Mathematical analysis and numerical approximation of a general linearized poro-hyperelastic model

We describe the behavior of a deformable porous material by means of a porohyperelastic 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. (C) 2020 The Authors. Published by Elsevier Ltd.

Automated summary

This paper discusses a previously proposed model for a deformable porous material, focusing on a linearized version of the model, mathematical analysis, and numerical approximation. The analysis particularly emphasizes the well-posedness of the model and the stability of the numerical discretization scheme, with attention to how the choice of polynomial degree for approximation depends on the porosity of the mixture.