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					Numerical Simulations of the Fluid-Structure Coupling
              in Physiological Vessels
                        Mini-Workshop I


                                   a
                     Miguel A. Fern´ndez
                     INRIA Rocquencourt, projet REO
                       miguel.fernandez@inria.fr


                                       e
               CRM Spring School, Montr´al
                     May 16, 2005




                            a
              Miguel A. Fern´ndez      Mini-Workshop I
Motivations




  It is generally admitted that local blood flow behavior (shear stress) plays a key role in the
  understanding of several cardiovascular diseases

  Artery walls are compliant

  We aim at simulating the mechanical interaction between the blood flow and the arterial wall
  [press] [defor] [caro] [anev]

  Difficulties:
      Large displacements: geometrical non-linearities, moving computational domain
      Numerical stability: blood and wall densities are close
      Computational cost: strong research effort to improve the coupling algorithms
                                       FSI cost       fluid cost + solid cost




                                              a
                                Miguel A. Fern´ndez    Mini-Workshop I
Outline



1   Motivations
2   Part I: Model Equations
        The Structure Equations
        The Fluid Equations: ALE formulation
        The Coupled Problem
        Global Energy Balance

3   Part II: Explicit and Implicit Coupling
        Explicit Coupling
        Numerical Stability: Added Mass Effect
        Implicit Coupling
        Fixed-Point and Newton Methods

4   Part III: Semi-Implicit Coupling
        Remarks on Explicit and Implicit Coupling
        A New Coupling Scheme
        Numerical Stability




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
                   Part I

Model Equations: Kinematics




                a
  Miguel A. Fern´ndez   Mini-Workshop I
Problem Setting

                                        Ωs (t)


                                            Γw (t)

                                Γin                             Γout
                                           Ωf (t)




      Ωf (t): fluid domain (filled with a viscous fluid, blood),
      Ωs (t): solid current configuration (arterial wall)
     Γw (t): fluid-solid interface, where we enforce
                  continuity of velocity
                  equilibrium of stress




                                            a
                              Miguel A. Fern´ndez    Mini-Workshop I
Problem Setting

                                        Ωs (t)


                                            Γw (t)

                                Γin                             Γout
                                           Ωf (t)




      Ωf (t): fluid domain (filled with a viscous fluid, blood),
      Ωs (t): solid current configuration (arterial wall)
      Γw (t): fluid-solid interface, where we enforce
                   continuity of velocity
                   equilibrium of stress


  The problem
  Determine Ωf (t), velocity and stress within the fluid and the solid.



                                            a
                              Miguel A. Fern´ndez    Mini-Workshop I
Solid Kinematics: Lagrange Formulation (I)

  Let Ωs be reference material configuration, for instance Ωs = Ωs (0).
       0                                                   0
                                                      xs (·, t)



               Ωs                     Γw
                                       0                                    Γw (t)   Ωs (t)
                0




  We introduce the solid deformation xs as an application

                                          xs : Ωs × R+ −→ R3 .
                                                0

  that maps Ωs into Ωs (t), i.e. xs (Ωs ) = Ωs (t).
             0                        0
    def
  x = xs (x0 , t) gives the position, at time t, of the material point x0 ∈ Ωs .
                                                                             0
  In other words, xs tracks the material points motion.




                                              a
                                Miguel A. Fern´ndez       Mini-Workshop I
 Solid Kinematics: Lagrange Formulation (II)

We introduce the following notations:
                             def
    displacement: ds (x0 , t) = xs (x0 , t) − x0
    velocity:
                                                                 s
                                                        ˙ def ∂d
                                                       ds =
                                                              ∂t
    acceleration:
                                                              2 s
                                                       ¨ def ∂ d
                                                       ds =
                                                             ∂t 2
                                     def
    deformation gradient: F (ds ) =          0x
                                                  s   =I+      0d
                                                                    s

    jacobian: J(ds ) = det F (ds )
    solid density in the reference configuration: ρs (x0 )
    second Piola-Kirchoff stress tensor S(ds ), related to ds through a appropriate consitutive law.
    For instance, for an hyper-elastic material we have

                                   ∂W (E (ds ))                     def   1“                     ”
                       S(ds ) =                 ,          E (ds ) =         F (ds )T F (ds ) − I ,
                                      ∂E                                  2
    with W a given elastic energy density and E (ds ) the Green-Lagrange stress tensor.
    unit normal vector on ∂Ωs : ns
                            0    0




                                                   a
                                     Miguel A. Fern´ndez       Mini-Workshop I
Solid Equations: Lagrange Formulation

  From the Momentum conservation, we have that ds solves:
                                ¨
                             ρs ds − div0 (F (ds )S(ds )) = 0,                   in    Ωs ,
                                                                                        0

  Boundary conditions:

                                 ds = 0,       on       ΓD ,
                                                         0

                 F (ds )S(ds )ns = 0,
                               0               on       ΓN ,
                                                         0
                     s       s
                 F (d )S(d       )ns
                                   0   = −gfluid ,        on       Γw
                                                                   0     (equilibrium of stress).

                                                         ΓN
                                                          0


                                  ΓD
                                   0              Ωs
                                                   0                 Γw
                                                                      0               ΓD
                                                                                       0


                                                          ΓN
                                                           0




                                                a
                                  Miguel A. Fern´ndez          Mini-Workshop I
 Solid Equations: Variational Formulation


         Test functions space:
                                                       n                                           o
                              def1               def
                          V s = HΓD (Ωs ) =
                                      0                    vs ∈ H 1 (Ωs ) :
                                                                      0
                                                                                      s
                                                                                     v|ΓD = 0
                                       0                                                 0




         Solid variational formulation
         Find ds (t) ∈ V s such that
              Z                   Z                                        Z
                      ¨
                   ρs d s · v s +   F (ds )S(ds ) :          0d
                                                                  s
                                                                      =−            gfluid · vs ,       ∀vs ∈ V s .
               Ωs
                0                Ωs
                                  0                                            Γw
                                                                                0


Idea of the proof. Multiply by vs ∈ V s , integrate by parts and take into account the boundary
conditions.




                                                    a
                                      Miguel A. Fern´ndez         Mini-Workshop I
Fluid Equations: Some Notation

  We assume the fluid to be homogeneous, Newtonian and incompressible.
  We introduce the notations:
      density: ρf
      dynamic viscosity: µ
      velocity: u(x, t), x ∈ Ωf (t)
      pressure: p(x, t)
      Cauchy stress tensor: σ(u, p), given by
                                              σ(u, p) = −pI + 2µε(u)
      strain rate tensor: ε(u), defined as
                                                   def   1h           i
                                             ε(u) =         u + ( u)T
                                                         2
      unit normal vector on ∂Ωf (t): n
      unit normal vector on ∂Ωf : n0
                              0




                                               a
                                 Miguel A. Fern´ndez      Mini-Workshop I
Fluid Equations: Eulerian Formulation (I)


                                          Ωf (t)
                             Γin          Γw (t)                    Γout




  Incompressible Navier-Stokes equations in Eulerian formulation:
                       „             «
                         ∂u
                    ρf      + u · u − 2µ div ε(u) + p = 0, in                   Ωf (t),
                         ∂t
                                                              div u = 0,   in   Ωf (t).

