【作者】 黄婷婷；

【导师】 柳清伙；

【作者基本信息】 厦门大学 ， 电子与通信工程， 2021， 硕士

【摘要】 电磁勘探法常用于重构地下目标结构,已在金属/非金属矿、油气、页岩气、地下水勘探等领域得到广泛应用。采用数值仿真技术,不仅能准确探测复杂地下情况,还能有效减少勘探成本。由于谱元法具有有限元的灵活性和谱方法的精度高特点,能实现数值误差随阶数指数收敛特性,开始广泛应用于电磁勘探。然而,传统的谱元法在低频时,由于系统的质量矩阵远小于刚度矩阵,导致系统矩阵存在奇异性,引起较大数值解误差甚至可能无法收敛,即低频崩溃。因此,为了在电磁勘探中避免低频崩溃问题,本文将对矢量亥姆霍兹方程的谱元法施加两种类型的约束条件,都能有效地克服低频崩溃现象,为本文的电磁散射问题的研究提供准确的数值解。在电磁勘探中,接收机中的电磁信号既含有背景场中的一次场数据,也包含地质信息的二次场数据。但由于一次场幅值远大于二次场信号,为较准确地获得地下介质的电性结构,需要有效去除背景场的影响。在低频电磁仿真中,电磁信号分解为背景场(一次场)和散射场(二次场),背景场由任意分层介质的各向异性格林函数解析求解,而散射场则利用谱元法求解。为了避免谱元法中低频崩溃问题,本文对矢量亥姆霍兹方程引入拉格朗日乘数法形式强加高斯定理,推导出带有散度约束的混合谱元法。该方法不仅能克服散射场的低频崩溃,还能避免源附近的奇异性,同时也允许激励源处于计算区域外,提高仿真精度和计算效率。虽然基于拉格朗日乘数法能克服低频崩溃,但同时也引入了额外的自由度,增加矩阵求解的复杂度。为了进一步提高仿真效率,本文将引入基于tree-cotree分解的混合方法。利用tree-cotree分解的混合谱元法将单元分解成tree边和cotree边。对tree边和cotree边分别采用节点基函数的梯度和棱边基函数进行展开,能自动满足散度约束方程,使得混合方法中的自由度与传统方法保证一致,大大提高了低频问题的仿真效率。数值结果验证了该方法相比传统方法和拉格朗日乘数法形式的混合谱元法的优越性。更多还原

【Abstract】 Electromagnetic exploration methods are often used to reconstruct underground target structures,and have been widely used in metal/non-metallic mines,oil and gas,shale gas,groundwater exploration and other fields.Using numerical simulation technology can not only accurately detect complex underground conditions,but also effectively reduce exploration costs.Because the spectral element method has the characteristics of flexibility of the finite element method and high precision of the spectral method,it can realize the exponential convergence characteristics of the numerical error with the order,and it has been widely used in electromagnetic prospecting.However,at low frequencies,because the mass matrix of the system is much smaller than the stiffness matrix,the system matrix is singular,the traditional spectral element method causes large numerical solution errors or even failure to converge,that is,low-frequency breakdown.Therefore,in order to avoid the problem of low-frequency breakdown in electromagnetic exploration,this article will impose two types of constraints on the spectral element method of the vector Helmholtz equation,which can effectively overcome the phenomenon of low-frequency breakdown and provide accurate numerical solutions for the study of the electromagnetic scattering problem in this article.In electromagnetic exploration,the electromagnetic signal in the receivers contain both the primary field data in the background field and the secondary field data of geological information.However,since the amplitude of the primary field is much larger than that of the secondary field signal,in order to obtain the electrical structure of the underground medium more accurately,it is necessary to effectively remove the influence of the background field.In the low-frequency electromagnetic simulation,the electromagnetic signal is decomposed into the background field(primary field)and the scattered field(secondary field).The background field is evaluated analytically by the anisotropic Green’s function of any layered medium,and the scattered field is solved by the spectral element method.In order to avoid the problem of low frequency breakdown in the spectral element method,this paper introduces the Lagrange multiplier method to the vector Helmholtz equation and imposes the Gauss theorem,and derives the mixed spectral element method with divergence constraints.This method can not only overcome the low-frequency breakdown of the scattered field,but also avoid the singularity near the source and allow the transmitters to be located outside the computational domain,improving the simulation accuracy and calculation efficiency.Although the Lagrange multiplier method can overcome the low-frequency breakdown,it also introduces additional degrees of freedom and increases the complexity of matrix solving.This paper will introduce a mixed method based on tree-cotree splitting for the purpose of further improving the simulation efficiency.The mixed spectral element method of tree-cotree splitting is used to decompose the elements into tree edges and cotree edges.The tree edge and cotree edge are expanded with the gradient of the node basis function and the edge basis function respectively,which can automatically satisfy the divergence constraint equation,so that the degree of freedom in the mixed method is consistent with the traditional method,and the simulation efficiency of low-frequency problems is greatly improved.The numerical results verify the superiority of this method compared with the traditional method and the Lagrange multiplier method of mixed spectral element method.更多还原