Parameter-robust discretization and preconditioning of Biot's consolidation model

Biot's consolidation model in poroelasticity has a number of applications in science, medicine, and engineering. The model depends on various parameters, and in practical applications these parameters ranges over several orders of magnitude. A current challenge is to design discretization techniques and solution algorithms that are well behaved with respect to these variations. The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems. The approach taken here is to consider the stability of the problem in non-standard or weighted Hilbert spaces and employ the operator preconditioning approach. We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement. The parameters of interest are small time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.Comment: 24 page

Weakly imposed symmetry and robust preconditioners for Biot's consolidation model

We discuss the construction of robust preconditioners for finite element approximations of Biot's consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger-Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.Comment: 21 page

A mixed finite element method for nearly incompressible multiple-network poroelasticity

In this paper, we present and analyze a new mixed finite element formulation of a general family of quasi-static multiple-network poroelasticity (MPET) equations. The MPET equations describe flow and deformation in an elastic porous medium that is permeated by multiple fluid networks of differing characteristics. As such, the MPET equations represent a generalization of Biot's equations, and numerical discretizations of the MPET equations face similar challenges. Here, we focus on the nearly incompressible case for which standard mixed finite element discretizations of the MPET equations perform poorly. Instead, we propose a new mixed finite element formulation based on introducing an additional total pressure variable. By presenting energy estimates for the continuous solutions and a priori error estimates for a family of compatible semi-discretizations, we show that this formulation is robust in the limits of incompressibility, vanishing storage coefficients, and vanishing transfer between networks. These theoretical results are corroborated by numerical experiments. Our primary interest in the MPET equations stems from the use of these equations in modelling interactions between biological fluids and tissues in physiological settings. So, we additionally present physiologically realistic numerical results for blood and tissue fluid flow interactions in the human brain

The effects of hypoxia, ischaemia and inotropic interventions on mammalian cardiac muscle

Variations in intracellular Ca2+ concentration ([Ca2+]i) are responsible for much of the modulation of the force of contraction in cardiac muscle. However, in many situations, changes in the sensitivity of the contractile proteins to Ca2+ also contribute to alterations of force. If [Ca2+]i and force production are measured simultaneously, it is possible to identify the contribution of these processes to the changes in the force of contraction produced by various interventions. Two interventions of clinical importance are hypoxia and ischaemia. Their effects were studied in thin ventricular papillary muscles, which were microinjected with the photoprotein aequorin in order to measure [Ca2+]i. During an initial exposure to hypoxia, the systolic rise of [Ca2+]i (the Ca2+ transient) often showed a moderate increase, but on subsequent exposures this increase disappeared and eventually the Ca2+ transient declined on exposure to anoxia. This decline could be converted back to an increase by exposure to an elevated glucose concentration, suggesting that the response was dependent on the metabolic state of the muscle. This was confirmed in a parallel series of experiments in which glycogen and lactate production were measured in Langendorff-perfused hearts exposed to hypoxia. Ischaemia has previously been difficult to study in isolated cardiac muscle. A new model was developed which allowed ischaemia to be mimicked in an isolated papillary muscle while [Ca2+]i was measured. Ischaemia caused a dramatic decline in tension, and a large increase in the amplitude of the Ca2+ transients developed over several minutes. These changes could be mimicked by the application of 20 mM lactic acid. After long exposures to ischaemia, resting tension developed and the Ca2+ transients declined. Repeated exposures to ischaemia caused an early fall in the Ca2+ transients, and the early development of resting tension and raised resting Ca2+. Thus, many of the effects of hypoxia and ischaemia seem to be attributable to an intracellular acidosis due to lactic acid accumulation, which is more severe in ischaemia due to the lack of flow. The severity of the acidosis depends on the initial metabolic state of the tissue. Although past laboratory studies have concentrated on isometric contraction, in vivo the heart has phases of both isotonic as well as isometric contraction, and modern use of cardiac myocytes means that unloaded preparations are now frequently studied. The effects of several positive and negative inotropic interventions on isotonic and isometric contraction were compared. For all interventions, the fractional effect on tension was 1.5 - 2 times larger than that on shortening, and this could be accounted for by the shape of the length- force relation for cardiac muscle in different inotropic states. Pimobendan is a new inotropic agent, which is thought to have both Ca2+ sensitising and phosphodiesterase inhibiting activity. Its effects on tension and Ca transients were studied. The results suggest that both these effects are active in living cardiac muscle, and that this combination may be advantageous for an inotropic agent. These experiments have demonstrated that changes in both the Ca2+ transient and Ca2+ sensitivity occur when cardiac muscle is exposed to hypoxia or ischaemia. The effects of interventions on isometric force and shortening are similar. Finally, drugs are now available which can alter Ca2+ sensitivity as well as the Ca2+ transient

Wond\u27ring

https://digitalcommons.library.umaine.edu/mmb-vp/4110/thumbnail.jp

Dissipative Quantum Hall Effect in Graphene near the Dirac Point

We report on the unusual nature of nu=0 state in the integer quantum Hall effect (QHE) in graphene and show that electron transport in this regime is dominated by counter-propagating edge states. Such states, intrinsic to massless Dirac quasiparticles, manifest themselves in a large longitudinal resistivity rho_xx > h/e^2, in striking contrast to rho_xx behavior in the standard QHE. The nu=0 state in graphene is also predicted to exhibit pronounced fluctuations in rho_xy and rho_xx and a smeared zero Hall plateau in sigma_xy, in agreement with experiment. The existence of gapless edge states puts stringent constraints on possible theoretical models of the nu=0 state.Comment: 4 pgs, 4 fg