动水压力波高阶双渐近时域平面透射边界I:理论推导
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摘要
为了准确模拟地震引起的动水压力波在半无限水库中传播引起的辐射阻尼,基于比例边界有限单元法构造了高阶双渐近时域平面透射边界。首先将描述半无限水库的连续波动方程转化为半离散的比例边界有限单元方程,然后通过特征模态变换将以动力刚度矩阵形式表达的控制方程解耦,获得模态动力刚度的连分式解答,再引入辅助变量将透射边界在频域的动力刚度表达等效为一个高阶矩阵方程,对其施加傅立叶逆变换后即得到时域透射边界条件。频域模态动力刚度的计算结果表明:该双渐近透射边界可以在全频范围内迅速逼近准确解,具有良好的收敛性能。
To simulate the radiation damping caused by hydrodynamic pressure wave propagation in semi-infinite reservoir due to earthquake,a high-order doubly asymptotic time-domain plane transmitting boundary is constructed based on the scaled boundary finite element method.Firstly,the continuous wave equation for semi-infinite reservoir is transformed into semi-discrete scaled boundary finite element equation.Applying the modal transformation,the governing equation in dynamic stiffness matrix is decoupled to obtain the continued fraction solution for modal dynamic stiffness.After introducing the auxiliary functions,the transmitting boundary formulated in frequency domain is constructed as a high-order matrix equation.Employing the inverse Fourier transformation,the time-domain transmitting boundary is obtained finally.The numerical results of modal dynamic stiffness show that the presented doubly asymptotic boundary has an excellent performance and can converge to exact solution quickly in the whole frequency domain.
To simulate the radiation damping caused by hydrodynamic pressure wave propagation in semi-infinite reservoir due to earthquake,a high-order doubly asymptotic time-domain plane transmitting boundary is constructed based on the scaled boundary finite element method.Firstly,the continuous wave equation for semi-infinite reservoir is transformed into semi-discrete scaled boundary finite element equation.Applying the modal transformation,the governing equation in dynamic stiffness matrix is decoupled to obtain the continued fraction solution for modal dynamic stiffness.After introducing the auxiliary functions,the transmitting boundary formulated in frequency domain is constructed as a high-order matrix equation.Employing the inverse Fourier transformation,the time-domain transmitting boundary is obtained finally.The numerical results of modal dynamic stiffness show that the presented doubly asymptotic boundary has an excellent performance and can converge to exact solution quickly in the whole frequency domain.
引文
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[9]Song C,Wolf J P.The scaled boundary finite-element method-alias consistent infinitesimal finite-element cellm ethod-for elastodynamics[J].Computer Methods in Applied Mechanics and Engineering,1997,147(3-4):329-355.
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[11]杜建国,林皋,谢清粮.一种新的求解坝面动水压力的半解析方法[J].振动与冲击,2009(3):31-34.
[12]Zienkiewicz O C,Taylor R L.The finite element method(5th ed)[M].Oxford:Butterworth-Heinemann,2000
[2]杜修力,王进廷.动水压力及其对坝体地震反应影响的研究进展[J].水利学报,2001(7):13-21.
[3]廖振鹏.工程波动理论导论[M].北京:科学出版社,2002.
[4]Zienkiewicz O C,Bettess P.Fluid-structure dynamic interaction and wave forces:An introduction to numericalt reatmen[tJ].International Journal for Numerical Methods in Engineering,1978,13(1):1-16.
[5]Sommerfeld A.Partial differential equations in physics[M].New York:Academic Press,1949.
[6]Givoli D.High-order local non-reflecting boundary conditions:a review[J].Wave Motion,2004,39(4):319-326.
[7]Prempramote S,Song C,Tin-Loi F,et al.High-order doubly asymptotic open boundaries for scalar wave equa?t ion[J].International Journal for Numerical Methods in Engineering,2009,79(3):340-374.
[8]Geers T L.Singly and doubly asymptotic computational boundarie[sC]//Proceedings of the IUTAM Symposium onComputational Methods for Unbounded Domains.Dordrecht:Kluwer Academic Publishers,1998:135-141.
[9]Song C,Wolf J P.The scaled boundary finite-element method-alias consistent infinitesimal finite-element cellm ethod-for elastodynamics[J].Computer Methods in Applied Mechanics and Engineering,1997,147(3-4):329-355.
[10]林皋,杜建国.基于SBFEM的坝-库水相互作用分析[J].大连理工大学学报,2005(5):723-729.
[11]杜建国,林皋,谢清粮.一种新的求解坝面动水压力的半解析方法[J].振动与冲击,2009(3):31-34.
[12]Zienkiewicz O C,Taylor R L.The finite element method(5th ed)[M].Oxford:Butterworth-Heinemann,2000
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