无限元方法及其应用
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摘要
无限元是几何上趋于无穷的单元,它是一种特殊的有限元,也是对有限元在求解无界域问题上的有效补充,并可实现与有限元间的无缝连接．无限元分为映射无限元和非映射无限元:映射无限元需要引入几何映射,在局部坐标系中构造插值形状函数,如Bettess元和Astley元;非映射无限元则直接在整体坐标系中构造插值形状函数,如Burnett元．本文评述求解无界域问题的无限元方法的研究现状和最新发展．首先介绍无限单元的概念和无限元方法的特点;围绕求解以Helmholtz方程控制的波动问题,评述几种常规无限单元的优劣,这些单元包括Bettess元、Astley元和Burnett元．然后介绍新近提出的广义无限元方法,以及与常规无限元方法的区别与联系．最后对无限元方法在各种问题中的应用做了总结．
An infinite element is the one that can handle a domain of infinity.It is a special finite element, and can have a seamless connection with convertional finite elements.It can be a mapped infinite element or a non-mapped infinite element.The former,such as Bettess element and Astley element,needs geometry mapping and shape functions in terms of local coordinates,while for the latter,the shape functions are directly expressed in terms of global coordinates.This paper reviews the state-of-the-art and recent advances of the infinite element method for unbounded domains.First,the concept and features of the infinite element method are introduced.Then,taking the wave problems governed by the Helmholtz equation as an example,several conventional infinite elements such as the Bettess element,the Astley element and the Burnett element are compared and reviewed.Next,we introduce the generalized infinite element and its relation to the conventional infinite elements.Finally,the applications of the infinite element to various problems are summarized.
An infinite element is the one that can handle a domain of infinity.It is a special finite element, and can have a seamless connection with convertional finite elements.It can be a mapped infinite element or a non-mapped infinite element.The former,such as Bettess element and Astley element,needs geometry mapping and shape functions in terms of local coordinates,while for the latter,the shape functions are directly expressed in terms of global coordinates.This paper reviews the state-of-the-art and recent advances of the infinite element method for unbounded domains.First,the concept and features of the infinite element method are introduced.Then,taking the wave problems governed by the Helmholtz equation as an example,several conventional infinite elements such as the Bettess element,the Astley element and the Burnett element are compared and reviewed.Next,we introduce the generalized infinite element and its relation to the conventional infinite elements.Finally,the applications of the infinite element to various problems are summarized.
引文
1 Bettess P,Zienkiewicz O C.Diffraction and refraction of sur- face waves using finite and infinite elements.International Journal for Numerical Methods in Engineering,1977,11: 1271～1290
2 Babuska I,Melenk J M.The partition of unity method.In- ternational Journal for Numerical Method in Engineering, 1997,40:727～758
3 Bettess P,Emson C,Chiam T C.A new mapped infinite element for exterior wave problems.In:Lewis R W,et al, eds.Numerical Methods in Coupled Systems.New York: John Wiley & Sons,1984.489～504
4 Zienkiewicz O C,Bando K,Bettess P,et al.Mapped in- finite elements for exterior wave problems.International Journal for Numerical Methods in Engineering,1985,21: 1229～1251
5 Bettess J A,Bettess P.A new mapped infinite wave element for general wave diffraction problems and its validation on the ellipse diffraction problem.Computer Methods in Ap- plied Mechanics and Engineering,1998,164:17～48
6 Astley R J,Macaulay G J.Mapped wave envelope elements for acoustical radiation and scattering.Journal of Sound and Vibration,1994,170(1):97～118
7 Astley R J,Eversman W.Finite element formulations for acoustical radiation.Journal of Sound and Vibration,1983, 88(1):47～64
8 Cremers L,Fyfe K R,Coyette J P.A variable order infi- nite acoustic wave envelope element.Journal of Sound and Vibration,1994,171(4):483～508
