多源条件下直流电阻率法有限元三维数值模拟中一种近似边界条件

An approximate boundary condition for FEM-based 3-D numerical simulation with multi-source direct current resistivity method

在多源直流电阻率法有限元三维数值模拟中,传统混合边界条件由于刚度矩阵与源点位置相关,无法实现线性方程组的快速回代求解,目前常用齐次边界条件或无限元边界进行替代,虽然实现了快速回代求解,但同时也降低了数值模拟的精度.为了实现快速回代求解,并确保数值模拟的计算精度,本文提出了一种近似边界条件方法.其核心思想是将与场源位置相关的边界系数矩阵从刚度矩阵中分离出来,使得分离后的刚度矩阵与场源位置无关.并将边界系数矩阵与边界处一次电场向量的乘积移到线性方程组右端源项中,当场源位置发生改变时,只需要重新计算右端源项就可以实现快速回代求解.理论模型数值计算表明,在水平地形条件下,本文边界条件数值精度优于混合边界条件;在起伏地形条件下,与齐次边界条件相比,本文边界条件数值结果与混合边界条件吻合度更高.

In finite element-based three-dimensional numerical simulation of direct current（DC）resistivity method,applying traditional mixed boundary condition cannot accomplish solving linear equations through recursion due to computation of stiffness coupling with source positions.Currently Neumann boundary condition or infinite element boundary condition is usually used instead.Although rapidly resolving,these two methods reduce the precision of numericalsimulation.To realize recursive resolving rapidly and ensure simulation precision,an approximate boundary condition is proposed to implement both fast recursive resolving and precise simulation.The key ideas underlying the method are to separate boundary coefficient matrix coupled with source positions from stiffness matrix so as to make the resultant stiffness matrix independent of source positions.Production of boundary coefficient matrix and primary electric field vectors on the boundary is then transferred to the right hand of linear equations.In doing so only right-hand source items need to be computed when source positions are changed.Synthetic tests show that the numerical simulation precision applying the newly proposed boundary condition is superior to the one using mixed boundary condition in the case of horizontal topography.Also,in the case of rugged topography,the simulation results,compared with the application of neumann boundary,are much closer to those with mixed boundary condition.