工程力学 ›› 2020, Vol. 37 ›› Issue (2): 23-33.doi: 10.6052/j.issn.1000-4750.2019.01.0099

• 基本方法 • 上一篇    下一篇

求解粘弹性问题的时域自适应等几何比例边界有限元法

何宜谦1, 王霄腾1, 祝雪峰2, 杨海天1, 薛齐文3   

  1. 1. 大连理工大学工程力学系/工业装备结构分析国家重点实验室, 辽宁, 大连 116024;
    2. 大连理工大学汽车工程学院, 辽宁, 大连 116024;
    3. 大连交通大学土木工程学院, 辽宁, 大连 116028
  • 收稿日期:2019-03-06 修回日期:2019-07-02 出版日期:2020-02-25 发布日期:2020-01-19
  • 通讯作者: 杨海天(1956-),男,浙江黄岩人,教授,博士,博导,从事计算力学和固体力学研究(E-mail:haitian@dlut.edu.cn). E-mail:haitian@dlut.edu.cn
  • 作者简介:何宜谦(1983-),男,辽宁兴城人,副教授,博士,硕导,从事比例边界元、反问题、损伤力学研究(E-mail:heyiqian@dlut.edu.cn);王霄腾(1991-),男,安徽旌德人,博士生,从事等几何比例边界元、多尺度研究(E-mail:790362804@qq.com);祝雪峰(1979-),男,河北隆尧人,副教授,博士,硕导,从事等几何分析、智能优化设计等研究(E-mail:xuefeng@dlut.edu.cn);薛齐文(1976-),男,湖北孝感人,教授,博士,博导,从事计算力学、反问题研究(Email:xueqiwen@djtu.edu.cn).
  • 基金资助:
    国家自然科学基金项目(11572077,11572068,11872015); 973项目(2015CB057804);工业装备结构分析国家重点实验室自主课题(S18402).

A TEMPORALLY PIECEWISE ADAPTIVE ISOGEOMETRIC SBFEM FOR VISCOELASTIC PROBLEMS

HE Yi-qian1, WANG Xiao-teng1, ZHU Xue-feng2, YANG Hai-tian1, XUE Qi-wen3   

  1. 1. State Key Lab of Structural Analysis for Industrial Equipment, Deptartment of Engineering Mechanics, Dalian University of Technology, Dalian, Liaoning 116024, China;
    2. School of Automotive Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China;
    3. School of Civil Engineering, Dalian Jiaotong University, Dalian, Liaoning 116028, China
  • Received:2019-03-06 Revised:2019-07-02 Online:2020-02-25 Published:2020-01-19

摘要: 提出一种基于分段时域自适应算法和等几何分析的求解粘弹性问题的数值方法。利用时域分段展开,建立了递推格式的比例边界元求解方程,环向比例边界采用等几何技术离散,在继承常规比例边界有限元半解析、便于处理应力奇异性/无限域问题等优点的同时,可更准确地描述几何边界,由此进一步提高了计算精度;在时域,通过分段时域自适应计算,保证不同时间步长下的计算精度。通过数值算例,从计算精度、收敛性等方面,对所提方法的有效性进行了验证。

关键词: 粘弹性, 等几何分析, 比例边界元方法, 时域自适应算法, 非均匀有理B样条

Abstract: A new numerical algorithm is presented to solve viscoelastic problems by combining a temporally piecewise adaptive technique with the isogeometric analysis. By expanding variables at a discretized time interval, a series of recursive scaled boundary finite element method equations in spatial domain is established with non-uniform rational B-spline discretization in the circumferential direction. The proposed algorithm not only takes advantages of conventional scaled boundary finite element method in dealing with singularity and unbounded domains, but also provides a more accurate geometric description of the boundary, resulting in more accurate special solutions. A piecewise adaptive process is utilized to fully maintain a steady accuracy in the time domain for different sizes of time step. Numerical examples are presented to demonstrate the effectiveness of the proposed model in term of computational accuracy and convergence.

