海洋声学中缓变分层波导的声波传播计算
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摘要
在声学、电磁学、地震学和一些其他的应用中,有许多在具有分层缓变波导中大尺度的波传播的问题。如果使用直接方法,譬如有限元法、有限差分法等,往往会导致求解大规模的线性方程组,这使得使得算法在内存和计算量的花费非常大,计算效率不高。在本文中,我们针对具有弯曲界面或者弯曲边界的波导,提出了适合于步进计算一般的坐标变换变换的方法,在坐标变换后的坐标系中,使用基于DtN(Dirichlet to Neumann)映射的正向基本解算子步进方法(FFOMM)和逆向基本解算子步进方法(IFOMM)分别求解Helmholtz方程正问题和反问题。
在海洋声学中,海洋环境可以看作是一种分层缓变的具有多个弯曲内部界面和多层介质的声波导。在这样的波导中如果要使用步进方法大步长的求解Helmholtz方程,首先需要用适合于步进计算坐标变换将波导中弯曲的界面或者弯曲边界拉直。本文中给出两种坐标变换的方法:第一种是局部的解析正交坐标变换法,局部解析坐标变换用一个解析的积分方程联系新旧坐标,使用牛顿法求解这一积分方程实现新旧坐标之间的转换。坐标变换必然相应的引起方程的变换,我们使用的方程变换将Helmholtz方程转化成为一个可以用步进方法求解的形式。对于具有一个平的顶部、平的底部、n个弯曲的内部界面的二维波导,我们给出了在变换方程系数在各个介质层一般形式的解析表达式,从而完成方程变换。然而,局部解析坐标变换方法需要满足在两个相邻界面之间存在一条水平直线的介质可分性条件。针对不满足可分性条件的情况,我们又提出了一种使用经典求解常微分方程的Runge-Kutta方法计算坐标变换的数值坐标变换方法,这一方法不仅具有更广的适用范围,而且不必引入辅助界面,从而更好的逼近原始问题。
对于变换之后的计算坐标系中的正问题,我们使用正向基本解步进方法来求解方程。FFOMM基于DtN算子的单向重建,将边界值问题转化为初始值问题,对于DtN算子和基本解算子满足的方程使用大步长方法离散在传播方向上的变量,用截断特征值展开方法逼近这些算子。FFOMM对于求解大尺度缓变波导中波的传播问题非常有效。
在具有分层缓变特性的波导中的波的大尺度传播问题中,反问题有非常重要的实际意义。一般说来,海洋声学中实际的反问题都很复杂。求解海洋中反问题一般使用两类方法:一种是基于正向方法(如FFOMM)的迭代法,另一种为反向方法,反向方法的运算量和计算速度都相比正向方法有很大的优势。基于FFOMM,我们提出了逆向基本解算子步进方法。在IFOMM中使用波导中存在的传播模的个数作为正则化参数,用截断的奇异值分解方法求解步进过程中出现的病态线性方程。数值计算证明,这个方法用于求解Helmholtz方程的反边界值有效,精确并且对于初始波源传播部分稳定,对于海洋声波中实际的反问题研究有着一定的意义。
In acoustics, electro-magnetism, seismic migration and other applications, there are many large scale wave propagation problems. A direct numerical computing like finite-element, finite-difference could be very expensive since these methods often result in a very large linear system. The forward fundamental operator marching method (FFOMM) which is developed by Lu Y.Y. etc based on Dirichlet-to-Neumann (DtN) reformulation is a highly efficient numerical marching scheme for solving Helmoholtz equation in a large scale domain with curved interfaces or boundaries. However, this FFOMM previously suggested can only march in a large range step size in some simple waveguides, such as horizontal stratified waveguide, the waveguide with one curved interface or one curved boundary. If there are more curved interfaces or curved boundaries in the domain of waveguide, then the FFOMM has to march in a very short step unless some treatments are done.
In the real world, some waveguides, like waveguides in ocean acoustics, often have multilayered stratified structure with some curved interfaces or boundaries. To efficiently compute wave propagation along the range with some marching methods, flattening the interfaces or boundaries and transforming equation are needed. In this paper, an analytical local orthogonal.coordinate transform and an analytical equation transform are constructed to flatten the interfaces and change the Helmholtz equation as a solvable form. For a waveguide with a flat top, a flat bottom and n curved interfaces, the coefficients of transformed Helmholtz equation is given in a closed formulation. However, when distance of two adjoint interfaces is so closely that there is not a horizontal straight-line to divide the corresponding layer into two parts, the analytical local orthogonal transform method will not be feasible. For this reason, a numerical coordinate transform and equation transform based on the classical Rugger-Kutta method are developed to solve Helmholtz equation by some marching methods in the situation.
In the transformed horizontal stratified waveguide, the FFOMM uses a large range step method to discretize the range variable and compute the forward problem of Helmholtz equation efficiently. The analytical transform method and the numerical transform method are particularly useful for long range wave propagation problems in slowly varying waveguides with multilayered medium structure. Furthermore, the methods can also be applied for the wave propagation problems in acoustical waveguides associated with varied density.
