土木与环境工程学报  2019, Vol. 41 Issue (5): 109-114   PDF    
二维耦合热弹性动力学问题的无网格自然邻接点Petrov-Galerkin法
李庆华 , 陈莘莘     
华东交通大学 土木建筑学院, 南昌 330013
摘要:为了更有效地求解二维耦合热弹性动力学问题,对无网格自然邻接点Petrov-Galerkin法在此类问题中的应用进行了研究,并发展了相应的计算方法。该方法建立试函数时可以只依赖于一组离散的节点,有效地避免了复杂的网格划分和网格畸变的影响。相对于常用的移动最小二乘而言,自然邻接点插值不涉及复杂的矩阵求逆运算,更不需要任何人为参数。由于运动方程和瞬态热传导方程相互影响,这些方程必须联立求解。采用Newmark法求解空间离散后得到的二阶常微分方程组,进而可直接获得温度场和位移场的数值结果。
关键词无网格法    自然邻接点插值    耦合热弹性动力学    Petrov-Galerkin法    
Meshless natural neighbour Petrov-Galerkin method for two-dimensional dynamic coupled thermoelasticity problem
Li Qinghua , Chen Shenshen     
School of Civil Engineering and Architecture, East China Jiaotong University, Nanchang 330013, P. R. China
Abstract: In order to solve the two-dimensional dynamic coupled thermoelasticity problem more effectively, a novel numerical method based on the meshless natural neighbour Petrov-Galerkin method is proposed in this study. Only a group of scattered nodes are required in this method, to construct approximation function and therefore complex meshing and disadvantage of mesh distortion are effectively eliminated. In comparison with the moving least-squares (MLS) approximation used widely in meshless methods, the natural neighbour interpolation requires no complex matrix inversions and no artificial intermediate parameters. The equations of motion and transient heat conduction equations of the coupled thermoelasticity interaction on each other and therefore these equations must be solved simultaneously. After spatially discretization, a series of second-order ordinary differential algebraic equations is obtained, which is solved by the Newmark method to obtain the numerical temperature and displacement field directly.
Keywords: meshless method    natural neighbour interpolation    dynamic coupled thermoelasticity    Petrov-Galerkin method    

当结构受到温变,一般会产生热应力,并且热应力是物体破坏的一个重要因素[1,2]。对受热结构进行分析时,解耦方法可先由热传导方程求出温度分布,再由热弹性方程求解位移和应力。但是,解耦方法没有考虑结构变形对温度场的影响[3]。事实上,热弹性力学中最基本的问题就是耦合热弹性问题。在耦合热弹性问题中,温度和变形会相互影响,温度场和应变场的耦合项必须体现在热传导方程中。为了求解温度、位移和应力,必须联立求解热传导方程和热弹性运动方程。相对于非耦合热弹性问题,耦合热弹性问题求解更困难。

热应力问题的数值方法主要基于发展较为成熟的有限元法[4,5]和边界元法[6,7,8]。近年来,部分学者尝试采用无网格法[9,10,11,12]求解热应力问题。无网格法不仅能够避免网格生成的复杂过程,而且在节点分布不规则时,损失的计算精度较小,从而日益得到重视[13,14]。近十多年来发展起来的无网格法―无网格自然邻接点Petrov-Galerkin法[15,16]不仅允许加权函数和试函数取自不同的函数空间[17],而且克服了本质边界条件不易施加的困难。此方法中,所有的积分都在中心为所考虑点的多边形子域上进行,而且多边形子域的构造十分方便。目前,无网格自然邻接点Petrov-Galerkin法在很多领域都得到广泛应用[18,19,20]。本文采用自然邻接点插值对温度和位移分别插值,与局部加权余量法结合,提出了适用于耦合热弹性动力学问题的无网格自然邻接点Petrov-Galerkin法。最后,通过数值算例验证了本文方法应用于耦合热弹性动力学问题分析的有效性和合理性。

1 自然邻接点插值

在求解域内给定M个离散节点,其集合为N={x1, x2, …, xM}。对任一节点xI,其一阶Voronoi结构可定义为

$ {T_I} = \{ \mathit{\boldsymbol{x}} \in {R^2}:d(\mathit{\boldsymbol{x}}, {\mathit{\boldsymbol{x}}_I}) < \mathit{d}{\rm{(}}\mathit{\boldsymbol{x, }}{\mathit{\boldsymbol{x}}_J}{\rm{), }}\forall J \ne I\} $ (1)

