Wave propagation in slowly varying waveguides using a finite element approach
Wave propagation in slowly varying waveguides using a finite element approach
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DOI：
10.1016/j.jsv.2018.11.004
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发表时间：
2019-03
影响因子：
4.7
通讯作者：
A. Fabro;N. Ferguson;B. Mace
中科院分区：
文献类型：
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作者：
A. Fabro;N. Ferguson;B. Mace
这项工作研究了一维波导中的结构波传播，其属性沿传播轴随机变化，特别是当属性变化足够慢以致反向散射可以忽略不计时，即使净变化很大。基于波的方法通常应用于均匀波导，但 WKB（Wentzel、Kramers 和 Brillouin 之后）近似可用于在相位和振幅变化方面找到波解的合适概括，但仅限于解析解。提出了波和有限元 (WFE) 方法，将 WKB 方法的适用性扩展到运动方程无法解析解的情况。波数表示为沿波导位置的函数。随后使用高斯-勒让德求积方案来获得相位变化，同时使用功率守恒来计算波幅。 WFE 方法用于评估每个积分点的波数。此外，空间相关的随机性可以通过随机场属性包含在公式中，在本文中通过 Karhunen-Loève 展开来表达。数值示例与标准有限元方法和可用的分析解决方案进行了比较。与完整的 FE 或分析解决方案相比，它们表现出良好的一致性，并且仅需要少量 WFE 评估，为波导中的高效随机分析提供了合适的框架。
This work investigates structural wave propagation in one dimensional waveguides with randomly varying properties along the axis of propagation, specifically when the properties vary slowly enough such that there is negligible backscattering, even if the net change is large. Wave-based methods are typically applied to homogeneous waveguides but the WKB (after Wentzel, Kramers and Brillouin) approximation can be used to find a suitable generalisation of the wave solution in terms of the change of phase and amplitude but is restricted to analytical solutions. A wave and finite element (WFE) approach is proposed to extend the applicability of the WKB method to cases where no analytical solution of the equations of motion is available. The wavenumber is expressed as a function of the position along the waveguide. A Gauss-Legendre quadrature scheme is subsequently used to obtain the phase change, while the wave amplitude is calculated using conservation of power. The WFE method is used to evaluate the wavenumbers at each integration point. Moreover, spatially correlated randomness can be included in the formulation by random field properties and in this paper is expressed by a Karhunen-Loève expansion. Numerical examples are compared to a standard FE approach and to available analytical solutions. They show good agreement when compared to either a full FE or analytical solution and require only a few WFE evaluations, providing a suitable framework for efficient stochastic analysis in waveguides.