Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach

We develop a computational model to study the interaction of a fluid with a poroelastic material. The coupling of Stokes and Biot equations represents a prototype problem for these phenomena, which feature multiple facets. On one hand it shares common traits with fluid-structure interaction. On the other hand it resembles the Stokes-Darcy coupling. For these reasons, the numerical simulation of the Stokes-Biot coupled system is a challenging task. The need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners are typical shortcomings of classical discretization methods applied to this problem. The application of loosely coupled time advancing schemes mitigates these issues because it allows to solve each equation of the system independently with respect to the others. In this work we develop and thoroughly analyze a loosely coupled scheme for Stokes-Biot equations. The scheme is based on Nitsche's method for enforcing interface conditions. Once the interface operators corresponding to the interface conditions have been defined, time lagging allows us to build up a loosely coupled scheme with good stability properties. The stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem. The error of the resulting method is also analyzed, showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes. In order to restore good approximation properties, while maintaining the computational efficiency of the loosely coupled approach, we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach. Both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand

Refactorization of Cauchy's method: a second-order partitioned method for fluid-thick structure interaction problems

This work focuses on the derivation and the analysis of a novel, strongly-coupled partitioned method for fluid-structure interaction problems. The flow is assumed to be viscous and incompressible, and the structure is modeled using linear elastodynamics equations. We assume that the structure is thick, i.e., modeled using the same number of spatial dimensions as fluid. Our newly developed numerical method is based on generalized Robin boundary conditions, as well as on the refactorization of the Cauchy's one-legged `theta-like' method, written as a sequence of Backward Euler-Forward Euler steps used to discretize the problem in time. This family of methods, parametrized by theta, is B-stable for any theta in [0.5,1] and second-order accurate for theta=0.5+O(tau), where tau is the time step. In the proposed algorithm, the fluid and structure subproblems, discretized using the Backward Euler scheme, are first solved iteratively until convergence. Then, the variables are linearly extrapolated, equivalent to solving Forward Euler problems. We prove that the iterative procedure is convergent, and that the proposed method is stable provided theta in [0.5,1]. Numerical examples, based on the finite element discretization in space, explore convergence rates using different values of parameters in the problem, and compare our method to other strongly-coupled partitioned schemes from the literature. We also compare our method to both a monolithic and a non-iterative partitioned solver on a benchmark problem with parameters within the physiological range of blood flow, obtaining an excellent agreement with the monolithic scheme

A FLUID-STRUCTURE INTERACTION MODEL CAPTURING LONGITUDINAL DISPLACEMENT IN ARTERIES: MODELING, COMPUTATIONAL METHOD, AND COMPARISON WITH EXPERIMENTAL DATA

The focus of this thesis is on numerical modeling of fluid-structure interaction (FSI) problems with application to hemodynamics. Recent in vivo studies, utilizing ultrasound contour and speckle tracking methods, have identified significant longitudinal wall displacements and viscoelastic arterial wall properties over a cardiac cycle. Existing computational models that use thin structure approximations of arterial walls have so far been limited to elastic models that capture only radial wall displacements. In this thesis, we present a new model and a novel loosely coupled partitioned numerical scheme modeling fluid-structure interaction (FSI) in blood flow allowing non-zero longitudinal displacement. In this work arterial walls are modeled by a linearly viscoelastic, cylindrical Koiter shell model capturing both radial and longitudinal displacement. Fluid flow is modeled by the Navier-Stokes equations for an incompressible, viscous fluid. The two are fully coupled via kinematic and dynamic coupling conditions. The proposed numerical scheme is based on a new modified Lie operator splitting that decouples the fluid and structure sub-problems in a way that leads to a loosely coupled scheme that is unconditionally stable. This was achieved by a clever use of the kinematic coupling condition at the fluid and structure sub-problems, leading to an implicit coupling between the fluid and structure velocities. The proposed scheme is a modification of the recently introduced “kinematically coupled scheme” for which the newly proposed modified Lie splitting significantly increases the accuracy. In this work it is shown that the new scheme, called the kinematically coupled β-scheme, is unconditionally stable for all β ∈ [0, 1]. The performance and accuracy of the scheme are studied on a series of instructive examples including a comparison with a monolithic scheme proposed by Quaini and Quarteroni in [77]. It is shown that the accuracy of our scheme is comparable to that of the monolithic scheme, while our scheme retains all the main advantages of partitioned schemes. The results of the computational model are compared with in vivo measurements of the common carotid artery wall motion, and with data capturing stenosed coronary arteries, showing excellent agreement.Mathematics, Department o