无网格伽辽金法及其在动力学问题中的应用
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摘要
无网格法是一门新兴的数值计算方法,和有限元方法相比,它在对系统进行分析时可以消除或者部分消除单元网格的约束,从而在处理如高速碰撞,大变形,裂纹扩展等类型的问题时显示出了其优越性。在动力学问题中,常常伴随着上面所述的各种问题,因此采用无网格法对动力学问题进行分析显然是适合的。
本文介绍了无网格方法的基本理论,并着重对无网格伽辽金法(Element-Free Galerkin Method,以下简称EFG法)的流程体系和影响因素进行了分析和讨论；然后,基于EFG法的理论推导了采用无网格法离散动力学问题系统方程的过程,给出了振动问题无网格系统方程的形式和求解方法；最后,将无网格法应用于动力学中的另一个重要领域——接触碰撞问题,详细介绍了碰撞问题的无网格分析方法,并提出了采用无网格与有限元结合的方法分析大型船舶碰撞问题的观点,为大型船舶碰撞问题的研究提出了新的思路。
本文编制了无伽辽金法的计算机程序,应用程序对一些算例进行了分析,算例的类型涵盖了静力学问题,振动问题以及接触碰撞问题。计算结果与影响因素分析充分证明无网格算法的可靠性和所编制程序的正确性,在理论上和应用上都具有一定的价值。通过本文的论述,证明了将无网格法应用于振动问题和碰撞问题是完全可行的,无网格法在动力学领域中具有广泛的应用前景。
Meshless method is a newly arisen kind of numerical method. Compare with Finite Element Method (FEM), it can eliminate or partly eliminate the bond lead by element mesh during the process of analyzing. For which it shows the advantage in dealing with issues such as high-speed impact, large deformation, dynamic crack growth, etc. Problems likes above usually come about in dynamics area so that it seems obviously adapted to make analysis of dynamics by using meshless method.
This paper firstly makes a summarization in meshless method and process system and influencing factors of Element-Free Galerkin Method (EFG) are analyzed and discussed based on the meshless method theories; And then from the same way,meshless-form function and its solving processes of vibration problem are derived; At last this paper makes application of meshless method into another important area of dynamics-contact impact problems, and detailedly introduces the analyzing process of contact impact problems by using meshless method. Furthermore a new thoughts is given that it is a good way to combine the meshless and FE methods to solving large-scale ship collision problems.
In this paper, some examples containing statics, vibration, and contact impact problems are analyzed by the EFG program. The analysis results sufficiently proof both reliability of the method and correctness of the program. So it is completely feasible to make use of meshless method in analysis of vibration and contact impact problems and the prospect is broad.
本文介绍了无网格方法的基本理论,并着重对无网格伽辽金法(Element-Free Galerkin Method,以下简称EFG法)的流程体系和影响因素进行了分析和讨论；然后,基于EFG法的理论推导了采用无网格法离散动力学问题系统方程的过程,给出了振动问题无网格系统方程的形式和求解方法；最后,将无网格法应用于动力学中的另一个重要领域——接触碰撞问题,详细介绍了碰撞问题的无网格分析方法,并提出了采用无网格与有限元结合的方法分析大型船舶碰撞问题的观点,为大型船舶碰撞问题的研究提出了新的思路。
本文编制了无伽辽金法的计算机程序,应用程序对一些算例进行了分析,算例的类型涵盖了静力学问题,振动问题以及接触碰撞问题。计算结果与影响因素分析充分证明无网格算法的可靠性和所编制程序的正确性,在理论上和应用上都具有一定的价值。通过本文的论述,证明了将无网格法应用于振动问题和碰撞问题是完全可行的,无网格法在动力学领域中具有广泛的应用前景。
Meshless method is a newly arisen kind of numerical method. Compare with Finite Element Method (FEM), it can eliminate or partly eliminate the bond lead by element mesh during the process of analyzing. For which it shows the advantage in dealing with issues such as high-speed impact, large deformation, dynamic crack growth, etc. Problems likes above usually come about in dynamics area so that it seems obviously adapted to make analysis of dynamics by using meshless method.
This paper firstly makes a summarization in meshless method and process system and influencing factors of Element-Free Galerkin Method (EFG) are analyzed and discussed based on the meshless method theories; And then from the same way,meshless-form function and its solving processes of vibration problem are derived; At last this paper makes application of meshless method into another important area of dynamics-contact impact problems, and detailedly introduces the analyzing process of contact impact problems by using meshless method. Furthermore a new thoughts is given that it is a good way to combine the meshless and FE methods to solving large-scale ship collision problems.
In this paper, some examples containing statics, vibration, and contact impact problems are analyzed by the EFG program. The analysis results sufficiently proof both reliability of the method and correctness of the program. So it is completely feasible to make use of meshless method in analysis of vibration and contact impact problems and the prospect is broad.
