多层非饱和孔隙介质动力响应的奇异边界法研究

Numerical simulation for the dynamic response of unsaturated porous media is of great significance to geotechnical engineering, ocean engineering, petroleum engineering and the other fields. However, for the dynamic analysis of multilayered unsaturated poroelastic media, the traditional numerical methods often encounter the issues of selecting the artificial boundary for 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 at origin, which avoids the numerical computation of singular integrals. Then, this proposal makes the first attempt to apply the SBM to obtaining the dynamic response of unsaturated porous media. Subsequently, the fast multipole method (FMM) is employed to deal with the dense matrix generated by the SBM. And then the FMM-SBM is utilized to obtain the dynamic response of unsaturated porous media with large scale domain or complex shaped boundary. Preconditioning techniques for the matrix are investigated, which are used to accelerate the iterative rate of coupled matrix generated from the multilayered unsaturated poroelastic problem. The purpose of this proposal is to develop a fast multipole SBM, which is high-performance and stable, for the dynamic analysis of multilayered unsaturated porous media, and thus provide an alternative numerical technique for many practical engineering problems.

非饱和孔隙介质动力响应的数值模拟研究对岩土工程、海洋工程和石油工程等众多领域具有重要的意义。传统的数值算法在模拟多层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。针对这些问题，本项目拟发展无网格、易编程、仅需边界配点的边界奇异法，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，并首次将奇异边界法用于模拟非饱和孔隙介质的动力响应问题；结合快速多极子算法，发展奇异边界法稠密矩阵方程的快速求解技术，实现大尺度区域或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；研究矩阵预处理技术，加快多层非饱和孔隙介质问题中生成耦合矩阵迭代收敛速度。目标是发展高效、稳定、能有效模拟多层非饱和孔隙介质动力响应的快速多极奇异边界法，为众多实际工程领域提供一种新型的数值算法。

非饱和孔隙介质中弹性波传播的数值模拟研究对石油工程、海洋工程以及地球物理等领域具有重要的意义。在本项目中，我们首次将一种前处理比较简单、无需处理奇异和近奇异积分的边界型无网格配点法，即奇异边界法，用于计算非饱和孔隙介质动力响应问题。. 针对非饱和孔隙介质中的瞬态波传播问题，首先将时域控制方程转化成频域方程，然后采用波分解法推导了频域控制方程的基本解，并分析了基本解的奇异性。随后借助于已有的Laplace方程和弹性静力学问题的源点强度因子，推导了非饱和孔隙介质频域动力响应问题的源点强度因子，从而建立了奇异边界法的计算模型。最后采用改进的快速傅里叶变换法，即指数窗口法，将频域结果转换成时域响应。对于区域尺度较大、边界形状较复杂或频率较高等计算量较大的问题，将自适应交叉近似算法用于压缩奇异边界法所生成的系数矩阵，从而有效地降低了奇异边界法的计算量和存储量，最终在普通的计算机上实现了数十万自由度的非饱和孔隙介质波传播的数值模拟。对于多层介质问题，研究了矩阵的性态，并成功地将快速算法用于加速奇异边界法。. 基于以上的研究内容，团队成员编写了相关的程序，模拟了不同动力荷载下各种计算区域中非饱和孔隙介质的动力响应，并发表了9篇研究论文。该项目的研究为很多涉及非饱和孔隙介质的工程应用提供了一种新的数值模拟工具。