饱和土动力学有限元分析的改进稳定分步算法

A modified stabilized fractional-step algorithmfor finite element analysis in saturated soil dynamics

基于Biot理论,控制饱和变形多孔介质中固相位移u和孔隙压力pw演变的场方程的空间半离散化导致u-pw型混合有限元方程。在固体颗粒和孔隙液体不可压缩以及零渗透性情况下,基本未知量u,pw的近似插值函数必须满足Babuska-Brezzi条件或者与之等价的Zienkiewicz和Taylor分片试验。采用相同低阶u-pw插值的有限元(如线性三角形单元和双线性四边形单元)不能满足B-B条件。分步算法作为一种稳定技术的引入可以绕开B-B条件,但现有分步算法在瞬态问题中仍存在虚假数值振荡和不稳定现象。本文在现有分步算法的基础上引入迭代过程,有效地缓解和克服了数值振荡现象,使低阶u-pw单元得以正常应用。应用双线性四节点u-pw单元的数值结果表明了所提出的包含迭代过程的改进分步算法的有效性。

Based on the Biot theory, the semi-discretization procedure of the field equations governing evolutions of the displacements u of the solid skeleton and the pore water pressure p_w in deforming saturated porous media results in the u-p_w mixed finite element formulations. In the limit of in compressibility of water and soil grains and zero permeability, the interpolation approximations for the primary variables u and p_w have to fulfill either the Babuska-Brezzi condition or the patch test proposed by Zienkiewicz and Taylor. It is known that the use of elements with equal low order interpolation for u and p_w fails to satisfy the above requirements. The fractional step algorithm introduced as a stabilization technique can circumvent the restrictions to the interpolation functions imposed by the Babuska-Brezzi condition. However, numerical results given by the existing fractional step algorithm show that the spurious oscillation and instability phenomenon in time step integration process still may occur for dynamic problems. A modified version of fractional step algorithm is proposed in the present paper by the introduction of a simple iteration procedure into the existing fractional step algorithm. The numerical difficulties mentioned above can be then effectively alleviated and even eliminated to allow the use of equal low order elements with success for dynamic problems. The numerical results demonstrate the effectiveness and good performance of the proposed modified version of fractional step algorithm.