Energy-based operator splitting approach for the time discretization of coupled systems of partial and ordinary differential equations for fluid flows: The Stokes case

The goal of this work is to develop a novel splitting approach for the numerical solution of multiscale problems involving the coupling between Stokes equations and ODE systems, as often encountered in blood flow modeling applications. The proposed algorithm is based on a semi-discretization in time based on operator splitting, whose design is guided by the rationale of ensuring that the physical energy balance is maintained at the discrete level. As a result, unconditional stability with respect to the time step choice is ensured by the implicit treatment of interface conditions within the Stokes substeps, whereas the coupling between Stokes and ODE substeps is enforced via appropriate initial conditions for each substep. Notably, unconditional stability is attained without the need of subiterating between Stokes and ODE substeps. Stability and convergence properties of the proposed algorithm are tested on three specific examples for which analytical solutions are derived

Efficient solving strategies for incompressible Navier-Stokes equations for large scale simulations using the open source software Feel++

International audienceOver the past few decades, the computational fluid dynamics has evolved to become an important tool for the description of the complex multi-physics, multi-scale phenomena characterizing blood flow. Its reliability depends both on the verification of the numerical methods and on the validation of the mathematical models. The aim of the first part of the talk is to present a preconditioning framework for the linear system arising from the finite element discretizations and time advancing finite difference schemes of the 3D steady and unsteady Navier-Stokes equations. In particular, we are interested in preconditioners based on an algebraic factorization of the system's matrix which exploit its block structure, such as the Pressure Convection-Diffusion (PCD) preconditioner , the SIMPLE preconditioner or the LSC preconditioner, see [Elman et al. (2014)]. A comparison between the efficiency of these preconditioners is ascertain by testing them over the 3D backward facing step benchmark. The iteration counts using the PCD preconditioner are independent of mesh size and high order finite elements and mildly dependent on Reynolds numbers for steady flow problems which is not the case for the other preconditioners. In the second part of the talk we describe a framework for the solution of flow problems relevant to biomechanics strongly supported by the aforementioned solving strategies. We assess the efficiency of this framework through experimental data for fluid flow in a nozzle model with rigid boundaries, a device designed to reproduce acceleration, deceleration and recirculation, features commonly encountered in medical devices [Stewart et al. (2012)]. The flow rates were chosen to cover laminar (Re = 500), transient (Re = 2000) and turbulent (Re = 3500) regimes. The numerical results displayed during the presentation are obtained using the open-source library Feel++ (Finite Element method Embedded Language in C++, www.feelpp.org). Figure 1: Computational Fluid Dynamics FDA Benchmark at Re = 2000

Hemodynamic simulations in the cerebral venous network: A study on the influence of different modeling assumptions

International audienceBlood flow computations in complex geometries are of major interest in various cardio-vascular applications. However, deriving an appropriate computational model is still an open issue and a central question is how to incorporate and quantify uncertainties due to different modeling assumptions. The present work is intended as a first step in this direction, in the particular case of blood flow in the cerebral venous system. After a careful evaluation of the influence of the computational methodology, the study investigates the impact on the velocity field and the wall shear stress of three inflow boundary conditions, two strategies for treating the outflow boundary condition and two different viscosity models. The results demonstrate that the effect of setting the inflow boundary condition on the forces created by blood flow, is likely greater than for other modeling assumptions, the other important factor being the blood viscosity model, especially in wall shear stress computations. They suggest that improvements on the one hand on the mathematical and computational approach, and on the other hand on available data for their treatment are needed

From medical imaging to numerical simulations

International audienceIn the last 20 years there have been lots of progress in 3D medical imaging (such as Magnetic Resonance Imaging, MRI, and X-ray Computed Tomography, CT) and in particular in modalities to visualise vascular structures. The resulting images have been successfully used in various clinical applications, in particular for cerebrovascular pathologies (e.g., neurosurgery planning; stenoses, aneurysm or thrombosis quantification; arteriovenous malformation detection and follow-up, etc.). The complexity of the processing and analysis of these images (size, information vs noise, artifacts, etc) led to the development of imaging tools such as vessel filtering, segmentation and quantification. There is however, until now, no database of synthetic images and associated ground-truths (segmented data) available in cerebrovascular images contrary to morphological brain image analysis (e.g. brainweb).In the ANR Vivabrain project, we combine the skills of several communities: computer science, applied mathematics, biophysics, and medicine to remedy the aforementioned observation. In particular we focus on complex multi-disciplinary problems such as (i) the handling of inter-individual cerebrovascular variability, (ii) the generation of computational meshes, (iii) the simulation of blood flows in the complete cerebrovascular system 3D+time (3D+t) including calibration and validation and (iv) the accurate simulation of the physical processes involved in MRA acquisition sequences in order to finally obtain realistic virtual angiographic images

