三角谱元法及其在地震正演模拟中的应用

谱元法（SEM）是基于有限元（FEM）的一种算法，在地震正演模拟中应用广泛，但是大部分研究都是基于四边形网格下的谱元法.本文给出了2阶谱元法在三角网格中（TSEM）的基本原理，包括Lagrange形函数的构建，数值积分公式的选取.在此基础上，分析了2阶TSEM方法的数值频散特性以及稳定性条件，并引入三角网格下3阶有限元方法进行分析对比，数值算例的结果证明2阶TSEM相比于3阶FEM具有更高的计算精度，以及更宽松的稳定条件.最后，本文将TSEM方法应用于中国西部地区的两个含溶洞介质的地质模型中，数值模拟结果表明TSEM方法能够有效地模拟复杂结构的介质，有助于对地震波场传播特性的认识.

Spectral element method with triangular mesh and its application in seismic modeling

Spectral element method (SEM) is a very powerful method for seismic modeling with computational efficiency and accuracy. In 2D case, most people prefer to employ the quadrilateral mesh for simulation, and less work has been done to study the SEM in triangular mesh (TSEM). In this paper, we introduced a second-order TSEM scheme, and demonstrate how to circumvent the difficulties in triangular mesh. After that, the dispersive performance and stability condition for second-order TSEM is analyzed. To make a comparison, a third-order finite element method scheme with triangular mesh (TFEM) is employed as a reference. The analyses show that the TSEM scheme is more accurate than the TFEM scheme with larger stability factor. To assess the performance of TSEM, a numerical experiment is presented, and the computational results confirm the above analyses. In the last part, the TSEM is applied to study the wave propagation in two geological models containing caves.