Biot固结方程的有限元方法及快速算法研究

项目名称： Biot固结方程的有限元方法及快速算法研究

项目编号： No.11501473

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 陈罗平

作者单位： 西南交通大学

项目金额： 18万元

中文摘要： 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。

中文关键词： Biot固结方程；有限元方法；多重网格算法；预处理子；Krylov子空间迭代法

英文摘要： 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.

英文关键词： Biot's consolidation equations;finite element methods;multigrid method;preconditioner;Krylov subspace iterative method