Biot固结方程的有限元方法及快速算法研究

Biot's consolidation equations are one of the most important project in soil mechanics. They describe the consolidation processes of a porous elastic medium which is saturated by a certain fluid when pressed by some external forces. Biot's consolidation equations have a wide range of applications such as in architecture, environmental and biomechanics fields. However, it is very difficult to get the classical solutions of the Biot's consolidation equations except for some special initial and boundary conditions. Therefore, it is of great significance to study the numerical solutions of Biot's consolidation equations. The goal of this work is to develop stable finite element discretizations for Biot's consolidation equations and fast solvers for the corresponding algebraic system of the equations. First, by applying the finite element exterior calculus framework, we will develop stable finite element discretizations for Biot's consolidation equations. Second, we will construct uniform convergent fast solvers that independent with the mesh and time step sizes for the algebraic system of the equations by studying the matrix structures, mainly including multigrid algorithms and preconditioned Krylov subspace iterative methods. Finally, we would study the finite element discretization methods and the fast solvers for the system when the operator grad-div dominates because of the value of Poisson ratio.

Biot固结方程是土力学的重要课题之一。它描述了含流体的多孔弹性介质在外部荷载作用下的固结过程。该模型在建筑，环境及生物力学等领域具有非常重要且广泛的应用价值。然而，除特定初边值条件下，一般很难求出Biot固结方程的解析解。因此，研究Biot固结方程的数值解具有非常重要的意义。本项目旨在研究Biot固结方程稳定的有限元离散格式及相应代数方程组的快速算法。研究内容包括：(1) 通过利用有限元外微分理论框架，构造Biot固结方程稳定的有限元离散格式；(2) 通过分析相应离散系统的系数矩阵结构，设计该矩阵与离散参数无关的一致收敛的快速算法，主要包括多重网格算法和预处理Krylov子空间迭代算法；(3) 研究当Poisson比取值导致方程中算子grad-div占优时的有限元离散方法和快速算法。

Biot固结方程是土力学的重要课题之一。它描述了含流体的多孔弹性介质在外部荷载作用下的固结过程。该模型在建筑，环境及生物力学等领域具有非常重要且广泛的应用价值。除特定的初边值条件下，一般很难求出Biot固结方程的解析解。因此，研究Biot固结方程的数值解具有非常重要的意义。本项目主要研究Biot固结方程稳定的有限元离散格式及相应代数方程组的快速算法。研究成果主要包括三部分，第一通过利用有限元外微分理论框架，构造了Biot固结方程在拟一致剖分网格下的有限元离散格式。 第二，通过分析相应离散系统的系数矩阵结构，设计了该离散系统与空间及时间离散参数无关的一致收敛的快速求解算法，包括多重网格算法和预处理Krylov子空间迭代算法。 第三， 研究了当Poisson比取值导致方程中算子grad-div占优时的有限元离散方法和快速算法等。