SOME PRECONDITIONING TECHNIQUES FOR SOLVING BIOT MODEL

求解Biot模型的若干预处理技术

郑植

11930645

硕士

数学

李景治

2021-05-17

2021-05-17

南方科技大学

深圳

There are many solid materials or structures composed of pores in our life. We usu- ally use “poroelasticity” to describe the interaction between fluid flow and solid defor- mation in porous media. We mainly introduce a generalized linear poroelastic model, the famous Biot model in mechanics, and construct iterative methods to solve it. We in- troduce in detail the composition and structure of each equation in the Biot model and list the physical meaning of each parameter. In order to overcome the elastic locking phenomenon encountered in the classical solution process, this thesis introduces an inter- mediate variable. The new form of the Biot model can be regarded as a combination of a generalized Stokes equation and a reaction-diffusion equation. We deduce the weak for- mulation of the Biot equations and introduce the finite element spaces to discretize them by finite element method. Then we obtain the corresponding discrete bilinear form and reformulate the original Biot problem into a 3-by-3 block tridiagonal matrix problem.Note that the matrix problem is similar to the double saddle point problem, we con- sider applying the Krylov subspace method to solve such a problem. First, based on matrix splitting, we use the alternating direction iteration method to design an iterative al- gorithm and relative preconditioner for the Biot model after variable substitution. When the block matrix satisfies certain conditions, the unconditional convergence of the itera- tive algorithm can be proved. In order to further improve the computational efficiency, we propose two relaxed preconditioners which have better spectral distribution and nu- merical performance. We analyze the spectrum of the preconditioned matrix and find the special property of relaxed one’s eigenvalue distribution. We also introduce two pre- conditioners based on the LU decomposition, namely block diagonal preconditioner and block triangular preconditioner. Two sub-blocks of these preconditioners are expressed as pressure and total pressure Schur complements. However, for this system, it is ex- tremely time-consuming for the computer to directly inverse the high-dimensional Schur complements. Therefore, we approximate the two Schur complements in an analytic way by using the Fourier analysis method to improve the efficiency of numerical computa- tion. Finally, through numerical experiments, we apply the preconditioners given in the thesis to accelerate the GMRES method under left preconditioning and give the numerical performance of different preconditioners with different parameters. All preconditioners can accelerate the residual convergence and improve the eigenvalue distribution of the coefficient matrix, but the effects vary considerably. The Fourier approximation form of the block triangular preconditioner performs best. We test the robustness of the Poisson ratio 𝜈 and permeability coefficient 𝜅 of the Biot model. From numerical results, we also verify the theoretical prediction of the theorems and inferences in the thesis.

生活中有很多由孔隙组成的固体材料或结构，通常我们会用“多孔弹性”来描述多孔介质中的流体流动与固体变形之间的相互作用。本文考虑研究一种广义的线性孔弹性模型：力学中著名的Biot模型，并构造迭代算法求解该问题。我们详细地介绍了Biot模型中各个方程的组成结构以及其中参数的实际物理意义。为了解决常规求解过程中遇到的弹性锁紧现象，本文选择引入一个中间变量，Biot模型的新表现形式可以被看作是一个广义 Stokes问题与一个反应扩散问题的组合。我们给出该方程组的弱形式，并引入有限元空间对其进行有限元离散，得到相应的离散双线性形式，将原Biot问题重新构造为3$\times$3分块三对角的矩阵问题。注意到该矩阵问题与双鞍点问题相似，我们考虑使用Krylov子空间方法来求解该问题。首先，基于矩阵分裂，我们使用交替方向迭代法为变量替代后的Biot模型设计了迭代方法和预处理器。当分块矩阵满足一定条件时，可证明迭代算法的无条件收敛。为了进一步提高运算效率，我们给出了两种松弛格式的预处理器，使其具有更好的谱分布和数值性能。我们证明了松弛预处理矩阵具体的特征值分布。我们还介绍了两个基于LU分解的预处理器，分别是分块对角预处理器和分块下三角预处理器，这两个预处理器的两个子块是由压力与总压力的Schur补表示的。但是对于该系统，计算机直接对高维的Schur补求逆是非常耗时的，因此我们通过使用傅立叶分析方法以解析的方式对Schur补进行了近似处理，以此提升数值计算效率。最后，通过多次数值试验，我们将文中给出的几种预处理器应用于左加速GMRES迭代方法，并给出了不同参数下不同预处理器的数值表现。所有预处理器均能加速残量收敛，改善系数矩阵的谱分布，但效果不相同，分块下三角预处理器的傅立叶逼近形式综合表现最佳。我们进一步测试了Biot模型的泊松比和渗透系数的鲁棒性。通过数值试验，我们还验证了文中的定理和推论的正确性。

Biot model Iterative method Preconditioner Convergence analysis Spectrum

Biot模型 迭代方法 预处理器 收敛性分析 谱分布

英语

独立培养

南方科技大学

Zheng Z. SOME PRECONDITIONING TECHNIQUES FOR SOLVING BIOT MODEL[D]. 深圳. 南方科技大学,2021.