9 Cremers L,Fyfe K R.On the use of variable order infinite wave envelope elements for acoustic radiation and scatter- ing.The Journal of The Acoustical Society of America, 1995,97(4):2028～2040
10 Astley R J.Transient wave envelope elements for wave problems.Journal of Sound and Vibration,1996,192(1): 245～261
11 Burnett D S.A three-dimensional acoustic infinite element based on a prolate spheroidal multi-pole expansion.The Journal of the Acoustical Society of America,1994,96(5): 2798～2816
12 Burnett D S,Holford R L.Prolate and oblate spheroidal acoustic infinite elements.Computer Methods in Applied Mechanics and Engineering,1998,158:117～141
13 1996 Bell Labs Fellows,http://www.bell-labs.com/pr/awa- rds/fellow/1996
14 Burnett D S,Holford R L.An ellipsoidal acoustic infinite element.Computer Methods in Applied Mechanics and En- gineering,1998,164:49～76
15 Astley R J.Mapped spheroidal wave-envelope elements for unbounded wave problems.International Journal for Nu- merical Methods in Engineering,1998,41:1235～1254
16 Li L X,Sun J S,Sakamoto H.On the virtual acoustical source in mapped infinite element.Journal of Sound and Vibration,2003,261:945～951
17 Zienkiewicz O C,Taylor R L.The Finite Element Method (4th ed.Volume 1,Basic Formulation and linear Problems). London:McGraw-Will Book Company(UK)Limited,1989
18 Moes N,Dolbow J,Belytschko T.A finite element method for crack growth without remeshing.International Journal for Numerical Methods in Engineering,1999,46:131～150
19 Daux C,Moes N,Dolbow J,Sukumar N,Belytschko T.Ar- bitrary branched and intersecting cracks with the extended finite element method.International Journal for Numerical Methods in Engineering,2000,48:1741～1760
20 Li L X,Han X P,Xu S Q.Study on the degeneration of quadrilateral element to triangular element.Commu- nications in Numerical Methods in Engineering,2004,20: 671～679
21 Li L X,Kunimatsu S,Han X P,Xu S Q.The analysis of interpolation precision of quadrilateral elements.Finite El- ements in Analysis & Design,2004,41(1):91～108
22李录贤,王铁军．扩展有限元法(XFEM)及其应用．力学进展, 2005,35(1):5～20
23 Li L X,Sun J S,Sakamoto H.An integration technique in Burnett infinite element.In:Proceedings of the ASME Noise Control and Acoustics Division-2003.New York: American Society of Mechanical Engineers,2003.57～62
24 Li L X,Kunimatsu S,Sun J S,Sakamoto H.A new conju- gated mapped infinite element.Journal of Computational Acoustics,2004,12(4):543～570
25 Li L X,Sun J S,Sakamoto H.A generalized infinite ele- ment for acoustic radiation.ASME Journal of Vibration and Acoustics,2005,127:2～11
26 Bettess P.Infinite Elements.UK:Penshaw Press,1992
27 Astley R J,Eversman W.Wave envelope and infinite ele- ment schemes for fan noise radiation from turbofan inlets. AIAA-83-0709,1983
28赵崇斌,张楚汉,张光斗．用无穷元模拟半无限平面弹性地基．清华大学学报,1986,26(1):51～64
29 Zhang C,Zhao C.Coupling method of finite and infinite elements for strip foundation wave problems.Earthquake Engineering and Structural Dynamics,1987,15:839～851
30赵崇斌．拱坝地基无限域模拟与地震波输入研究:[博士论文]．北京:清华大学,1989
31燕柳斌．用三维映射无限元模拟重力坝地基．水利学报,1991, 10:7～12
32燕柳斌．用映射无限元模拟直立式沉箱基础．红水河,1997,16(2): 25～27
33燕柳斌．结构分析的有限元及无限元法．武汉:武汉工业大学出版社,1998
34向前．无限元参与地基基础-坝体动力相互作用分析:[硕士论文]．广西:广西大学,2004
35解家毕．带桩多高层建筑共同作用分析方法及其在纠偏中的应用: [博士论文]．武汉:武汉大学,2004
36 Li L X,Sun J S,Sakamoto H.A modified infinite element method for acoustic radiation.Journal of Computational Acoustics,2002,10(1):113～121
37 Li L X,Sun J S,Sakamoto H.Application of the modified in- finite element method to two-dimensional transient acoustic radiation.Applied Acoustics,2003,64:55～70
38 Li L X,Kunimatsu S,Sun J S.Numerical experiments of the generalized mapped infinite element.In:The 18~(th)Inter- national Congress on Acoustics(ICA 2004),Kyoto,Japan.Ⅰ-133～Ⅰ-136
39 Li L X,Wang AQ.On the“mapped spheroidal”infinite el- ement for unbounded wave problems.In:Liu G R,et al, eds.Computational Methods.Netherlands:Springer,2006. 481～485
40王爱琴．弹性力学无界问题的无限元解法:[硕士论文]．西安:西安交通大学,2006
