基于简谐波动模型的法向透射边界及其向任意波动的推广
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摘要
基于外行简谐波动模型、多项式插值理论和具有解析表达式的外插序列,导出了N阶透射边界及其误差项与收敛的充分条件。利用外插序列的时空坐标满足同一直线方程的几何特征,完成了简谐波动N阶透射边界向任意外行波动阶透射边界的推广。依据外插序列的解析表达式,导出了后者误差项的近似表达式和收敛性的充分条件。N阶透射边界的主部与N阶多次透射公式相同,但误差项的表达式不同,精度与收敛性的分析方法与结果也不同。提出了与虚拟透射边界相关的频率放大倍数概念,认为它们是影响虚拟透射边界精度的关键因素。
Based on outgoing harmonic wave model,polynomial interpolation theory and extrapolation sequence with analytical expression,the N-order transmitting boundary and its error-term and the sufficien condition of convergence were derived.Using the geometric features that spatiotemporal coordinate of extrapolation sequence satisfies the same linear equation,the extension from N-order transmitting boundary for harmonic waves to that for arbitrary outgoing wave was completed.According to analytical expression of extrapolation sequence,the approximate expression of error-term and the sufficien condition of convergence for the latter were derived.The principal part of N-order transmitting boundary is similar to N-order multi-transmitting formula,but the expression of error-term is different,both analysis method and result of the precision and convergence are different too.New conceptions of frequency amplification were proposed which related to virtual transmitting boundary,and they are considered as the key factors effecting precision of virtual transmitting boundary.
Based on outgoing harmonic wave model,polynomial interpolation theory and extrapolation sequence with analytical expression,the N-order transmitting boundary and its error-term and the sufficien condition of convergence were derived.Using the geometric features that spatiotemporal coordinate of extrapolation sequence satisfies the same linear equation,the extension from N-order transmitting boundary for harmonic waves to that for arbitrary outgoing wave was completed.According to analytical expression of extrapolation sequence,the approximate expression of error-term and the sufficien condition of convergence for the latter were derived.The principal part of N-order transmitting boundary is similar to N-order multi-transmitting formula,but the expression of error-term is different,both analysis method and result of the precision and convergence are different too.New conceptions of frequency amplification were proposed which related to virtual transmitting boundary,and they are considered as the key factors effecting precision of virtual transmitting boundary.
引文
[1]廖振鹏.工程波动理论导论[M].2版.北京:科学出版社,2002.LiaoZhenpeng.Introduction to wave motion theories for engineering[M](2nd ed).Beijing:Science Press,2002.
[2]Clayton R,Engquist B.Absorbing boundary conditions for acoustic and elastic wave equations[J].Bull.Seism.Soc.Am.,1977,67(6):1529-1540.
[3]Hingdon.R L,Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation[J].Math.Comp.,1986 47(1):437-459.
[4]杜修力,赵密.一种新的高阶弹簧-阻尼-质量边界无限域圆柱对称波动问题[J].力学学报,2009,41(2):207-215.Du Xiuli,Zhao Mi.A novel high-order spring-dashpot-mass boundary for cylindrical-symmetry wave motions in infinite domain[J].ActaMechanica Sinica,2009,41(2):207-215.
[5]刘晶波,王振宇,杜修力,等.波动问题的三维时域黏弹性人工边界[J].工程力学,2005,22(6):46-51.Liu Jingbo,Wang Zhenyu,Du Xiuli,et al.Three-dimensional visco-elastic artificial boundaries in time domain for wave motion problems[J].Engineering Mechanics,2005,22(6):46-51.
[6]张晓志,谢礼立.连续傍轴近似公式及其多种离散形式[J].地震工程与工程振动,2004,24(5):1-6.Zhang Xiaozhi,Xie Lili.Continuous paraxial approximation formula and it’s multiple discrete forms[J].Earthquake Engineering and EngineeringVibration,2010,30(5):9-13.
[7]张晓志,屈成忠,龚靖.局部平面波假定与优化傍轴透射边界[J].地震工程与工程振动,2010,30(5):9-13.Zhang Xiaozhi,Qu Chengzhong,Gong Jing.Local plane wave supposition and best-fitting artificial transmitting boundary[J].Earthquake Engi-neering and Engineering Vibration,2010,30(5):9-13.
[2]Clayton R,Engquist B.Absorbing boundary conditions for acoustic and elastic wave equations[J].Bull.Seism.Soc.Am.,1977,67(6):1529-1540.
[3]Hingdon.R L,Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation[J].Math.Comp.,1986 47(1):437-459.
[4]杜修力,赵密.一种新的高阶弹簧-阻尼-质量边界无限域圆柱对称波动问题[J].力学学报,2009,41(2):207-215.Du Xiuli,Zhao Mi.A novel high-order spring-dashpot-mass boundary for cylindrical-symmetry wave motions in infinite domain[J].ActaMechanica Sinica,2009,41(2):207-215.
[5]刘晶波,王振宇,杜修力,等.波动问题的三维时域黏弹性人工边界[J].工程力学,2005,22(6):46-51.Liu Jingbo,Wang Zhenyu,Du Xiuli,et al.Three-dimensional visco-elastic artificial boundaries in time domain for wave motion problems[J].Engineering Mechanics,2005,22(6):46-51.
[6]张晓志,谢礼立.连续傍轴近似公式及其多种离散形式[J].地震工程与工程振动,2004,24(5):1-6.Zhang Xiaozhi,Xie Lili.Continuous paraxial approximation formula and it’s multiple discrete forms[J].Earthquake Engineering and EngineeringVibration,2010,30(5):9-13.
[7]张晓志,屈成忠,龚靖.局部平面波假定与优化傍轴透射边界[J].地震工程与工程振动,2010,30(5):9-13.Zhang Xiaozhi,Qu Chengzhong,Gong Jing.Local plane wave supposition and best-fitting artificial transmitting boundary[J].Earthquake Engi-neering and Engineering Vibration,2010,30(5):9-13.
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