【摘要】：大地電磁正演領域,目前流行的數值計算方法包括有限差分法和有限元法,該類數值計算方法都是基于網格插值求解計算域物理場,對復雜地電模型的網格剖分比較繁瑣,計算精度嚴重依賴于單元剖分的形狀和大小。為克服這些困難,無網格法作為有網格法的一種重要補充和發(fā)展具有重要的研究意義和實用價值。本文重點研究基于全局弱式無網格法的大地電磁二維正演問題。首先從大地電磁場二維問題所滿足的Maxwell方程組出發(fā),利用變分問題與邊值問題的等價性,對大地電磁二維問題滿足的微分方程進行推導并得到其相應的變分形式。利用一系列離散場節(jié)點對求解域進行離散,采用移動最小二乘法(MLS)構造積分點支持域形函數。根據Galerkin法原理,推導出與大地電磁法二維邊值問題對應的無網格全局弱式矩陣表達式。采用含背景網格的高斯積分將計算域離散化,求得最終總體矩陣的離散表達式。后續(xù)通過多個一維層狀模型數值模擬計算,研究了相關重要參數,如高斯點數目,無量綱支持域尺寸等對計算精度及計算效率的影響,并確定其參數最優(yōu)值,選取高斯點數量為2~3、無量綱支持域尺寸a為1.1~1.3。確定最優(yōu)參數值后,分別用有限元法和全局弱式無網格法計算幾個典型一維模型和二維模型,通過對比計算結果及計算精度,驗證本文中提出的全局弱式無網格法的有效性及在處理復雜模型上的優(yōu)越性。最后,根據冬瓜山銅礦實際情況設計了二維地質模型,并用本文提出的全局弱式無網格法對其進行數值模擬計算,其計算結果對低阻礦體實際形態(tài)反應較好,進一步證明了無網格算法在地球物理電磁波數值模擬中具有重要的意義。
[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.
【學位授予單位】：長安大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：P631.325

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