双层介质声传播问题的标准解及有限元解

文章通过双层介质中的声传播问题，研究了有限元方法在水下声场计算中的应用。基于传统的Galerkin方法推导出水下声场的有限元方程，采用四节点四边形单元离散求解物理域，可选择辐射边界条件、DtN (Dirichletto Neumann)非局部算子、完美匹配层来处理出射声场，得到有限元解。为了验证该有限元模型，需要高精度的参考解。水平不变均匀介质中的声传播问题存在解析解，但双层介质问题不存在解析解。因此，对于双层介质声传播问题，使用波数积分法推导出标准解。分别考虑了有限深度和无限深度双层介质两种情况，并进行了数值模拟。数值结果表明，文章所提的有限元模型与参考解非常吻合。此外，还发现当某号简正波的本征值非常接近割线时，简正波模型KRAKEN难以准确计算该号简正波的本征值，从而声场计算结果存在明显误差；但是有限元方法不需要计算本征值，所以当KRAKEN模型出现此类问题时，有限元方法仍能给出准确的声场计算结果，表明有限元方法在普适性方面优于简正波方法。

Benchmark solution and finite element solution of sound propagation in a two-layer medium

In this paper, the sound propagation problem in a two-layer medium is studied and a finite element model for un-derwater sound field calculation is proposed. The finite element equation is derived by the traditional Galerkin method, and the four-node quadrilateral element is used to discretize the computation domain. Then the traditional radiation boundary condition, the DtN (Dirichlet to Neumann) non-local operator and perfectly matched layer are adopted to process the outgoing sound field and obtain the finite element solutions, and the differences between them are compared. In order to verify the finite element model, a benchmark solution is needed. Analytical solutions of sound propagation exist in a homogeneous medium. However, in a two-layer medium, there is no analytical solu-tion. Therefore, for the sound field calculation in a two-layer medium, the wavenumber integration method is used to provide benchmark solutions. The numerical simulations of sound field in a two-layer medium with limited and unlimited depths are conducted respectively. The results show that the finite element model is in good agreement with the benchmark solution. In addition, it is found that the normal mode model KRAKEN may be difficult to calculate the eigenvalue of the normal mode accurately when the eigenvalue of a normal mode is very close to the secant, so the error of sound field calculation result is significant; but the finite element method does not need to calculate the eigenvalues. Thus, the finite element model proposed in this paper can solve the sound field calculation problem where the KRAKEN model is hard to solve accurately, which means that the practicability of finite element method is better than that of the normal mode method.