41 Kumar P.Static infinite element formulation.ASCE Jour- nal of Structural Engineering,1985,111(11):2355～2372
42 Bettess P.Short-wave scattering: problems and techniques. Philosophical Transactions of the Royal Society A-Math- ematical Physical and Engineering Sciences,2004,362: 421～443
43 Ortiz P.Finite elements using a plane-wave basis for scat- tering of surface water waves.Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineer- ing Sciences,2004,362:525～540
44 Astley R J.Infinite elements for wave problems:a review of current formulations and an assessment of accuracy.In- ternational Journal for Numerical Methods in Engineering, 2000,49:951～976
45 Gerdes K.A review of infinite element methods for exterior Helmholtz problems.Journal of Computational Acoustics, 2000,8,(1):43～62
46应隆安．无限元方法．北京:北京大学出版社,1992
47 Ying L.Error Estimates of the finite element method ap- proximating smooth solutions to 3-D couservation laws.Ad- vances in Mathematics,2002,31(5):451～460
48应隆安．一个求解多维守恒律方程组的二阶显式有限元格式．计算数学,2001,23(1):321～332
49应隆安．多维守恒律的二阶显式有限元格式及收敛性．中国科学A辑,2000,30(5):388～400
50应隆安．双曲型守恒律方程组的Godunov格式中离散激波的渐近稳定性．高等学校计算数学学报,1999,(4):319～332
51 Ying L.Infinite element method for problems on unbounded and multiply connected domains.Acta Mathematica Scien- tia,2001,21B(4):440～452
52辛海涛,李玉龙,马轩祥等．无限元方法及其在烤瓷熔附金属全冠应力分析中的应用．中国生物医学工程学报,2004,23(5):461～466
2 Babuska I,Melenk J M.The partition of unity method.In- ternational Journal for Numerical Method in Engineering, 1997,40:727～758
3 Bettess P,Emson C,Chiam T C.A new mapped infinite element for exterior wave problems.In:Lewis R W,et al, eds.Numerical Methods in Coupled Systems.New York: John Wiley & Sons,1984.489～504
4 Zienkiewicz O C,Bando K,Bettess P,et al.Mapped in- finite elements for exterior wave problems.International Journal for Numerical Methods in Engineering,1985,21: 1229～1251
5 Bettess J A,Bettess P.A new mapped infinite wave element for general wave diffraction problems and its validation on the ellipse diffraction problem.Computer Methods in Ap- plied Mechanics and Engineering,1998,164:17～48
6 Astley R J,Macaulay G J.Mapped wave envelope elements for acoustical radiation and scattering.Journal of Sound and Vibration,1994,170(1):97～118
7 Astley R J,Eversman W.Finite element formulations for acoustical radiation.Journal of Sound and Vibration,1983, 88(1):47～64
8 Cremers L,Fyfe K R,Coyette J P.A variable order infi- nite acoustic wave envelope element.Journal of Sound and Vibration,1994,171(4):483～508
9 Cremers L,Fyfe K R.On the use of variable order infinite wave envelope elements for acoustic radiation and scatter- ing.The Journal of The Acoustical Society of America, 1995,97(4):2028～2040
10 Astley R J.Transient wave envelope elements for wave problems.Journal of Sound and Vibration,1996,192(1): 245～261
11 Burnett D S.A three-dimensional acoustic infinite element based on a prolate spheroidal multi-pole expansion.The Journal of the Acoustical Society of America,1994,96(5): 2798～2816
12 Burnett D S,Holford R L.Prolate and oblate spheroidal acoustic infinite elements.Computer Methods in Applied Mechanics and Engineering,1998,158:117～141
13 1996 Bell Labs Fellows,http://www.bell-labs.com/pr/awa- rds/fellow/1996
14 Burnett D S,Holford R L.An ellipsoidal acoustic infinite element.Computer Methods in Applied Mechanics and En- gineering,1998,164:49～76
15 Astley R J.Mapped spheroidal wave-envelope elements for unbounded wave problems.International Journal for Nu- merical Methods in Engineering,1998,41:1235～1254
16 Li L X,Sun J S,Sakamoto H.On the virtual acoustical source in mapped infinite element.Journal of Sound and Vibration,2003,261:945～951
17 Zienkiewicz O C,Taylor R L.The Finite Element Method (4th ed.Volume 1,Basic Formulation and linear Problems). London:McGraw-Will Book Company(UK)Limited,1989
18 Moes N,Dolbow J,Belytschko T.A finite element method for crack growth without remeshing.International Journal for Numerical Methods in Engineering,1999,46:131～150