  Boundary conditions:

                    σ(u, p)n = gin−out ,       on      Γin−out ,
                                ˙
                           u = ds ,     on    Γ (t)w
                                                          (continuity of velocities),




                                           a
                             Miguel A. Fern´ndez       Mini-Workshop I
Fluid Equations: Eulerian Formulation (II)

                                                Γw(tn+1)
                     w
                    Γ (tn )
                                            j           i
                                                i
                                        j
                                                                          Ωf (tn )
                                                            Ωf (tn+1)
      Remarks
         In the sequel we will deal with fluid meshes following the interface motion
         in order (accurate interface fluid load).
         In such case, the approximation of

                                                    ∂u
                                                       (x, t),
                                                    ∂t
         becomes troublesome. This can be overcome by the introduction of a
         change of variables (the ALE map)




                                            a
                              Miguel A. Fern´ndez       Mini-Workshop I
Fluid Domain Description (I)

              def
  We set Ωf = Ωf (0) as a reference fluid domain.
          0
                                                   xf (·, t)

                                                                           Ωf (t)
                      Ωf
                       0             Γw                                    Γw (t)
                                      0



  Assume we know the interface motion through the solid deformation x s w .
                                                                      |Γ            0

  We introduce the fluid domain deformation xf as an injective application

                                            xf : Ωf × R+ −→ R3 ,
                                                  0

  such that
                                    xf w = xs w ,
                                     |Γ     |Γ          xf (Ωf ) = Ωf (t).
                                                             0
                                        0          0

  In other words xf = Ext(xs w ), with Ext a given extension operator.
                           |Γ0


                            Remark
                            Inside Ωf the map xf is arbitrary
                                    0


                                               a
                                 Miguel A. Fern´ndez     Mini-Workshop I
Fluid Domain Description (II)

  Thus, xf tracks the motion of the fluid domain. But, in general, it doesn’t track the motion
  of the fluid particles.
  It is useful to introduce:
                                                   def
      the fluid domain displacement: df (x0 , t) = xf (x0 , t) − x0

                    Example of extension operator:
                                                         8    f                          f
                                                         > −∆d = 0,
                                                         >                       in     Ω0 ,
                                                         <
                                                                f            s
                                   Harmonic lift:              d =d ,            on     Γw ,
                                                                                         0
                                                         >
                                                         >
                                                                df = 0,
                                                         :
                                                                                 on     Γin−out ,

      the fluid domain velocity: w, defined by

                                                     def     ∂df      ∂xf
                                                w =                 =
                                                             ∂t |x0   ∂t          |x0

                                         def
      the deformation gradient: F (df ) =       0x
                                                     f
                                                         =I+        0d
                                                                         f

      and the jacobian: J(df ) = det F (df )




                                                a
                                  Miguel A. Fern´ndez          Mini-Workshop I
Fluid Domain Description (III)

      Remarks
         In general, w(x0 , t) = u(xf (x0 , t), t) for x0 ∈ Ωf .
                                                             0
         However, since by definition xf = xs on Γw , we have
                                                 0

                                     ∂df      ∂ds
                                            =        ,       on    Γw ,
                                                                    0
                                     ∂t |x0   ∂t |x0

         and, therefore, w = u on Γw
                                   0




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
 Time Derivatives: Eulerian vs. ALE (I)

Let u = u(x, t) be a vector field defined on Ωf (t), ∀t > 0.
    For x ∈ Ωf (t), the Arbitrary Lagrange-Euler (ALE) derivative of u is defined by

                                       ∂u            def ∂
                                              (x, t) =      u(xf (x0 , t), t),
                                       ∂t |x0            ∂t
             ˆ          ˜−1
    with x0 = xf (·, t)     (x), i.e. x = xf (x0 , t).


 Proposition (ALE vs. Eulerian)

 Let x ∈ Ωf (t) with x = xf (x0 , t) and x0 ∈ Ωf . Then
                                               0

                      ∂u                                                   ∂u
                              (x, t) = w(x0 , t) ·        u(x, t) +            (x, t)     ,
                       ∂t |x0                                               ∂t
                      |     {z     }                                       | {z }
                     ALE derivative                                   Eulerian derivative

        def ˙
 with w = df .

Idea of the proof. Use chain rule.




                                                  a
                                    Miguel A. Fern´ndez    Mini-Workshop I
Why the terminology ALE?




  If w = 0, i.e. the fluid domain does not changes in time, we have

                               ∂u                 ∂u
                                       (x, t) =       (x, t)     ,
                                ∂t |x0             ∂t
                               |     {z     }     | {z }
                              ALE derivative Eulerian derivative
  hence we recover the classical Eulerian derivative
  if w = u, i.e the map xf tracks the fluid particles motion, we have

                        ∂u                                   ∂u
                                (x, t) = u(x, t) · u(x, t) +    (x, t) ,
                         ∂t |x0                              ∂t
                        |     {z     }   |            {z             }
                       ALE derivative material or Lagrangian derivative
  hence we recover the classical Lagrangain or material derivative




                                            a
                              Miguel A. Fern´ndez   Mini-Workshop I
Fluid Equations: ALE Formulation


                                         Ωf (t)
                            Γin          Γw (t)                 Γout




  Incompressible Navier-Stokes equations in ALE formulation:
                   „                     «
                     ∂u
                ρf          + (w − u) · u − 2µ div ε(u) + p = 0,             in   Ωf (t),
                     ∂t |x0
                                                                div u = 0,   in   Ωf (t).

  Boundary conditions:

                                  σ(u, p)n = gin ,      on      Γin
                                  σ(u, p)n = gout ,        on    Γin
                                                                w
                                            u = w,    on     Γ (t)




                                          a
                            Miguel A. Fern´ndez   Mini-Workshop I
 Fluid Equations: ALE Variational Formulation (I)


         Test functions space
                         n                                                                                                 o
                V f (t) = vf : Ωf (t) → R3 : vf = ˆ ◦ [xf (·, t)]−1 ,
                                                  v                                                           1
                                                                                                         ˆ ∈ HΓw (Ωf )
                                                                                                         v         0
                                                                                                                  0


In particular, for each vf ∈ V f we have

                                                                    ∂vf
                                                                           = 0.
                                                                    ∂t |x0


