We describe the behavior of a deformable porous material by means of a poro-hyperelastic model that has been previously proposed in Chapelle and Moireau (2014) under general assumptions for mass and momentum balance and isothermal conditions for a two-component mixture of fluid and solid phases. In particular, we address here a linearized version of the model, based on the assumption of small displacements. We consider the mathematical analysis and the numerical approximation of the problem. More precisely, we carry out firstly the well-posedness analysis of the model. Then, we propose a numerical discretization scheme based on finite differences in time and finite elements for the spatial approximation; stability and numerical error estimates are proved.

Particular attention is dedicated to the study of the saddle-point structure of the problem, that turns out to be interesting because velocities of the fluid phase and of the solid phase are combined into a single quasi-incompressibility constraint. Our analysis provides guidelines to select the componentwise polynomial degree of approximation of fluid velocity, solid displacement and pressure, to obtain a stable and robust discretization based on Taylor–Hood type finite element spaces. Interestingly, we show how this choice depends on the porosity of the mixture, i.e. the volume fraction of the fluid phase.

我们通过孔隙-超弹性模型描述了可变形多孔材料的行为，该模型先前在Chapelle和Moireau（2014）中提出，是在流体和固体两组分混合物的质量和动量平衡以及等温条件的一般假设下阶段。特别地，我们在此基于小位移的假设来处理模型的线性化版本。我们考虑问题的数学分析和数值近似。更准确地说，我们首先进行模型的适定性分析。然后，我们提出了一种基于时间有限差分和空间近似的数字离散方案。证明了稳定性和数值误差估计。

特别要注意研究该问题的鞍点结构，这很有趣，因为流体相和固相的速度被组合为一个准不可压缩约束。我们的分析为选择流体速度，固体位移和压力的近似多项式多项式提供了指导，以基于Taylor-Hood型有限元空间获得稳定且鲁棒的离散化。有趣的是，我们展示了这种选择如何取决于混合物的孔隙率，即流体相的体积分数。