Biot固结有限元方程组的病态规律分析

针对Biot固结有限元方程组的病态问题，采用正交试验和量纲分析法探讨病态性的变化规律及其影响因素。主要考虑单元平均尺寸、时间步长、压缩性和渗透性等4个因素的影响。分别提出2组相互独立的相似准则，在此基础上得到2个新的无量纲量，并将其用于分析Biot固结有限元方程组系数矩阵的条件数的变化规律。结合算例进行参数分析结果表明，对病态程度的敏感性由大到小依次为单元平均尺寸、时间步长、压缩性和渗透性；单元平均尺寸太小会使病态加剧；随着2个无量纲量的增大，病态程度呈现单调增加的趋势。其结果为深化对Biot固结数值分析病态问题的认识提供有益参考。

Study on ill-conditioned problems of Biot's consolidation finite element equations

Aimed at the ill-conditioned problem of Biot’s consolidation finite element method (BCFEM), both orthogonal test and dimensional analysis method are used to study the ill-conditioning regularity and its influencing factors. The influencing factors such as the average size of finite element, the length of timestep, the compressibility of soil and the permeability of soil are taken into account. Two groups of independent similarity criteria are proposed, on the basis of which, two novel dimensionless variables are obtained to analyze the variation of the condition number of BCFEM coefficient matrix. Through the parametric study of an illustrative case, the results show that the average size of finite element, the length of timestep, the compressibility of soil and the permeability of soil are in the order of priority of sensitivity. The smaller the averaged size of finite element, the worse the ill-conditioning situation of BCFEM. With the increase of the presented dimensionless variables, the ill-conditioning number shows a tendency to increasing, etc. The conclusions are helpful to study the ill-conditioned problem of BCFEM.