Key words: viscoelasticity, isogeometric analysis, scaled boundary finite element method, temporally piecewise adaptive algorithm, non-uniform rational B-spline

中图分类号: 

  • TB121
[1] Christensen R M, Freund L B. Theory of viscoelasticity[M]. New York:Academic Press, 1982:1-5.
[2] 董忠红, 吕彭民. 移动荷载下粘弹性层状沥青路面动力响应模型[J]. 工程力学, 2011, 28(12):153-159. Dong Zhonghong, Lü Pengmin. Time domain calculation method based on hysteretic damping model[J]. Engineering Mechanics, 2011, 28(12):153-159. (in Chinese)
[3] 冯希金, 危银涛, 李志超, Kaliske Michael. 未硫化橡胶非线性粘弹性本构模型研究[J]. 工程力学, 2016, 33(7):212-219. Feng Xijin, Wei Yintao, Li Zhichao, Kaliske Michael. Research on nonlinera viscoelastic constitutive model for uncured rubber[J]. Engineering Mechanics, 2016, 33(7):212-219. (in Chinese)
[4] 张泷, 刘耀儒, 杨强, 吕庆超. 基于不可逆内变量热力学的岩石材料粘弹-粘塑性本构方程[J]. 工程力学, 2015, 32(9):34-41. Zhang Long, Liu Yaoru, Yang Qiang, Lü Qingchao. A viscoelastic-viscoplastic constitutive equation of rock based on irreversible intrnal state varible thermo-dynamics[J]. Engineering Mechanics, 2015, 32(9):34-41. (in Chinese)
[5] Duan J B, Lei Y J, Li D K. Fracture analysis of linear viscoelastic materials using triangular enriched crack tip elements[J]. Finite Elements in Analysis & Design, 2011, 47(10):1157-1168.
[6] Zhang H H, Rong G, Li L X. Numerical study on deformations in a cracked viscoelastic body with the extended finite element method[J]. Engineering Analysis with Boundary Elements, 2010, 34(6):619-624.
[7] Zhang H H, Li L X. Modeling inclusion problems in viscoelastic materials with the extended finite element method[J]. Finite Elements in Analysis and Design, 2010, 45:721-729.
[8] Sawant S, Muliana A. A thermo-mechanical viscoelastic analysis of orthotropic materials[J]. Composite Structures, 2008, 83(1):61-72.
[9] Yang B J, Kim B R, Lee H K. Predictions of viscoelastic strain rate dependent behavior of fiber-reinforced polymeric composites[J]. Composite Structures, 2012, 94(4):1420-1429.
[10] Moita J S, Araujo A L, Martins P, et al. A finite element model for the analysis of viscoelastic sandwich structures[J]. Computers & Structures, 2011, 89(21/22):1874-1881.
[11] 朱媛媛, 胡育佳, 程昌钧. 无网格方法在平面粘弹性力学问题中的应用[J]. 力学季刊, 2006, 27(3):404-412. Zhu Yuayuan, Hu Yujia, Cheng Changjun. Application of element-free galerkin method on plane problems of visco-easticity[J]. Chinese Quarterly of Mechanics, 2006, 27(3):404-412. (in Chinese)
[12] Barrett K E, Gotts A C. FEM for one- and two-dimensional viscoelastic materials with spherical and rotating domains using FFT[J]. Computers & Structures, 2004, 82(2/3):181-192.
[13] Lei Y J, Duan J B, Li D K, et al. Crack problems in a viscoelastic medium using enriched finite element method[J]. International Journal of Mechanical Sciences, 2012, 58(1):34-46.
[14] Han Z, Yang H, Liu L. Solving viscoelastic problems with cyclic symmetry via a precise algorithm and EFGM[J]. Acta Mechanica Sinica, 2006, 22(2):170-176.
[15] Ashrafi H, Shariyat M, Khalili S M R, et al. A boundary element formulation for the heterogeneous functionally graded viscoelastic structures[J]. Applied Mathematics and Computation, 2013, 225:246-262.
[16] Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis:CAD, finite elements, NURBS, exact geometry and mesh refinement[J]. Computer Methods in Applied Mechanics & Engineering, 2005, 194(39/40/41):4135-4195.
[17] Simpson R N, Bordas S P A, Lian H, et al. An isogeometric boundary element method for elastostatic analysis:2D implementation aspects[J]. Computers & Structures, 2013, 118(6):2-12.
[18] Erfan S, Shirko F. Isogeometric analysis:vibration analysis, Fourier and wavelet spectra[J]. Journal of Theoretical and Applied Vibration and Acoustics, 2017, 3(2):145-164.
[19] Song C, Wolf J P. The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for elastodynamics[J]. Computer Methods in Applied Mechanics & Engineering, 1997, 147(3/4):329-355.
[20] He Y Q, Yang H T. Solving viscoelastic problems by combining SBFEM and a temporally piecewise adaptive algorithm[J]. Mechanics of Time-Dependent Materials, 2017(2/3):1-17.
[21] Natarajan S, Wang J C, Song C, et al. Isogeometric analysis enhanced by the scaled boundary finite element method[J]. Computer Methods in Applied Mechanics and Engineering, 2015, 283(1):733-762.
[22] 张勇, 林皋, 胡志强, 钟红. 基于等几何分析的比例边界有限元方法[J]. 计算力学学报, 2012, 29(3):433-438. Zhang Yong, Lin Gao, Hu Zhiqiang, Zhong Hong. Scaled boundary finite element method based on isogeometric analysis[J]. Chinese Journal of Computational Mechanics, 2012, 29(3):433-438. (in Chinese)
[23] 张勇, 林皋, 胡志强. 比例边界等几何分析方法Ⅰ:波导本征问题[J]. 力学学报, 2012, 44(2):382-392. Zhang Yong, Lin Gao, Hu Zhiqiang. Scaled boundary isogeometric analysis and its application I:Eigenvalue problem of waveguide[J]. Theoretical and Applied Mechanics, 2012, 44(2):382-392. (in Chinese)
[24] Lin G, Li P, Liu J, Zhang P Ch. Transient heat conduction analysis using the NURBS-enhanced scaled boundary finite element method and modified precise integration method[J]. Acta Mechanica Solida Sinica, 2017, 30(5):445-464.
[25] Wang W Y, Peng Y, Wei Z J, Guo Z J, Yang Y. High performance analysis of liquid sloshing in horizontal circular tanks with internal body by using IGA-SBFEM[J]. Engineering Analysis with Boundary Elements, 2019, 101:1-16.
[26] Liu J, Li J B, Li P. New Application of the Isogeometric Boundary Representations Methodology with SBFEM to Seepage Problems in Complex Domains[J]. Computers & Fluids, 2018:241-255.
[27] Li P, Liu J, Lin G, Zhang P Ch, Yang G T. A NURBS-based scaled boundary finite element method for the analysis of heat conduction problems with heat fluxes and temperatures on side-faces[J]. International Journal of Heat and Mass Transfer, 2017, 113:764-779.
[28] Li P, Liu J, Lin G, Zhang P Ch, Xu B. A combination of isogeometric technique and scaled boundary method for the solution of the steady-state heat transfer problems in arbitrary plane domain with Robin boundary[J]. Engineering Analysis with Boundary Elements, 2017, 82:43-56.
[29] 杨海天, 李哈汀. 时域自适应算法求解弹性地基薄板的动力问题[J]. 应用力学学报, 2012, 29(2):164-169, 239. Yang Hatian, Li Hating. Dynamic analysis of plate on elastic foundation via self-adaptive precise algorithm in the time domain[J]. Chinese Journal of Applied Mechanics, 2012, 29(2):164-169, 239. (in Chinese)
