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

Published on Jun 1, 2021in Computers & Mathematics With Applications3.37
· DOI :10.1016/J.CAMWA.2020.07.025
Nicolás Barnafi1
Estimated H-index: 1
(Polytechnic University of Milan),
Nicolas A. Barnafi2
Estimated H-index: 2
(Polytechnic University of Milan)
+ 1 AuthorsAlfio Quarteroni83
Estimated H-index: 83
(Polytechnic University of Milan)
Abstract 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.
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We address numerical solvers for a poromechanics model particularly adapted for soft materials, as it generally respects thermodynamics principles and energy balance. Considering the multi-physics nature of the problem, which involves solid and fluid species, interacting on the basis of mass balance and momentum conservation, we decide to adopt a solution strategy of the discrete problem based on iterative splitting schemes. As the model is similar (but not equivalent to) the Biot poromechanics ...