多尺度有限元法及其应用研究进展
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摘要
简单回顾多尺度有限元法的基本思想与发展历程,探讨多尺度有限元法基函数的构造及其边界条件的选取,详细介绍该方法在断裂力学、结构健康监测、地下水流模拟、材料科学等领域中的应用,并对多尺度有限元法的发展前景进行展望。
A large number of natural sciences and engineering application problems with multi-scale features.This paper briefly reviews the multiscale finite element method development process,describes the typical approach in several areas,such as fracture mechanics,structural health monitoring,underground water flow simulation,materials science applications,pointed out the advantages and disadvantages of each,and finally multiscale finite element method for the development prospect.
A large number of natural sciences and engineering application problems with multi-scale features.This paper briefly reviews the multiscale finite element method development process,describes the typical approach in several areas,such as fracture mechanics,structural health monitoring,underground water flow simulation,materials science applications,pointed out the advantages and disadvantages of each,and finally multiscale finite element method for the development prospect.
引文
[1]BABUSKA I,CALOZ G,OSBOM E.Special finite elementmethods for a class of second order elliptic problems withrough coefficients[J].SIAM J Numer Anal,1994,31(4):945-981.
[2]HOU T Y,WU Xiaohui.A multiscale finite element methodfor elliptic problems in composite materials and porous media[J].J Comp Phy,1997,134(1):169-189.
[3]HOU T Y,WU Xiaohui,CAI Zhi qiang.Convergence of amultiscale finite element method for elliptic problems withrapidly oscillating coefficients[J].Math Comp,1999,68:913-943.
[4]EFENDIEV Y R,HOU T Y,WU Xiaohui.Convergence of anonconforming multiscale finite element method[J].SIAM JNumer Anal,2000,37(3):888-910.
[5]HOU T Y,WU Xiaohui,ZHANG Yu.Removing the cellresonance error in the multiscale finite element method via aPetrovGalerkin fonnulation[J].Commun Math Sci,2004,2(2):185-205.
[6]EFENDIEV Y,GINTING V,HOU T Y,et al.Accuratemultiscale finite element methods for twophase flowsimulations[J].J Comp Phy,2006,220(1):155-174.
[7]CHEN Zhi ming,HOU T Y.A mixed multiscale finite elementmethod for elliptic problems with oscillating coeffieients[J].Math Comp,2002,72:541-576.
[8]CHEN Zhi ming,YUE Xingye.Numerical homogenization ofwell singularities in the flow transport through heterogeneousporous media[J].Multiscale Model Simul,2003,1(2):260-303.
[9]CHEN Jinru,CUI Junzhi.A multiscale finite element methodfor elliptic problems with highly oscillatory coefficients[J].Appl Numer Math,2004,50(1):1-13.
[10]ALLAIRE G,BRIZZI R.A multiscale finite element method fornumerical homogenization[J].Multiscale Model Simul,2005,4(3):790-812.
[11]CHU J,EFENDIEV Y,GINTING V,et al.Flow basedoversampling technique for multiscale finite element methods[J].Adv Water Resour,2008,31(4):599-608.
[12]AARNES J E,STEIN K,LIE K A.A hierarchical multiscalemethod for twophase flow based upon mixed finite elementsand nonuniform grids[J].Multiscale Model Simul,2006,5(2):337-363.
[13]ZHANG Hongwu,ZHANG Sheng,GUO Xu,et al.Mutiplespatial and temporal scales method for numerical simulation ofnonclassical heat conduction problems:one dimensional case[J].Int J Solids Struct,2005,42(3/4):877-899.
[14]ZHANG Hongwu,ZHANG Sheng,BI Jinying,et al.Thermomechanical analysis of periodic multiphase materials by amultiscale asymptotic homogenization approach[J].Int JNumer Meth Eng,2007,69(1):87-113.
[15]付振东.非均质饱和多孔介质准静态行为分析的耦合多尺度及耦合升尺度有限元法[D].大连:大连理工大学,2010.
[16]陆新征,林旭川,叶列平.多尺度有限元建模方法及其应用[J].华中科技大学学报:城市科学版,2008,25(4):76-80.
