104 research outputs found
A Finite Element Approach For Modeling Biomembranes In Incompressible Power-Law Flow
We present a numerical method to model the dynamics of inextensible
biomembranes in a quasi-Newtonian incompressible flow, which better describes
hemorheology in the small vasculature. We consider a level set model for the
fluid-membrane coupling, while the local inextensibility condition is relaxed
by introducing a penalty term. The penalty method is straightforward to
implement from any Navier-Stokes/level set solver and allows substantial
computational savings over a mixed formulation. A standard Galerkin finite
element framework is used with an arbitrarily high order polynomial
approximation for better accuracy in computing the bending force. The PDE
system is solved using a partitioned strongly coupled scheme based on
Crank-Nicolson time integration. Numerical experiments are provided to validate
and assess the main features of the method
Numerical Approach Based on the Composition of One-Step Time-Integration Schemes For Highly Deformable Interfaces
In this work, we propose a numerical approach for simulations of large
deformations of interfaces in a level set framework. To obtain a fast and
viable numerical solution in both time and space, temporal discretization is
based on the composition of one-step methods exhibiting higher orders and
stability, especially in the case of stiff problems with strongly oscillatory
solutions. Numerical results are provided in the case of ordinary and partial
differential equations to show the main features and demonstrate the
performance of the method. Convergence properties and efficiency in terms of
computational cost are also investigated
Mathematical modelling of active contraction in isolated cardiomyocytes
We investigate the interaction of intracellular calcium spatio-temporal variations with the self-sustained contractions in cardiac myocytes. A consistent mathematical model is presented considering a hyperelastic description of the passive mechanical properties of the cell, combined with an active-strain framework to explain the active shortening of myocytes and its coupling with cytosolic and sarcoplasmic calcium dynamics. A finite element method based on a Taylor-Hood discretization is employed to approximate the nonlinear elasticity equations, whereas the calcium concentration and mechanical activation variables are discretized by piecewise linear finite elements. Several numerical tests illustrate the ability of the model in predicting key experimentally established characteristics including: (i) calcium propagation patterns and contractility, (ii) the influence of boundary conditions and cell shape on the onset of structural and active anisotropy and (iii) the high localized stress distributions at the focal adhesions. Besides, they also highlight the potential of the method in elucidating some important subcellular mechanisms affecting, e.g. cardiac repolarizatio
An adaptive finite element method for the modeling of the equilibrium of red blood cells
International audienceThis contribution is concerned with a the numerical modeling of an isolated red blood cell (RBC), and more generally of phospholipid membranes. We propose an adaptive Eulerian finite element approximation, based on the level set method, of a shape optimization problem arising in the study of RBC's equilibrium. We simulate the equilibrium shapes that minimize the elastic bending energy under prescribed constraints of fixed volume and surface area. An anisotropic mesh adaptation technique is used in the vicinity of the cell's membrane to enhance the robustness of the method. Efficient time and spatial discretizations are considered and implemented. We address in detail the main features of the proposed method and finally we report several numerical experiments in the two-dimensional and the three-dimensional axisymmetric cases. The effectiveness of the numerical method is further demonstrated through numerical comparisons with semi-analytical solutions provided by a reduced order model
A Necklace Model for Vesicles Simulations in 2D
International audienceThe aim of this paper is to propose a new numerical model to simulate 2D vesicles interacting with a newtonian fluid. The inextensible membrane is modeled by a chain of circular rigid particles which are maintained in cohesion by using two different type of forces. First, a spring force is imposed between neighboring particles in the chain. Second, in order to model the bending of the membrane, each triplet of successive particles is submitted to an angular force. Numerical simulations of vesicles in shear flow have been run using Finite Element Method and the FreeFem++[1] software. Exploring different ratios of inner and outer viscosities, we recover the well known "Tank-Treading" and "Tumbling" motions predicted by theory and experiments. Moreover, for the first time, 2D simulations of the "Vacillating-Breathing" regime predicted by theory in [2] and observed experimentally in [3] are done without special ingredient like for example thermal fluctuations used in [4]
Mathematical modelling of active contraction in isolated cardiomyocytes
We investigate the interaction of intracellular calcium spatio-temporal variations with the self-sustained contractions in cardiac myocytes. A consistent mathematical model is presented considering a hyperelastic description of the passive mechanical properties of the cell, combined with an active-strain framework to explain the active shortening of myocytes and its coupling with cytosolic and sarcoplasmic calcium dynamics. A finite element method based on a Taylor-Hood discretization is employed to approximate the nonlinear elasticity equations, whereas the calcium concentration and mechanical activation variables are discretized by piecewise linear finite elements. Several numerical tests illustrate the ability of the model in predicting key experimentally established characteristics including: (i) calcium propagation patterns and contractility, (ii) the influence of boundary conditions and cell shape on the onset of structural and active anisotropy and (iii) the high localized stress distributions at the focal adhesions. Besides, they also highlight the potential of the method in elucidating some important subcellular mechanisms affecting, e.g. cardiac repolarization
Fully Eulerian finite element approximation of a fluid-structure interaction problem in cardiac cells
Integrated Heart - Coupling multiscale and multiphysics models for the simulation of the cardiac function
Mathematical modelling of the human heart and its function can expand our understanding of various cardiac
diseases, which remain the most common cause of death in the developed world. Like other physiological
systems, the heart can be understood as a complex multiscale system involving interacting phenomena at the
molecular, cellular, tissue, and organ levels. This article addresses the numerical modelling of many aspects
of heart function, including the interaction of the cardiac electrophysiology system with contractile muscle
tissue, the sub-cellular activation-contraction mechanisms, as well as the hemodynamics inside the heart
chambers. Resolution of each of these sub-systems requires separate mathematical analysis and specially
developed numerical algorithms, which we review in detail. By using specific sub-systems as examples, we
also look at systemic stability, and explain for example how physiological concepts such as microscopic force
generation in cardiac muscle cells, translate to coupled systems of differential equations, and how their stability
properties influence the choice of numerical coupling algorithms. Several numerical examples illustrate
three fundamental challenges of developing multiphysics and multiscale numerical models for simulating
heart function, namely: (i) the correct upscaling from single-cell models to the entire cardiac muscle, (ii) the
proper coupling of electrophysiology and tissue mechanics to simulate electromechanical feedback, and (iii)
the stable simulation of ventricular hemodynamics during rapid valve opening and closure
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