全局弱式无网格法在大地电磁二维正演中的应用研究
发布时间：2018-11-23 16:39
【摘要】：大地电磁正演领域,目前流行的数值计算方法包括有限差分法和有限元法,该类数值计算方法都是基于网格插值求解计算域物理场,对复杂地电模型的网格剖分比较繁琐,计算精度严重依赖于单元剖分的形状和大小。为克服这些困难,无网格法作为有网格法的一种重要补充和发展具有重要的研究意义和实用价值。本文重点研究基于全局弱式无网格法的大地电磁二维正演问题。首先从大地电磁场二维问题所满足的Maxwell方程组出发,利用变分问题与边值问题的等价性,对大地电磁二维问题满足的微分方程进行推导并得到其相应的变分形式。利用一系列离散场节点对求解域进行离散,采用移动最小二乘法(MLS)构造积分点支持域形函数。根据Galerkin法原理,推导出与大地电磁法二维边值问题对应的无网格全局弱式矩阵表达式。采用含背景网格的高斯积分将计算域离散化,求得最终总体矩阵的离散表达式。后续通过多个一维层状模型数值模拟计算,研究了相关重要参数,如高斯点数目,无量纲支持域尺寸等对计算精度及计算效率的影响,并确定其参数最优值,选取高斯点数量为2~3、无量纲支持域尺寸a为1.1~1.3。确定最优参数值后,分别用有限元法和全局弱式无网格法计算几个典型一维模型和二维模型,通过对比计算结果及计算精度,验证本文中提出的全局弱式无网格法的有效性及在处理复杂模型上的优越性。最后,根据冬瓜山铜矿实际情况设计了二维地质模型,并用本文提出的全局弱式无网格法对其进行数值模拟计算,其计算结果对低阻矿体实际形态反应较好,进一步证明了无网格算法在地球物理电磁波数值模拟中具有重要的意义。
[Abstract]:In the field of magnetotelluric forward modeling, the popular numerical methods include finite difference method and finite element method. These numerical methods are based on grid interpolation to solve the physical field in computational domain. The accuracy of calculation depends heavily on the shape and size of the unit. In order to overcome these difficulties, meshless method as an important supplement and development of grid method has important significance and practical value. This paper focuses on the magnetotelluric two-dimensional forward modeling based on the global weak meshless method. Starting from the Maxwell equations satisfied by the two-dimensional magnetotelluric problem and using the equivalence between the variational problem and the boundary value problem, the differential equations of the two-dimensional magnetotelluric problem are derived and the corresponding variational forms are obtained. A series of discrete field nodes are used to discretize the solution domain, and the moving least square method (MLS) is used to construct the integral points to support the domain shape function. According to the principle of Galerkin method, the meshless global weak matrix expression corresponding to the two-dimensional boundary value problem of magnetotelluric method is derived. The computation domain is discretized by Gao Si integral with background mesh, and the discrete expression of the final total matrix is obtained. Through numerical simulation of several one-dimensional layered models, the effects of some important parameters, such as Gao Si number and dimensionless support domain size, on the calculation accuracy and efficiency are studied, and the optimum values of the parameters are determined. The number of Gao Si points is 2 ~ 3 and the dimension of dimensionless support domain a is 1.1 ~ (1.3). After determining the optimal parameter value, the finite element method and the global weak meshless method are used to calculate several typical one-dimensional and two-dimensional models respectively. The effectiveness of the global weak meshless method proposed in this paper and its superiority in dealing with complex models are verified. Finally, according to the actual situation of Dongguashan Copper Mine, a two-dimensional geological model is designed, and the global weak meshless method proposed in this paper is used to numerically simulate the orebody. The calculated results have a good response to the actual shape of the low resistivity orebody. It is further proved that the meshless algorithm is of great significance in the numerical simulation of geophysical electromagnetic waves.
【学位授予单位】：长安大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：P631.325
[Abstract]:In the field of magnetotelluric forward modeling, the popular numerical methods include finite difference method and finite element method. These numerical methods are based on grid interpolation to solve the physical field in computational domain. The accuracy of calculation depends heavily on the shape and size of the unit. In order to overcome these difficulties, meshless method as an important supplement and development of grid method has important significance and practical value. This paper focuses on the magnetotelluric two-dimensional forward modeling based on the global weak meshless method. Starting from the Maxwell equations satisfied by the two-dimensional magnetotelluric problem and using the equivalence between the variational problem and the boundary value problem, the differential equations of the two-dimensional magnetotelluric problem are derived and the corresponding variational forms are obtained. A series of discrete field nodes are used to discretize the solution domain, and the moving least square method (MLS) is used to construct the integral points to support the domain shape function. According to the principle of Galerkin method, the meshless global weak matrix expression corresponding to the two-dimensional boundary value problem of magnetotelluric method is derived. The computation domain is discretized by Gao Si integral with background mesh, and the discrete expression of the final total matrix is obtained. Through numerical simulation of several one-dimensional layered models, the effects of some important parameters, such as Gao Si number and dimensionless support domain size, on the calculation accuracy and efficiency are studied, and the optimum values of the parameters are determined. The number of Gao Si points is 2 ~ 3 and the dimension of dimensionless support domain a is 1.1 ~ (1.3). After determining the optimal parameter value, the finite element method and the global weak meshless method are used to calculate several typical one-dimensional and two-dimensional models respectively. The effectiveness of the global weak meshless method proposed in this paper and its superiority in dealing with complex models are verified. Finally, according to the actual situation of Dongguashan Copper Mine, a two-dimensional geological model is designed, and the global weak meshless method proposed in this paper is used to numerically simulate the orebody. The calculated results have a good response to the actual shape of the low resistivity orebody. It is further proved that the meshless algorithm is of great significance in the numerical simulation of geophysical electromagnetic waves.
【学位授予单位】：长安大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：P631.325
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