二维耦合热弹性动力学问题的无网格自然邻接点Petrov-Galerkin法

                      2019-11-26 12:11:40 土木建筑与环境工程 2019年5期

                      李庆华 陈莘莘

                      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

                      中图分类号:TP301.6   文献标志码:A   文章编号:2096-6717(2019)05-0109-06

                      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[6-8]。近年来,部分学者尝试采用无网格法[9-12]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,無网格自然邻接点Petrov-Galerkin法在很多领域都得到广泛应用[18-20]。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

                      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。

                      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和图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和图5可以看出,本文数值解与解析解[22]吻合得很好。

                      [12] ZHENG B J, GAO X W, YANG K, et al. A novel meshless local Petrov-Galerkin method for dynamic coupled thermoelasticity analysis under thermal and mechanical shock loading [J]. Engineering Analysis with Boundary Elements, 2015, 60: 154-161.

                      [13] 高欣, 段庆林, 李书卉, 等. 裂纹问题的一致性高阶无网格法[J]. 计算力学学报, 2018, 35(3): 275-282.

                      GAO X, DUAN Q L, LI S H, et al. Consistent high order meshfree method for crack problems [J]. Chinese Journal of Computational Mechanics, 2018, 35(3): 275-282.(in Chinese)

                      [14] 王东东, 张汉杰, 梁庆文. 等几何修正准凸无网格法[J]. 计算力学学报, 2016, 33(4): 605-612.

                      WANG D D, ZHANG H J, LIANG Q W. Isogeometric refined quasi-convex meshfree method [J]. Chinese Journal of Computational Mechanics, 2016, 33(4): 605-612. (in Chinese)

                      [15] CAI Y C, ZHU H H. A meshless local natural neighbour interpolation method for stress analysis of solids [J]. Engineering Analysis with Boundary Elements, 2004, 28(6): 607-613.

                      [16] WANG K, ZHOU S J, SHAN G J. The natural neighbour Petrov-Galerkin method for elasto-statics [J]. International Journal for Numerical Methods in Engineering, 2005, 63(8): 1126-1145.

                      [17] ATLURI S N, ZHU T. A new meshless local petrov-galerkin (MLPG) approach in computational mechanics [J]. Computational Mechanics, 1998, 22(2): 117-127.

                      [18] 李順利, 龙述尧, 李光耀, 等. 自然邻接点局部Petrov-Galerkin法求解中厚板弯曲问题[J]. 湖南大学学报(自然科学版), 2011, 38(1): 53-57.

                      LI S L, LONG S Y, LI G Y, et al. Natural neighbor petrov-galerkin method for moderately thick plates [J]. Journal of Hunan University(Natural Sciences), 2011, 38(1): 53-57.(in Chinese)

                      [19] 王凯, 周慎杰, 聂志峰, 等. 基于局部自然邻近无网格法的形状优化[J]. 机械工程学报, 2009, 45(10): 185-191.

                      WANG K, ZHOU S J, NIE Z F, et al. Shape optimization based on the local natural neighbor Petrov-Galerkin method [J]. Journal of Mechanical Engineering, 2009, 45(10): 185-191.(in Chinese)

                      [20] 陈莘莘, 李庆华, 刘永胜. 轴对称动力学问题的无网格自然邻接点Petrov-Galerkin法 [J]. 振动与冲击, 2015, 34(3): 61-65.

                      CHEN S S, LI Q H, LIU Y S. Meshless natural neighbour Petrov-Galerkin method for axisymmetric dynamic problems [J]. Journal of Vibration and Shock, 2015, 34(3): 61-65. (in Chinese)

                      [21] 张亚辉, 林家浩. 结构动力学基础[M]. 辽宁 大连: 大连理工大学出版社,2007.

                      ZHANG Y H, LIN J H. Fundamentals of structural dynamics [M]. Dalian, Liaoning: Dalian University of Technology Press, 1993. (in Chinese)

                      [22] CARSLAW H S, JAEGER J C. Conduction of Heat in Solids [M]. Clarendon: Oxford University Press, 1959.

                      (编辑 王秀玲)

                      (function(){ var bp = document.createElement('script'); var curProtocol = window.location.protocol.split(':')[0]; if (curProtocol === 'https') { bp.src = 'https://zz.bdstatic.com/linksubmit/push.js'; } else { bp.src = 'http://push.zhanzhang.baidu.com/push.js'; } var s = document.getElementsByTagName("script")[0]; s.parentNode.insertBefore(bp, s); })();