引文
[1]Lucy L B.A numerical approach to the testing of the fission hypothesis [J]. The Astron J, 1977,8(12):013-1024.
[2]Monaghan J J. Smoothed particle hydrodynamics [J]. Annual Eeview Astronomics and Astrophysics,1992,30:543-574.
[3]Dyka C T. Addressing tension instability in SPH methods [R]. Technical Report NRL/MR/6384, NRL,1994.
[4]Chen J K, Beraun J E, Jih C J, An improvement for tensile instability in smoothed particle hydrodynamics [J]. Comput Methods Appl Mech Engrg,2000,184:67-85.
[5]Vignjevic R, Campbell J, Libersky L.A treatment of zero-energy modes in the smoothed particle hydrodynamics method. Comput. Methods Appl. Mech.Energy.,2000,184:67-85.
[6]Lancaster P, Salkauskas K. Surfaces generated by moving least-squares methods, Math Comput,1981,37(155):141-158.
[7]Nayroles B, Touzot G, Villon P. Generalizing the finite element method:diffuse approximation and diffuse elements. Computational Mechanics,1992,10(5):307-318.
[8]Belytscho T, Lu Y Y, Gu L. Element free Galerkin methods [J]. Int J Numer Methods Engrg,1994,37:229-256.
[9]Liu W K, Jun S, Adee J, Belytschko T. Reproducing kernel particle methods. Int J Numer Methods Engrg,1995,38(10):1655-1679.
[10]Liu W K, Chen Y J. Wavelet and multiple scale reproducing kernel method. Int J Numer Methods Fluise,1995,21(10):901-931.
[11]Liu W K, JunS, Li S F et al. ReProducing kernel particle methods for structural dynamics. Int. J.Num.Meth. Engng.,1995,38:1655-1679.
[12]Onate E, Idelsohn S, Zienkiewicz O O, Talor R L. A finite point method in computational mechanics application to convective transport and fluid flow. Int J Numer Methods Engrg,1996,39(22):3839-3866.
[13]Oden J T, Duarte C A M, Zienkiewicz O C A Hp adaptive method using clouds. Comput Methods Appl Mech Engrg,1998,153(2):117-126.
[14]Melenk J M, Babuska I. The Partition of Unity Finite Element Method:Basic Theory and Applications. Computer Methods in Applied Mechanics and Engineering.139:289-314.
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[17]Liszka T J, Duarte C A M, Tworzydlo W W. Hp-meshless clouds method. Comput Methods Appl Engrg,1996,139(3):263-288.
[18]Atluri S N, Sladek J et al. The local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity. Computational Mechanics 25,2000:180-198
[19]Atluri S N, Zhu T. A new meshless local Pretrov-Galerkin approach in computational mechanics. Computational Mechanics,1998,22(2):113-127.
[20]周维垣,寇小东.无单元法及其工程应用[J].力学学报,1998,30(2)：193-202.
[21]宋康祖,陆明万.无网格方法综述.清华大学工程力学、数学、热物理学术会议论文集,北京：清华大学出版社,1998.340-345.
[22]张雄,胡炜等.加权最小二乘无网格法.力学学报,35(4),2003：425-431
[23]宋康祖,陆明万,张雄.固体力学中的无网格方法[J].力学进展,2000,30(1)：55-65.
[24]Liu G R. Mesh Free Methods:Moving Beyond the Finite Element Method. CRC press, Boca Raton, USA.
[25]Liu G R. Comparisons of two meshfree local point interpolation methods for structural analyses. Computational Mechanics,2002,29:107-121.
[26]Liu G R, Gu Y T. Assessment and applications of point interpolation methods for computational mechanics. International Journal for Numerical Mrthods in Engineering,59: 1373-1397.
[27]Wang J G, Liu G R. Radial point interpolation method for elastoplastic problems. Proc. of the 1st Int. Conf. On Structural Stability and Dynamics, Dec.7-9,2000, Taipei,703-708, 2000.
[28]Liu G R, Gu Y T. A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids [J]. Sound and Vibration,246(1),29-46.
[29]Kansa E J. Multiquadrics-A Scattered Data Approximation Scheme with Applications to Computational Fluid dynamics. Computers Math. Applic.,19(8/9),127-145.
[30]Sharan M, Kansa E J, Gupta S. Application of the Multiquadric Method for Numerical Solution of Elliptic Partial Differential Equations. Applied Mathematics and Computation, 84,275-302.
[31]Schaback R, Wendland H. Characterzation and construction of radial basis functions. Multivatiate Approximation and Applications. Cambridge University Press.
[32]Liu G R, Dai Y K, Lim K M, Gu Y T. A point interpolation mesh free method for static and frequency analysis of two-dimensional piezolectric structures. Compu- -tational Mechanics.