         Fluid variational formulation
         Find u(t) ∈ H 1 (Ωf (t)) with u|Γw (t) = w|Γw (t) and p(t) ∈ L2 (Ωf (t)) such that
                   Z                           Z                                      Z
             d
                                ρf u · v f −                ρf (div w)u · vf +                     ρf (w − u) ·       u · vf
             dt        Ωf (t)                      Ωf (t)                                 Ωf (t)
                        Z                                   Z                         Z
             + 2µ                    ε(u) : ε(vf ) −                     p div vf =                  g · vf ,   ∀vf ∈ V f (t),
                            Ωf (t)                              Ωf (t)                    Γin−out
             Z
                          q div u = 0,         ∀q ∈ L2 (Ωf (t)),
                 Ωf (t)




                                                             a
                                               Miguel A. Fern´ndez             Mini-Workshop I
 Fluid Equations: ALE Variational Formulation (II)

Idea of the proof. By multiplying by vf ∈ V f , integrating by parts and taking into account the
boundary conditions. Here we only consider the mass term:
              Z                         Z
                          ∂u                           ∂u
                       ρf        · vf =      ρf J(df )         · vf
                Ωf (t)    ∂t |x0          Ωf
                                           0
                                                       ∂t |x0
                                        Z           “                ” Z ∂J(df )
                                              ∂
                                      =               ρf J(df ) · vf −                        ρf u · v f
                                          Ωf ∂t |x0
                                           0                                  Ωf
                                                                               0
                                                                                    ∂t |x0
                                             Z                        Z
                                         d
                                      =           J(df )ρf u · vf −          J(df )(div w)ρf u · vf
                                        d t Ωf  0                       Ωf 0
                                             Z                    Z
                                         d
                                      =             ρf u · v f −           (div w)ρf u · vf .
                                        d t Ωf (t)                  Ωf (t)

where we used the fact that
                                         ∂J(df )
                                                  = J(df ) div w.
                                           ∂t |x0




                                                  a
                                    Miguel A. Fern´ndez   Mini-Workshop I
 Fluid Load at the Interface: Variational Residual

Proposition
Let (u, p) solution of the previous fluid sub-problem, v s ∈ V s and
                                                              1
                                                                     1
                                                   L : H 2 (Γw ) −→ H∂Ωf −Γw (Ωf ),
                                                             0                 0
                                                                                        0    0


a given lift operator. Let          vf   =   L(vs )       ◦   [xf (·, t)]−1    (obviously vf ∈ V f (t)), then
                                                                                             /
   Z                       Z                                           Z                           Z
                                                                  d
            gfluid · vs =                σ(u, p)n · vs =                             ρf u · v f −                ρf (div w)u · vf
       Γw
        0                      Γw (t)                             dt       Ωf (t)                      Ωf (t)
                                             Z                                                   Z                               Z
                                         +                ρf (w − u) ·         u · vf + 2µ                     ε(u) : ε(vf ) −                p div vf .
                                                 Ωf (t)                                               Ωf (t)                         Ωf (t)

Idea of the proof. Integration by parts.

Remarks:
       We solve the fluid equations with test functions vanishing on the interface.
       We compute the forcing term with test functions not vanishing on the interface. This
       explains the terminology “variational residual”
       This approach has a straightforward discrete counterpart.



                                                                 a
                                                   Miguel A. Fern´ndez              Mini-Workshop I
The Coupled Problem: Strong Formulation


  Fluid:
                    „                              «
                        ∂u
               ρf              + (u − w) ·     u       − 2µ div ε(u) +             p = 0,    in    Ωf (t),
                        ∂t |x0
                                                                             div u = 0,      in    Ωf (t),
                                                             σ(u, p)n = gin−out ,            on    Γin−out ,

  Solid:
                                 ¨
                              ρs ds − div0 (F (ds )S(ds )) = 0,              in      Ωs ,
                                                                                      0

                                                                       ds = 0,        on    ΓD ,
                                                                                             0
                                                         s        s
                                                   F (d )S(d          )ns
                                                                        0   = 0,      on    ΓN ,
                                                                                             0

  Coupling conditions: geometry, velocity and stress

                                          df = Ext(ds w ),
                                                    |Γ                 in     Ωf ,
                                                                               0       Ωf (t) = (I + df )(Ωf ),
                                                                                                           0
                                                              0

                                                   u = w(df ),         on     Γw (t),
                                                       f −T
           F (ds )S(ds )n0 = J(df )σ(u, p)F (d )              n0 ,     on     Γw .
                                                                               0




                                                 a
                                   Miguel A. Fern´ndez        Mini-Workshop I
The Coupled Problem: Variational Formulation


  Fluid domain motion:

                                 df = Ext(ds w ),                       ˙
                                                                   w = df ,           Ωf (t) = (I + df )(Ωf ).
                                           |Γ         0
                                                                                                          0


  Fluid: Find u(t) ∈ H 1 (Ωf (t)) with u|Γw (t) = w|Γw (t) and p(t) ∈ L2 (Ωf (t)) such that
                   Z                            Z                                        Z
             d
                                ρf u · v f −                 ρf (div w)u · vf +                       ρf (w − u) ·      u · vf
             dt        Ωf (t)                       Ωf (t)                                   Ωf (t)
                        Z                                    Z                           Z
              + 2µ                   ε(u) : ε(vf ) −                      p div vf =                     g · vf ,    ∀vf ∈ V f (t),
                            Ωf (t)                               Ωf (t)                      Γin−out
             Z
                          q div u = 0,          ∀q ∈ L2 (Ωf (t)),
                 Ωf (t)



  Solid: Find ds (t) ∈ V s such that
              Z                   Z                                                          Z
                      ¨
                   ρs d s · v s +   F (ds )S(ds ) :                          0d
                                                                                  s
                                                                                      =−              gfluid · vs ,    ∀vs ∈ V s .
                  Ωs
                   0                       Ωs
                                            0                                                    Γw
                                                                                                  0




                                                         a
                                           Miguel A. Fern´ndez                Mini-Workshop I
The Coupled Problem: Variational Formulation

  Fluid domain motion:

                                 df = Ext(ds w ),                      ˙
                                                                  w = df ,           Ωf (t) = (I + df )(Ωf ).
                                           |Γ        0
                                                                                                         0


  Fluid: Find u(t) ∈ H 1 (Ωf (t)) with u|Γw (t) = w|Γw (t) and p(t) ∈ L2 (Ωf (t)) such that
                   Z                           Z                                         Z
             d
                                ρf u · v f −                ρf (div w)u · vf +                        ρf (w − u) ·    u · vf
             dt        Ωf (t)                      Ωf (t)                                    Ωf (t)
                        Z                                   Z                           Z
             + 2µ                    ε(u) : ε(vf ) −                     p div vf =                     g · vf ,   ∀vf ∈ V f (t),
                            Ωf (t)                              Ωf (t)                      Γin−out
             Z
                          q div u = 0,          ∀q ∈ L2 (Ωf (t)),
                 Ωf (t)



