三维直流电阻率自然单元-无限元耦合数值模拟

3DDC resistivity numerical modeling by natural-infinite element coupling method

直流电阻率数值模拟对于实测电阻率数据反演、解释地下真实电阻率结构具有重要意义.为解决复杂地电模型剖分以及处理截断边界时面临的困难,本文提出一种新的三维自然单元-无限元耦合法.该算法在自然单元区域灵活布设自然节点以填充复杂模型,并引入无限单元替代截断边界条件以促使刚度矩阵与电源位置无关.首先推导了稳定电流场基本方程;然后介绍了二、三维自然单元法及其形函数的构建过程;再引入三维多向映射无限单元并实现自然单元-无限元的有效耦合;最后通过4个数值模型,结合传统有限单元法、有限元-无限元耦合法以及COMSOL Multiphysics软件的计算结果,验证了算法的正确性及有效性.数值结果表明,本文所提算法满足精度要求,能有效减小计算区域,无需二次更新刚度矩阵,对复杂模型适应性强.

Numerical modeling of direct current(DC)resistivity models plays an essential role in the inversion of measured resistivity data and the interpretation of underground resistivity structure.In order to solve the difficulties of complex geoelectric models in the subdivision and conventional numerical algorithms in dealing with truncated boundaries,a new 3D natural-infinite element coupling method was proposed based on the natural element method and infinite element method.The coupled algorithm behaves well on filling or subdividing complex models by flexibly arranging natural nodes in the study area.The coupled infinite element helps to avoid loading truncated boundary conditions and keep the sparse stiffness matrix remain symmetric and independent of the current source.At first,we deduced the fundamental equation of the quasi-static DC field.Then,the 2Dand 3Dnatural element methods were introduced,and the formula of the 3Dshape function and its derivative were provided according to the chain derivative method of compound functions.Besides,we used the 3Dmulti-directional mapping infinite element to act as the boundary element,which helps to construct the 3D natural-infinite element coupling method.Finally,four numerical examples were tested and compared with the numerical result by the conventional finite element method,finite-infinite element method,and COMSOL Multiphysics software.The numerical results show that the coupled method has the advantages of having high accuracy,helping narrow down the study area,needing no update of the stiffness matrix,and facilitating the subdivision of complex models.