[33]Lancaster P, Salkauskas K. Surfaces generated by moving least squares methods. Math. Comput.37,141-158.
[34]Cleveland W S. Visualizing Data, AT&T Bell Laboratories, Murray Hill, NJ.
[35]Belytschko T, Krongauz Y, Organ D, Liu W K. Smoothing and accelerated computations in the element free Galerkin method [J].Comp. And Appl. Math.74,111-126.
[36]徐次达.新计算力学加权残值法.上海：同济大学出版社,1997.
[37]Krok J, Orkisz J. Aunified approach to the FE generalized variational FD method for nonlinear mechanics. Concept and numerical approach.353-362. Springer-Verlag.
[38]Beissel S, Belytsehko T. Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech.Engrg.1996,139:49-74.
[39]Belytsehko T, Organ D, Krongnauz Y. A coupled finite element-element free Galerkin method. ComPut. Mech.1995,17:186-195.
[40]王勖成.有限单元法.清华大学出版社.2003,7.
[41]王彬.《振动分析及应用》.清华大学出版社,2001：69-71.
[42]李庆扬等.《数学分析》.海潮出版社,1992：219-221.
[43]J.H.金斯伯格(著),白化同,李俊宝(译).机械与结构振动—理论与应用,中国宇航出版社,2005：184-186.
[44]王勖成,邵敏.有限单元法基本原理和数值方法.清华大学出版社,1997：331-333.
[45]米勒PC,席伦WO著,曾子平,向豪英等译.线性振动—多自由度振动系统理论分析方法.天津大学出版社,1989：165-166.
[46]Johnson K L. Contact Mechanics. Press Syndicate of the University of Cambridge,1985.
[47]Liu G, Zhu J, Yu L et al. Elasto-Plastic contact of rough surfaces. STLE Annual Meeting, Orlando, FL, May 20-24,2001. Tribology Transactions,2001,44:437-443.
[48]姜晋庆.接触体应力分析.西北工业大学八○一教研室,1985年8月.
[49]Owen D R, Hinton E. Finite elements in plasticity:theory and practice. Pineridge Press Limited Swansea, U. K.1980.
[50]Wang F J, Chen J G, Yao Z H. A contact searching algorithm for contact-impact problem, Int. J. Numer. Methods. Engn,2000,16(4):374-382.
[2]Monaghan J J. Smoothed particle hydrodynamics [J]. Annual Eeview Astronomics and Astrophysics,1992,30:543-574.
[3]Dyka C T. Addressing tension instability in SPH methods [R]. Technical Report NRL/MR/6384, NRL,1994.
[4]Chen J K, Beraun J E, Jih C J, An improvement for tensile instability in smoothed particle hydrodynamics [J]. Comput Methods Appl Mech Engrg,2000,184:67-85.
[5]Vignjevic R, Campbell J, Libersky L.A treatment of zero-energy modes in the smoothed particle hydrodynamics method. Comput. Methods Appl. Mech.Energy.,2000,184:67-85.
[6]Lancaster P, Salkauskas K. Surfaces generated by moving least-squares methods, Math Comput,1981,37(155):141-158.
[7]Nayroles B, Touzot G, Villon P. Generalizing the finite element method:diffuse approximation and diffuse elements. Computational Mechanics,1992,10(5):307-318.
[8]Belytscho T, Lu Y Y, Gu L. Element free Galerkin methods [J]. Int J Numer Methods Engrg,1994,37:229-256.
[9]Liu W K, Jun S, Adee J, Belytschko T. Reproducing kernel particle methods. Int J Numer Methods Engrg,1995,38(10):1655-1679.
[10]Liu W K, Chen Y J. Wavelet and multiple scale reproducing kernel method. Int J Numer Methods Fluise,1995,21(10):901-931.
[11]Liu W K, JunS, Li S F et al. ReProducing kernel particle methods for structural dynamics. Int. J.Num.Meth. Engng.,1995,38:1655-1679.
[12]Onate E, Idelsohn S, Zienkiewicz O O, Talor R L. A finite point method in computational mechanics application to convective transport and fluid flow. Int J Numer Methods Engrg,1996,39(22):3839-3866.
[13]Oden J T, Duarte C A M, Zienkiewicz O C A Hp adaptive method using clouds. Comput Methods Appl Mech Engrg,1998,153(2):117-126.
[14]Melenk J M, Babuska I. The Partition of Unity Finite Element Method:Basic Theory and Applications. Computer Methods in Applied Mechanics and Engineering.139:289-314.
[15]Babuska I., Melenk J M, The partition of unity method. Int. J. Numer. Meth. Engrg, 40,1997:727-758
[16]Onate E, Idelsohn S, Zienkiewicz O O, Talor R L. A finite point method in computational mechanics application to convective transport and fluid flow. Int J Numer Methods Engrg,1996,39(22):3839-3866.