It is noticed that the inverse problem is also very important for large scale wave propagation problems in acoustics or electro magnetics. In our work, a called "inverse fundamental operator marching method" (IFOMM) based on the Dirichlet-to-Neumann map is suggested for solving the large scale inverse boundary value problem associated with the two dimensional Helmholtz equation in a range dependent waveguide. This method solves the badly ill-conditioned matrix equations arising in the marching progress by truncated singular value decomposition with the number of propagation modes in the waveguide chosen as the regularization parameter. Numerical examples show that the proposed method is computationally efficient, highly accurate with respect to the propagating part of a starting field.
在海洋声学中,海洋环境可以看作是一种分层缓变的具有多个弯曲内部界面和多层介质的声波导。在这样的波导中如果要使用步进方法大步长的求解Helmholtz方程,首先需要用适合于步进计算坐标变换将波导中弯曲的界面或者弯曲边界拉直。本文中给出两种坐标变换的方法:第一种是局部的解析正交坐标变换法,局部解析坐标变换用一个解析的积分方程联系新旧坐标,使用牛顿法求解这一积分方程实现新旧坐标之间的转换。坐标变换必然相应的引起方程的变换,我们使用的方程变换将Helmholtz方程转化成为一个可以用步进方法求解的形式。对于具有一个平的顶部、平的底部、n个弯曲的内部界面的二维波导,我们给出了在变换方程系数在各个介质层一般形式的解析表达式,从而完成方程变换。然而,局部解析坐标变换方法需要满足在两个相邻界面之间存在一条水平直线的介质可分性条件。针对不满足可分性条件的情况,我们又提出了一种使用经典求解常微分方程的Runge-Kutta方法计算坐标变换的数值坐标变换方法,这一方法不仅具有更广的适用范围,而且不必引入辅助界面,从而更好的逼近原始问题。
对于变换之后的计算坐标系中的正问题,我们使用正向基本解步进方法来求解方程。FFOMM基于DtN算子的单向重建,将边界值问题转化为初始值问题,对于DtN算子和基本解算子满足的方程使用大步长方法离散在传播方向上的变量,用截断特征值展开方法逼近这些算子。FFOMM对于求解大尺度缓变波导中波的传播问题非常有效。
在具有分层缓变特性的波导中的波的大尺度传播问题中,反问题有非常重要的实际意义。一般说来,海洋声学中实际的反问题都很复杂。求解海洋中反问题一般使用两类方法:一种是基于正向方法(如FFOMM)的迭代法,另一种为反向方法,反向方法的运算量和计算速度都相比正向方法有很大的优势。基于FFOMM,我们提出了逆向基本解算子步进方法。在IFOMM中使用波导中存在的传播模的个数作为正则化参数,用截断的奇异值分解方法求解步进过程中出现的病态线性方程。数值计算证明,这个方法用于求解Helmholtz方程的反边界值有效,精确并且对于初始波源传播部分稳定,对于海洋声波中实际的反问题研究有着一定的意义。
In acoustics, electro-magnetism, seismic migration and other applications, there are many large scale wave propagation problems. A direct numerical computing like finite-element, finite-difference could be very expensive since these methods often result in a very large linear system. The forward fundamental operator marching method (FFOMM) which is developed by Lu Y.Y. etc based on Dirichlet-to-Neumann (DtN) reformulation is a highly efficient numerical marching scheme for solving Helmoholtz equation in a large scale domain with curved interfaces or boundaries. However, this FFOMM previously suggested can only march in a large range step size in some simple waveguides, such as horizontal stratified waveguide, the waveguide with one curved interface or one curved boundary. If there are more curved interfaces or curved boundaries in the domain of waveguide, then the FFOMM has to march in a very short step unless some treatments are done.
In the real world, some waveguides, like waveguides in ocean acoustics, often have multilayered stratified structure with some curved interfaces or boundaries. To efficiently compute wave propagation along the range with some marching methods, flattening the interfaces or boundaries and transforming equation are needed. In this paper, an analytical local orthogonal.coordinate transform and an analytical equation transform are constructed to flatten the interfaces and change the Helmholtz equation as a solvable form. For a waveguide with a flat top, a flat bottom and n curved interfaces, the coefficients of transformed Helmholtz equation is given in a closed formulation. However, when distance of two adjoint interfaces is so closely that there is not a horizontal straight-line to divide the corresponding layer into two parts, the analytical local orthogonal transform method will not be feasible. For this reason, a numerical coordinate transform and equation transform based on the classical Rugger-Kutta method are developed to solve Helmholtz equation by some marching methods in the situation.
In the transformed horizontal stratified waveguide, the FFOMM uses a large range step method to discretize the range variable and compute the forward problem of Helmholtz equation efficiently. The analytical transform method and the numerical transform method are particularly useful for long range wave propagation problems in slowly varying waveguides with multilayered medium structure. Furthermore, the methods can also be applied for the wave propagation problems in acoustical waveguides associated with varied density.
It is noticed that the inverse problem is also very important for large scale wave propagation problems in acoustics or electro magnetics. In our work, a called "inverse fundamental operator marching method" (IFOMM) based on the Dirichlet-to-Neumann map is suggested for solving the large scale inverse boundary value problem associated with the two dimensional Helmholtz equation in a range dependent waveguide. This method solves the badly ill-conditioned matrix equations arising in the marching progress by truncated singular value decomposition with the number of propagation modes in the waveguide chosen as the regularization parameter. Numerical examples show that the proposed method is computationally efficient, highly accurate with respect to the propagating part of a starting field.
引文
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