Efficient 3D vector finite element modeling for TEM based on absorbing boundary condition
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摘要:
瞬变电磁法的三维有限元正演通常采用齐次边界条件，为满足该边界条件，需要构建较大尺寸的模型，这降低了正演问题的求解速度.针对该问题，本文采用吸收边界条件代替齐次边界条件，以缩小模型体积，加快正演速度：首先，从时间域麦克斯韦方程组出发，推导了基于库伦规范的矢量势的微分控制方程，结合一阶吸收边界条件推导了相应的的弱形式方程；在此基础上采用一阶四面体矢量单元进行单元分析、Newmark法进行时间离散，实现了瞬变电磁法的快速三维正演.通过均匀半空间模型的解析解，H型地电断面的CR1Dmod解和相应模型有限元解的对比，验证了本文算法的正确性.均匀半空间模型分别采用吸收边界条件和齐次边界条件的正演结果对比表明：吸收边界条件确实可以提高三维正演的精度或者缩小模型尺寸、加快计算速度.
Abstract:The homogeneous boundary condition is commonly used in the 3D FEM forward modeling of TEM. In order to satisfy the approximation of this boundary condition, it is necessary to construct a large-scale model, which reduces the solving speed of the forward problem. Aiming at the problem, in this paper, absorbing boundary condition was used to reduce the size of the model and accelerate the solution speed instead of homogeneous boundary condition. Firstly, the differential equation of magnetic vector potential based on Coulomb's gauge was derived from Maxwell's equations in time domain, combining the first-order absorbing boundary condition, the corresponding weak form equation was deduced and simplified. Then, by using the first-order tetrahedral vector element to carry out element analysis, Newmark method for time discretization, an efficient TEM 3D modeling method was realized. The proposed algorithm was validated by comparing the analytical solution of homogeneous half space, the CR1Dmod solution of H-type geoelectric section, with their corresponding finite element solution. The further comparison between the modeling results of absorbing boundary condition and homogeneous boundary condition showed that the absorbing boundary condition could improve the accuracy of 3D modeling or reduce the size of the model and accelerate the solution speed.
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Key words:
- TEM /
- Forward modeling /
- Vector finite element /
- Absorbing boundary condition
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图 1
四面体矢量单元的节点和棱边示意图
Figure 1.
Sketch map of nodes and edges in a tetrahedral vector unit
图 4
均匀半空间数值解及其相对误差(σ=0.01 S·m-1, 6 km×6 km×6 km)
Figure 4.
Numerical solution and relative errors of homogeneous half-space (σ=0.01 S·m-1, 6 km×6 km×6 km)
图 5
均匀半空间数值解及其相对误差(σ=0.01 S·m-1, 10 km×10 km×10 km)
Figure 5.
Numerical solution and relative errors of homogeneous half-space (σ=0.01 S·m-1, 10 km×10 km×10 km)
图 6
H型地电断面的有限元解和CR1Dmod解及其相对误差
Figure 6.
FEM solution and CR1Dmod solution of H type geoelectric section and their relative errors
图 7
水平低阻异常体模型示意图
Figure 7.
Sketchmap of horizontal low resistivity anomalous body model
图 9
水平低阻异常体的瞬变电磁响应时间切片
Figure 9.
Time slice of TEM response of horizontal low resistivity anomalous body model
图 10
XZ平面0.05~0.06 ms电流密度变化
Figure 10.
Change of current density in XZ plane from 0.05 ms to 0.06 ms
图 11
倾斜低阻异常体模型示意图
Figure 11.
Sketch map of sloping low resistivity anomalous body model
图 12
O点水平和倾斜低阻异常体模型的瞬变电磁响应
Figure 12.
TEM response of models of horizontal and sloping low resistivity anomalous body at point O
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