Development of special finite elements for two-dimensional elastic wave problems

为了使用有限元法对波动问题建模，通常将域离散化，使得每个波长大约有十个节点。然而，如果波长短和/或计算域大，则这样的过程在计算上是昂贵的并且不切实际。拟议的研究的目的是开发弹性波问题的有限元能够包含许多波长每个节点间距，而不是每个波长的许多元素。这将通过将平面波基分解应用于弹性波方程来实现。这些元素将允许我们放松的传统要求，约10个节点，每个波长，因此解决弹性波问题，而不细化网格的计算域在每个频率相比，传统的有限元网格，建议的有限元能够减少所需的变量总数，以解决一个问题高达90%。这将允许弹性波问题的快速和准确的解决方案，并将导致在计算机模拟时间和内存方面的巨大节省。

In order to model wave problems using the finite element method, it is usual to discretise the domain such that there are about ten nodal points per wavelength. However, such a procedure is computationally expensive and impractical if the wavelength is short and/or the computational domain is large. The aim of the proposed research is to develop finite elements for elastic wave problems capable of containing many wavelengths per nodal spacing rather than many elements per wavelength. This will be achieved by applying the plane wave basis decomposition to the elastic wave equation. These elements will allow us to relax the traditional requirement of around ten nodal points per wavelength and therefore solve elastic wave problems without refining the mesh of the computational domain at each frequency.Compared with conventional finite element meshes, the proposed finite elements are capable of reducing the total number of variables needed to solve a problem by up to 90%. This will allow fast and accurate solutions of elastic wave problems, and will lead to huge savings in terms of computer simulation time and memory.