Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 50

1.3 Numerical Analysis

As we have seen previously the coupling conditions at the interface are of two types:
kinematic (equality of the velocities in the case of a viscous flow) and dynamic (the
action-reaction principle). From a numerical point of view the most direct way to
satisfy both these two conditions is to solve the coupled fluid-structure problem
thanks to a unique solver based on a global weak formulation. One obtains then
schemes that are called monolithic . Examples of such approach are numerous.
Without being exhaustive, one can refer to [ 101 , 144 ] based on an ALE formula-
tion of the fluid equations, [ 4 , 36 ] based on fictitious domain method or to [ 65 ].
By construction both the kinematic and the dynamic conditions are satisfied and
it leads to strongly coupled schemes. These methods conserve the energy at the
fluid-structure interface and are consequently usually stable. Nevertheless they are
not really modular and do not allow to take easily advantage of the specificities of
each sub-problem. From the numerical analysis point of view, we refer to [ 44 , 92 ,
115 , 148 ], for stability and convergences studies on fully coupled schemes.
With the stagerred schemes the fluid and the structure are computed by two
specific solvers and one question is then: how to couple these two solvers efficiently?
Afirststrategyistousea weakly coupled or explicit schemes where the fluid and the
structure equations are solved once per time step. The advantage of such methods
is that they are cheap since they require the resolution of the fluid and the structure
only once per time step. These methods have been wildly used for the simulation of
compressible flows (see, for instance, [ 134 - 136 ]). We refer to [ 51 ] for a state of the
art of such methods.
Nevertheless the kinematic condition and the dynamic condition are not both
satisfied and it may lead to numerical instabilities in particular in the case of strong
added mass effect, as already mentioned in the previous section and as we will see
in Sect. 1.3.1 . These instabilities were seen in the case of blood flows in arteries,
where the incompressible flow interacts with a structure whose density is closed
to the fluid density. The role played by the added mass was underlined in [ 116 ]
and formalized in [ 25 ] (and later on in [ 72 ] for more general time discretization
schemes). We will see, on the toy model introduced in Sect. 1.2.1 that an explicit
scheme, based on a Dirichlet to Neumann splitting, is unconditionally unstable if
the fluid-structure density ratio is less than a given quantity (that may depend on
the geometrical characteristics of the considered problem). It is strongly linked to
the fluid incompressibility.
Consequently, strongly coupled or implicit schemes have been further developed
and used to obtain stable numerical methods. These schemes are naturally stable
since they conserve the energy balance at the interface. Nevertheless they are costly
since they require many requests to each solver at each time step. Thus, in the
early 2000 year efforts have been made to accelerate the convergence and the
efficiency of fixed point procedures [ 37 , 116 , 127 , 128 , 131 ]orofinexactNewton
methods [ 101 , 124 , 125 , 156 ] or exact Newton methods [ 58 - 61 ]. Moreover more
recently methods have been introduced based on Robin-Robin decomposition [ 6 ]