式中:d(x, xI)表示点x与节点xI之间的距离。

为计算Sibson插值形函数,需进一歩定义二阶Voronoi结构。

$ \begin{array}{l} {T_{IJ}} = \{ \mathit{\boldsymbol{x}} \in {R^2}:d\left( {\mathit{\boldsymbol{x}}, {\mathit{\boldsymbol{x}}_I}} \right) < d\left( {\mathit{\boldsymbol{x}}, {\mathit{\boldsymbol{x}}_J}} \right) < \\ d(\mathit{\boldsymbol{x}}, {\mathit{\boldsymbol{x}}_K}), \forall J \ne I \ne K\} \end{array} $ (2)

图 1所示为点x的一阶和二阶Voronoi结构。根据Sibson插值的定义[15],点x的插值函数为

图 1x的一次和二次Voronoi结构 Fig. 1 First-order and second-order Voronoi cells about x

$ {\varphi _I}\left( \mathit{\boldsymbol{x}} \right) = {A_I}\left( \mathit{\boldsymbol{x}} \right)/A\left( \mathit{\boldsymbol{x}} \right) $ (3)

式中:AI(x)为点x的二阶Voronoi结构TxI的面积;A(x)为点x的一阶Voronoi结构Tx的面积。

定义了各节点的插值函数后,点x的温度函数类似于有限元法可写为

$ \theta \left( \mathit{\boldsymbol{x}} \right) = \sum\limits_{I = 1}^n {{\varphi _I}\left( \mathit{\boldsymbol{x}} \right){\theta _I}} $ (4)

式中:θI(I=1, 2, L, n)是点x周围自然邻接点的节点温度。

2 控制方程的弱形式及其离散化

设线性耦合热弹性动力学问题的区域为ΩΓΩ的边界。热弹性力学的应力、位移与温度之间的关系为[1]

$ {\sigma _{ij}} = 2\mu \;{\varepsilon _{ij}} + \lambda {\varepsilon _{kk}}{\delta _{ij}} - \beta {\rm{ }}\theta {\delta _{ij}} $ (5)

式中:σij为应力; εij为应变,λμ为拉梅常数; β为热应力系数; θ为温度变化值。小变形情况下,应变εij与位移ui的几何关系为

$ {\varepsilon _{ij}} = ({u_{i, j}} + {u_{j, i}})/2 $ (6)

在经典的热弹性理论中,运动方程和热传导方程可表示为[1]

$ {\sigma _{ij, j}} + {b_i} - \rho \;{{\ddot u}_i} = 0 $ (7)
$ k{\theta _{, ii}} - \rho {\rm{ }}c\dot \theta - \beta \;{\theta _0}{{\mathit{\dot u}}_{i, i}} + Q = 0 $ (8)

式中:ρ为质量密度; $\ddot u$i为加速度; $\dot u$i, i为体积应变率; bi为体力;$\dot \theta $表示温度对时间的导数; θ0为参考温度;Q为热源; k为导热系数; c为比热容。在边界Γ上,边界条件包括力学和热学两种,数学表达式为

$ {u_i}\left( {\mathit{\boldsymbol{x}}, t} \right) = {{\bar u}_i}\left( {\mathit{\boldsymbol{x}}, t} \right), \mathit{\boldsymbol{x}} \in {\mathit{\Gamma }_u} $ (9)
$ {\mathit{t}_i}\left( {\mathit{\boldsymbol{x}}, t} \right) = {{\bar t}_i}\left( {\mathit{\boldsymbol{x}}, t} \right), \mathit{\boldsymbol{x}} \in {\mathit{\Gamma }_t} $ (10)
$ \theta \left( {\mathit{\boldsymbol{x}}, t} \right) = \bar \theta \left( {\mathit{\boldsymbol{x}}, t} \right), \mathit{\boldsymbol{x}} \in {\mathit{\Gamma }_\theta } $ (11)
$ k{\theta _{, i}}{n_i} = \bar q, \mathit{\boldsymbol{x}} \in {\mathit{\Gamma }_q} $ (12)

