Three-Dimensional Electromagnetic Modeling Using Spectral Element Method With Tetrahedral Grid

In recent years,the electrical exploration technology has played an increasingly important role in geological survey,resource exploration,environment and engineering,etc.In the research of electrical exploration,to obtain modeling for fine structures or large-scale of geological problem,an efficient numerical simulation strategy is urgently needed,which is of great significance for large-scale and precise inversions.In terms of numerical simulation technology,the finite-element method is good for its flexible grids,while the spectral method has the characteristics of high precision.The spectral element method combines the advantages of the two methods,which divides the physical domain into finite elements and uses high-order orthogonal polynomials to construct the basis function in each element.It can not only simulate complex models,but also avoid the numerical oscillation caused by traditional high-order interpolation,and can stably improve the accuracy of numerical simulation.The spectral element method has been paid more and more attention in electrical exploration.At present,the spectral element method applied in electrical exploration is mostly based on structured hexahedral grids,and the corresponding basis functions include Gauss-Lobatto-Legendre(GLL)or Gauss-Lobatto-Chebyshev(GLC).Hower,most rocks and ores in nature have different shapes and even sharp corners.In addition,irregular terrain is one of the factors that cannot be ignored in forward modeling.Due to its existence,the current field is distorted,which affects the electrical distribution of underground media.For these irregular interfaces,the simple structured grids cannot achieve accurate fitting.Thus,the unstructured tetrahedral spectral element method can be used to accurately fit these complex structures by controlling the distribution and size of the grids,and can achieve higher accuracy by modifying the order of the basis function,so as to better reflect the real geological conditions.Starting from the weighted residual Galerkin method,this paper will address in detail the technical content of high-order spectral element method based on tetrahedral grids.The core theory mainly includes the generation of spectral interpolation basis function and the construction of interpolation nodes.In this thesis,the orthogonal basis functions based on tetrahedral spectral element method is established by using PKD polynomials.Since the basis function is defined in the reference coordinate system of right angle tetrahedron,affine transformation needs to be introduced to ensure that the relative positions of interpolation nodes are kept unchanged.Moreover,the coupling correlation between the coordinates of tetrahedral elements makes the operation of integral and differentiation difficult,so the collapse transformation is used to convert it into the related calculation in hexahedral elements.In addition,the reasonable interpolation nodes are particularly important for the stability and accuracy of numerical simulation.Generally,it is necessary to consider the interpolation attributes and the number of interpolation nodes.With that,this paper uses the Warp & Blend point set.Since there is no explicit analytical expression in the integral operation of Proriol-Koornwinder-Dubiner(PKD)polynomial,a numerical integration is used to ensure accuracy.In this paper,the calculation of the coefficient matrix adopts the parallel strategy and is loaded into compressed storage to form a global matrix,which reduces the memory consumption.Finally,the multifrontal direct solver MUMPS is used to fast decompose large sparse matrix and solve the linear equations system.Based on the theoretical researches,this paper makes an in-depth study on the forward modeling of DC resistivity method.Based on the boundary value problem of DC resistivity method,the governing equation for unstructured spectral element method is established.By comparison with the semi-analytical solution for a layered earth model,it is concluded that the relative error of numerical modeling decreases significantly with the increase of the order of the basis functions,and in places it can even reach exponential convergence.Then,taking a cubic anomalous model as example,the relationship between the calculation accuracy and the efficiency of the adaptive refining algorithm and the high-order spectral element method is discussed.The spectral element method in this paper can achieve higher accuracy with fewer degree of freedom(Do Fs).Moreover,in the numerical simulation,if the dual-track scheme of mesh refinement and order increment is adopted,the accuracy can be further improved.X After the flexible tetrahedral grids are used to simulate the complex model,the influence of terrain effect can be studied.The topographic effect is a superposition of a potential distortion and a geometry effect,which can be removed by terrain correction.Anisotropy has always been an important topic in geophysical research.In the field of electrical exploration,most of the underground media are assumed to be isotropic,however,the anisotropy is commonly existing in nature,which brings difficulties to the data interpretation.Therefore,in this paper the unstructured spectral element method is used to study the anisotropic media.The conductivity tensor and Euler rotation are used to describe arbitrarily anisotropic medium.This paper also discusses identification strategies of electrical anisotropy in detail: the strike and tendency of layering in the anisotropic media can be identified by using the polarity measurement of pole-pole array,while the nonlinear equatorial dipole-dipole array is more sensitive to the identification of anisotropic fractures.Finally,the effectiveness of terrain correction is checked again in the model research of topographic earth.The research of the unstructured spectral element method in this paper is based on nodal element technology.The DC electric field naturally meets the continuity condition,so it can well solve the forward problem in DC resistivity modeling.However,the edge element is usually used to solve the continuity problem in frequencydomain airborne EM.In this paper,the electric and magnetic field are represented by the combination of vector potential A and scalar potential ?.In this way,the Maxwell equations are simplified to the equations for the potentials A and ?.At the same time,the existence of spurious modes is eliminated by using Coulomb gauge.This paper further compares the electromagnetic field based on the edge element with the potential method based on nodal element.In terms of the speed and accuracy,it is concluded that the algorithm in this paper has more advantages in improving accuracy and efficiency.Then,we studies the responses of the abnormal body and terrain for frequency-domain airborne EM system through multiple models.This paper systematically studies the forward modeling theory of unstructured spectral element method,and realizes the forward simulation of DC resistivity method for isotropic and anisotropic earth models,and frequency-domain airborne EM.It will lay a good foundation for high-precision forward and inverse modeling in electrical exploration.