  Solid: Find ds (t) ∈ V s such that
            Z                 Z
                                                                                         `                 ´
                   ¨
               ρs d s · v s +   F (ds )S(ds ) :                           0d
                                                                               s
                                                                                   = − Rf u, p, df ; L(vs ) ,          ∀vs ∈ V s .
             Ωs
              0                         Ωs
                                         0
                                                                                       |        {z         }
                                                                                           variational
                                                                                         fluid residual



                                                           a
                                             Miguel A. Fern´ndez               Mini-Workshop I
Global Energy Equality




Proposition
Assume that u = 0 on Γin−out and that the solid is an hyper-elastic material. Then, we have
the following energy equality
                Z                Z             Z               ! Z
            d            ρf 2         ρs ˙s 2
                           |u| +        |d | +      W (E (ds )) +         2µ|ε(u)|2 = 0
            dt    Ωf (t) 2         Ωs 2
                                    0            Ωs
                                                  0                Ωf (t)




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Global Energy Equality




Proposition
Assume that u = 0 on Γin−out and that the solid is an hyper-elastic material. Then, we have
the following energy equality
               „Z                 Z             Z                « Z
            d            ρf 2          ρs ˙s 2
                           |u| +         |d | +      W (E (ds )) +          2µ|ε(u)|2 = 0
            dt    Ωf (t) 2          Ωs 2
                                     0            Ωs
                                                   0                 Ωf (t)
                |              {z            } |       {z      }   |        {z      }
                            Kinetic                  Elastic          Dissipated
                            energy                   energy               power




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
 Global Energy Equality




 Proposition
 Assume that u = 0 on Γin−out and that the solid is an hyper-elastic material. Then, we have
 the following energy equality
                „Z                 Z             Z                « Z
             d            ρf 2          ρs ˙s 2
                            |u| +         |d | +      W (E (ds )) +          2µ|ε(u)|2 = 0
             dt    Ωf (t) 2          Ωs 2
                                      0            Ωs
                                                    0                 Ωf (t)
                 |              {z            } |       {z      }   |        {z      }
                             Kinetic                  Elastic          Dissipated
                             energy                   energy               power

                                                                      ˙
Idea of the proof. Multiply the fluid and solid equations by u(t) and ds , respectively, integrate by
parts and take into account the boundary conditions.




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
                             Part II

Time Discretization: Explicit and Implicit Coupling




                           a
             Miguel A. Fern´ndez   Mini-Workshop I
Time Discretization

  Fluid domain motion:

                                 df = Ext(ds w ),                      ˙
                                                                  w = df ,           Ωf (t) = (I + df )(Ωf ).
                                           |Γ                                                            0
                                                     0



  Fluid : Find u(t) ∈ H 1 (Ωf (t)) with u|Γw (t) = w|Γw (t) and p(t) ∈ L2 (Ωf (t)) such that
                   Z                           Z                                        Z
             d
                                ρf u · v f −                ρf (div w)u · vf +                       ρf (w − u) ·    u · vf
             dt        Ωf (t)                      Ωf (t)                                   Ωf (t)
                        Z                                   Z                           Z
             + 2µ                    ε(u) : ε(vf ) −                     p div vf =                    g · vf ,   ∀vf ∈ V f (t),
                            Ωf (t)                              Ωf (t)                      Γin−out
             Z
                          q div u = 0,          ∀q ∈ L2 (Ωf (t)),
                 Ωf (t)



  Solid : Find ds (t) ∈ V s such that
            Z                  Z
                                                                                        `                 ´
                   ¨
                ρs d s · v s +   F (ds )S(ds ) :                          0d
                                                                               s
                                                                                   = −Rf u, p, df ; L(vs ) ,         ∀vs ∈ V s .
              Ωs
               0                        Ωs
                                         0




                                                           a
                                             Miguel A. Fern´ndez               Mini-Workshop I
Time Discretization

  Fluid domain motion:

                        ˜              n+1                          df,n+1 − df,n
           df,n+1 = Ext(ds |Γw ),                   wn+1 =                        ,              Ωf,n+1 = (I + df,n+1 )(Ωf ).
                                                                                                                         0
                                         0                               ∆t
  Fluid : Find u(t) ∈ H 1 (Ωf (t)) with u|Γw (t) = w|Γw (t) and p(t) ∈ L2 (Ωf (t)) such that
                   Z                           Z                                        Z
             d
                                ρf u · v f −                ρf (div w)u · vf +                       ρf (w − u) ·    u · vf
             dt        Ωf (t)                      Ωf (t)                                   Ωf (t)
                        Z                                   Z                           Z
             + 2µ                    ε(u) : ε(vf ) −                     p div vf =                    g · vf ,   ∀vf ∈ V f (t),
                            Ωf (t)                              Ωf (t)                      Γin−out
             Z
                          q div u = 0,          ∀q ∈ L2 (Ωf (t)),
                 Ωf (t)



  Solid : Find ds (t) ∈ V s such that
            Z                  Z
                                                                                        `                 ´
                   ¨
                ρs d s · v s +   F (ds )S(ds ) :                          0d
                                                                               s
                                                                                   = −Rf u, p, df ; L(vs ) ,         ∀vs ∈ V s .
              Ωs
               0                        Ωs
                                         0




                                                           a
                                             Miguel A. Fern´ndez               Mini-Workshop I
Time Discretization

  Fluid domain motion:

                         ˜         n+1                    df,n+1 − df,n
            df,n+1 = Ext(ds |Γw ),            wn+1 =                    ,             Ωf,n+1 = (I + df,n+1 )(Ωf ).
                                                                                                              0
                                    0                          ∆t

  Fluid (Implicit Euler): Find un+1 ∈ H 1 (Ωf,n+1 ) with un+1
                                                          |Γw,n+1
                                                                     n+1
                                                                  = w|Γw,n+1 and
  p n+1 ∈ L2 (Ωf,n+1 ) such that
                 Z                          Z                    Z
              1                           1
                          ρf un+1 · vf −          ρf u n · v f +           ρf (div wn+1 )un+1 · vf
             ∆t Ωf,n+1                    ∆t Ωf,n                  Ωf,n+1
                Z                                               Z
             +           ρf (wn+1 − un+1 ) · un+1 · vf + 2µ               ε(un+1 ) : ε(vf )
                  Ωf,n+1                                          Ωf,n+1
                Z                       Z
             −           p n+1 div vf =         gn+1 · vf , ∀vf ∈ V f (tn+1 ),
                   Ωf,n+1                         Γin−out

  Solid : Find   ds (t)   ∈   Vs  such that
            Z                     Z
                                                                             `                 ´
                      ¨
                   ρs d s · v s +     F (ds )S(ds ) :          0d
                                                                    s
                                                                        = −Rf u, p, df ; L(vs ) ,      ∀vs ∈ V s .
              Ωs
               0                    Ωs
                                     0




                                                       a
                                         Miguel A. Fern´ndez        Mini-Workshop I
Time Discretization

  Fluid domain motion:

                            ˜     n+1                    df,n+1 − df,n
               df,n+1 = Ext(ds |Γw ),        wn+1 =                    ,              Ωf,n+1 = (I + df,n+1 )(Ωf ).
                                                                                                              0
                                   0                          ∆t

  Fluid (Implicit Euler): Find un+1 ∈ H 1 (Ωf,n+1 ) with un+1
                                                          |Γw,n+1
                                                                     n+1
                                                                  = w|Γw,n+1 and
  p n+1 ∈ L2 (Ωf,n+1 ) such that
                 Z                          Z                    Z
              1                           1
                          ρf un+1 · vf −          ρf u n · v f +           ρf (div wn+1 )un+1 · vf
             ∆t Ωf,n+1                    ∆t Ωf,n                  Ωf,n+1
                Z                                               Z
             +           ρf (wn+1 − un+1 ) · un+1 · vf + 2µ               ε(un+1 ) : ε(vf )
                  Ωf,n+1                                          Ωf,n+1
                Z                       Z
             −           p n+1 div vf =         gn+1 · vf , ∀vf ∈ V f (tn+1 ),
                       Ωf,n+1                    Γin−out

  Solid (Leap-Frog): Find         ds,n+1     ∈ V s such that
     Z                                                  Z
                   ds,n+1 − 2ds,n + ds,n−1
              ρs                           · vs +                F (ds,n+1 )S(ds,n+1 ) :        0d
                                                                                                     s
         Ωs
          0
                           (∆t)2                            Ωs
                                                             0
                                                                      `                              ´
                                                                 = −Rf un+1 , p n+1 , df,n+1 ; L(vs ) ,       ∀vs ∈ V s .



                                                      a
                                        Miguel A. Fern´ndez         Mini-Workshop I
 Coupling Strategies: Explicit/Implicit

Key point:
                  ˜ n+1
Give a prediction ds |Γw of the interface displacement.
                      0




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
 Coupling Strategies: Explicit/Implicit

Key point:
                  ˜ n+1
Give a prediction ds |Γw of the interface displacement.
                      0



Two possibilities:




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
 Coupling Strategies: Explicit/Implicit

Key point:
                  ˜ n+1
Give a prediction ds |Γw of the interface displacement.
                        0



Two possibilities:
                        ˜ n+1
     Explicit coupling: ds |Γw given in terms of the interface displacement at the previous
                               0
     time-steps: ds,n , ds,n−1 , . . .
          0th-order interface prediction:
                                                               ˜ n+1     s,n
                                                               ds |Γw = d|Γw .
                                                                    0        0
          1st-order interface prediction:
                                                                                  s,n−1
                                                                          ds,n − d|Γw
                                                                           |Γw
                                               ˜ n+1
                                               ds |Γw = ds,n + ∆t            0        0
                                                                                          .
                                                         |Γw
                                                    0           0                ∆t
          ...




                                                       a
                                         Miguel A. Fern´ndez        Mini-Workshop I
 Coupling Strategies: Explicit/Implicit

Key point:
                  ˜ n+1
Give a prediction ds |Γw of the interface displacement.
                        0



Two possibilities:
                        ˜ n+1
     Explicit coupling: ds |Γw given in terms of the interface displacement at the previous
                               0
     time-steps: ds,n , ds,n−1 , . . .
          0th-order interface prediction:
                                                               ˜ n+1     s,n
                                                               ds |Γw = d|Γw .
                                                                       0        0
          1st-order interface prediction:
                                                                                     s,n−1
                                                                             ds,n − d|Γw
                                                                              |Γw
                                               ˜ n+1
                                               ds |Γw = ds,n + ∆t               0        0
                                                                                             .
                                                         |Γw
                                                    0              0                ∆t
          ...
     Implicit coupling: No prediction, we simply set

                                                        ˜ n+1
                                                        ds |Γw = ds,n+1 .
                                                               0  |Γw       0




                                                       a
                                         Miguel A. Fern´ndez           Mini-Workshop I
Explicit Coupling: Detailed Algorithm

1   Compute fluid domain displacement and velocity:

                                                               df,n+1 − df,n
                 df,n+1 = Ext(ds,n |Γw ),        wn+1 =                      ,           Ωf,n+1 = (I + df,n+1 )(Ωf ).
                                                                                                                 0
                                       0                            ∆t
2   Solve fluid in the new domain Ωf,n+1 :
                  Z                          Z                    Z
               1                           1
                           ρf un+1 · vf −          ρf u n · v f +           ρf (div wn+1 )un+1 · vf
              ∆t Ωf,n+1                    ∆t Ωf,n                  Ωf,n+1
                 Z                                               Z
              +           ρf (wn+1 − un+1 ) · un+1 · vf + 2µ               ε(un+1 ) : ε(vf )
                   Ωf,n+1                                          Ωf,n+1
                 Z                       Z
              −           p n+1 div vf =         gn+1 · vf , ∀vf ∈ V f (tn+1 ),
                         Ωf,n+1                     Γin−out
                                     n+1
                     un+1 |Γw,n+1 = w|Γw,n+1 .

3   Solve solid with the new fluid forcing term Rf
       Z                                                   Z
                     ds,n+1 − 2ds,n + ds,n−1
                ρs                           · vs +                 F (ds,n+1 )S(ds,n+1 ) :       0d
                                                                                                       s
           Ωs
            0
                             (∆t)2                             Ωs
                                                                0
                                                                         `                              ´
                                                                    = −Rf un+1 , p n+1 , df,n+1 ; L(vs ) ,      ∀vs ∈ V s .

4   Next time step

                                                         a
                                           Miguel A. Fern´ndez         Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:
    This algorithm is very attractive:

                                   FSI cost ≈ fluid cost + solid cost




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:
    This algorithm is very attractive:

                                   FSI cost ≈ fluid cost + solid cost

    We have a fully uncoupled problem.




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:
    This algorithm is very attractive:

                                   FSI cost ≈ fluid cost + solid cost

    We have a fully uncoupled problem.
    Velocity continuity is not exactly enforced (due to extrapolation)
                                                  s,n−1
                                          ds,n − d|Γw
                                           |Γw
                                                                    ds,n+1 − ds,n
                                                                     |Γw      |Γw
                               un+1
                                |Γw
                                      =         0       0
                                                                =       0        0
                                                                                      .
                                  0                ∆t                      ∆t
                                          |         {z     }        |       {z    }
                                              extrapolated              interface
                                                interface                velocity
                                                 velocity




                                                a
                                  Miguel A. Fern´ndez       Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:
    This algorithm is very attractive:

                                   FSI cost ≈ fluid cost + solid cost

    We have a fully uncoupled problem.
    Velocity continuity is not exactly enforced (due to extrapolation)
                                                  s,n−1
                                          ds,n − d|Γw
                                           |Γw
                                                                    ds,n+1 − ds,n
                                                                     |Γw      |Γw
                               un+1
                                |Γw
                                      =         0       0
                                                                =       0        0
                                                                                      .
                                  0                ∆t                      ∆t
                                          |         {z     }        |       {z    }
                                              extrapolated              interface
                                                interface                velocity
                                                 velocity



