多孔介质弹性问题的单元微分法

Element differential method for poroelastic problems

考虑Biot固结理论,建立了求解多孔介质中流体渗透与固体力学耦合问题的数值模型,并通过新型强形式有限单元法(单元微分法)对该问题进行分析计算。相比于弱形式算法,单元微分法能够通过直接对控制方程进行离散,不需要数值积分计算,因此该算法在对多场耦合问题进行求解时拥有较为简单的离散格式,且其计算系数矩阵时表现出极高的效率。该数值算法使用的是有限元中的拉格朗日单元,与强形式的无网格法相比,能够获得相对更精确且更稳定的计算结果。通过引入单元微分法以及隐式时间迭代格式,能够快速地计算出多孔介质耦合方程中各时间步的位移及孔压值。选取两个经典的数值模型,一个是一维Terzaghi柱模型,另一个是二维饱和土带模型。针对这两个问题,分别通过与解析解和有限元法结果相对比,验证了该算法的精度和稳定性。

A numerical model for solving the coupling problem of fluid flow and solid mechanics in porous media is established based on the Biot's consolidation theory,and the numerical analysis and calculation are carried out by using a new strong-form finite element method(element differential method,EDM).By comparing with the weak-form methods,the control equation for poroelastic problems can be discretized directly by the element differential method without any numerical integration calculation.Therefore,the method has a relatively simple discrete format when solving the multi-field coupling problem,and it shows high efficiency when calculating the coefficient matrix.The numerical method uses the Lagrange element in the finite element method,which can obtain relatively accurate and stable results compared with the strong-form meshless method.By introducing the element differential method and the implicit time iteration scheme,the displacement and pore pressure of each time step in the porous media can be calculated directly.Two classical numerical models are selected,one is the one-dimensional Terzaghi column model,and the other is the two-dimensional saturated soil zone model.For these two problems,the accuracy and stability of the proposed are verified by comparing with the results of analytical solution and finite element method.