瞬变电磁BEDS-FDTD三维正演新算法及稳定性验证  

3Dtransient electromagnetics forward modeling using BEDS-FDTD and its stability verification

作　　者：柳尚斌 孙怀凤[1,2,3] 李文翰 王震 杨洋[1,2,3] LIU ShangBin;SUN HuaiFeng;LI WenHan;WANG Zhen;YANG Yang(Geotechnical and Structural Engineering Research Center,Shandong University,Jinan 250061,China;Laboratory of Earth Electromagnetic Exploration,Shandong University,Jinan 250061,China;Advanced Exploration and Transparent City Innovation Centre,Shandong Research Institute of Industrial Technology,Jinan 250061,China;Beijing Research Institute of Shandong University,Beijing 100086,China)

机构地区：[1]山东大学岩土与结构工程研究中心,济南250061 [2]山东大学地球电磁探测研究所,济南250061 [3]山东省工业技术研究院先进勘探与透明城市协同创新中心,济南250061 [4]山东大学北京研究院,北京100086

出　　处：《地球物理学报》2023年第2期841-853,共13页Chinese Journal of Geophysics

基　　金：国家自然科学基金项目(42074145)资助。

摘　　要：常规FDTD的时间步长需满足Courant-Friedrich-Lewy(CFL)稳定性条件,导致时间迭代步过多,正演非常耗时.针对上述问题,本文采用Backward Euler(BE)差分方法近似Maxwell方程组中场对时间的一阶导数,使时间步长突破CFL限制;引入Direct Splitting(DS)策略将电磁场分量解耦,并将大型稀疏矩阵降阶和重构为一系列低阶且主对角占优的三对角矩阵,加快方程求解效率;而且,为了减少模型尺寸,针对上述改进的全新方程,本文采用双线性变换方法(bilinear transform, BT)推导了复频率偏移完全匹配层(complex frequency shifted perfectly matched layer, CFS-PML)吸收边界,形成瞬变电磁三维正演全新算法:BEDS-FDTD.首先采用von Neumann方法测试了新算法在有耗介质中、非均匀时间步长下的稳定性.之后,将BEDS-FDTD算法用于模型实验,并将模拟结果与层状模型的半解析解进行了对比,实验结果证明了新算法的计算精度能够满足需要;同时对新算法的计算效率进行了分析,当采用Tesla A100 GPU计算时,50×50×50网格数目的模型仅用10 s,即使网格数目增加到200×200×200,也仅需224 s.最后,将BEDS-FDTD算法用于复杂三维模型计算.Conventional FDTD algorithm requires the time steps to satisfy the Courant-Friedrich-Lewy(CFL) stability condition, which leads to too many time iteration steps and makes the 3D TEM forward modeling very time-consuming. In this paper, the Backward Euler(BE) difference method is used to approximate the time derivative in Maxwell′s equations. Then the Direct Splitting(DS) strategy is introduced to decouple the electromagnetic field components and to reduce the order and reconstruct the large sparse matrices into a series of triple diagonal matrices of low order and dominated by the main diagonal, making the solution of the equation more efficient. Moreover, in order to reduce the model size, for the above-mentioned improved new equation, this paper adopts the bilinear transform(BT) method to theoretically deduce the complex frequency shifted perfectly matched layer(CFS-PML) absorption boundary. A new algorithm for transient electromagnetic 3D forward modeling is formed: BEDS-FDTD. First, the stability of the BEDS-FDTD algorithm in a lossy medium under non-uniform time steps is proved using the von Neumann method. Then, to verify the accuracy of the algorithm, the calculation results of BEDS-FDTD are compared with the semi-analytical solution of the layered model, and it is verified that the accuracy of the new algorithm meets the requirements of simulation calculation. After that, the performance of the new algorithm is tested. When using the Tesla A100 designed for high-performance computing, BEDS-FDTD only takes 10s to simulate a model with 50×50×50 cells, and even if the number of cells is increased to 200×200×200,the new algorithm takes less than 4 minutes(224 s). Finally, the BEDS-FDTD algorithm is applied to a complex three-dimensional model.

关 键 词：瞬变电磁 三维正演 BEDS CFS-PML 无条件稳定性 

分 类 号：P631[天文地球—地质矿产勘探]