Geometrical multiscale model of an idealized left ventricle with fluid-structure interaction effects coupled to a one-dimensional viscoelastic arterial network

A geometrical multiscale model for blood flow through an ideal left ventricle and the main arteries is presented. The blood flow in the three-dimensional idealized left ventricle is solved through a monolithic fluid-structure interaction solver. To account for the interaction between the heart and the circulatory system the heart flow is coupled through an ideal valve with a network of viscoelastic one-dimensional models representing the arterial network. The geometrical multiscale approach used in this work is based on the exchange of averaged/integrated quantities between the fluid problems. The peripheral circulation is modelled by zero-dimensional windkessel terminals. We demonstrate that the geometrical multiscale model is (i) highly modular in that component models can be easily replaced with higher-fidelity ones whenever the user has a specific interest in modelling a particular part of the system, (ii) passive in that it reaches a stable limit cycle of flow rate and pressure in a few heartbeat cycles when driven by a periodic force acting on the epicardium, and (iii) capable of operating at physiological regimes.

Modular vs. non-modular preconditioners for fluid-structure systems with large added-mass effect

Annalisa Quaini, Alfio Quarteroni

In this article we address the numerical simulation of fluid-structure interaction (FSI) problems featuring large added-mass effect. We analyze different preconditioners for the coupled system matrix obtained after space- time discretization and linearization of the FSI problem. The classical Dirichlet-Neumann preconditioner has the advantage of ``modularity'' because it allows to reuse existing fluid and structure codes with minimum effort (simple interface communication). Unfortunately, its performance is very poor in case of large added-mass effects. Alternatively, we consider two non-modular approaches. The first one consists in preconditioning the coupled system with a suitable diagonal scaling combined with an ILUT preconditioner. The system is then solved by a Krylov method. The drawback of this procedure is that the combination of fluid and structure codes to solve the coupled system is not straightforward. The second non-modular approach we consider is a splitting technique based on an inexact block-LU factorization of the linear FSI system. The resulting algorithm computes the fluid velocity separately from the coupled pressure-structure system at each iteration, reducing the computational cost. Independently of the preconditioner, the efficiency of semi-implicit algorithms (i.e., those that treat geometric and fluid nonlinearities in an explicit way) is highlighted and their performance compared to the one of implicit algorithms. All the methods are tested on three-dimensional blood-vessel systems. The algorithm combining the non-modular ILUT preconditioner with Krylov methods proved to be the fastest.

2008

Algorithms for the coupling of one-dimensional arterial networks with three-dimensional fluid-structure interaction problems

Simone Deparis, Adelmo Cristiano Innocenza Malossi

This work is focused on the development of a geometrical multiscale framework for modeling the human cardiovascular system. This approach is designed to deal with different geometrical and mathematical models at the same time, without any preliminary hypotheses on the layout of the general multiscale problem. This flexibility allows to set up a complete arterial tree model of the circulatory system, assembling first a network of one-dimensional models, described by non-linear hyperbolic equations, and then replacing some elements with more detailed (and expensive) three-dimensional models, where the Navier-Stokes equations are coupled with structural models through fluid-structure interaction algorithms. The coupling between models of different scale and type is addressed imposing the conservation equations in terms of averaged/integrated quantities (i.e., the flow rate and the normal component of the traction vector); in particular, three coupling strategies have been explored for the fluid problem. In all the cases, these strategies lead to small non-linear interface problems, which are solved using classical iterative algorithms.

2011

Algorithms for fluid-structure interaction problems arising in hemodynamics

Annalisa Quaini

We discuss in this thesis the numerical approximation of fluid-structure interaction (FSI) problems with a particular concern (albeit not exclusive) on hemodynamics applications. Firstly, we model the blood as an incompressible fluid and the artery wall as an elastic structure. To solve the coupled problem, we propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, the fluid velocity is computed separately from the coupled pressure-structure velocity system at each iteration, hence reducing the computational cost. This approach leads to two different families of methods which extend to FSI problems schemes that were previously adopted for pure fluid problems. The algorithms derived from inexact factorization methods are compared with other schemes based on two preconditioners for the FSI system. The first one is the classical Dirichlet-Neumann preconditioner, which has the advantage of modularity (i.e. it allows to reuse existing fluid and structure codes with minimum effort). Unfortunately, its performance is very poor in case of large added-mass effect, as it happens in hemodynamics. Alternatively, we consider a non-modular approach which consists in preconditioning the coupled system with a suitable diagonal scaling combined with an ILUT preconditioner. The system is then solved by a Krylov method. The drawback of this procedure is the loss of modularity. Independently of the preconditioner, the efficiency of semi-implicit algorithms is highlighted. All the methods are tested on two and three-dimensional blood-vessel systems. The algorithm combining the non-modular ILUT preconditioner with Krylov methods proved to be the fastest. However, modular and inexact factorization based methods should not be disregarded because they can considerably benefit from code parallelization, unlike the ILUT-Krylov approach. Finally, we improve the structure model by representing the vessel wall as a linear poroelastic medium. Our non-modular approach and the partitioned procedures arising from a domain decomposition viewpoint are extended to fluid-poroelastic structure interactions. Their numerical performance are analyzed and compared on simplified blood-vessel systems.

EPFL2009

Algorithms for fluid-structure interaction problems arising in hemodynamics

Annalisa Quaini

EPFL2009