19 Daux C,Moes N,Dolbow J,Sukumar N,Belytschko T.Ar- bitrary branched and intersecting cracks with the extended finite element method.International Journal for Numerical Methods in Engineering,2000,48:1741～1760
20 Li L X,Han X P,Xu S Q.Study on the degeneration of quadrilateral element to triangular element.Commu- nications in Numerical Methods in Engineering,2004,20: 671～679
21 Li L X,Kunimatsu S,Han X P,Xu S Q.The analysis of interpolation precision of quadrilateral elements.Finite El- ements in Analysis & Design,2004,41(1):91～108
22李录贤,王铁军．扩展有限元法(XFEM)及其应用．力学进展, 2005,35(1):5～20
23 Li L X,Sun J S,Sakamoto H.An integration technique in Burnett infinite element.In:Proceedings of the ASME Noise Control and Acoustics Division-2003.New York: American Society of Mechanical Engineers,2003.57～62
24 Li L X,Kunimatsu S,Sun J S,Sakamoto H.A new conju- gated mapped infinite element.Journal of Computational Acoustics,2004,12(4):543～570
25 Li L X,Sun J S,Sakamoto H.A generalized infinite ele- ment for acoustic radiation.ASME Journal of Vibration and Acoustics,2005,127:2～11
26 Bettess P.Infinite Elements.UK:Penshaw Press,1992
27 Astley R J,Eversman W.Wave envelope and infinite ele- ment schemes for fan noise radiation from turbofan inlets. AIAA-83-0709,1983
28赵崇斌,张楚汉,张光斗．用无穷元模拟半无限平面弹性地基．清华大学学报,1986,26(1):51～64
29 Zhang C,Zhao C.Coupling method of finite and infinite elements for strip foundation wave problems.Earthquake Engineering and Structural Dynamics,1987,15:839～851
30赵崇斌．拱坝地基无限域模拟与地震波输入研究:[博士论文]．北京:清华大学,1989
31燕柳斌．用三维映射无限元模拟重力坝地基．水利学报,1991, 10:7～12
32燕柳斌．用映射无限元模拟直立式沉箱基础．红水河,1997,16(2): 25～27
33燕柳斌．结构分析的有限元及无限元法．武汉:武汉工业大学出版社,1998
34向前．无限元参与地基基础-坝体动力相互作用分析:[硕士论文]．广西:广西大学,2004
35解家毕．带桩多高层建筑共同作用分析方法及其在纠偏中的应用: [博士论文]．武汉:武汉大学,2004
36 Li L X,Sun J S,Sakamoto H.A modified infinite element method for acoustic radiation.Journal of Computational Acoustics,2002,10(1):113～121
37 Li L X,Sun J S,Sakamoto H.Application of the modified in- finite element method to two-dimensional transient acoustic radiation.Applied Acoustics,2003,64:55～70
38 Li L X,Kunimatsu S,Sun J S.Numerical experiments of the generalized mapped infinite element.In:The 18~(th)Inter- national Congress on Acoustics(ICA 2004),Kyoto,Japan.Ⅰ-133～Ⅰ-136
39 Li L X,Wang AQ.On the“mapped spheroidal”infinite el- ement for unbounded wave problems.In:Liu G R,et al, eds.Computational Methods.Netherlands:Springer,2006. 481～485
40王爱琴．弹性力学无界问题的无限元解法:[硕士论文]．西安:西安交通大学,2006
41 Kumar P.Static infinite element formulation.ASCE Jour- nal of Structural Engineering,1985,111(11):2355～2372
42 Bettess P.Short-wave scattering: problems and techniques. Philosophical Transactions of the Royal Society A-Math- ematical Physical and Engineering Sciences,2004,362: 421～443
43 Ortiz P.Finite elements using a plane-wave basis for scat- tering of surface water waves.Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineer- ing Sciences,2004,362:525～540
44 Astley R J.Infinite elements for wave problems:a review of current formulations and an assessment of accuracy.In- ternational Journal for Numerical Methods in Engineering, 2000,49:951～976
45 Gerdes K.A review of infinite element methods for exterior Helmholtz problems.Journal of Computational Acoustics, 2000,8,(1):43～62
46应隆安．无限元方法．北京:北京大学出版社,1992
47 Ying L.Error Estimates of the finite element method ap- proximating smooth solutions to 3-D couservation laws.Ad- vances in Mathematics,2002,31(5):451～460
48应隆安．一个求解多维守恒律方程组的二阶显式有限元格式．计算数学,2001,23(1):321～332
49应隆安．多维守恒律的二阶显式有限元格式及收敛性．中国科学A辑,2000,30(5):388～400
50应隆安．双曲型守恒律方程组的Godunov格式中离散激波的渐近稳定性．高等学校计算数学学报,1999,(4):319～332
51 Ying L.Infinite element method for problems on unbounded and multiply connected domains.Acta Mathematica Scien- tia,2001,21B(4):440～452
52辛海涛,李玉龙,马轩祥等．无限元方法及其在烤瓷熔附金属全冠应力分析中的应用．中国生物医学工程学报,2004,23(5):461～466
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