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

为了用有限元方法模拟波动问题，通常将区域离散，使每个波长大约有十个节点。然而，如果波长很短和/或计算域很大，这种方法在计算上是昂贵的和不切实际的。所提出的研究目的是发展弹性波问题的有限元，使其能够在每个节点间隔中包含多个波长，而不是每个波长中包含多个元素。这将通过对弹性波动方程应用平面波基分解来实现。这些元素将使我们能够放松每个波长大约10个节点的传统要求，从而解决弹性波问题，而无需在每个频率上细化计算域的网格。与传统的有限元网格相比，所提出的有限元网格能够将求解问题所需的变量总数减少高达90%。这将允许快速和准确地解决弹性波问题，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.