基于无单元Galerkin法探地雷达正演模拟

Forward simulation of Ground Penetrating Radar based on the element-free Galerkin method

无单元Galerkin法采用滑动最小二乘法拟合场函数,只需节点无需单元,具有前处理简单、精度高、解高次连续等优点,被用于求解探地雷达(GPR)正问题.本文从Maxwell方程出发,推导了GPR正演需满足的波动方程;详细介绍了滑动最小二乘法形函数的构造方法.针对EFGM不满足插值条件导致强加边界条件的处理变复杂的特性,采用罚因子法对强加边界条件进行了处理;同时为了消除EFGM进行GPR正演模拟时来自截断边界处的超强反射,采用透射边界条件把GPR波在截断边界处的反射波透射出去,进而压制了来自截断边界处的反射波.然后,编制了EFGM的GPR正演模拟Matlab程序,应用该程序对典型GPR地电模型进行了正演模拟,并把该正演剖面图与基于线性插值FEM正演剖面图进行了对比,结果表明了EFGM用于GPR正演计算的正确性及有效性,并且在相同节点数条件下,EFGM比矩形剖分的FEM的精度要高,更有利于指导雷达剖面的数据解译.

Forward simulation of Ground Penetrating Radar （GPR） is conducted by the element- free Galerkin method （EFGM） which has advantages such as simple preprocessing, high precision and solution of the high-order continuous derivatives, so it only requires nodes without elements and adopts the moving least-squares （MLS） fitting field function. The wave equation that the GPR ~orward simulation needs to satisfy is derived starting with the Maxwell equation, and then the constructing method of the MLS shape function is presented in detail. To address the complicated processing of the imposed boundary conditions, a penalty factor is employed to process the imposed boundary condition because the EFGM cannot meet the interpolation condition. Meanwhile, to eliminate ultra-intense reflection from the truncated boundary when the EFGM is employed in the GPR forward simulation, the truncated boundary condition is used to transmit the reflected waves from the truncated boundary, so as to restrain these reflected waves. A Matlab program for GPR forward simulation based on the EFGM is compiled and used in the forward simulation for the typical GPR geoelectric model. A comparison of the resultant profile from the simulation with the FEM forward-simulation profile based on linear interpolation shows that the EFGM simulation has better continuity and smoothness, and the consistency of the simulation results indicates the correctness and effectiveness of the EFGM used for GPR forward computation.