                     Two major questions:
                          Is energy well balanced at the discrete level?.
                          Is this scheme stable?. [movie1] [movie2]




                                                a
                                  Miguel A. Fern´ndez       Mini-Workshop I
 Explicit Coupling: A Priori Energy Estimate (I)

Proposition
Assume that un+1 = 0 on Γin−out , the structure is an hyper-elastic material and
                 »Z           Z       – Z
               1                                                        ∂q
                           q−        q =          div wn+1 q, ∀q with          = 0.
              ∆t    Ωf,n+1      Ωf,n       Ωf,n+1                       ∂t |x0

Then, the following energy inequality holds
        "Z                      Z                Z        ˛               ˛2 Z         ˛               ˛2 #
   1                ρf n+1 2           ρf n 2          ρs ˛ ds,n+1 − ds,n ˛         ρs ˛ ds,n − ds,n−1 ˛
                      |u   | −           |u | +           ˛
                                                          ˛
                                                                          ˛ −
                                                                          ˛
                                                                                       ˛
                                                                                       ˛
                                                                                                       ˛
                                                                                                       ˛
   ∆t        Ωf,n+1 2             Ωf,n 2           Ωs 2
                                                     0
                                                                 ∆t              Ωs 2
                                                                                  0
                                                                                              ∆t
                         "Z                      Z                  # Z
                      1
                   +           W (E (ds,n+1 )) −       W (E (ds,n )) +          2µ|ε(un+1 )|2
                      ∆t    Ωs
                             0                     Ωs
                                                    0                    Ωf,n+1

                                                    `                                 ds,n+1 − ds,n ´
                                                − Rf un+1 , p n+1 , df,n+1 ; L(un+1 −              ) ≤0
                                                                                           ∆t




                                                    a
                                      Miguel A. Fern´ndez   Mini-Workshop I
 Explicit Coupling: A Priori Energy Estimate (I)

Proposition
Assume that un+1 = 0 on Γin−out , the structure is an hyper-elastic material and
                 »Z           Z       – Z
               1                                                        ∂q
                           q−        q =          div wn+1 q, ∀q with          = 0.
              ∆t    Ωf,n+1      Ωf,n       Ωf,n+1                       ∂t |x0

Then, the following energy inequality holds
        "Z                      Z                Z        ˛               ˛2 Z         ˛               ˛2 #
   1                ρf n+1 2           ρf n 2          ρs ˛ ds,n+1 − ds,n ˛         ρs ˛ ds,n − ds,n−1 ˛
                      |u   | −           |u | +           ˛
                                                          ˛
                                                                          ˛ −
                                                                          ˛
                                                                                       ˛
                                                                                       ˛
                                                                                                       ˛
                                                                                                       ˛
   ∆t        Ωf,n+1 2             Ωf,n 2           Ωs 2
                                                     0
                                                                 ∆t              Ωs 2
                                                                                  0
                                                                                              ∆t
                         "Z                      Z                  # Z
                      1
                   +           W (E (ds,n+1 )) −       W (E (ds,n )) +          2µ|ε(un+1 )|2
                      ∆t    Ωs
                             0                     Ωs
                                                    0                    Ωf,n+1

                                                     `                                   ds,n+1 − ds,n ´
                                                − Rf un+1 , p n+1 , df,n+1 ; L(un+1 −                  ) ≤0
                                                                                              ∆t
                                                  |                          {z                         }
                                                  Z
                                                               n+1     n+1         n+1    ds,n+1 − ds,n
                                                           σ(u      ,p     )n · (u     −                )
                                                    Γw,n+1                                     ∆t




                                                    a
                                      Miguel A. Fern´ndez   Mini-Workshop I
 Explicit Coupling: A Priori Energy Estimate (I)

Proposition
Assume that un+1 = 0 on Γin−out , the structure is an hyper-elastic material and
                 »Z           Z       – Z
               1                                                        ∂q
                           q−        q =          div wn+1 q, ∀q with          = 0.
              ∆t    Ωf,n+1      Ωf,n       Ωf,n+1                       ∂t |x0

Then, the following energy inequality holds
        "Z                      Z                 Z        ˛                ˛2 Z           ˛               ˛2 #
   1                ρf n+1 2           ρf n 2           ρs ˛ ds,n+1 − ds,n ˛            ρs ˛ ds,n − ds,n−1 ˛
                      |u   | −           |u | +            ˛
                                                           ˛
                                                                            ˛ −
                                                                            ˛
                                                                                           ˛
                                                                                           ˛
                                                                                                           ˛
                                                                                                           ˛
   ∆t        Ωf,n+1 2             Ωf,n 2            Ωs 2
                                                      0
                                                                  ∆t                Ωs 2
                                                                                      0
                                                                                                  ∆t
                         "Z                      Z                   # Z
                      1
                   +           W (E (ds,n+1 )) −        W (E (ds,n )) +            2µ|ε(un+1 )|2
                      ∆t    Ωs
                             0                     Ωs0                     Ωf,n+1
                                           
                                artificial           `                                    ds,n+1 − ds,n ´
                                              − Rf un+1 , p n+1 , df,n+1 ; L(un+1 −                    ) ≤0
                                  power                                                       ∆t
                                                |                           {z                          }
                                                 Z
                                                               n+1    n+1         n+1     ds,n+1 − ds,n
                                                           σ(u     ,p     )n · (u      −                )
                                                   Γw,n+1                                      ∆t




                                                      a
                                        Miguel A. Fern´ndez   Mini-Workshop I
 Explicit Coupling: A Priori Energy Estimate (II)

Fluid domain displacement and velocity:

                                                        df,n+1 − df,n
                   df,n+1 = Ext(ds,n ),
                                 |Γw
                                           wn+1 =                     ,        Ωf,n+1 = (I + df,n+1 )(Ωf ).
                                                                                                       0
                                   0                         ∆t

Fluid subproblem: Find un+1 ∈ H 1 (Ωf,n+1 ) with un+1|Γw,n+1
                                                                     n+1
                                                                = w|Γw,n+1 and p n+1 ∈ L2 (Ωf,n+1 ) such
that
               Z                          Z                       Z
            1                           1
                        ρf un+1 · vf −            ρf u n · v f +           ρf (div wn+1 )un+1 · vf
           ∆t Ωf,n+1                    ∆t Ωf,n+1                   Ωf,n+1
              Z                                                Z
            +          ρf (wn+1 − un+1 ) · un+1 · vf + 2µ               ε(un+1 ) : ε(vf )
                Ωf,n+1                                           Ωf,n+1
              Z                       Z
            −          p n+1 div vf =         gn+1 · vf , ∀vf ∈ V f (tn+1 ),
                      Ωf,n+1                  Γin−out

Solid subproblem: Find ds,n+1 ∈ V s such that
   Z                                               Z
                 ds,n+1 − 2ds,n + ds,n−1
            ρs                           · vs +             F (ds,n+1 )S(ds,n+1 ) :    0d
                                                                                            s
       Ωs
        0
                         (∆t)2                         Ωs
                                                        0
                                                                      `                              ´
                                                                 = −Rf un+1 , p n+1 , df,n+1 ; L(vs ) ,   ∀vs ∈ V s .




