独立覆盖流形法的本质边界条件施加方法
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摘要
目前,数值流形方法、无网格法等新的数值计算方法存在本质边界条件不易严格施加的问题。针对笔者前期提出的独立覆盖流形法,通过一个悬臂梁的例子,系统地分析了本质边界条件施加问题。采用多项式覆盖函数,提出了改进边界覆盖函数和直接设定独立覆盖函数2种方法,不仅严格满足边界条件,而且能保证边界附近的近似函数逼近真实解。这2种方法避免了常用罚函数法中的罚数取值对计算结果和方程性态的影响问题,而且只需令部分自由度不参与计算就能实现,操作简单。通过设置覆盖函数来施加边界条件的方式可供其他新方法借鉴。
At present,some new numerical methods such as Numerical Manifold Method( NMM) and Meshless Method are facing with the difficulty of strictly applying essential boundary conditions. Through a case study of a cantilever beam,the application of essential boundary conditions is systematically analyzed in NMM based on independent covers previously proposed by the authors. On the basis of polynomial cover functions,two methods are presented: one is the improved method of boundary cover functions; and the other is the method of setting independent cover functions. The boundary conditions are strictly satisfied,and the approximate functions near the boundaries are guaranteed to approach the real solutions. The proposed methods are refrained from the influence of penalty number in common penalty method on computational results and linear equation conditions. Moreover,the implementation is very simple,for it just needs some degrees of freedom not involved in the computation. The proposed approach of applying boundary conditions by setting cover functions has a reference value for other new methods.
At present,some new numerical methods such as Numerical Manifold Method( NMM) and Meshless Method are facing with the difficulty of strictly applying essential boundary conditions. Through a case study of a cantilever beam,the application of essential boundary conditions is systematically analyzed in NMM based on independent covers previously proposed by the authors. On the basis of polynomial cover functions,two methods are presented: one is the improved method of boundary cover functions; and the other is the method of setting independent cover functions. The boundary conditions are strictly satisfied,and the approximate functions near the boundaries are guaranteed to approach the real solutions. The proposed methods are refrained from the influence of penalty number in common penalty method on computational results and linear equation conditions. Moreover,the implementation is very simple,for it just needs some degrees of freedom not involved in the computation. The proposed approach of applying boundary conditions by setting cover functions has a reference value for other new methods.
引文
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[14]苏海东,颉志强.梁的独立覆盖分析方法[J].长江科学院院报.doi:10.11988/ckyyb.20161045.
[15]钱伟长,叶开沅.弹性力学[M].北京:科学出版社,1956:213-242.
[16]申向东,王正中.材料力学[M].北京:中国水利水电出版社,2012.
[17]ZHENG Hong,LI Wei,DU Xiu-li.Exact Imposition of Essential Boundary Condition and Material Interface Continuity in Galerkin-based Meshless Methods[J].International Journal for Numerical Methods in Engineering,2017,110(4/5/6/7):637-660.
[18]林绍忠,祁勇峰,苏海东.数值流形法中覆盖函数的改进形式及其应用[J].长江科学院院报,2006,23(6):55-58.
[19]徐芝纶.弹性力学(上册)[M].第2版.北京:人民教育出版社,1982:361-363.
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[2]BELYTSCHKO T,KRONGAUZ Y,ORGAN D,et al.Meshless Methods:An Overview and Recent Developments[J].Computer Methods in Applied Mechanics and Engineering,1996,139(1/2/3/4):3-47.
[3]STROUBOULIS T,COPPS K,BABUSKA I.The Generalized Finite Element Method[J].Computer Methods in Applied Mechanics and Engineering,2001,190(32):4081-4192.
[4]SHI G H.Manifold Method of Material Analysis[C]∥U.S.Army Research Office.Transactions of the Ninth Army Conference on Applied Mathematics and Computing.Minneapolis,Minnesota,U.S.A,June 18-21,1991:51-76.
[5]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 and Engineering,2005,194(39/40/41):4135-4195.
[6]张雄,刘岩,马上.无网格法的理论及应用[M].力学进展,2009,39(1):1-36.
[7]李录贤,刘书静,张慧华,等.广义有限元方法研究进展[J].应用力学学报,2009,26(1):96-108.
[8]苏海东,颉志强,龚亚琦,等.基于独立覆盖的流形法收敛性及覆盖网格特性[J].长江科学院院报,2016,33(2):131-136.
[9]苏海东,龚亚琦,颉志强,等.基于矩形独立覆盖初步实现结构静力分析的自动计算[J].长江科学院院报,2016,33(2):144-150.
[10]BABUSKA I,MELENK J M.The Partition of Unity Method[J].International Journal for Numerical Methods in Engineering,1997,40(4):727-758.
[11]祁勇峰,苏海东,崔建华.部分重叠覆盖的数值流形方法初步研究[J].长江科学院院报,2013,30(1):65-70.
[12]SU Hai-dong,QI Yong-feng,GONG Ya-qi,et al.Preliminary Research of Numerical Manifold Method Based on Partly Overlapping Rectangular Covers[C]∥DDA Commission of International Society for Rock Mechanics.Proceedings of the 11th International Conference on Analysis of Discontinuous Deformation(ICADD11).Fukuoka,Japan,August 27-29,2013,London:Taylor&Francis Group,2013:341-347.
[13]苏海东,祁勇峰,龚亚琦,等.任意形状覆盖的数值流形方法初步研究[J].长江科学院院报,2013,30(12):91-96.
[14]苏海东,颉志强.梁的独立覆盖分析方法[J].长江科学院院报.doi:10.11988/ckyyb.20161045.
[15]钱伟长,叶开沅.弹性力学[M].北京:科学出版社,1956:213-242.
[16]申向东,王正中.材料力学[M].北京:中国水利水电出版社,2012.
[17]ZHENG Hong,LI Wei,DU Xiu-li.Exact Imposition of Essential Boundary Condition and Material Interface Continuity in Galerkin-based Meshless Methods[J].International Journal for Numerical Methods in Engineering,2017,110(4/5/6/7):637-660.
[18]林绍忠,祁勇峰,苏海东.数值流形法中覆盖函数的改进形式及其应用[J].长江科学院院报,2006,23(6):55-58.
[19]徐芝纶.弹性力学(上册)[M].第2版.北京:人民教育出版社,1982:361-363.
[20]林绍忠.用预处理共轭梯度法求解有限元方程组及程序设计[J].河海大学学报,1998,26(3):112-115.