[17]孙戬.多尺度下裂纹断裂过程区力学特性分析[D].西安:西安科技大学,2009.
[18]瞿伟廉,何钟山,刘嘉.动力荷载作用下杆系钢结构节点疲劳裂纹扩展断裂破坏的分析方法[J].土木工程学报,2010,43(12):78-86.
[19]黄其青,谢伟.基于有限元重合网格法的等大共面的三维表面裂纹交互因子研究[J].西北工业大学学报,2009,27(1):105-109.
[20]李兆霞,李爱群,陈鸿天,等.大跨桥梁结构以健康监测和状态评估为目标的有限元模拟[J].东南大学学报:自然科学版,2003,33(5):562-572.
[21]丁幼亮,李爱群,缪长青,等.大跨桥梁结构损伤诊断与安全评估的多尺度有限元模拟研究[J].地震工程与工程振动,2006,26(2):66-72.
[22]孙正华,李兆霞,陈鸿天.结构多尺度有限元模型修正试验研究[J].特种结构,2008,25(4):61-64.
[23]孙正华,李兆霞,陈鸿天.结构行为一致多尺度有限元模型修正及验证[J].东南大学学报:自然科学版,2009,39(1):85-90.
[24]薛禹群,叶淑君,谢春红,等.多尺度有限元法在地下水模拟中的应用[J].水利学报,2004,35(7):7-13.
[25]贺新光.非均质多孔介质中水流问题的多尺度数值模拟[D].北京:中国农业大学,2006.
[26]林琳,杨金忠,方跃骏,等.多尺度有限元法在地下水拟三维数值模拟中的应用[J].中国农村水利水电,2005(12):10-12.
[27]贺新光,任理.求解非均质多孔介质中非饱和水流问题的一种自适应多尺度有限元方法:Ⅰ数值格式[J].水利学报,2009,40(1):38-51.
[28]贺新光,任理.求解非均质多孔介质中非饱和水流问题的一种自适应多尺度有限元方法:Ⅱ数值结果[J].水利学报,2009,40(2):138-144.
[29]张鸣翠.LCM数值模拟的自适应多尺度有限元法[J].科协论坛,2010(9):76-77.
[30]周妍.多尺度有限元法在复合材料液态成型模拟中的应用[D].武汉:武汉理工大学,2009.
[31]曹磊,徐广为,沈连女官.用多尺度方法分析热喷涂层应力[J].材料科学与工艺,2006,14(1):22-24.
[32]黄均平,彭向和.基于能量统一格式的多尺度有限元法[J].计算机辅助工程,2010,19(4):38-43.
[2]HOU T Y,WU Xiaohui.A multiscale finite element methodfor elliptic problems in composite materials and porous media[J].J Comp Phy,1997,134(1):169-189.
[3]HOU T Y,WU Xiaohui,CAI Zhi qiang.Convergence of amultiscale finite element method for elliptic problems withrapidly oscillating coefficients[J].Math Comp,1999,68:913-943.
[4]EFENDIEV Y R,HOU T Y,WU Xiaohui.Convergence of anonconforming multiscale finite element method[J].SIAM JNumer Anal,2000,37(3):888-910.
[5]HOU T Y,WU Xiaohui,ZHANG Yu.Removing the cellresonance error in the multiscale finite element method via aPetrovGalerkin fonnulation[J].Commun Math Sci,2004,2(2):185-205.
[6]EFENDIEV Y,GINTING V,HOU T Y,et al.Accuratemultiscale finite element methods for twophase flowsimulations[J].J Comp Phy,2006,220(1):155-174.
[7]CHEN Zhi ming,HOU T Y.A mixed multiscale finite elementmethod for elliptic problems with oscillating coeffieients[J].Math Comp,2002,72:541-576.
[8]CHEN Zhi ming,YUE Xingye.Numerical homogenization ofwell singularities in the flow transport through heterogeneousporous media[J].Multiscale Model Simul,2003,1(2):260-303.
