Two-dimensional magnetotelluric modeling using the Meshfree Local Petrov-Galerkin method
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摘要:
有限差分法和有限单元法在大地电磁场数值模拟中已经得到了广泛的应用，但其数值结果的精度在很大程度上依赖于网格的离散程度.当模拟起伏地形、弯曲界面等复杂地电模型大地电磁场响应时，常常需要花费大量的时间以便得到较合理的离散网格.无网格局部Petrov-Galerkin法（MLPG）不同于有限差分法和有限元法，其形函数和权函数脱离了网格的束缚.本文详细推导了二维大地电磁场边值问题的弱式形式，并将其离散为局部积分域内的表达形式.通过模拟二维海洋地电模型大地电磁场响应，并与结构网格有限元结果进行对比，验证了本文算法和程序的正确性及精度.设计了一个含有弯曲界面的二维地电模型，讨论了不同离散网格对MLPG无网格法模拟结果的影响，并与结构有限元法结果进行了比较，结果表明MLPG无网格法模拟结果受离散网格影响较小.最后利用MLPG无网格法计算了两个海洋起伏地形模型的大地电磁响应，讨论了海底起伏地形对大地电磁响应的影响.
Abstract:The finite difference method (FDM) and the finite element method (FEM) have been widely used in magnetotelluric (MT) modelling, but their accuracy heavily depends on discretized grids. For a complex model with rough topography or curved interfaces, it might take a lot of time to generate a proper mesh for fitting the topography and the interfaces.A new numerical simulation method, called Meshfree Local Petrov-Galerkin method (MLPG), has been proposed to solve this problem.The shape function and the weight function of MLPG are only related to the distance between nodes.This method, at root, overcomes the drawback of the conventional FDM or FEM method that depends on elements. In this work, we transformed the magnetotelluric boundary value problem into a weak form by using the weighted residual method and obtained its discrete form in a local integral domain. By simulating the MT responses of a 2-D conductivity model and comparing the numerical solutions with those from the structured FEM, we have verified the validity and accuracy of this new algorithm. Numerical examples show that the MLPG method is less affected by a grid than the structured FEM. In terms of the MLPG algorithm, we have simulated MT responses of two marine conductivity models with topography and analyzed the modeling results.
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Key words:
- Magnetotelluric /
- Meshfree method /
- Marine terrain relief
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图 1
二维大地电磁研究区域示意图
Figure 1.
Schematic illustration of 2-D magnetotelluric modeling domain
图 2
MLPG法的问题域、局部子域、权函数域、支持域示意图
Figure 2.
Schematic illustration of the problem domain, integral domain, weight function domain and support domain of MLPG
图 3
位于介质分界面上的节点局部积分域分解图
Figure 3.
Schematic illustration of the integral domain decomposition when a node is located on an material interface
图 5
MLPG无网格法数值解与矩形网格有限元数值解的对比
Figure 5.
Comparison between the numerical solutions obtained by the MLPG and those by the FEM
图 6
MLPG无网格法100m网格距时节点和高斯点的分布
Figure 6.
Distribution of nodes and Gauss points in MLPG grid with spacing100 m
图 8
MLPG无网格法200 m网格距时节点和高斯点的分布
Figure 8.
Distribution of nodes and Gauss points in MLPG grid with spacing 200 m
图 10
MLPG无网格法250 m网格距时节点和高斯点的分布
Figure 10.
Distribution of nodes and Gauss points in MLPG grid with spacing 250 m
图 12
MLPG无网格法和结构有限元法在1.0 Hz下3套不同离散网格模拟结果对比
Figure 12.
Comparison of MT responses (1.0 Hz) obtained by the MLPG andstructured FEmethodwith three different grids
图 13
MLPG无网格法和结构有限元法在0.1 Hz下3套不同离散网格模拟结果对比
Figure 13.
Comparison of MT responses (0.1 Hz) obtained by MLPG and structured FE method with three different grids
图 16
二维海洋梯形地堑模型节点分布
Figure 16.
Node distribution of 2-D marine trapezoidal grabenmodel
图 18
二维海洋梯形地堑模型在1.0 Hz和0.1 Hz下大地电磁响应
Figure 18.
MT responses of 2D marine trapezoidal graben model at 1.0 Hz and 0.1 Hz
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