有限一维弹性波动问题的公理化求解及其在自由场中的应用
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摘要
在保证级数的一致收敛的前提下,可以利用Laplace变换对有限一维空间弹性动力学边值问题给出公理化的严格求解过程,此过程能够作为Lavrentieff与Hilbert等著名学者的工作的补充。这一解答能够应用于岩土工程中的自由场计算问题,并且能够从原则上避免截断误差。在针对特定问题的计算中,该方法的效率明显高于有限元,并且在一定程度上弥补了现有的自由场专用计算软件不能计算纯弹性体的缺点。同时,作为解析解算法,该方法对于数值算法的精度分析有一定意义。
The axiomatic solution process for a one dimensional finite space elastic dynamic boundary value problem can be given by using Laplace transform under the premise of uniform convergence of series, and this process can be a supplement of the work of famous scholars such as Lavrentieff and Hilbert. This solution can be used for the free field analysis in geotechnical engineering, and it can also avoid truncation error in principle. The method in this paper has significantly higher efficiency than that of the finite element method in the calculation for a specific problem, and it can make up for the disadvantage of the existing free field dedicated calculation software which cannot calculate pure elastic bodies to a certain extent. Also, as an analytical algorithm, the method has some significance for the precision analysis of numerical algorithm.
The axiomatic solution process for a one dimensional finite space elastic dynamic boundary value problem can be given by using Laplace transform under the premise of uniform convergence of series, and this process can be a supplement of the work of famous scholars such as Lavrentieff and Hilbert. This solution can be used for the free field analysis in geotechnical engineering, and it can also avoid truncation error in principle. The method in this paper has significantly higher efficiency than that of the finite element method in the calculation for a specific problem, and it can make up for the disadvantage of the existing free field dedicated calculation software which cannot calculate pure elastic bodies to a certain extent. Also, as an analytical algorithm, the method has some significance for the precision analysis of numerical algorithm.
引文
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[2]陈炜昀,夏唐代,黄睿,周新民.P1波在非饱和土地基表面的反射特性[J].工程力学,2013,30(9):56―62.Chen Weiyun,Xia Tangdai,Huang Rui,Zhou Xinmin.Reflection characteristics of P1 waves at the free boundary of unsaturated soil[J].Engineering Mechanics,2013,30(9):56―62.(in Chinese)
[3]Stein E M,Shakarchi R.Fourier Analysis:An Introduction[M].Beijing:World Publishing Corporation,2013:188.
[4]黄大奎,舒慕曾.数学物理方法[M].北京:高等教育出版社;柏林:施普林格出版社,2001:327―337.Huang Dakui,Shu Muzeng.Method of Mathematical Physics[M].Beijing:China Higher Education Press;Berlin:Heidelberg:Springer-Verlag,2001:327―337.(in Chinese)
[5]Folland G B.Fourier Analysis and Its Applications[M].Beijing:China Machine Press,2005:281―286.
[6]拉夫连季耶夫M A,沙巴特БB.复变函数论方法[M].北京:高等教育出版社,2006:448.Lavrentieff M A,Shabat B V.Methods of functions of a complex variable[M].Beijing:Higher Education Press,2006:448.(in Chinese)
[7]Luo X B,Aref A J,Dargush G F.Analysis and optimal design of layered structures subjected to impulsive loading[J].Computers and Structures,2009,87(9):543―551.
[8]柯朗R,希尔伯特D.数学物理方法[M].北京:科学出版社,2011:405―418.Courant R,Hilbert D.Methods of mathematical physics[M].Beijing:Science Press,2011:405―418.(in Chinese)
[9]Mikusinski.Operational Calculus[M].New York:Pergamon Press,1959:508.
[10]Asmar N H.Partial Differential Equations and Boundary Value Problems with Fourier Series,2/E[M].Beijing:China Machine Press,2012:481.
[11]Palka B P.An Introduction to Complex Function Theory[M].Beijing:World Publishing Corporation,2012:251.
[12]扈培础.复变函数敎程[M].北京:科学出版社,2008:62.Hu Peichu.Complex Function Tutorial[M].Beijing:Science Press,2008,62.(in Chinese)
[13]赵建丛,黄文亮.复变函数与积分变换[M].上海:华东理工大学出版社,2008:159―163.Zhao Jiancong,Huang Wenliang.Complex function and integral transform[M].Shanghai:East China University of Science and Technology Press,2008:159―163.(in Chinese)
[14]谢定义.土动力学[M].北京:高等教育出版社,2011:409―422.Xie Dingyi.Soil dynamics[M].Beijing:Higher Education Press,2011:409―422.(in Chinese)
[15]刘晶波,王文晖,赵冬冬.地下结构横截面地震反应拟静力计算方法对比研究[J].工程力学,2013,30(1):105―111.Liu Jingbo,Wang Wenhui,Zhao Dongdong.Comparison of the pseudo-static methods for seismic analysis of the underground structures[J].Engineering Mechanics,2013,30(1):105―111.(in Chinese)
[16]马宏伟,吴斌.弹性动力学及其数值方法[M].北京:中国建材工业出版社,2000:24―27.Ma Hongwei,Wu Bin.Elastic dynamics and its numerical method[M].Beijing:China Building Materials Press,2000:24―27.(in Chinese)
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