利用谱元法计算SNREI地球的表面负荷变形

Spectral element method to load deformation in a SNREI Earth

谱元法作为一种基于变分原理的数值方法,已经成为地球动力学数值模拟的重要手段.本文旨在对Martinec(2000)正演模拟表面负荷问题的谱-有限元方法进行改进.简单地介绍了谱元法计算负荷边值问题的相关理论,并推导了中性分层液体中的平衡方程以及其对应的弱形式解.在地球半径方向上,采取高阶样条函数作为负荷解的基函数,并利用高斯-勒让德方法进行数值积分.均质地球模型数值试验表明,相较于Martinec的方法,本文方法可以有效地加快数值解的收敛速度.最后,利用PREM模型计算了SNREI地球的负荷勒夫数.数值计算结果表明本文方法的收敛性和稳定性都非常好.本文方法获得的负荷勒夫数与采用传统龙格-库塔积分方法获得的结果相对误差在0.01%的量级.

As a numerical method based on variational principle,the spectral element method (SEM) has become an important means of numerical simulation of many geodynamic problems. This paper aims to improve the spectral-finite element method of Martinec (2000) to surface load forward problem. The related theoretical formulas of calculating the load deformation boundary value problem by spectral element method are briefly introduced,and the equilibrium equation in the neural stratified liquid and the corresponding weak form solution are derived. In the direction of the Earth′s radius,the high-order spline functions are used as bases,and the numerical integration is performed by using the Gauss-Legendre method. The results of homogeneous earth model show that the proposed method can effectively speed up the convergence of numerical solutions compared with Martinec′s method. Finally,the load Love numbers of the PREM model is calculated,which also show that this new method has good stability and convergence. The relative error between the results by our method and by traditional Runge-Kutta method is less than 0.01%.