Uncertainty propagation and sensitivity analysis: results from the Ocular Mathematical Virtual Simulator

International audienceWe propose an uncertainty propagation study and a sensitivity analysis with the Ocular Mathematical Virtual Simulator, a computational and mathematical model that predicts the hemodynamics and biomechanics within the human eye. In this contribution, we focus on the effect of intraocular pressure, retrolaminar tissue pressure and systemic blood pressure on the ocular posterior tissue vasculature. The combination of a physically-based model with experiments-based stochastic input allows us to gain a better understanding of the physiological system, accounting both for the driving mechanisms and the data variability

Mathematical modeling, analysis and simulations for fluid mechanics and their relevance to in silico medicine

This manuscript gathers my contributions focused on developing new mathematical and computational methods for analyzing biological flows as complex multiphysics and multiscale phenomena. The description of the underlying mechanisms stems from the basic principles of fluid dynamics and is translated into systems of partial or ordinary differential equations. The overall objective of this work is the study of these equations atthe continuous and discrete levels, their coupling and the development of a reliable and efficient computational framework to implement various numerical methods to approximate the solutions to these problems. The numerical simulations incorporate realistic geometries, are thoroughly validated against experimental data and target specific biomedical applications. The first chapter focuses on three-dimensional models,in which the motion of a biofluid in a complex, realistic geometry is governed by the incompressible Navier-Stokes equations in a domain with inlet and outlet boundaries, the main application in view being the study of the cerebral venous network. The purpose of Chapter 2 is two-fold: (i) first, we present contributions towards the elaboration of several reduced 0d models describing the coupled dynamics of different biofluids in the eye-cerebral system; (ii) second, we describe a new splitting strategy for the numerical solving of coupled systems of partial and ordinary differential equations for fluid flows. In Chapter 3, the fluid dynamics description from the previous chapters is enriched to take into account the combined effects of flow and different structures, from a multiphysics perspective. The resulting framework is subsequently utilized for simulating blood flow inthe aorta, in view of a specific biomedical application and for the numerical simulation of particulate flows, with an emphasis on the issue of how to handle contacts

End matter : Oxley Memorial Library of Queensland

Membres du jury: Dominique CHAPELLE (rapporteur), Philippe G. CIARLET (directeur), Doina CIORANESCU, Yvon MADAY, Cristinel MARDARE, Annie RAOULT (rapporteur).The aim of this thesis is to study several questions which arise in the theory of elasticity, by using methods of mathematical analysis and differential geometry. In the one-dimensional case, related to the study of elastic wires, we prove some existence, uniqueness and stability results for a curve in Sobolev spaces. Then, we treat the general case of an immersion of arbitrary dimension and co-dimension of a submanifold in an Euclidean space. We show that the classical existence and uniqueness result for such an immersion can be extended up to the boundary of the submanifold, under a specific, but mild, regularity assumption on this set. Moreover, we show that the mapping constructed in this fashion is locally Lipschitz-continuous with respect to suitable topologies. Finally, we reconsider the study of elastic wires, to obtain some linear and nonlinear Korn inequalities for curves in dimension 3.Le but de cette thèse est d'étudier des questions issues de la théorie de l'élasticité en utilisant des méthodes d'analyse mathématique et de géométrie différentielle. Dans le cas mono-dimensionnel, qui est lié à l'étude des fils élastiques, nous prouvons des résultats d'existence, d'unicité et de stabilité d'une courbe dans des espaces de Sobolev. Nous traitons ensuite le cas général d'une immersion de dimension et de co-dimension quelconques d'une sous-variété dans l'espace euclidien. Nous montrons ainsi que le résultat classique d'existence et d'unicité d'une telle immersion peut être étendu jusqu'au bord de la sous-variété, sous une hypothèse de régularité peu restrictive sur celui-ci. En outre, nous montrons que l'application ainsi construite est localement lipschitzienne pour les topologies appropriées. Enfin, nous revenons à l'étude des fils élastiques, pour obtenir des inégalités de Korn linéaires et non linéaires pour les courbes en dimension 3