14 research outputs found
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