Hybrid Multiscale Finite Volume method for two-phase flow in porous media

We present a novel hybrid (or multiphysics) algorithm, which couples pore-scale and Darcy descriptions of two-phase flow in porous media. The flow at the pore-scale is described by the Navier?Stokes equations, and the Volume of Fluid (VOF) method is used to model the evolution of the fluid?fluid interface. An extension of the Multiscale Finite Volume (MsFV) method is employed to construct the Darcy-scale problem. First, a set of local interpolators for pressure and velocity is constructed by solving the Navier?Stokes equations; then, a coarse mass-conservation problem is constructed by averaging the pore-scale velocity over the cells of a coarse grid, which act as control volumes; finally, a conservative pore-scale velocity field is reconstructed and used to advect the fluid?fluid interface. The method relies on the localization assumptions used to compute the interpolators (which are quite straightforward extensions of the standard MsFV) and on the postulate that the coarse-scale fluxes are proportional to the coarse-pressure differences. By numerical simulations of two-phase problems, we demonstrate that these assumptions provide hybrid solutions that are in good agreement with reference pore-scale solutions and are able to model the transition from stable to unstable flow regimes. Our hybrid method can naturally take advantage of several adaptive strategies and allows considering pore-scale fluxes only in some regions, while Darcy fluxes are used in the rest of the domain. Moreover, since the method relies on the assumption that the relationship between coarse-scale fluxes and pressure differences is local, it can be used as a numerical tool to investigate the limits of validity of Darcy's law and to understand the link between pore-scale quantities and their corresponding Darcy-scale variables.

Analysis of a finite volume element method for the Stokes problem

In this paper we propose a stabilized conforming finite volume element method for the Stokes equations. On stating the convergence of the method, optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations. A superconvergence result is also derived by using a postprocessing projection method. In particular, the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries. Finally, some numerical experiments that confirm the predicted behavior of the method are provided.

Evolutionary and convergence stability for continuous phenotypes in finite populations derived from two-allele models.

The evolution of a quantitative phenotype is often envisioned as a trait substitution sequence where mutant alleles repeatedly replace resident ones. In infinite populations, the invasion fitness of a mutant in this two-allele representation of the evolutionary process is used to characterize features about long-term phenotypic evolution, such as singular points, convergence stability (established from first-order effects of selection), branching points, and evolutionary stability (established from second-order effects of selection). Here, we try to characterize long-term phenotypic evolution in finite populations from this two-allele representation of the evolutionary process. We construct a stochastic model describing evolutionary dynamics at non-rare mutant allele frequency. We then derive stability conditions based on stationary average mutant frequencies in the presence of vanishing mutation rates. We find that the second-order stability condition obtained from second-order effects of selection is identical to convergence stability. Thus, in two-allele systems in finite populations, convergence stability is enough to characterize long-term evolution under the trait substitution sequence assumption. We perform individual-based simulations to confirm our analytic results.

Random mating with a finite number of matings.

Random mating is the null model central to population genetics. One assumption behind random mating is that individuals mate an infinite number of times. This is obviously unrealistic. Here we show that when each female mates a finite number of times, the effective size of the population is substantially decreased.

Compressed Sensing Single-Breath-Hold CMR for Fast Quantification of LV Function, Volumes, and Mass.

OBJECTIVES: The purpose of this study was to compare a novel compressed sensing (CS)-based single-breath-hold multislice magnetic resonance cine technique with the standard multi-breath-hold technique for the assessment of left ventricular (LV) volumes and function. BACKGROUND: Cardiac magnetic resonance is generally accepted as the gold standard for LV volume and function assessment. LV function is 1 of the most important cardiac parameters for diagnosis and the monitoring of treatment effects. Recently, CS techniques have emerged as a means to accelerate data acquisition. METHODS: The prototype CS cine sequence acquires 3 long-axis and 4 short-axis cine loops in 1 single breath-hold (temporal/spatial resolution: 30 ms/1.5 × 1.5 mm(2); acceleration factor 11.0) to measure left ventricular ejection fraction (LVEFCS) as well as LV volumes and LV mass using LV model-based 4D software. For comparison, a conventional stack of multi-breath-hold cine images was acquired (temporal/spatial resolution 40 ms/1.2 × 1.6 mm(2)). As a reference for the left ventricular stroke volume (LVSV), aortic flow was measured by phase-contrast acquisition. RESULTS: In 94% of the 33 participants (12 volunteers: mean age 33 ± 7 years; 21 patients: mean age 63 ± 13 years with different LV pathologies), the image quality of the CS acquisitions was excellent. LVEFCS and LVEFstandard were similar (48.5 ± 15.9% vs. 49.8 ± 15.8%; p = 0.11; r = 0.96; slope 0.97; p < 0.00001). Agreement of LVSVCS with aortic flow was superior to that of LVSVstandard (overestimation vs. aortic flow: 5.6 ± 6.5 ml vs. 16.2 ± 11.7 ml, respectively; p = 0.012) with less variability (r = 0.91; p < 0.00001 for the CS technique vs. r = 0.71; p < 0.01 for the standard technique). The intraobserver and interobserver agreement for all CS parameters was good (slopes 0.93 to 1.06; r = 0.90 to 0.99). CONCLUSIONS: The results demonstrated the feasibility of applying the CS strategy to evaluate LV function and volumes with high accuracy in patients. The single-breath-hold CS strategy has the potential to replace the multi-breath-hold standard cardiac magnetic resonance technique.

Development and experimental validation of a finite element model of total ankle replacement.

Total ankle replacement remains a less satisfactory solution compared to other joint replacements. The goal of this study was to develop and validate a finite element model of total ankle replacement, for future testing of hypotheses related to clinical issues. To validate the finite element model, an experimental setup was specifically developed and applied on 8 cadaveric tibias. A non-cemented press fit tibial component of a mobile bearing prosthesis was inserted into the tibias. Two extreme anterior and posterior positions of the mobile bearing insert were considered, as well as a centered one. An axial force of 2kN was applied for each insert position. Strains were measured on the bone surface using digital image correlation. Tibias were CT scanned before implantation, after implantation, and after mechanical tests and removal of the prosthesis. The finite element model replicated the experimental setup. The first CT was used to build the geometry and evaluate the mechanical properties of the tibias. The second CT was used to set the implant position. The third CT was used to assess the bone-implant interface conditions. The coefficient of determination (R-squared) between the measured and predicted strains was 0.91. Predicted bone strains were maximal around the implant keel, especially at the anterior and posterior ends. The finite element model presented here is validated for future tests using more physiological loading conditions.

An iterative Multiscale Finite-Volume algorithm converging to the exact solution

The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained).