基于Lamb问题的谱元法和有限元法模拟精度比较

在地震动数值模拟方法中，谱元法和有限元法是应用较广泛的两种方法。基于经典的Lamb问题模型，首先推导给出地表竖向位移的解析解答。然后分别利用常用的四阶谱元法和线性有限元法，模拟了地表脉冲力源作用下模型的位移响应。考虑有意义的最短波长内的采样点个数及单元高宽比的变化，对比了两种方法的模拟精度；结果表明:对于谱元法，观测点与波源之间需至少包含两个网格，在此条件下，最短波长内包含一个网格（最短波长内5个采样点）时，数值解与解析解的误差小于1%，已达很高的精度；对于有限元法，最短波长内需包含10个网格时才能达到这一精度。此外，在满足网格尺寸要求的前提下，单元水平向与垂直向尺寸的比值在1∶1到5∶1的范围内时，谱元法和有限元法的模拟精度均变化不大。因此，单位波长内采样点个数相同时，谱元法的模拟精度比有限元法高的多，同时，在一定范围内两种方法的模拟结果对于宽高比的变化不敏感。

Comparison of simulation precision between spectral element method and finite element method based on lamb problem

Spectral element method (SEM) and finite element method (FEM) are widely used in the numerical simulation of ground motion. Based on the classical Lamb problem model, the analytical solution of the vertical displacement of the surface is firstly derived. Then the fourth-order SEM and linear FEM are respectively used to simulate the displacement response of the model under the action of the surface impulsive force source. The simulation precisions of the two methods are compared considering the variation of the number of sampling points in the shortest wave length and the aspect ratio. The results are as follows. For the spectral element method, at least two meshes should be contained between the observation point and the wave source. Under this condition, when the shortest wave length contains one grid (5 sampling points within the shortest wave length), the error between numerical solution and analytical solution is less than 1%, which has achieved a relatively high accuracy. For the finite element method, this precision can be achieved when 10 grids are included in the shortest wavelength. In addition, the simulation accuracies of both SEM and FEM have little change when the ratio of horizontal to vertical grid size is within the range of 1 to 5. Therefore, the simulation accuracy of SEM is much higher than that of FEM when the number of sampling points in the shortest wave length is the same. And the simulation results of the two methods are not sensitive to the variation of the aspect ratio within a certain range.