式中:uiti分别表示给定的位移和面力; θq分别表示给定的温度和热通量; ni为边界Γ的外法线方向余弦。此外,相应的初始条件为

$ \theta \left( {\mathit{\boldsymbol{x}}, 0} \right) = {\theta _0}, \mathit{\boldsymbol{x}} \in \mathit{\Omega } $ (13)
$ \mathit{\boldsymbol{u}}\left( {\mathit{\boldsymbol{x}}, 0} \right) = {\mathit{\boldsymbol{u}}_0}, \mathit{\boldsymbol{x}} \in \mathit{\Omega } $ (14)
$ \mathit{\boldsymbol{v}}\left( {\mathit{\boldsymbol{x}}, 0} \right) = {\mathit{\boldsymbol{v}}_0}, \mathit{\boldsymbol{x}} \in \mathit{\Omega } $ (15)

式中:θ0为初始温度; u0v0分别为初始位移和初始速度。

在子域ΩIs上,式(7)和式(8)的局部弱形式可分别表示为

$ \int {_{\Omega _s^I}{\mathit{v}_\mathit{I}}({\sigma _{ij, j}} + {b_i} - \rho \;{{\ddot u}_i}){\rm{d}}\mathit{\Omega } = 0} $ (16)
$ \int {_{\Omega _s^I}{\mathit{v}_\mathit{I}}(\mathit{k}{\theta _{, \mathit{ii}}} - \rho \;c\dot \theta - \rho \;{\theta _0}{{\dot u}_{i, i}} + Q){\rm{d}}\mathit{\Omega } = 0} $ (17)

式中:vI为加权函数。对式(16)左边积分的第1项进行分部积分并利用散度定理后,可得

$ \begin{array}{l} \int {_{\Omega _s^I}({\mathit{v}_I}\rho {{\ddot u}_i} + {\mathit{v}_{I, j}}{\sigma _{ij}}){\rm{d}}\mathit{\Omega }} - \\ \int {_{\mathit{\Gamma }_s^I}{v_I}{t_i}{\rm{d}}\mathit{\Gamma }} - \int {_{\mathit{\Omega }_s^I}{v_I}{b_i}{\rm{d}}\mathit{\Omega = }{\rm{0}}} \end{array} $ (18)

式中:ΓIs为子域ΩIs的边界,通常由内边界ΓIsi和局部边界位于整体边界上给定面力边界条件的ΓIst和给定位移边界条件的ΓIsu三部分组成。在无网格自然邻接点Petrov-Galerkin法中,多边形子域ΩIs是由共享节点I的Delaunay三角形TIi构成,如图 2所示。选取FEM三角形线性单元的形函数NI为权函数,式(18)可简化为

图 2 局部多边形子域 Fig. 2 The local polygonal sub-domains

$ \begin{array}{l} \int {_{\Omega _s^I}({\mathit{N}_I}\rho {{\ddot u}_i} + {N_{I, j}}{\sigma _{ij}}){\rm{d}}\mathit{\Omega }} - \\ \int {_{\mathit{\Gamma }_{st}^I}{N_I}{{\bar t}_i}{\rm{d}}\mathit{\Gamma }} - \int {_{\mathit{\Omega }_s^I}{N_I}{b_i}{\rm{d}}\mathit{\Omega = }{\rm{0}}} \end{array} $ (19)

以此类推,式(17)可变为

$ \begin{array}{l} \int {_{\Omega _s^I}({\mathit{N}_I}\rho {\rm{c}}\dot \theta + {N_I}\beta {\theta _0}{{\mathit{\dot u}}_{\mathit{i, i}}} + {\mathit{N}_{I, i}}\mathit{k}\theta {, _i}){\rm{d}}\mathit{\Omega }} - \\ \int {_{\mathit{\Gamma }_{sq}^I}{N_I}\bar q{\rm{d}}\mathit{\Gamma }} - \int {_{\mathit{\Omega }_s^I}{N_I}\mathit{Q}{\rm{d}}\mathit{\Omega = }{\rm{0}}} \end{array} $ (20)

由于只对空间域进行离散,求解域Ω内的位移u(x, t)和温度

$ \left\{ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{u}}\left( {\mathit{\boldsymbol{x}}, t} \right) = \sum\limits_{I = 1}^\mathit{n} {{\varphi _I}\left( \mathit{\boldsymbol{x}} \right){\mathit{\boldsymbol{u}}_I}\left( t \right)} }\\ {\mathit{\theta }\left( {\mathit{\boldsymbol{x}}, t} \right) = \sum\limits_{I = 1}^\mathit{n} {{\varphi _I}\left( \mathit{\boldsymbol{x}} \right){\theta _I}\left( t \right)} } \end{array}} \right. $ (21)

