Finite Element Methods for Large-Strain Poroelasticity/Chemotaxis Models Simulating the Formation of Myocardial Oedema

Abstract

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 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 finite element schemes.

Coupling chemotaxis and growth poromechanics for the modelling of feather primordia patterning

“…The nonlinear tolerances are taken identical to the 2D case. For the linear solvers, we use a GMRES with the nested Schur preconditioner recently proposed in [3], with an absolute tolerance of 1E-12 and a relative one of 1E-2. The use of such a big relative tolerance yields a simplified inexact-Newton method [21] that results in more nonlinear iterations that require much less time.…”

“…The poromechanics block is more difficult, which is reflected in the complexity of the preconditioner under consideration. It is based on the block preconditioner proposed in [3] for largestrain poromechanics, where we use a lower Schur complement block factorisation with the fields u and (ψ, p). For such a block we consider the action of an AMG preconditioner, whereas for the corresponding Schur complement block we consider instead a sparse representation given by an adhoc extension of the fixed-stress splitting scheme proposed in [5].…”

“…We highlight that the proposed preconditioner represents an improvement over [3], as we employ only preconditioner actions on each block, which allows us to use a GMRES linear solver instead of a flexible GMRES (which can be substantially more demanding in terms of computational cost).…”

“…This phase separation persists even up to the cellular level, as there is the cytoskeleton and the cytoplasm. For this reason, poroelastic models have become very pervasive in the modelling of soft living tissue, such as in oedema formation [3,28], cardiac perfusion [47,2], lung characterisation [7], and brain injury [46].…”

Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning

“…This phase separation persists even up to the cellular level, as there is the cytoskeleton and the cytoplasm. For this reason, poroelastic models have become very pervasive in the modelling of soft living tissue, such as in oedema formation [8,9], cardiac perfusion [10,11], lung characterisation [12], and brain injury [13].…”