基于动力响应的非饱和多孔介质参数反演的强格式边界型离散方法研究

Parameter inversion of porous media is of great significance to geotechnical engineering, ocean engineering, petroleum engineering and the other fields. The frequently-used saturated porous media can not describe the realistic physical and mechanical properties and thus affects the identified results. In addition, for the dynamic analysis of unsaturated poroelastic media, the traditional numerical methods often encounter the issues of selecting the artificial boundary for an infinite domain, singular or nearly singular integration, and expensive computation resulting from dense-matrix. To overcome these issues, this proposal is aimed at developing a boundary collocation method, the singular boundary method (SBM), which is meshless and easy-to-program. We will investigate the new approach to isolate the singularity of the fundamental solution of the considered problem, which avoids the numerical computation of singular integrals. Then, this proposal attempts to apply the SBM to obtaining the dynamic response of unsaturated porous media. Subsequently, the adaptive cross approximation (ACA) is employed to deal with the dense matrix generated by the SBM. And then the ACA-SBM is utilized to obtain the dynamic response of unsaturated porous media with large scale domain or complex shaped boundary. Reasonable optimal methods for the parameter inversion are investigated, which are used to ensure the stability and convergence of the inversion. The purpose of this proposal is to develop an inverse scheme based on the ACA-SBM and a reasonable optimal method, which is high-performance and stable, for the parameter inversion of unsaturated porous media, and thus provide an alternative numerical technique for many practical engineering problems.

非饱和多相多孔介质参数反演在于岩石工程和地质勘探等许多领域有重要的应用。有限元和边界元方法模拟非饱和孔隙介质动力响应问题时会遇到无限域的人工边界选取、奇异与近奇异积分计算以及稠密矩阵计算量大等难题。基于申请者已有的工作基础，本项目拟发展非饱和多孔介质参数反演的无网格、易编程、仅需边界配点的边界奇异法。关键科学问题是研究消除非饱和孔隙介质波动方程基本解源点奇异性的新方法，避免奇异积分的数值计算，并用于模拟非饱和孔隙介质的动力响应问题；结合自适应交叉近似算法，发展奇异边界法稠密矩阵方程的快速求解技术，实现大尺度区域、复杂边界形状的非饱和孔隙介质动力响应问题的快速求解；研究合适的非线性反演计算方法，确保反演计算的收敛性和稳定性。目标是为反演大尺度复杂几何域情况下的非饱和孔隙介质参数，提出一个新的高效数值模拟方案。

非饱和多孔介质的动力学行为与岩石工程和地质勘探等诸多领域相关。本项目的研究内容主要集中在以下方面：1）建立模拟非饱和孔隙介质动力响应问题的奇异边界法计算模型，即基于傅里叶变换，将时域动力学问题转换为频域动力学问题，通过Hörmander方法推导基本解，然后借助窗函数法和奇异边界法求解频域问题，再通过傅里叶逆变换技术得到时域问题的数值解；2）基于零场方程，提出了一种适用于Neumann边界的源点强度因子的解析公式，减少奇异边界法求解源点强度因子的复杂度，有助于非饱和孔隙介质中波传播问题奇异边法的便捷实施；3）考虑边界上可能存在的边界接触特性，研究了基于投影算法的奇异边界法，并将之用于求解Signorini问题和无摩擦接触问题，在土体浸润线问题和岩石接触问题中均能获得应用；4）研究了局域型的奇异边界法，用于加速大规模和多尺度非饱和多孔介质动力学正问题的计算效率。基于以上工作，本项目已发表SCI论文4篇，协助培养博士研究生一名。

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