将式(21)代入式(19)和式(20),可得耦合热弹性动力学问题的离散控制方程为

$ {\mathit{\boldsymbol{M}}^u}\mathit{\boldsymbol{\ddot u}}\left( t \right) + {\mathit{\boldsymbol{K}}^u}\mathit{\boldsymbol{u}}\left( t \right) + {\mathit{\boldsymbol{H}}^u}\mathit{\boldsymbol{\theta }}\left( t \right) = {\mathit{\boldsymbol{F}}^u}\left( t \right) $ (22)
$ {\mathit{\boldsymbol{C}}^\theta }\mathit{\boldsymbol{\dot \theta }}\left( t \right) + {\mathit{\boldsymbol{K}}^\theta }\mathit{\boldsymbol{\theta }}\left( t \right) + {\mathit{\boldsymbol{G}}^\theta }\mathit{\boldsymbol{\dot u}}\left( t \right) = {\mathit{\boldsymbol{F}}^\theta }\left( t \right) $ (23)

式中:

$ \mathit{\boldsymbol{M}}_{IJ}^\mathit{u} = \int {_{\Omega _s^I}\rho {N_I}{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_J}{\rm{d}}\mathit{\Omega }} $ (24)
$ \mathit{\boldsymbol{K}}_{IJ}^\mathit{u} = \int {_{\Omega _s^I}{\mathit{\boldsymbol{V}}_I}\mathit{\boldsymbol{D}}{\mathit{\boldsymbol{B}}_J}{\rm{d}}\mathit{\Omega }} $ (25)
$ \mathit{\boldsymbol{H}}_{IJ}^\mathit{u} = - \int {_{\Omega _s^I}{\mathit{\boldsymbol{W}}_I}\beta {\varphi _J}{\rm{d}}\mathit{\Omega }} $ (26)
$ \mathit{F}_I^\mathit{u}\left( t \right) = \int {_{\mathit{\Gamma }_{st}^I}{\mathit{N}_I}\mathit{\bar t}\left( t \right){\rm{d}}\mathit{\Gamma }} + \int {_{\mathit{\Omega }_s^I}{\mathit{N}_I}\mathit{\boldsymbol{b}}\left( t \right){\rm{d}}\mathit{\Omega }} $ (27)
$ \mathit{C}_{IJ}^\theta = \int {_{\Omega _s^I}\rho \mathit{c}{N_I}{\varphi _J}{\rm{d}}\mathit{\Omega }} $ (28)
$ \mathit{K}_{IJ}^\theta = \int {_{\Omega _s^I}k} \left( {\frac{{\partial {N_1}}}{{\partial x}}\frac{{\partial {\varphi _J}}}{{\partial \mathit{x}}} + \frac{{\partial {N_1}}}{{\partial y}}\frac{{\partial {\varphi _J}}}{{\partial \mathit{y}}}} \right){\rm{d}}\mathit{\Omega } $ (29)
$ \mathit{\boldsymbol{G}}_{IJ}^\theta = \int {_{\Omega _s^I}{N_I}\beta {\theta _0}{\mathit{\boldsymbol{P}}_J}{\rm{d}}\mathit{\Omega }} $ (30)
$ \mathit{F}_I^\theta \left( t \right) = \int {_{\mathit{\Gamma }_{sq}^I}{\mathit{N}_I}\bar q\left( t \right){\rm{d}}\mathit{\Gamma }} + \int {_{\mathit{\Omega }_s^I}{\mathit{N}_I}Q\left( t \right){\rm{d}}\mathit{\Omega }} $ (31)

式中:

