本文選題：矢量有限元 + 三維�。� 參考：《吉林大學(xué)》2016年碩士論文

【摘要】：大地電磁測深法是一種頻率域電磁測深法,它以天然存在的交變電磁場為場源進行深部地質(zhì)研究,因此野外施工比較方便,可以減小成本,另外這種方法還有不會被高阻層屏蔽以及對低阻地層的分辨率比較高等優(yōu)點,所以它在地球物理勘探中得到了廣泛的應(yīng)用。進行大地電磁的正演數(shù)值模擬,能讓我們對各種不同電性介質(zhì)下場的規(guī)律有個很好的認識,為最終要實現(xiàn)的反演提供基礎(chǔ)和保證。本文應(yīng)用矢量有限元方法,實現(xiàn)了帶地形三維大地電磁的正演數(shù)值模擬。首先,從MAXWELL方程組出發(fā),根據(jù)三維大地電磁所滿足的邊界條件,利用加權(quán)余量法推導(dǎo)了與三維大地電磁邊值問題等價的變分方程。采用規(guī)則的六面體單元對計算區(qū)域進行剖分,使用矢量插值函數(shù)進行單元分析,插值函數(shù)簡單,精度高,滿足法向方向場的不連續(xù)性,能夠避免節(jié)點有限元法中“偽解”的問題,可以不用做散度校正。最后形成的單元剛度矩陣巨大而且非常稀疏,考慮到計算速度和內(nèi)存占用的問題,將總體系數(shù)矩陣中采用一維非零元素存儲,將不完全喬列斯基分解與雙復(fù)共軛梯度算法相結(jié)合對方程進行求解,在保證計算速度的同時,高效、穩(wěn)定地收斂。其次,建立均勻半空間模型和層狀模型,并與解析解進行對比,電阻率和相位的計算精度都很高,從而驗證了算法及程序的正確。在驗證的過程中發(fā)現(xiàn),利用傳統(tǒng)的三維大地電磁邊界條件時,精度不高,計算速度慢,針對此情況還需給出計算區(qū)域前后左右四個側(cè)面的邊界條件。另外也認識到網(wǎng)格剖分對計算結(jié)果的影響很大,得到了一些關(guān)于網(wǎng)格剖分的結(jié)論。在此基礎(chǔ)上,本文又給出了三維異常體(包含高阻體和低阻體)模型,分別計算了XY模式和YX模式下異常體模型的視電阻率和阻抗相位的異常響應(yīng),并且分析了它們的響應(yīng)特征。最后,利用矢量有限元方法,實現(xiàn)了帶地形的三維大地電磁正演模擬,為了驗證結(jié)果,分別建立了一個大家常用的三維梯形山峰模型和山谷模型。計算了頻率f=2Hz時XY模式與YX模式下的視電阻率和阻抗相位平面圖,以及山峰地形上不同測線的響應(yīng)曲線,并對結(jié)果進行了分析;為了研究不同頻率下地形對大地電磁響應(yīng)的影響規(guī)律,對2Hz、50Hz和2000Hz三個頻點在兩種不同模式下進行數(shù)值模擬,并對其響應(yīng)規(guī)律進行了討論;為了確定是否可以用二維地形的模擬結(jié)果去近似三維地形,將沿著X剖面(Y=0)三維模擬結(jié)果與二維模擬結(jié)果進行了比較,在這里它們的橫截面是一樣的。將模擬結(jié)果進行對比,得到如下結(jié)論:XY模式下三維地形的響應(yīng)結(jié)果與TM模式下二維地形的響應(yīng)結(jié)果比較接近,YX模式下三維地形的響應(yīng)結(jié)果與TE模式下二維地形的響應(yīng)結(jié)果的差別較大。
[Abstract]:Magnetotelluric sounding is a frequency-domain electromagnetic sounding method. It takes the natural alternating electromagnetic field as the source for deep geological research, so the field construction is convenient and the cost can be reduced. In addition, this method has the advantages of not being shielded by high resistivity layer and high resolution of low resistivity formation, so it has been widely used in geophysical exploration. The forward numerical simulation of magnetotelluric can give us a good understanding of the law of the end of various electric media and provide the foundation and guarantee for the final inversion. In this paper, the forward numerical simulation of 3D magnetotelluric with terrain is realized by using vector finite element method. Firstly, according to the boundary conditions of 3D magnetotelluric, the variational equation equivalent to 3D magnetotelluric boundary value problem is derived by using weighted residual method from MAXWELL equations. The regular hexahedron element is used to divide the calculation area, and the vector interpolation function is used for element analysis. The interpolation function is simple and accurate, which satisfies the discontinuity of the normal direction field. The problem of "pseudo solution" in the finite element method can be avoided, and the divergence correction can be avoided. The resulting element stiffness matrix is large and sparse. Considering the problem of computing speed and memory occupation, the one-dimensional nonzero element is used in the total coefficient matrix. The incomplete Cholesky decomposition and the double complex conjugate gradient algorithm are combined to solve the equation, which ensures the calculation speed and converges efficiently and stably. Secondly, the uniform half-space model and the layered model are established, and compared with the analytical solution, the calculation accuracy of resistivity and phase is very high, which verifies the correctness of the algorithm and the program. In the process of verification, it is found that the accuracy is not high and the calculation speed is slow when the traditional 3D magnetotelluric boundary condition is used. In addition, it is recognized that mesh generation has a great influence on the calculation results, and some conclusions about mesh generation are obtained. On the basis of this, the 3D anomalous body model (including high resistivity and low resistivity) is given, and the abnormal responses of apparent resistivity and impedance phase of anomalous volume model under XY mode and YX mode are calculated, respectively. Their response characteristics are analyzed. Finally, the 3D magnetotelluric forward modeling with terrain is realized by using vector finite element method. In order to verify the results, a commonly used three-dimensional trapezoidal peak model and a valley model are established respectively. The apparent resistivity and impedance phase plane maps of XY mode and YX mode at frequency f=2Hz are calculated, and the response curves of different surveying lines on the mountain topography are also calculated, and the results are analyzed. In order to study the influence of terrain on magnetotelluric response at different frequencies, the two different modes of 2Hz / 50Hz and 2000Hz are numerically simulated, and the response law is discussed. In order to determine whether it is possible to approximate the three-dimensional terrain with the simulation results of two-dimensional terrain, the results of three-dimensional simulation along the X section are compared with the results of two-dimensional simulation, where their cross sections are the same. Compare the simulation results, The results obtained are as follows: the response results of 3D terrain in XY model and 2D terrain in TM mode are close to those of 3D terrain in TM-mode and the response result of 3D terrain in te mode is different from that in TM-mode.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：P631.325

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