基于非结构外推多重网格法的带地形二维大地电磁各向异性有限元正演  

Two-dimensional magnetotelluric anisotropic finite element modeling with undulating terrains based on unstructured extrapolation multigrid method

作　　者：潘克家[1] 韩旭 王鹏德 王晋轩 肖晓[2] 张林成 PAN Kejia;HAN Xu;WANG Pengde;WANG Jinxuan;XIAO Xiao;ZHANG Lincheng(School of Mathematics and Statistics,Central South University,Changsha,Hunan 410083,China;School of Geosciences and Info-Physics,Central South University,Changsha,Hunan 410083,China;School of Information and Electronic Engineering,Hunan City University,Yiyang,Hunan 413002,China)

机构地区：[1]中南大学数学与统计学院,湖南长沙410083 [2]中南大学地球科学与信息物理学院,湖南长沙410083 [3]湖南城市学院信息与电子工程学院,湖南益阳413002

出　　处：《石油地球物理勘探》2023年第2期454-468,共15页Oil Geophysical Prospecting

基　　金：国家自然科学基金面上项目“极端地质条件下三维大地电磁正反演关键技术研究”(42274101);“近场源频率域电磁测深机理与方法研究”(42074087);湖南省自然科学基金杰出青年项目“外推瀑布式多网格法及其在三维地电磁场计算中的应用”(2018JJ1042)、“基于非结构化有限元—无限元耦合的可控源电磁法三维正演研究”(2021JJ40024);中南大学研究生自主探索创新项目“基于四面体网格的外推多网格法及其在三维起伏地形大地电磁正演中的应用”(1053320213258)联合资助。

摘　　要：基于非结构化网格的大地电磁各向异性有限元正演模拟若采用经典迭代法,收敛速度慢,不适用于复杂地电模型的正演计算。多重网格法作为经典迭代法的改进算法,是求解椭圆方程离散线性系统的一种有效算法。然而,经典多重网格法依赖于嵌套的正交网格,无法直接应用于非结构化网格问题的求解。笔者提出一种基于半结构化网格,即通过对一个初始的非结构网格进行逐层二分加密,并利用外推瀑布式多重网格算法(EXCMG)快速求解各向异性介质中二维大地电磁有限元正演大规模复线性系统。EXCMG算法运用三角形网格上的外推和高次插值技术,构造新的多网格延拓算子,通过两层粗网格上的数值解构造下层密网上有限元解的高阶逼近,作为多网格磨光算子BiCGStab的迭代初值,加速迭代收敛。对国际标准测试模型(COMMEMI-2D1和COMMEMI-2D4)运用该方法进行测试,得到的视电阻率和相位相对误差均在1%以内,求解时间较BiCGStab、聚合型代数多网格法(AGMG)大幅缩短。EXCMG算法具有较好的拓展性和适应性,可处理复杂地电模型、任意起伏地形和各向异性问题。Magnetotelluric anisotropic finite element forward modeling based on unstructured grids has a slow convergence rate if the classical iterative method is used,and it is not suitable for the forward calculation of complex geoelectric models.The multigrid method,as an improvement to the classical iterative method,is an effective algorithm for solving discrete linear systems of elliptic equations.However,the classical multigrid method relies on nested orthogonal grids and cannot be directly applied to solve unstructured grid problems.Therefore,the authors propose the concept of semi-structured grids.In other words,an initial unstructured grid receives binary encryption level by level,and an extrapolation cascadic multigrid method(EXCMG)is used to quickly solve the large-scale complex linear system derived from two-dimensional magnetotelluric finite element forward modeling in anisotropic media.The EXCMG uses extrapolation and high-order interpolation techniques on a triangular grid to construct a new multigrid prolongation operator and construct the high-order approximation to the finite element solution on refined grids with numerical solutions on two coarse meshes,which is then used as the iterative initial value of the multigrid smoothing operator,namely,BiCGStab,so as to accelerate its convergence.The method is tested with international standard test models(COMMEMI-2D1 and COMMEMI-2D4),and the relative differences of the apparent resistivity and phase are within 1%.In addition,the solution time of the method is significantly reduced compared with that of BiCGStab and aggregation algebraic multigrid method(AGMG).The EXCMG shows excellent scalability and adaptability and can handle complex geoelectric models,arbitrary undulating terrains,and anisotropic problems.

关 键 词：大地电磁正演 有限元 非结构化网格 外推瀑布式多重网格算法 起伏地形 

分 类 号：P631[天文地球—地质矿产勘探]