$ {\mathit{\boldsymbol{V}}_I}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{N_{I, \mathit{x}}}}&0&{{N_{I, y}}}\\ 0&{{N_{I, y}}}&{{N_{I, \mathit{x}}}} \end{array}} \right] $ (32)
$ {\mathit{\boldsymbol{B}}_J}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{\varphi _{J, x}}}&0\\ 0&{{\varphi _{J, {\rm{y}}}}}\\ {{\varphi _{J, y}}}&{{\varphi _{J, x}}} \end{array}} \right] $ (33)
$ {\mathit{\boldsymbol{W}}_I} = {\left[ {\begin{array}{*{20}{c}} {{N_{I, \mathit{x}}}}&{{N_{I, \mathit{y}}}} \end{array}} \right]^{\rm{T}}} $ (34)
$ {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_J}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{\varphi _J}}&0\\ 0&{{\varphi _J}} \end{array}} \right] $ (35)
$ {\mathit{\boldsymbol{P}}_J} = \left[ {\begin{array}{*{20}{c}} {{\varphi _{\mathit{J}, \mathit{x}}}}&{{\varphi _{\mathit{J}, \mathit{y}}}} \end{array}} \right] $ (36)
$ \mathit{\boldsymbol{D}}{\rm{ = }}\frac{E}{{1 - {v^2}}}\left[ {\begin{array}{*{20}{c}} 1&\mathit{v}&0\\ v&1&0\\ 0&0&{\frac{{1 - v}}{2}} \end{array}} \right]\left( {{\rm{plane}}\;{\rm{stress}}} \right) $ (37)
$ \beta {\rm{ = }}\frac{{\alpha E}}{{1 - v}}\left( {{\rm{plane}}\;{\rm{stress}}} \right) $ (38)

对于平面应变问题,需把E换成E/(1-v2),v换成v/(1-v),α换成(1+v)α。式(22)和式(23)可合并为

$ \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{M}}^\mathit{u}}}&0\\ 0&0 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\ddot u}}}\\ {\mathit{\boldsymbol{\ddot \theta }}} \end{array}} \right\} + \left[ {\begin{array}{*{20}{c}} 0&0\\ {{\mathit{\boldsymbol{G}}^\theta }}&{{\mathit{\boldsymbol{C}}^\theta }} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\dot u}}}\\ {\mathit{\boldsymbol{\dot \theta }}} \end{array}} \right\} + \\ \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{K}}^u}}&{{\mathit{\boldsymbol{H}}^u}}\\ 0&{{\mathit{\boldsymbol{K}}^\theta }} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{u}}\\ \theta \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{F}}^u}}\\ {{\mathit{\boldsymbol{F}}^\theta }} \end{array}} \right\} \end{array} $ (39)

式(39)即为施加边界条件后的系统耦合微分方程组,可采用Newmark方法[21]进行求解。

3 数值算例

为了验证所提方法的有效性,考虑如图 3所示的单位面积方板,该问题为平面应变状态下的一个经典算例。初始时刻板的温度和位移均为0,板上边受到突加的热载荷,另外3边均绝热,且无法向位移。弹性模量E=1,泊松比v=0.3,导热系数k=1,密度ρ=1,比热容c=1,热膨胀系数α=0.02。计算中,采用15×15个节点将方板离散,时间步长取为2.0×10-3

图 3 突加热载荷的方板 Fig. 3 A suddenly heated unit square plate

当不考虑惯性项和耦合项时,此问题属于准静态热弹性力学问题。图 4图 5分别给出了方板y轴上不同坐标处的温度和竖向位移变化情况。从图 4图 5可以看出,本文数值解与解析解[22]吻合得很好。

图 4 温度随时间的变化 Fig. 4 Temporal variation of the temperature

图 5 竖向位移随时间的变化 Fig. 5 Temporal variation of the vertical displacement

当考虑惯性项时,为了便于对耦合和非耦合情况下的计算结果进行比较,引入如下修正的耦合系数[12]

$ \eta = \frac{{\left( {1 + v} \right){\alpha ^2}E{\theta _0}}}{{\left( {1 - v} \right)\left( {1 - 2v} \right)\rho c}} $ (40)

式中:耦合系数η的取值范围一般为0.01~0.2。相关温度取为θ0=100,则对应的耦合系数为η=0.186。图 6图 7分别为y轴上不同坐标处耦合项对温度和位移的影响。显然,耦合项对温度的影响很大,但对位移的影响可忽略不计。

图 6 耦合效应对温度的影响 Fig. 6 Coupling effects on the temperature

图 7 耦合效应对竖向位移的影响 Fig. 7 Coupling effects on the vertical displacement

4 结论

无网格自然邻接点Petrov-Galerkin法是一种简单适用,且效率和精度均十分优良的无网格分析方法。在采用自然邻接点插值对位移和温度分别插值的基础上,将FEM三角形线性单元的形函数作为加权函数,采用加权余量法详细推导了二维耦合热弹性动力学问题的无网格自然邻接点Petrov-Galerkin法计算公式。数值算例表明,所提的二维耦合热弹性动力学问题求解方法可行。

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