Fourier谱方法求解二维波动方程及动态仿真多列波的干涉现象
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摘要
本文主要介绍了Fourier谱求解二维波动方程及动态仿真多列波的干涉实验现象。该研究有助于探究海洋工程的海水涨潮实验、球面波全息干涉呈像、以及地震波相互干涉后的损坏强度探测等细节化仿真实验。Fourier谱方法主要是通过快速Fourier变换(FFT),使得待求解方程转换到频率域上,借用降阶的思想将PDE问题简化为ODE方程组,再使用变步长的龙格库塔算法(Runge-Kutta)求解该微分方程组,然后使用Fourier逆变换(IFFT)将之还原到原来的空间域。通过与传统的有限元、有限差分等数值方法进行比较,得到结论:Fourier谱在逼近二维波动方程方面具有求解时间短、收敛快、占用内存少、光滑性更好等特点;从原理上讲,该方法在求解周期性的PDE问题方面更具有优势,FFT在方域上的计算复杂度为O(N2 logN),这也是计算快的原因;此外,根据波的性质可知,波的传播快慢与初边值条件,还有传播介质密切相关。本文给出了FFT结合Runge-Kutta算法动态仿真二维弹性波的求解步骤,以及误差分析理论,也揭示了波的动态叠加与传播原理,为波的求解和应用提供了重要参考。
This paper mainly introduces the Fourier spectrum method to solve 2 Dwave equation and interference experiment phenomenon of dynamic simulation of multi-wave.This research is helpful to explore the detailed simulation experiments of seawater rising tide experiments,spherical wave holographic interference image,and damage intensity detection after seismic wave interference.The Fourier spectrum method transform the equation to be solved into frequency domain form by using fast fourier transformation.Based on the idea of reduced order,the partial differential equation problem is simplified to the ordinary differential equation equation group,which is solved by the variable step size Runge-Kutta algorithm.Then the solutions are restored to the original spatial domain by using invert fast fourier transformation.Comparing with the traditional finite element method and finite difference method,the fourier spectrum can approximate the two-dimensional wave equation with short solving time,fast convergence,low memory and better smoothness.In principle,this method is more advantageous in solving periodic PDE problems.The computational complexity of FFT on the square domain is O(N2 log N),which is also the reason for fast calculation.In addition,according to the nature of wave,the propagation speed of wave is closely related to the initial value and the density of propagation medium.In this paper,the steps of solving the two-dimensional elastic wave,which is dynamically simulated by adopting FFT with Runge-Kutta algorithm,and the error analysis theory are given.The dynamic superposition and propagation principle of waves is revealed to provide significant references for the solution and application of waves.
This paper mainly introduces the Fourier spectrum method to solve 2 Dwave equation and interference experiment phenomenon of dynamic simulation of multi-wave.This research is helpful to explore the detailed simulation experiments of seawater rising tide experiments,spherical wave holographic interference image,and damage intensity detection after seismic wave interference.The Fourier spectrum method transform the equation to be solved into frequency domain form by using fast fourier transformation.Based on the idea of reduced order,the partial differential equation problem is simplified to the ordinary differential equation equation group,which is solved by the variable step size Runge-Kutta algorithm.Then the solutions are restored to the original spatial domain by using invert fast fourier transformation.Comparing with the traditional finite element method and finite difference method,the fourier spectrum can approximate the two-dimensional wave equation with short solving time,fast convergence,low memory and better smoothness.In principle,this method is more advantageous in solving periodic PDE problems.The computational complexity of FFT on the square domain is O(N2 log N),which is also the reason for fast calculation.In addition,according to the nature of wave,the propagation speed of wave is closely related to the initial value and the density of propagation medium.In this paper,the steps of solving the two-dimensional elastic wave,which is dynamically simulated by adopting FFT with Runge-Kutta algorithm,and the error analysis theory are given.The dynamic superposition and propagation principle of waves is revealed to provide significant references for the solution and application of waves.
引文
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[14]陈山,杨顶辉,邓小英.四阶龙格-库塔方法的一种改进算法及地震波场模拟[J].地球物理学报,2010,53(5):1196-1206.CHEN Shan,YANG Dinghui,DENG Xiaoying.An improved algorithm of the fourth-order Runge-Kutta.method wave-field simulation[J].Chinese Journal of Geophysics,2010,53(5):1196-1206.(in Chinese)
[15]BURDEN R L,FAIRES J D.数值分析[M].7版.冯烟利,朱海燕,译.北京:高等教育出版社,2005:182-568.BURDEN R L,FAIRES J D.Numerical analysis[M].7th ed.FENG Yanli,ZHU Hai,trans.Beijing:Advanced education press,2005:182-568.(in Chinese)
[16]TREFETHEN L N.Pseudospectra of liuear operators[J].SIAM Review,1997,39(3):383-406.
[17]BRIGGS W L,HENSON V E.The DFT:an Owner’s manual for the discrete Fourier transform[J].Journal of the American Statistical Association,1995,94(445):56-86.
