Annalisa Quaini

Splitting Methods Based on Algebraic Factorization for Fluid-Structure Interaction

We discuss in this paper the numerical approximation of fluid-structure interaction (FSI) problems dealing with strong added-mass effect. We propose new semi-implicit algorithms based on inexact block-

A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method

LU

Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction

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, reducing the computational cost. We investigate explicit-implicit decomposition through algebraic splitting techniques originally designed for the FSI problem. This approach leads to two different families of methods which extend to FSI the algebraic pressure correction method and the Yosida method, two schemes that were previously adopted for pure fluid problems. Furthermore, we have considered the inexact factorization of the fluid-structure system as a preconditioner. The numerical properties of these methods have been tested on a model problem representing a blood-vessel system.

Validation of an open source framework for the simulation of blood flow in rigid and deformable vessels

We address the numerical simulation of fluid-structure interaction problems characterized by a strong added-mass effect. We propose a semi-implicit coupling scheme based on an algebraic fractional-step method. The basic idea of a semi-implicit scheme consists in coupling implicitly the added-mass effect, while the other terms (dissipation, convection and geometrical nonlinearities) are treated explicitly. Thanks to this kind of explicit–implicit splitting, computational costs can be reduced (in comparison to fully implicit coupling algorithms) and the scheme remains stable for a wide range of discretization parameters. In this paper we derive this kind of splitting from the algebraic formulation of the coupled fluid-structure problem (after finite-element space discretization). From our knowledge, it is the first time that algebraic fractional step methods, used thus far only for fluid problems in computational domains with rigid boundaries, are applied to fluid-structure problems. In particular, for the specific semi-implicit method presented in this work, we adapt the Yosida scheme to the case of a coupled fluid-structure problem. This scheme relies on an approximate LU block factorization of the matrix obtained after the discretization in time and space of the fluid-structure system. We analyze the numerical performances of this scheme on 2D fluid-structure simulations performed with a simple 1D structure model.

A Modular, Operator Splitting Scheme for Fluid-Structure Interaction Problems with Thick Structures

The interaction between a fluid and a poroelastic structure is a complex problem that couples the Navier-Stokes equations with the Biot system. The finite element approximation of this problem is involved due to the fact that both subproblems are indefinite. In this work, we first design residual-based stabilization techniques for the Biot system, motivated by the variational multiscale approach. Then, we state the monolithic Navier-Stokes/Biot system with the appropriate transmission conditions at the interface. For the solution of the coupled system, we adopt both monolithic solvers and heterogeneous domain decomposition strategies. Different domain decomposition methods are considered and their convergence is analyzed for a simplified problem. We compare the efficiency of all the methods on a test problem that exhibits a large added-mass effect, as it happens in hemodynamics applications.

Numerical approximation of a control problem for advection-diffusion processes

We discuss in this paper the validation of an open source framework for the solution of problems arising in hemodynamics. The proposed framework is assessed through experimental data for fluid flow in an idealized medical device with rigid boundaries and a numerical benchmark for flow in compliant vessels. The core of the framework is an open source parallel finite element library that features several algorithms to solve both fluid and fluid-structure interaction problems. The numerical results for the flow in the idealized medical device (consisting of a conical convergent, a narrow throat, and a sudden expansion) are in good quantitative agreement with the measured axial components of the velocity and pressures for three different flow rates corresponding to laminar, transitional, and turbulent regimes. We emphasize the crucial role played by the accuracy in performing numerical integration, mesh, and time step to match the measurements. The numerical fluid-structure interaction benchmark deals with the propagation of a pressure wave in a fluid-filled elastic tube. The computed pressure wave speed and frequency of oscillations, and the axial velocity of the fluid on the tube axis are close to the values predicted by the analytical solution associated with the benchmark. A detailed account of the methods used for both benchmarks is provided.

Validation of a 3D computational fluid-structure interaction model simulating flow through an elastic aperture

We present an operator-splitting scheme for fluid-structure interaction (FSI) problems in hemodynamics, where the thickness of the structural wall is comparable to the radius of the cylindrical fluid domain. The equations of linear elasticity are used to model the structure, while the Navier-Stokes equations for an incompressible viscous fluid are used to model the fluid. The operator splitting scheme, based on Lie splitting, separates the elastodynamics structure problem, from a fluid problem in which structure inertia is included to achieve unconditional stability. We prove energy estimates associated with unconditional stability of this modular scheme for the full nonlinear FSI problem defined on a moving domain, without requiring any sub-iterations within time steps. Two numerical examples are presented, showing excellent agreement with the results of monolithic schemes. First-order convergence in time is shown numerically. Modularity, unconditional stability without temporal sub-iterations, and simple implementation are the features that make this operator-splitting scheme particularly appealing for multi-physics problems involving fluid-structure interaction.

Symmetry breaking and preliminary results about a Hopf bifurcation for incompressible viscous flow in an expansion channel

Two different approaches are proposed to enhance the efficiency of the numerical resolution of optimal control problems governed by a linear advection-diffusion equation. In the framework of the Galerkin-Finite Element (FE) method, we adopt a novel a posteriori error estimate of the discretization error on the cost functional; this estimate is used in the course of a numerical adaptive strategy for the generation of efficient grids for the resolution of the optimal control problem. Moreover, we propose to solve the control problem by adopting a reduced basis (RB) technique, hence ensuring rapid, reliable and repeated evaluations of input-output relationship. Our numerical tests show that by this technique a substantial saving of computational costs can be achieved.

Numerical characterization of hemodynamics conditions near aortic valve after implantation of Left Ventricular Assist Device.

This work presents a validation of a fluid-structure interaction computational model simulating the flow conditions in an in vitro mock heart chamber modeling mitral valve regurgitation during the ejection phase during which the trans-valvular pressure drop and valve displacement are not as large. The mock heart chamber was developed to study the use of 2D and 3D color Doppler techniques in imaging the clinically relevant complex intra-cardiac flow events associated with mitral regurgitation. Computational models are expected to play an important role in supporting, refining, and reinforcing the emerging 3D echocardiographic applications. We have developed a 3D computational fluid-structure interaction algorithm based on a semi-implicit, monolithic method, combined with an arbitrary Lagrangian-Eulerian approach to capture the fluid domain motion. The mock regurgitant mitral valve corresponding to an elastic plate with a geometric orifice, was modeled using 3D elasticity, while the blood flow was modeled using the 3D Navier-Stokes equations for an incompressible, viscous fluid. The two are coupled via the kinematic and dynamic conditions describing the two-way coupling. The pressure, the flow rate, and orifice plate displacement were measured and compared with numerical simulation results. In-line flow meter was used to measure the flow, pressure transducers were used to measure the pressure, and a Doppler method developed by one of the authors was used to measure the axial displacement of the orifice plate. The maximum recorded difference between experiment and numerical simulation for the flow rate was 4%, the pressure 3.6%, and for the orifice displacement 15%, showing excellent agreement between the two.

Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology

This computational study shows, for the first time, a clear transition to two-dimensional Hopf bifurcation for laminar incompressible flows in symmetric plane expansion channels. Due to the well-known extreme sensitivity of this study on computational mesh, the critical Reynolds numbers for both the known symmetry-breaking (pitchfork) bifurcation and Hopf bifurcation were investigated for several layers of mesh refinement. It is found that under-refined meshes lead to an overestimation of the critical Reynolds number for the symmetry breaking and an underestimation of the critical Reynolds number for the Hopf bifurcation.