                                                        a
                                          Miguel A. Fern´ndez       Mini-Workshop I
 Explicit Coupling: A Priori Energy Estimate (III)

Idea of the proof. We take appropriate test functions and then we add the fluid and the solid
variational formulations. In particular,
    for the solid discrete variational formulation:
                                                         ds,n+1 − ds,n
                                              vs =                     ,
                                                              ∆t


                   Remark
                    s
                   vΓw = 0 and hence vs ∈ V s .
                     0

    for the fluid discrete variational formulation:

                           vf = un+1 − L(v|Γw ) − L(un+1 − v|Γw ) ∈ V f (tn+1 ).
                                          s
                                                     |Γw
                                                            s
                                                 0                0        0




                   Remark
                      1    f
                          v|Γw = 0 and hence vf ∈ V f .
                            0

                      2   Let us recall that un+1 = v|Γw .
                                              |Γw
                                                     s
                                                     0        0




                                                 a
                                   Miguel A. Fern´ndez       Mini-Workshop I
Explicit Coupling: Let’s Summarize




                                      a
                        Miguel A. Fern´ndez   Mini-Workshop I
Explicit Coupling: Let’s Summarize


      Observations:
          Explicit coupling is cheap

                              FSI cost ≈ fluid cost + solid cost




                                           a
                             Miguel A. Fern´ndez   Mini-Workshop I
Explicit Coupling: Let’s Summarize


      Observations:
          Explicit coupling is cheap

                                FSI cost ≈ fluid cost + solid cost

          Numerical experiments (haemodynamics) show hat explicit coupling can be
          unstable
          Energy estimate with an artificial power term
                       Z
                                                                  ds,n+1 − ds,n
                                    σ(un+1 , p n+1 )n · (un+1 −                 ).
                           Γw,n+1                                      ∆t




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Explicit Coupling: Let’s Summarize


      Observations:
          Explicit coupling is cheap

                                FSI cost ≈ fluid cost + solid cost

          Numerical experiments (haemodynamics) show hat explicit coupling can be
          unstable
          Energy estimate with an artificial power term
                       Z
                                                                  ds,n+1 − ds,n
                                    σ(un+1 , p n+1 )n · (un+1 −                 ).
                           Γw,n+1                                      ∆t

          Explicit coupling is stable, and widely used, in aeroelasticity (see Farhat,
          Piperno)




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Explicit Coupling: Let’s Summarize


      Observations:
          Explicit coupling is cheap

                                FSI cost ≈ fluid cost + solid cost

          Numerical experiments (haemodynamics) show hat explicit coupling can be
          unstable
          Energy estimate with an artificial power term
                       Z
                                                                  ds,n+1 − ds,n
                                    σ(un+1 , p n+1 )n · (un+1 −                 ).
                           Γw,n+1                                      ∆t

          Explicit coupling is stable, and widely used, in aeroelasticity (see Farhat,
          Piperno)


                 Major questions:
                      What is the source of instabilities?
                      Is there a hidden stability condition?



                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
A 2D Simplified Model


                       Γw                         η

              Γin                                                   Γout
                                             Ωf
                               symmetry axis




                                     a
                       Miguel A. Fern´ndez        Mini-Workshop I
A 2D Simplified Model


                             Γw                          η

                   Γin                                                            Γout
                                                    Ωf
                                      symmetry axis

  Solid: string model (small displacements)

                                  ρs hs η + Lη = p|Γw ,
                                        ¨                          in      Γw ,




                                            a
                              Miguel A. Fern´ndez        Mini-Workshop I
A 2D Simplified Model

                               Γw                           η

                    Γin                                                              Γout
                                                       Ωf
                                         symmetry axis

  Solid: string model (small displacements)
                                     ρs hs η + Lη = p|Γw ,
                                           ¨                          in      Γw ,
  with
      η: vertical displacement
      hs : thickness
                                                     2
      L: linear operator (for instance Lη = aη − b ∂ η )
                                                    ∂x




                                               a
                                 Miguel A. Fern´ndez        Mini-Workshop I
A 2D Simplified Model


                              Γw                         η

                   Γin                                                            Γout
                                                    Ωf
                                       symmetry axis

  Solid: string model (small displacements)

                                     ρs hs η + Lη = p|Γw ,
                                           ¨                       in      Γw ,

  Fluid: we keep a fixed fluid domain and we neglect viscous and convective effects
    8 ∂u
    > ρf                      Ωf ,
    > ∂t + p = 0,
    >
    >
                         in
    >
    <
            div u = 0,   in   Ωf ,
    >
    >
    >
    >
    >               ˙
            u · n = η,   on   Γw ,
    :
         p = pin−out ,   on   Γin−out .




                                            a
                              Miguel A. Fern´ndez        Mini-Workshop I
A 2D Simplified Model


                              Γw                         η

                   Γin                                                             Γout
                                                    Ωf
                                       symmetry axis

  Solid: string model (small displacements)

                                     ρs hs η + Lη = p|Γw ,
                                           ¨                       in      Γw ,

  Fluid: we keep a fixed fluid domain and we neglect viscous and convective effects
    8 ∂u
    > ρf                      Ωf ,
                                                         8
    > ∂t + p = 0,
    >
    >
                         in                              >
                                                         >                        −∆p = 0,   in   Ωf ,
    >
    <                                                    >
                                                         <
            div u = 0,   in   Ωf ,                         ∂p       ∂u
                                              =⇒              = −ρf    · n = −ρf η ,
                                                                                 ¨           on   Γw ,
    >
    >                           w                        > ∂n
                                                         >          ∂t
    >
    >               ˙
            u · n = η,   on   Γ ,                        >
                                                         :
    >
    :                                                                 p = pin−out ,          on   Γin−out .
         p = pin−out ,   on   Γin−out .




                                            a
                              Miguel A. Fern´ndez        Mini-Workshop I
A 2D Simplified Model

                                Γw                         η

                   Γin                                                               Γout
                                                      Ωf
                                         symmetry axis
  Solid: string model (small displacements)
                                       ρs hs η + Lη = p|Γw ,
                                             ¨                       in      Γw ,
  Fluid: we keep a fixed    fluid domain and we neglect viscous and convective effects
    8 ∂u
    > ρf                        Ωf ,
                                                           8
    > ∂t + p = 0,
    >
    >
                           in                              >
                                                           >                        −∆p = 0,   in   Ωf ,
    >
    <                                                      >
                                                           <
             div u = 0,    in   Ωf ,                         ∂p       ∂u
                                                =⇒              = −ρf    · n = −ρf η ,
                                                                                   ¨           on   Γw ,
    >
    >                             w                        > ∂n
                                                           >          ∂t
    >
    >                 ˙
              u · n = η,   on   Γ ,                        >
                                                           :
    >
    :                                                                   p = pin−out ,          on   Γin−out .
          p = pin−out ,    on   Γin−out .


