Crank-Nicolson FDTD 3D forward modeling for the transient electromagnetic field
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摘要:
时域有限差分（FDTD）方法使用Yee网格剖分电磁场的空间采样, 通过时间步迭代实现电磁场数值模拟, 具有内存消耗低、计算简单等特点, 常用于瞬变电磁三维正演.然而, 常规FDTD方法的时间迭代步长Δt受Courant-Friedrich-Lewy（CFL）条件严格限制, 过多的迭代次数以及过密的采样往往导致计算速度慢、累积误差不断增大.本文提出一种不受CFL条件约束的无条件稳定隐式差分算法Crank-Nicolson FDTD（CN-FDTD）用于瞬变电磁三维正演.基于Crank-Nicolson差分方法对Maxwell方程组重新离散, 空间网格仍然采用Yee元胞, 时间步进采用在整时间步电场、磁场同时采样的策略, 建立无条件稳定FDTD格式, 突破CFL条件限制.与常规FDTD交替采样相比, CN-FDTD电场、磁场同时采样的策略构成的隐式差分格式, 需要求解大型稀疏矩阵方程组.通常, 瞬变电磁三维正演模型中产生的矩阵阶数往往较大, 需要占用大量内存和求解时间.为解决上述问题, 采用Crank-Nicolson-cycle-sweep-uniform（CNCSU-FDTD）方法近似求解CN-FDTD方程, 在保证求解精度的同时, 计算效率大幅提高.在边界条件处理上, 采用双线性变换推导了复频率参数完全匹配层（CFS-PML）吸收边界.采用均匀半空间模型、四类三层模型进行精度验证, 发现CN-FDTD三维正演结果与解析解、线性数字滤波解吻合较好.之后, 与接触带上的低阻复杂模型进行对比, 结果显示CN-FDTD正演结果与矢量有限元、有限体积法以及FDTD计算结果吻合较好.在此基础上, 研究了时间步放大对CN-FDTD计算精度的影响, 发现最大时间步放大到常规FDTD的3200倍时才会在晚期出现较明显的误差.在一台CPU为Intel Core i5-7300HQ的笔记本电脑单线程计算条件下, 模拟到关断后30 ms仅需要50 min.在进行并行化后, 将有望实现复杂模型分钟级的三维正演, 从而为三维反演提供可靠、快速的正演方法.
Abstract:The time-domain finite-difference (FDTD) method defines the spatial position of the electromagnetic field using the Yee grid and can simulate the propagation of the electromagnetic field with low memory cost. It is often used in three-dimensional forward modeling of transient electromagnetics. However, the time-step size of the conventional FDTD is strictly limited by the Courant-Friedrich-Lewy (CFL) stability condition. Too many iterations and intensive sampling result in an increase in calculation time and cumulative error. In this study, an implicit difference algorithm, called Crank-Nicolson FDTD (CN-FDTD), is proposed for forward modeling of three-dimensional transient electromagnetics. This algorithm is no longer constrained by the CFL condition and is unconditionally stable. The space grid still adopts the Yee cell. The electric field and magnetic field are simultaneously sampled in integer time steps to establish an unconditionally stable Crank-Nicolson scheme. Compared to conventional FDTD, CN-FDTD introduces an implicit difference with the strategy of sampling in space and time simultaneously. It needs to solve large sparse matrix equations at each step and will consume a lot of physical memory and time. Crank-Nicolson-cycle-sweep-uniform (CNCSU-FDTD) is applied to solve this problem. This approximation method greatly improves the calculation efficiency while ensuring the accuracy of the solution. The bilinear transformation method is used to derive the complex-frequency-shifted perfectly matched layer (CFS-PML) absorption boundary condition. Tests are conducted on the uniform half-space and three-layered models to examine the accuracy of CN-FCTD. The forward modeling results in this approach are in good agreement with the analytical solution and the linear digital filter solutions. The accuracy of the algorithm is further tested using a complex low-resistance model with a vertical contact zone. The CN-FDTD result is in good agreement with the finite element, finite volume, and original FDTD results. Then the effect of time-step magnification on the calculation accuracy of CN-FDTD is studied. When the time step is enlarged to 3200 times of the original FDTD algorithm, the obvious error appears in the late stage. All tests are conducted on a laptop with Intel Core i5-7300HQ CPU. It takes only 50 minutes to finish forward modeling on a complex model to 30 ms after the current is turned off with single-thread. By parallelization, it is expected to realize 3D forward modeling in minutes and provide a reliable and fast forward modeling algorithm for inversion.
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图 2
常规FDTD与CN-FDTD电场、磁场时间采样分布对比
Figure 2.
Comparison of original FDTD and CN-FDTD sampling in time for electric and magnetic fields
图 3
均匀半空间模型CN-FDTD与解析解对比
Figure 3.
Comparison of CN-FDTD numerical solution and analytical solution for Homogeneous half space model
图 4
典型三层模型CN-FDTD数值解与线性数字滤波解对比
Figure 4.
Comparison of digital filter solution and CN-FDTD for typical there-layered models
图 5
三维垂直接触带复杂模型(Li et al., 2017)
Figure 5.
Three-dimensional complex model with a vertical contact zone
图 6
三维垂直接触带复杂模型计算结果对比
Figure 6.
The numerical results of three-dimensional complex model
图 7
CN-FDTD在不同CFLNs下结果的对比
Figure 7.
Comparison of CN-FDTD numerical results under different CFLNs
图 8
CN-FDTD在不同CFLNs下Δt的变化对比
Figure 8.
Changes of Δt in the CN-FDTD under different CFLNs
表 1
层状模型参数
Table 1.
Parameters of the layered models
模型 ρ1(Ωm) ρ2(Ωm) ρ3(Ωm) h1(m) h2(m) A 20 50 500 40 100 H 100 10 100 60 40 K 100 1000 100 60 40 Q 100 10 1 40 100 
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表 2
不同CFLNs下CN-FDTD算法内存开销及计算时间
Table 2.
Memory overhead and computing time of CN-FDTD algorithm under different CFLNs
CFLN 总迭代次数 计算时间(min) 内存(Mb) 200 16220 70 192 400 13091 56 192 800 12038 51 192 1600 11656 50 192 3200 11528 49 192
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