(七节点)单元的情形.如图17所示为一受内压带圆孔 复合圆弧,外半径$R = 100$,内半径$r = 60$,其内部圆孔的直径$d = 10$.所受边界条件与上一算例类似,两端为对称边界条件,内壁面压强$p = 1000$. 材料的弹性模量$E$为$2\times10^5$,泊松比$\nu $为0.3. 按平面应力分析.

图17   受内压带圆孔复合圆弧

Fig. 17   Composite circular arch with circular holes under inner pressure

分别采用本文所提到的两种FECM方法进行计算. 由于目前还没有此问题的解析解,因此需要使用ANSYS进行计算对比.对于SFECM,由于问题的复杂性,采用较密节点分布并使用二十五节点高阶单元,由于模型中有些点处生成拉格朗日单元是不便的,因此在这些位置采用七节点三角形单元. 对于WFECM,仍采用九节点拉格朗日单元,所采用网格密度和ANSYS类似.图18中给出了两种FECM的网格分布. 图19中给出了SFECM中所使用的几个七节点三角形单元的具体形成方式.同样,对于模型中另外两个圆孔周围也有这种单元分布,在这里仅给出中间圆孔周围的单元形式.本算例的两种模型总节点数分别为30,368和7,728,ANSYS为22,020.

图18   两种FECM所采用的节点分布

Fig. 18   The nodes distribution of two method of FECM

图19   中间圆孔周围的三角形单元组成形式

Fig. 19   Triangular elements around a circular hole in the middle of the model

比较如图 17中红线上节点处的位移值和应力值.在图20中给出了3种方法所得的位移$u_{x}$和等效Mises应力曲线,经过比较 发现3种方法所得结果十分接近.

图 20   3种方法计算所得路径上的位移和应力值

Fig.20   The value of displacement and stress on the path calculated by three methods respectively

本算例验证了对于复杂几何外形,无论是强形式还是弱形式,本文方法均可以得到精确的结果.

本算例还验证了七节点三角形单元的准确性,发现在较难生成拉格朗日单元的位置,可使用三角形单元,而且仍得到十分精确的结果.

6 结论与讨论

本文介绍了强形式与弱形式两种自由单元法,用数值算例验证了其正确性与有效性,可得出下列结论:

(1) 自由单元法是一种单元配点法,对每个节点只需要一个独立的自由形成的局部单元进行配点,具有建模方便和容易实施并行计算的优点.

(2) 由于自由单元法只需要一个单元配点,因此对于内部点,为了获得较高的计算精度,所用的等参单元至少要有一个内部点(用于配点).

(3) 强形式的自由单元法,直接由控制方程和边界条件建立系统方程组,便于建立多物理场耦合问题的统一求解格式;可用任意形式的等参变量建立单元节点形函数,并可选用任意形式的等参单元. 缺点是,由于直接由控制方程配点要用到单元的二阶偏导数,因此需要至少二次的等参单元,如果要想获得较高的计算精度并要求节点数不要太多,则需要使用高次单元.

(4) 弱形式的自由单元法,由于通过了一次积分,只用到了单元的一阶偏导数,因此,采用同样阶次的等参单元,计算精度要比强形式的高.缺点是,对于不是全微分形式(守恒形式)的控制方程,则不易将其进行分部积分,较难建立多场耦合问题的统一求解格式.此外,由于要用到单元积分,因此所用的单元形函数,需要用从$-1$到$+1$的参变量来表征.

(5) 因为每个节点使用其独立的等参单元,不难想象全部节点的等参单元将形成互相重叠的网格,这对结果的显示会带来一定的不便.

(6) 尽管本文以二维弹性力学问题作为研究对象,但其方法和所推导的公式很容易推广到其它类型的问题.

附录

七节点三角形等参单元形函数

$N_1 (\xi ,\eta ) = \dfrac{1}{18}\xi \Big [ - 3 + \sqrt 3 + 3\xi - 3\eta (1 - 2\sqrt 3 + 2\sqrt 3 \eta + 2\sqrt 3 \xi )\Big ] $

$N_2 (\xi ,\eta ) = \dfrac{1}{18}\xi \Big [ 3 + \sqrt 3 + 3\xi - 3\eta (1 + 2\sqrt 3 + 2\sqrt 3 \eta + 2\sqrt 3 \xi ) \Big ] $

$ N_3 (\xi ,\eta ) = \dfrac{1}{18}\Big [ 3\eta (\sqrt 3 + 3\eta ) + \sqrt 3 \xi- 3\eta (1 + 2\sqrt 3 \eta )\xi - 6(1 + \sqrt 3 \eta )\xi ^2\Big ] $

$ N_4 (\xi ,\eta ) = \dfrac{1}{9}\Big [\xi ( - 2\sqrt 3 + 3\xi )+ 3\eta ^2(3 + 4\sqrt 3 \xi ) + 6\eta ( - \sqrt 3 + \xi + 2\sqrt 3 \xi ^2) \Big ] $

$ \begin{array}{l} N_5 (\xi ,\eta ) = \dfrac{2}{9}\xi \Big [3 - \sqrt 3 + 6\xi + 3\eta (1 + \sqrt 3 + 2\sqrt 3 \eta + 2\sqrt 3 \xi ) \Big ] \end{array} $

$ \begin{array}{l} N_6 (\xi ,\eta ) = \dfrac{2}{9}\xi \Big [ - 3 - \sqrt 3 + 6\xi + 3\eta (1 - \sqrt 3 + 2\sqrt 3 \eta + 2\sqrt 3 \xi ) \Big ] \end{array} $

$ \begin{array}{l} N_7 (\xi ,\eta ) = \dfrac{1}{2}(2 + \Big [\sqrt 3 - 3\eta )\eta + \sqrt 3 \xi - 3\eta (1 + 2\sqrt 3 \eta )\xi - 6(1 + \sqrt 3 \eta )\xi ^2\Big ] \end{array} $

The authors have declared that no competing interests exist.


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react-text: 116 This paper presents a fast multipole accelerated singular boundary method (SBM) to the solution of the large-scale three-dimensional Helmholtz equation at low frequency. By using a desingularization strategy to directly compute singular kernels in the strong-form collocation discretization, the SBM formulations are derived for the Dirichlet and Neumann problems. A fast multipole method (FMM)... /react-text react-text: 117 /react-text [Show full abstract]
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