The direct current(DC)resistivity method performs detection activities by artificially arranging power supply electrodes and measurement electrodes on the surfground.Nowadays,this method has been widely used in major national projects,such as underground mineral resource exploration and tunnel construction.In actual field work,the DC resistivity method is mainly applied to three-dimensional problems.Problems such as complex topography,multiple anomalous areas,and complex anomalous body boundaries have challenged the accuracy of the three-dimension DC resistivity method.If the forward accuracy is low,the interpretation of the inversion data in the later will make the inversion result inaccurate or even cause misunderstanding.Therefore,it is imperative to study high-precision and efficient 3D DC resistivity forward modeling.For the 3D DC resistivity forward modeling,we propose an hp-type adaptive finite element algorithm based on the h-type adaptive finite element algorithm and the p-type adaptive finite element algorithm.In this paper,we first derived the three-dimensional DC resistivity boundary value equation,and then used the Galerkin finite element method to convert the differential equation to an equivalent integral equation.Finally,the program was developed using C++ language.In the three-dimensional forward modeling,in order to highly fit the undulating surface and the surface of the complex anomaly,we use the unstructured mesh to discrete the solution area.The traditional h-type adaptive finite element algorithm only needs to calculate the posteriori error estimator of the finite element solution,and then base on this estimator to adaptively refine the mesh.The hp-type adaptive finite element algorithm proposed in this paper not only needs to calculate the posteriori error estimator but also needs to calculate the smoothness estimator of the solution on the cell.Based on these two estimators,the algorithm will decide which cells to perform h-refinement and which cells to implement p-refinement(that is,increase the order of the shape function by 1).The refinement strategy is: for a cell with a large error,if the smoothness of the finite element solution is high,the cell will be implement p-refinement;if the smoothness of the finite element solution is low,the cell will be implement h-refinement.The program uses the tensor product of one-dimensional polynomials to generate the shape function of any order in three-dimensional space.Numerical examples show that the hp-type adaptive finite element algorithm proposed in this paper has high accuracy,and the convergence speed of the finite element solution comparison with the h-type adaptive algorithm and the p-type adaptive algorithm shows that the algorithm proposed in this paper can greatly save the computing resources while improving the accuracy of the finite element solution.The terrain model verified that our algorithm can deal with the undulating terrain very well,and can obtain more accurate numerical results.In the process of 3D DC resistivity inversion,we applied the regularization theory to establish the inversion objective function,and used a nonlinear conjugate gradient algorithm to solve the minimum value of this objective function.We have initially completed the inversion study of the 3D DC resistivity method.In numerical examples,the inversion results obtained through different initial models are basically the same,which proves that our inversion procedure is relatively stable.In the comparison of the inversion results of the high-resistance anomaly model and the low-resistance anomaly model,it is found that the sensitivity of the inversion procedure for the high-resistance anomaly is lower than that of the low-resistance anomaly. |