[30] Yang H T, Liu Y. A combined approach of EFGM and precise algorithm in time domain solving viscoelasticity problems[J]. International Journal of Solids & Structures, 2003, 40(3):701-714.
[31] Deeks A J, Wolf J P. A virtual work derivation of the scaled boundary finite-element method for elastostatics[J]. Computational Mechanics, 2002, 28(6):489-504.
[32] Deeks A J, Augarde C. A meshless local Petrov-Galerkin scaled boundary method[J]. Computational Mechanics. 2005, 36(3):159-170.
[33] Liu G R, Trung N T. Smoothed finite element methods[M]. Boca Raton FL:CRC Press, 2010:584.
[34] 皮尔. 非均匀有理B样条[M]. 北京:清华大学出版社, 2010:103-131. Les Piegl. Non-Uniform ratuional B-spline[M]. Beijing:Tsinghua University Press, 2010:103-131. (in Chinese)
[1] 郭迎庆, 李阳, 徐赵东, 陈笑, 王军建. 采用电动式激振器的混合试验系统设计[J]. 工程力学, 2020, 37(1): 108-114.
[2] 卢啸, 吕泉林. 自复位粘弹性腹杆的力学原理与滞回性能研究[J]. 工程力学, 2019, 36(6): 138-146.
[3] 汪超, 谢能刚, 黄璐璐. 基于扩展等几何分析和混沌离子运动算法的带孔结构形状优化设计[J]. 工程力学, 2019, 36(4): 248-256.
[4] 程永锋, 朱照清, 卢智成, 张富有. 运动简谐振子作用下地基梁体系振动特性的半解析研究[J]. 工程力学, 2018, 35(7): 18-23.
[5] 石础, 罗宇, 胡志强. 考虑失效的非线性Burgers'海冰模型及其数值应用[J]. 工程力学, 2018, 35(7): 249-256.
[6] 夏阳, 廖科. 复杂三维曲梁结构的无闭锁等几何分析算法研究[J]. 工程力学, 2018, 35(11): 17-25.
[7] 庞林, 林皋, 钟红. 比例边界等几何方法在断裂力学中的应用[J]. 工程力学, 2016, 33(7): 7-14.
[8] 陈昌萍, 胡海涛. 热电耦合场中压电粘弹性微梁的静力坍塌分析[J]. 工程力学, 2016, 33(4): 43-48,66.
[9] 薛冰寒, 林皋, 胡志强, 张勇. 求解摩擦接触问题的IGA-B可微方程组方法[J]. 工程力学, 2016, 33(10): 35-43.
[10] 过斌, 葛建立, 杨国来, 吕加. 三维实体结构NURBS等几何分析[J]. 工程力学, 2015, 32(9): 42-48.
[11] 柳国环, 练继建, 燕翔. 差动和波浪力激励下海床-桩-墩-桥的地震弹塑性:原理、方法、程序与智能建模[J]. 工程力学, 2015, 32(6): 133-140.
[12] 李创第, 李暾, 葛新广, 邹万杰. 一般线性粘弹性阻尼器耗能结构瞬态响应的非正交振型叠加精确解[J]. 工程力学, 2015, 32(11): 140-149.
[13] 李皓玉, 杨绍普, 刘进, 司春棣. 移动分布荷载下层状粘弹性体系的动力响应分析[J]. 工程力学, 2015, 32(1): 120-127.
[14] 徐永生, 杨海天, 赵潇, 张国庆. 基于Kriging代理模型的二维分数阶粘弹性问题的数值求解[J]. 工程力学, 2013, 30(8): 23-28,34.
[15] 段玮玮,闻敏杰,李强. 饱和分数导数型粘弹性土层竖向振动放大效应[J]. 工程力学, 2013, 30(4): 235-240.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
[1] 邓明科, 李琦琦, 刘海勃, 景武斌. 高延性混凝土低矮剪力墙抗震性能试验研究及抗剪承载力计算[J]. 工程力学, 2020, 37(1): 63 -72 .
[2] 金浏, 王涛, 杜修力, 夏海. 钢筋混凝土悬臂梁剪切破坏及尺寸效应律研究[J]. 工程力学, 2020, 37(1): 53 -62 .
[3] 万志强, 陈建兵. 数据稀缺与更新条件下基于概率密度演化-测度变换的认知不确定性量化分析[J]. 工程力学, 2020, 37(1): 34 -42 .
[4] 金俊超, 佘成学, 尚朋阳. 基于Hoek-Brown准则的应变软化模型有限元数值实现研究[J]. 工程力学, 2020, 37(1): 43 -52 .
[5] 郑山锁, 荣先亮, 张艺欣, 董立国. 冻融损伤低矮RC剪力墙数值模拟方法[J]. 工程力学, 2020, 37(2): 70 -80 .
[6] 韩明君, 王伟兵, 李鸿瑞, 周朝逾, 马连生. 单圆弧波纹管膜片的非线性大变形分析[J]. 工程力学, 2020, 37(1): 26 -33 .
[7] 王元清, 顾浩洋, 廖小伟. 钢结构角焊缝低温抗剪疲劳性能的试验研究[J]. 工程力学, 2020, 37(1): 73 -79,134 .
[8] 李游, 李传习, 陈卓异, 贺君, 邓扬. 基于监测数据的钢箱梁U肋细节疲劳可靠性分析[J]. 工程力学, 2020, 37(2): 111 -123 .
[9] 朱宏平, 沈文爱, 雷鹰, 袁涌, 胡宇航, 张莹. 结构减隔震控制系统性能监测、评估与提升[J]. 工程力学, 2020, 37(1): 1 -16 .
[10] 袁驷, 孙浩涵. 二维自由振动问题的自适应有限元分析初探[J]. 工程力学, 2020, 37(1): 17 -25 .
X

近日,本刊多次接到来电,称有不法网站冒充《工程力学》杂志官网,并向投稿人收取高额费用。在此,我们郑重申明:

1.《工程力学》官方网站是本刊唯一的投稿渠道(原网站已停用),《工程力学》所有刊载论文必须经本刊官方网站的在线投稿审稿系统完成评审。我们不接受邮件投稿,也不通过任何中介或编辑收费组稿。

2.《工程力学》在稿件符合投稿条件并接收后会发出接收通知,请作者在接到版面费或审稿费通知时,仔细检查收款人是否为“《工程力学》杂志社”,千万不要汇款给任何的个人账号。请广大读者、作者相互转告,广为宣传!如有疑问,请来电咨询:010-62788648。

感谢大家多年来对《工程力学》的支持与厚爱,欢迎继续关注我们!

《工程力学》杂志社

2018年11月15日