[9]CHEN Jinru,CUI Junzhi.A multiscale finite element methodfor elliptic problems with highly oscillatory coefficients[J].Appl Numer Math,2004,50(1):1-13.
[10]ALLAIRE G,BRIZZI R.A multiscale finite element method fornumerical homogenization[J].Multiscale Model Simul,2005,4(3):790-812.
[11]CHU J,EFENDIEV Y,GINTING V,et al.Flow basedoversampling technique for multiscale finite element methods[J].Adv Water Resour,2008,31(4):599-608.
[12]AARNES J E,STEIN K,LIE K A.A hierarchical multiscalemethod for twophase flow based upon mixed finite elementsand nonuniform grids[J].Multiscale Model Simul,2006,5(2):337-363.
[13]ZHANG Hongwu,ZHANG Sheng,GUO Xu,et al.Mutiplespatial and temporal scales method for numerical simulation ofnonclassical heat conduction problems:one dimensional case[J].Int J Solids Struct,2005,42(3/4):877-899.
[14]ZHANG Hongwu,ZHANG Sheng,BI Jinying,et al.Thermomechanical analysis of periodic multiphase materials by amultiscale asymptotic homogenization approach[J].Int JNumer Meth Eng,2007,69(1):87-113.
[15]付振东.非均质饱和多孔介质准静态行为分析的耦合多尺度及耦合升尺度有限元法[D].大连:大连理工大学,2010.
[16]陆新征,林旭川,叶列平.多尺度有限元建模方法及其应用[J].华中科技大学学报:城市科学版,2008,25(4):76-80.
[17]孙戬.多尺度下裂纹断裂过程区力学特性分析[D].西安:西安科技大学,2009.
[18]瞿伟廉,何钟山,刘嘉.动力荷载作用下杆系钢结构节点疲劳裂纹扩展断裂破坏的分析方法[J].土木工程学报,2010,43(12):78-86.
[19]黄其青,谢伟.基于有限元重合网格法的等大共面的三维表面裂纹交互因子研究[J].西北工业大学学报,2009,27(1):105-109.
[20]李兆霞,李爱群,陈鸿天,等.大跨桥梁结构以健康监测和状态评估为目标的有限元模拟[J].东南大学学报:自然科学版,2003,33(5):562-572.
[21]丁幼亮,李爱群,缪长青,等.大跨桥梁结构损伤诊断与安全评估的多尺度有限元模拟研究[J].地震工程与工程振动,2006,26(2):66-72.
[22]孙正华,李兆霞,陈鸿天.结构多尺度有限元模型修正试验研究[J].特种结构,2008,25(4):61-64.
[23]孙正华,李兆霞,陈鸿天.结构行为一致多尺度有限元模型修正及验证[J].东南大学学报:自然科学版,2009,39(1):85-90.
[24]薛禹群,叶淑君,谢春红,等.多尺度有限元法在地下水模拟中的应用[J].水利学报,2004,35(7):7-13.
[25]贺新光.非均质多孔介质中水流问题的多尺度数值模拟[D].北京:中国农业大学,2006.
[26]林琳,杨金忠,方跃骏,等.多尺度有限元法在地下水拟三维数值模拟中的应用[J].中国农村水利水电,2005(12):10-12.
[27]贺新光,任理.求解非均质多孔介质中非饱和水流问题的一种自适应多尺度有限元方法:Ⅰ数值格式[J].水利学报,2009,40(1):38-51.
[28]贺新光,任理.求解非均质多孔介质中非饱和水流问题的一种自适应多尺度有限元方法:Ⅱ数值结果[J].水利学报,2009,40(2):138-144.
[29]张鸣翠.LCM数值模拟的自适应多尺度有限元法[J].科协论坛,2010(9):76-77.
[30]周妍.多尺度有限元法在复合材料液态成型模拟中的应用[D].武汉:武汉理工大学,2009.
[31]曹磊,徐广为,沈连女官.用多尺度方法分析热喷涂层应力[J].材料科学与工艺,2006,14(1):22-24.
[32]黄均平,彭向和.基于能量统一格式的多尺度有限元法[J].计算机辅助工程,2010,19(4):38-43.
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