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

A., Barnafi, Nicolás; Zunino, Paolo; Dedè, Luca; Quarteroni, Alfio

doi:10.1016/j.camwa.2020.07.025

Cited by 17 publications

(27 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this section we draw inspiration from the stability analysis of a system similar to (2.2), recently proved in [12]; and from the study of the linearised hyperelasticity equations, recently studied in [35]. We restrict the description to Neo-Hookean poroelastic solids with material parameters µ s , λ s , for which the effective first Piola-Kirchhoff stress tensor is…”

Section: Definition and Preliminariesmentioning

confidence: 99%

“…Even if many numerical methods for the approximation of linear poroelasticity and their convergence analysis can be found in the recent literature (see [12,[16][17][18]52] and the references therein), much less attention has been given to formulations addressing large-strain poromechanics. In this regard, recent works include an enriched Galerkin framework [26], stabilised finite elements [15,77] and hybrid finite elements as well [76].…”

Section: Introductionmentioning

confidence: 99%

“…The method uses a MINI element for the displacement-porosity pair [6] (see also [8,22] for the case of hyperelasticity), and linear Lagrangian elements for the remaining unknowns. As the stability of the fully nonlinear system is still a paramount task (and an open problem even for the segregated nonlinear poroelasticity without the reaction-diffusion coupling, see, e.g., [12]), we address the solvability of the linearised set of equations, focusing on the semidiscrete in-time formulation. For the corresponding analysis, we propose a fixed-point strategy, and then, Schauder fixed point theorem, together with the Fredholm alternative, the Babuška-Brezzi theory, the Lax-Milgram lemma, and classical theory of quasi-linear equations, allow us to assert unique solvability robustly with respect to the material parameters of the linearised poroelastic solid.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema

Nicolas¹,

Gómez-Vargas²,

Lourenço³

et al. 2021

Preprint

View full text Add to dashboard Cite

In this paper we propose a novel coupled poroelasticity-diffusion model for the formation of extracellular oedema and infectious myocarditis valid in large deformations, manifested as an interaction between interstitial flow and the immune-driven dynamics between leukocytes and pathogens. The governing partial differential equations are formulated in terms of skeleton displacement, fluid pressure, Lagrangian porosity, and the concentrations of pathogens and leukocytes. A five-field mixed-primal finite element scheme is proposed for the numerical approximation of the problem, and we provide the stability analysis for a simplified system emanating from linearisation. We also discuss the construction of an adequate, Schur complement based, nested preconditioner. The produced computational tests exemplify the properties of the new model and of the mixed schemes.

show abstract

Section: Definition and Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema

Nicolas¹,

Gómez-Vargas²,

Lourenço³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The analysis of the well-posedness, stability and numerical approximation of this class of poromechanics models is still largely open. Among recent advances, we highlight the development and analysis of an implicit time discretization preserving the dissipation-energy identity at the discrete level [18]; an energy-preserving implicit-explicit time discretization incorporating a (non-iterative) operator splitting, decoupling solid and flow computations [19]; an energy-stable space and time discretization for a linearized model with focus on quasi-incompressible solids [20]; and finally, a space and time discretization for the same linearized model, exploiting a generalized saddle point structure and ultimately suggesting the use of Taylor-Hood type finite elements [21].…”

Section: Introductionmentioning

confidence: 99%

Iterative splitting schemes for a soft material poromechanics model

Both¹,

A.²,

Radu³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

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 problem, we follow the abundant literature for solvers of the latter equations, developing two approaches that resemble the well known undrained and fixed-stress splits for the Biot model. A thorough convergence analysis of the proposed schemes is performed. In particular, the undrained-like split is developed and analyzed in the framework of generalized gradient flows, whereas the fixed-stress-like split is understood as block-diagonal L 2 -type stabilization and analyzed by means of a relative stability analysis. In addition, the application of Anderson acceleration is suggested, improving the robustness of the split schemes. Finally, we test these methods on different benchmark tests, and we also compare their performance with respect to a monolithic approach. Together with the theoretical analysis, the numerical examples provide guidelines to appropriately choose what split scheme shall be used to address realistic applications of the soft material poromechanics model.

show abstract

“…In addition to interactions between fluid networks, recently also the viscous forces acting within each network have come to the fore [6,14,12,31]. At its core, the effect of viscosity can be accounted for by replacing the Darcy approximation in the poroelasticity model by a Brinkman approximation [11,40].…”

mentioning

confidence: 99%

Robust approximation of generalized Biot-Brinkman problems

Hong¹,

Kraus²,

Kuchta³

et al. 2021

Preprint

View full text Add to dashboard Cite

The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical examples.

show abstract

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

Cited by 17 publications

References 25 publications

Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema

Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema

Iterative splitting schemes for a soft material poromechanics model

Robust approximation of generalized Biot-Brinkman problems

Contact Info

Product

Resources

About