Partitioned Solution of Geometrical Multiscale Problems for the Cardiovascular System:Models, Algorithms, and Applications

The aim of this work is the development of a geometrical multiscale framework for the simulation of the human cardiovascular system under either physiological or pathological conditions. More precisely, we devise numerical algorithms for the partitioned solution of geometrical multiscale problems made of different heterogeneous compartments that are implicitly coupled with each others. The driving motivation is the awareness that cardiovascular dynamics are governed by the global interplay between the compartments in the network. Thus, numerical simulations of stand-alone local components of the circulatory system cannot always predict effectively the physiological or pathological states of the patients, since they do not account for the interaction with the missing elements in the network. As a matter of fact, the geometrical multiscale method provides an automatic way to determine the boundary (more precisely, the interface) data for the specific problem of interest in absence of clinical measures and it also offers a platform where to study the interaction between local changes (due, for instance, to pathologies or surgical interventions) and the global systemic dynamics. To set up the framework an abstract setting is devised; the local specific mathematical equations (partial differential equations, differential algebraic equations, etc.) and the numerical approximation (finite elements, finite differences, etc.) of the heterogeneous compartments are hidden behind generic operators. Consequently, the resulting global interface problem is formulated and solved in a completely transparent way. The coupling between models of different dimensional scale (three-dimensional, one-dimensional, etc.) and type (Navier-Stokes, fluid-structure interaction, etc.) is addressed writing the interface equations in terms of scalar quantities, i.e., area, flow rate, and mean (total) normal stress. In the resulting flexible framework the heterogeneous models are treated as black boxes, each one equipped with a specific number of compatible interfaces such that (i) the arrangement of the compartments in the network can be easily manipulated, thus allowing a high level of customization in the design and optimization of the global geometrical multiscale model, (ii) the parallelization of the solution of the different compartments is straightforward, leading to the opportunity to make use of the latest high-performance computing facilities, and (iii) new models can be easily added and connected to the existing ones. The methodology and the algorithms devised throughout the work are tested over several applications, ranging from simple benchmark examples to more complex cardiovascular networks. In addition, two real clinical problems are addressed: the simulation of a patient-specific left ventricle affected by myocardial infarction and the study of the optimal position for the anastomosis of a left ventricle assist device cannula

On the continuity of mean total normal stress in geometrical multiscale cardiovascular problems

In this work an iterative strategy to implicitly couple dimensionally-heterogeneous blood flow models accounting for the continuity of mean total normal stress at interface boundaries is developed. Conservation of mean total normal stress in the coupling of heterogeneous models is mandatory to satisfy energetic consistency between them. Nevertheless, existing methodologies are based on modifications of the Navier-Stokes variational formulation, which are undesired when dealing with fluid-structure interaction or black box codes. The proposed methodology makes possible to couple one-dimensional and three-dimensional fluid-structure interaction models, enforcing the continuity of mean total normal stress while just imposing flow rate data or even the classical Neumann boundary data to the models. This is accomplished by modifying an existing iterative algorithm, which is also able to account for the continuity of the vessel area, when required. Comparisons are performed to assess differences in the convergence properties of the algorithms when considering the continuity of mean normal stress and the continuity of mean total normal stress for a wide range of flow regimes. Finally, examples in the physiological regime are shown to evaluate the importance, or not, of considering the continuity of mean total normal stress in hemodynamics simulations

Design and analysis of a Schwarz coupling method for a dimensionally heterogeneous problem

International audienceIn the present work, we study and analyze an efficient iterative coupling method for a dimensionally heterogeneous problem . We consider the case of 2-D Laplace equation with non symmetric boundary conditions with a corresponding 1-D Laplace equation. We will first show how to obtain the 1-D model from the 2-D one by integration along one direction, by analogy with the link between shallow water equations and the Navier-Stokes system. Then, we will focus on the design of an Schwarz-like iterative coupling method. We will discuss the choice of boundary conditions at coupling interfaces. We will prove the convergence of such algorithms and give some theoretical results related to the choice of the location of the coupling interface, and the control of the difference between a global 2-D reference solution and the 2-D coupled one. These theoretical results will be illustrated numerically.Dans ce document nous étudions et analysons et une méthode de couplage multidimensionnel itérative. Nous considérons le cas de l'équation de Laplace 2-D avec des conditions aux bords non symétriques, couplée avec une équation de Laplace 1-D correspondante. dans un premier temps nous montrons comment obtenir le modèle 1-D à partir du modèle 2-D par intégration verticale et par analogie avec la dérivation des équations de Saint-Venant à partir des équations de Navier-Stokes. Ensuite nous présentons un algorithme de couplage de type Schwarz. Nous discutons le choix des conditions aux interfaces de couplage. Nous démontrons la convergence de tels algorithmes et donnons quelques résultats théoriques sur le choix de la position des interfaces de couplage. Un résultat théorique sur le contrôle de l'erreur entre la solution globale 2-D de référence et la solution 2-D couplée sera aussi donné. Enfin nous illustrons ces résultats numériquement

A two-level time step technique for the partitioned solution of one-dimensional arterial networks

In this work a numerical strategy to address the solution of the blood flow in one-dimensional arterial networks through a topology-based decomposition is presented. Such decomposition results in the local analysis of the blood flow in simple arterial segments. Hence, iterative methods are used to perform the strong coupling among the segments, which communicate through non-overlapping interfaces. Specifically, two approaches are considered to solve the associated nonlinear interface problem: (i) the Newton method and (ii) the Broyden method. Moreover, since the modeling of blood flow in compliant vessels is tackled using explicit finite element methods, we formulate the coupling problem using a two-level time stepping technique. A local (inner) time step is used to solve the local problems in single arteries, meeting thus local stability conditions, while a global (outer) time step is employed to enforce the continuity of physical quantities of interest among the one-dimensional segments. Several examples of application are presented. Firstly a study about spurious reflexions produced at interfaces as a consequence of the two-level time stepping technique is carried out. Secondly, the application of the methodologies to physiological scenarios is presented, specifically addressing the solution of the blood flow in a model of the entire arterial network

Parallel meshing, discretization and computation of flow in massive discrete fracture networks

In the present work a message passing interface (MPI) parallel implementation of an optimization-based approach for the simulation of underground flows in large discrete fracture networks is proposed. The software is capable of parallel execution of meshing, discretization, resolution, and postprocessing of the solution. We describe how optimal scalability performances are achieved combining high efficiency in computations with an optimized use of MPI communication protocols. Also, a novel graph-topology for communications, called the multi-Master approach, is tested, allowing for high scalability performances on massive fracture networks. Strong scalability and weak scalability simulations on random networks counting order of 10^5 fractures are reported