[18]孙卫涛.弹性波动方程的有限差分数值方法[M].北京:清华大学出版社,2009:98-145.SUN Weitao.Finite difference numerical method for elastic wave equation[M].Beijing:Tsinghua University Press,2009:98-145.(in Chinese)
[19]ZHAO Zhixin,XU Jiren,RYUJI KUBOT A.Effects of soil amplification ratio and multiple wave interference for ground motion due to earthquake[J].Chinese sicence Bulletin,2004,49(22):2405-2414.
[20]危银涛,杨挺青,杜星文.橡胶类材料大变形本构关系及其有限元方法[J].固体力学学报,1999,20(4):281-289.WEI Yintao,YANG Tingqing,DU Xingwen.On the large deformation rubber-like materials constitutive laws and finite[J].Acta Mechanica solida sinica,1999,20(4):281-289.(in Chinese)
[2]EULER LEONHARD.General principles of the state of equilibrium of a fluid[J].Histoire de La cadémie Royale des Sciences et des Belles-Lettres de Berlin,Mémoires.1757,11:217-73.
[3]GRONWALL T H.Note on the derivatives with respect to a parameter of the solutions of a system of differential equations[J].Mathematische Annalen of Math,1919,20(4):292-296.
[4]LANCZOS C.Trigonometric interpolation of empirical and analytical functions[J].Journal of Mathematics and Physics,1938,17(1/2/3/4):123-199.
[5]COOLEY J W,TUKEY J W.An algorithm for the machine calculation of complex Fourier series[J].Mathematics of Computation,1965,19(90):297-301.
[6]TREFETHEN L N.Pseudospectra of linear operators(eng)[J].Siam Review,1997,39(3):383-406.
[7]ORSZAG S A.Comparison of pseudospectral and spectral approximation[J].Studies in Applied Mathematics,1972,51(3):253-259.
[8]PATERA A T.A spectral element method for fluid dynamics:Laminar flow in a channel expansion[J].Journal of Computational Physics,1984,54(3):468-488.
[9]SHEN Jie,TANG Tao,WANG Lilian.Spectral methods:algorithms,analysis and applications[M].Berlin:Springer Publishing Company,Incorporated,2011:115-350.
[10]李会元,乔海军.六边形Fourier谱方法[J].应用数学与计算数学学报,2013,27(1):147-162.LI Huiyuan,QIAO Haijun.Hexagonal Fourier spectral method[J].Communication on Applied Mathematics and Computation,2013,27(1):147-162.(in Chinese)
[11]KREISS H,OLIGER J.Stability of the Fourier method[J].Siam Journal on Numerical Analysis,1979,16(3):421-433.
[12]FORNBERG B.The pseudospectral method:comparisons with finite differences for the elastic wave equation[J].Geophysics,2012,52(4):483-501.
[13]刘占芳,郭原,唐少强,等.弹性应力波的双脉冲结构与平板冲击试验验证[J].应用数学和力学,2018,39(3):249-265.LIU Zhanfang,GUO Yuan,TANG Shaoqiang,et al.Dual pulse wave structural of elastic stress waves and plate impact verification[J].Applied Mathematics and Mechanics,2018,39(3):249-265.(in Chinese)
[14]陈山,杨顶辉,邓小英.四阶龙格-库塔方法的一种改进算法及地震波场模拟[J].地球物理学报,2010,53(5):1196-1206.CHEN Shan,YANG Dinghui,DENG Xiaoying.An improved algorithm of the fourth-order Runge-Kutta.method wave-field simulation[J].Chinese Journal of Geophysics,2010,53(5):1196-1206.(in Chinese)
[15]BURDEN R L,FAIRES J D.数值分析[M].7版.冯烟利,朱海燕,译.北京:高等教育出版社,2005:182-568.BURDEN R L,FAIRES J D.Numerical analysis[M].7th ed.FENG Yanli,ZHU Hai,trans.Beijing:Advanced education press,2005:182-568.(in Chinese)
[16]TREFETHEN L N.Pseudospectra of liuear operators[J].SIAM Review,1997,39(3):383-406.
[17]BRIGGS W L,HENSON V E.The DFT:an Owner’s manual for the discrete Fourier transform[J].Journal of the American Statistical Association,1995,94(445):56-86.
[18]孙卫涛.弹性波动方程的有限差分数值方法[M].北京:清华大学出版社,2009:98-145.SUN Weitao.Finite difference numerical method for elastic wave equation[M].Beijing:Tsinghua University Press,2009:98-145.(in Chinese)
[19]ZHAO Zhixin,XU Jiren,RYUJI KUBOT A.Effects of soil amplification ratio and multiple wave interference for ground motion due to earthquake[J].Chinese sicence Bulletin,2004,49(22):2405-2414.
[20]危银涛,杨挺青,杜星文.橡胶类材料大变形本构关系及其有限元方法[J].固体力学学报,1999,20(4):281-289.WEI Yintao,YANG Tingqing,DU Xingwen.On the large deformation rubber-like materials constitutive laws and finite[J].Acta Mechanica solida sinica,1999,20(4):281-289.(in Chinese)