多源条件下复杂模型直流电阻率无单元伽略金法三维快速正演

Due to the meshes based on the nodal link in formation is indispensable, the traditional numerical simulation methods are subject to certain constraints. The element-free Galerkin method (EFGM) is not dependent on the meshes based on the nodal link information, results in high flexibility and high precision. Which makes it became an emerging forward modeling method in the field of geophysics. The research of EFGM had made great progress. However, its forward process involves huge computational costs. How to cut the computational costs from the key technologies involved in EFGM, under the invariableness of model and accuracy, there is no effective solution so far. Based on the previous research work, this project focuses on the three-dimensional fast forward modeling of DC resistivity method forward modeling using EFGM. An adaptive background cell subdivision method, a hierarchic local support domain construction method and extended computational domain by coupling finite element method is proposed to adopt. and combining the CSR compression storage technology and the efficient Pardiso direct solver in multi-sources case, to cut the computational costs. The algorithm is further used to study the three-dimensional DC resistivity response mechanism of complex models, and the influence of different refined modeling degrees on the simulation results. The research of this project is significative greatly for promoting the practical application of high flexibility and high precision three-dimensional DC resistivity element free Galerkin method, and provides theoretical guidance for the practical three-dimensional DC resistivity exploration.

传统的数值模拟方法依赖于节点连接信息的单元而受到一定的约束。无单元伽略金法无需节点连接信息的单元剖分，具有高灵活性和高精度等特点，已成为地球物理领域的新兴正演模拟方法。三维无单元伽略金法正演过程涉及大型计算量，如何在模型和精度不变前提下，从无单元伽略金法涉及的关键技术中大幅减少正演计算量，尚无有效的解决方案。在前期研究工作基础上，本项目围绕直流电阻率无单元伽略金法三维快速正演问题开展研究。采用自适应背景积分单元方法、分级局部支持域构造方法及耦合有限元扩边减少正演计算量，多源条件下结合CSR压缩存储技术和高效Pardiso直接求解器，达到提高计算效率目的。进一步，利用该算法研究复杂地电模型三维直流电率响应机制、不同精细化建模程度对结果的影响规律。本项目研究对促进高灵活性、高精度三维直流电阻率无单元伽略金法实际应用有重要意义，为实际三维直流电阻率勘探提供理论指导。

传统数值模拟方法，例如有限单元法和有限差分法，依赖于节点连接信息的单元而受到一定的约束，特别是采用非结构化单元时需要借助网格剖分器。本项目利用加略金分析离散弱式积分方程，实现无需基于节点连接信息单元剖分的直流电阻率法无单元加略金法正演模拟，具有高灵活性和高精度等特点。本项目重点针对三维无单元伽略金法正演过程涉及大型计算量导致计算效率低的问题，在方法本身涉及的关键技术中提出了三个解决方案。方案1为自适应背景积分单元方法，实现背景积分单元根据节点分布自适应合理分布，总体上减少背景积分单元数量，减少正演计算量；方案2为分级局部支持域构造方法，实现积分点支持域能在层次分明的局部域内进行构造，避免每一个积分点均在全局域内搜索支持域，提高支持域构造效率；方案3为耦合有限单元法扩边及直接求解法求解方程，实现减少无单元法计算区域，并在多源情况下利用直接求解器实现一次系数矩阵分解多次回代高效求解方程组。结果表明该三方案能在保证高模拟精度的情况下，明显提高正演计算效率，对任意复杂的地电模型，可实现多源条件下高灵活性、高精度三维直流电阻率法无单元伽略金法快速正演模拟。此外，利用无单元法形函数处理直流电阻率法有限单元法中的边界计算，进一步减少了计算域要求，使得建模更方便；利用单位分解积分，实现了直流电阻率法无单元局部弱式法，使得不再需要全局域剖分的背景积分单元，使得算法更加灵活。该项目将为实际直流电阻率勘探提供更加精细化正演模拟理论。