基于全局弱式无单元法直流电阻率正演模拟

全局弱式无单元法是在有限单元法基础上发展起来的一种数值模拟方法,它采用局部支持域内的节点信息来构造形函数实现局部精确逼近,摆脱了单元,仅依赖于节点信息,具有预处理简单、模拟精度高、灵活性强的特点,适用于复杂地电条件下直流电阻率正演模拟.本文采用RPIM构造直流电阻率全局弱式无单元法形函数,利用RPIM形函数推导了直流电阻率全局弱式无单元法方程.然后,编制了直流电阻率全局弱式无单元法正演模拟Fortran程序,利用该程序对典型的地电模型进行了正演模拟,并将正演结果与基于线性插值的FEM正演结果及解析解进行对比,结果表明采用RPIM形函数的全局弱式无单元法用于直流电阻率正演模拟的正确性及有效性,且在同等条件下,全局弱式无单元法模拟精度高于矩形剖分的FEM,更有利于指导电法勘探的数据解译;利用该程序对复杂地电模型进行了正演模拟,结果表明全局弱式无单元法对复杂地电模型模拟效果良好,适应性强,灵活性高,可任意加密节点提高模拟精度.

A global weak form element free method for direct current resistivity forward simulation

We present a global weak form element free method (EFM) for simulation of direct current resistivity. EFM is a new numerical simulation method developed on the basis of finite element method (FEM). The key point of this method is the absence of elements and the nodes free from the elemental restraint, which makes it very flexible and the pre-processing simple. It utilizes the nodes of local support domain to construct shape functions to achieve the accurate approximations of the local domain. Approximations of EFM are of high order and boundary conditions are enforced simply, because the radial point interpolation method (RPIM) is used to construct shape function. Therefore, EFM is more suitable to simulate complex models than FEM. First, the boundary value problem and the corresponding variational problem of direct current resistivity forward simulation are derived starting from the partial differential equation of current field. Second, the construction of RPIM shape function is introduced in details. Third, equations of the global weak form EFM for direct current resistivity is derived in details based on RPIM shape function. Then, a Fortran program is written according to the equations. By forward simulation with this program, a homogeneous half-space model was used to verify our element free approach. At the same time, we compared the solutions of EFM and FEM in details which shows that the solutions of EFM are more accurate. Furthermore, the solutions indicate the correctness and effectiveness of the EFM for direct current resistivity forward simulation. Finally, we improve the simulation accuracy successfully by refining nodes arbitrarily, and the solutions of EFM forward simulation for complex geoelectric models show that EFM has a high degree of flexibility.