[17]Liszka T J, Duarte C A M, Tworzydlo W W. Hp-meshless clouds method. Comput Methods Appl Engrg,1996,139(3):263-288.
[18]Atluri S N, Sladek J et al. The local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity. Computational Mechanics 25,2000:180-198
[19]Atluri S N, Zhu T. A new meshless local Pretrov-Galerkin approach in computational mechanics. Computational Mechanics,1998,22(2):113-127.
[20]周维垣,寇小东.无单元法及其工程应用[J].力学学报,1998,30(2)：193-202.
[21]宋康祖,陆明万.无网格方法综述.清华大学工程力学、数学、热物理学术会议论文集,北京：清华大学出版社,1998.340-345.
[22]张雄,胡炜等.加权最小二乘无网格法.力学学报,35(4),2003：425-431
[23]宋康祖,陆明万,张雄.固体力学中的无网格方法[J].力学进展,2000,30(1)：55-65.
[24]Liu G R. Mesh Free Methods:Moving Beyond the Finite Element Method. CRC press, Boca Raton, USA.
[25]Liu G R. Comparisons of two meshfree local point interpolation methods for structural analyses. Computational Mechanics,2002,29:107-121.
[26]Liu G R, Gu Y T. Assessment and applications of point interpolation methods for computational mechanics. International Journal for Numerical Mrthods in Engineering,59: 1373-1397.
[27]Wang J G, Liu G R. Radial point interpolation method for elastoplastic problems. Proc. of the 1st Int. Conf. On Structural Stability and Dynamics, Dec.7-9,2000, Taipei,703-708, 2000.
[28]Liu G R, Gu Y T. A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids [J]. Sound and Vibration,246(1),29-46.
[29]Kansa E J. Multiquadrics-A Scattered Data Approximation Scheme with Applications to Computational Fluid dynamics. Computers Math. Applic.,19(8/9),127-145.
[30]Sharan M, Kansa E J, Gupta S. Application of the Multiquadric Method for Numerical Solution of Elliptic Partial Differential Equations. Applied Mathematics and Computation, 84,275-302.
[31]Schaback R, Wendland H. Characterzation and construction of radial basis functions. Multivatiate Approximation and Applications. Cambridge University Press.
[32]Liu G R, Dai Y K, Lim K M, Gu Y T. A point interpolation mesh free method for static and frequency analysis of two-dimensional piezolectric structures. Compu- -tational Mechanics.
[33]Lancaster P, Salkauskas K. Surfaces generated by moving least squares methods. Math. Comput.37,141-158.
[34]Cleveland W S. Visualizing Data, AT&T Bell Laboratories, Murray Hill, NJ.
[35]Belytschko T, Krongauz Y, Organ D, Liu W K. Smoothing and accelerated computations in the element free Galerkin method [J].Comp. And Appl. Math.74,111-126.
[36]徐次达.新计算力学加权残值法.上海：同济大学出版社,1997.
[37]Krok J, Orkisz J. Aunified approach to the FE generalized variational FD method for nonlinear mechanics. Concept and numerical approach.353-362. Springer-Verlag.
[38]Beissel S, Belytsehko T. Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech.Engrg.1996,139:49-74.
[39]Belytsehko T, Organ D, Krongnauz Y. A coupled finite element-element free Galerkin method. ComPut. Mech.1995,17:186-195.
[40]王勖成.有限单元法.清华大学出版社.2003,7.
[41]王彬.《振动分析及应用》.清华大学出版社,2001：69-71.
[42]李庆扬等.《数学分析》.海潮出版社,1992：219-221.
[43]J.H.金斯伯格(著),白化同,李俊宝(译).机械与结构振动—理论与应用,中国宇航出版社,2005：184-186.
[44]王勖成,邵敏.有限单元法基本原理和数值方法.清华大学出版社,1997：331-333.
[45]米勒PC,席伦WO著,曾子平,向豪英等译.线性振动—多自由度振动系统理论分析方法.天津大学出版社,1989：165-166.
[46]Johnson K L. Contact Mechanics. Press Syndicate of the University of Cambridge,1985.
[47]Liu G, Zhu J, Yu L et al. Elasto-Plastic contact of rough surfaces. STLE Annual Meeting, Orlando, FL, May 20-24,2001. Tribology Transactions,2001,44:437-443.
[48]姜晋庆.接触体应力分析.西北工业大学八○一教研室,1985年8月.
[49]Owen D R, Hinton E. Finite elements in plasticity:theory and practice. Pineridge Press Limited Swansea, U. K.1980.
[50]Wang F J, Chen J G, Yao Z H. A contact searching algorithm for contact-impact problem, Int. J. Numer. Methods. Engn,2000,16(4):374-382.