基于区域分解和矢量有限元的可控源电磁法三维并行快速正演

The algorithm efficiency and calculation accuracy of traditional methods of secondary field calculation and global encryption strategy is barely satisfactory in processing the undulating terrain, and the high gradient and singularity of local artificial sources that are found prevalent in 3D CSEM (Controlled Source Electromagnetic) FEM (Finite Element Modeling) direct modeling. Therefore, by introducing the algorithm on domain decomposition vector finite element, this project divides the large and complex CSEM 3D forward problems into several simple subfields, applies Robin transmission conditions on the interface between adjacent domains and hence solves the problems respectively according to the properties of the solving objects in the subfields. Such vector finite element algorithm of high parallelism can effectively solve “pseudo solution” in the node finite elements and provide a valid path to the MPI parallel solution of super-scale 3D CSEM. When being combined with the complete thinking which includes the adaptive strategy of unstructured grid and posteriori error estimation and the object-oriented adaptive refinement strategy, it can improve the whole accuracy and efficiency of CSEM 3D forward modeling and acquire the electromagnetic value at the measuring point position with the smallest amount of calculation, so as to realize the goal solving the complex 3D CSEM problem with the least amount of computation in a fast and accurate way.

针对CSEM三维有限元正演中普遍存在的局部人工源的奇异性、起伏地形难以处理、计算量庞大等问题，传统的基于结构化网格、全局加密策略的算法效率、计算精度差强人意。为此，申请者拟引入区域分解矢量有限元算法，该算法将复杂、大型的CSEM三维正问题拆分为若干个简单子域，在相邻区域的分界面上施加Robin传输条件，且根据各子域上求解目标体局部特性分别求解。这种具有高度并行度的矢量有限元算法，有效解决节点有限元存在的“伪解”问题，为超大规模三维CSEM的MPI并行求解提供了一条有效的途径。再结合非结构化网格、后验误差估计的自适应策略、面向目标体自适应加密等整套思路，整体提升CSEM三维正演精度与效率，力求以最小计算量求解测点位置上电磁场值，达到快速、高精度求解复杂三维CSEM问题的目的。

针对当前CSEM三维正演普遍存在起伏地形难以处理，计算量大等问题，本项目开展了基于区域分解和矢量有限元的可控源电磁法三维快速正演研究。主要研究了基于区域分解三维可控源电磁法矢量有限元算法，基于MPI区域分解法的并行处理技术，基于数值解残差的后验误差估计分析。引入非重叠型的区域分解法，在相邻区域的分界面上施加Robin传输条件，开展了基于区域分解矢量有限元计算方法研究。采用矢量有限元与非结构化网格结合的方式，推导出四面体单元上的有限元表达式，测试了不同线性方程组求解算法对区域分解算法的适应性，总结出最优化的求解算法，借助可靠的后验误差估计算法，获得可控源电磁法有限元的单元误差分布，最终构建出一套快速、高精度、适用于带地形的复杂各向异性可控源电磁法模型的三维正演的方法体系。此项目成果能应用于实际生产，特别是在提高工作效率方面，前景诱人。