树网格离散优化三维瞬变电磁有限体积正演算法

3D finite-volume forward modeling of transient electromagnetic using octree meshes

三维瞬变电磁正演算法作为研究处理与解释方法的重要基础,如何加速计算过程,减少计算机内存消耗尤为重要.为此,本文采用有限体积算法在八叉树(octree)网格上对时间域Maxwell方程组进行空间离散,相比于交错六面体网格,octree网格在局部区域网格细度相同的条件下可以显著减少细化区域外的网格数量,对复杂几何体边界的模拟更加灵活,而相比于非结构四面体网格,octree网格单元位置排列更加规律.通过octree网格离散三维正演模型使计算网格规模显著降低,减少了待求解方程的未知数,降低了物理内存消耗.空间离散后,瞬变电磁正演响应可以表示为关于初始磁场的矩阵指数函数,采用位移逆Krylov子空间模型降阶算法实现瞬变电磁场的求解,只需对系数矩阵进行一次矩阵分解和多次回代即可获得一系列时间序列的瞬变电磁场正演结果.本文算法在空间离散和方程求解两方面优化三维瞬变电磁正演过程,数值算例结果验证了本文算法的精度和高效性.

An efficient and stable 3D transient electromagnetic(TEM)simulation algorithm is the theoretical basis for processing and interpretation methods.Accelerating the calculation process and reduce computer memory consumption is particularly important.In this paper,we use the mimetic finite volume algorithm to discretize the time-domain Maxwell's equations on octree meshes.Compared with the staggered mesh,the octree mesh can significantly reduce the number of elements at the boundary with same grid fineness in the core area,and the number of grids at the boundary is more flexible for the simulation of complex geometric boundaries.Compared with unstructured tetrahedral grids,the location of octree cells is more regular.Thus,using octree meshes can reduce the unknown of equations to be solved and the memory consumption.After discretizing the time-domain Maxwell's equations by the mimetic finite volume method,the transient electromagnetic response can be expressed as a matrix exponential function with respect to the initial magnetic field.Using the shift-and-invert(SAI)Krylov subspace model reduction algorithm,a series of TEM modeling results can be obtained by performing only one matrix decomposition and hundreds of reverse substitutions on the displacement coefficient matrix.The algorithm in this paper optimizes the 3D TEM modeling process in both spatial discretization and equation solving.The numerical experiments demonstrated that the method is accurate and efficient.