              Why a new model?
                   Physics: reproduces propagation phenomena [movie]
                   Numerics: explicit coupling unstable [movie]

                                              a
                                Miguel A. Fern´ndez        Mini-Workshop I
The Added Mass Effect

  The simplified coupled problem:
                                      8
                                      >
                                      >     −∆p = 0,      in    Ωf ,
                                      >
                                      <
                                        ∂p
                           Fluid:          = −ρf η ,
                                                 ¨        on        Γw ,
                                      > ∂n
                                      >
                                      >
                                      :                                           (1)
                                            p = 0,        on        Γin−out .

                           Solid:    ρs hs η + Lη = p|Γw ,
                                           ¨                        in     Γw ,




                                          a
                            Miguel A. Fern´ndez   Mini-Workshop I
The Added Mass Effect

  The simplified coupled problem:
                                           8
                                           >
                                           >     −∆p = 0,          in    Ωf ,
                                           >
                                           <
                                             ∂p
                                Fluid:          = −ρf η ,
                                                      ¨            on        Γw ,
                                           > ∂n
                                           >
                                           >
                                           :                                               (1)
                                                 p = 0,            on        Γin−out .

                                Solid:    ρs hs η + Lη = p|Γw ,
                                                ¨                            in     Γw ,


                                          e
        Definition (Inverse Steklov-Poincar´ operator)
                                    1                  1
        The operator MA : H − 2 (Γw ) → H 2 (Γw ) is defined as follows: for each
                 1
        g ∈   H− 2   (Γw ) we define MA (g ) = q|Γw , with q ∈ H 1 (Ωf ) the solution of
                                        8
                                        > −∆q = 0,
                                        >                   in   Ωf ,
                                        >
                                        <
                                           ∂q
                                              = g,          on    Γw ,
                                        > ∂n
                                        >
                                        >
                                        :
                                            q = 0,          on    Γin−out .




                                               a
                                 Miguel A. Fern´ndez       Mini-Workshop I
The Added Mass Effect

  The simplified coupled problem:
                                           8
                                           >
                                           >     −∆p = 0,          in    Ωf ,
                                           >
                                           <
                                             ∂p
                                Fluid:          = −ρf η ,
                                                      ¨            on        Γw ,
                                           > ∂n
                                           >
                                           >
                                           :                                               (1)
                                                 p = 0,            on        Γin−out .

                                Solid:    ρs hs η + Lη = p|Γw ,
                                                ¨                            in     Γw ,


                                          e
        Definition (Inverse Steklov-Poincar´ operator)
                                    1                  1
        The operator MA : H − 2 (Γw ) → H 2 (Γw ) is defined as follows: for each
                 1
        g ∈   H− 2   (Γw ) we define MA (g ) = q|Γw , with q ∈ H 1 (Ωf ) the solution of
                                        8
                                        > −∆q = 0,
                                        >                   in   Ωf ,
                                        >
                                        <
                                           ∂q
                                              = g,          on    Γw ,
                                        > ∂n
                                        >
                                        >
                                        :
                                            q = 0,          on    Γin−out .
  Using this definition we have:
                                            p|Γw = MA (−ρf η ).
                                                           ¨



                                               a
                                 Miguel A. Fern´ndez       Mini-Workshop I
The Added Mass Effect

  The simplified coupled problem:
                                       8
                                       >
                                       >     −∆p = 0,      in    Ωf ,
                                       >
                                       <
                                         ∂p
                           Fluid:           = −ρf η ,
                                                  ¨        on        Γw ,
                                       > ∂n
                                       >
                                       >
                                       :                                           (1)
                                             p = 0,        on        Γin−out .

                           Solid:     ρs hs η + Lη = p|Γw ,
                                            ¨                        in     Γw ,

  Condensed solid problem: we have p|Γw = MA (−ρf η ), thus
                                                  ¨

                                    ρs hs η + Lη = MA (−ρf η ).
                                          ¨                ¨




                                           a
                             Miguel A. Fern´ndez   Mini-Workshop I
The Added Mass Effect

  The simplified coupled problem:
                                       8
                                       >
                                       >     −∆p = 0,      in    Ωf ,
                                       >
                                       <
                                         ∂p
                           Fluid:           = −ρf η ,
                                                  ¨        on        Γw ,
                                       > ∂n
                                       >
                                       >
                                       :                                           (1)
                                             p = 0,        on        Γin−out .

                           Solid:     ρs hs η + Lη = p|Γw ,
                                            ¨                        in     Γw ,

  Condensed solid problem: we have p|Γw = MA (−ρf η ), thus
                                                  ¨
                            ` s s    f
                                          ´
                             ρ h + ρ MA η + Lη = 0, in
                                            ¨                               Γw .   (2)




                                           a
                             Miguel A. Fern´ndez   Mini-Workshop I
The Added Mass Effect

  The simplified coupled problem:
                                      8
                                      >
                                      >     −∆p = 0,      in    Ωf ,
                                      >
                                      <
                                        ∂p
                           Fluid:          = −ρf η ,
                                                 ¨        on        Γw ,
                                      > ∂n
                                      >
                                      >
                                      :                                              (1)
                                            p = 0,        on        Γin−out .

                           Solid:    ρs hs η + Lη = p|Γw ,
                                           ¨                        in     Γw ,
  Condensed solid problem: we have p|Γw = MA (−ρf η ), thus
                                                  ¨
                         ` s s       f
                                            ´
                          ρ h +     ρ MA     η + Lη = 0, in
                                              ¨                               Γw .   (2)
                                    | {z }
                                 added mass




                                          a
                            Miguel A. Fern´ndez   Mini-Workshop I
The Added Mass Effect

  The simplified coupled problem:
                                         8
                                         >
                                         >     −∆p = 0,      in    Ωf ,
                                         >
                                         <
                                           ∂p
                             Fluid:           = −ρf η ,
                                                    ¨        on        Γw ,
                                         > ∂n
                                         >
                                         >
                                         :                                                        (1)
                                               p = 0,        on        Γin−out .

                             Solid:     ρs hs η + Lη = p|Γw ,
                                              ¨                        in     Γw ,
  Condensed solid problem: we have p|Γw = MA (−ρf η ), thus
                                                  ¨
                         ` s s       f
                                            ´
                          ρ h +     ρ MA     η + Lη = 0, in
                                              ¨                                  Γw .             (2)
                                    | {z }
                                 added mass

        Remarks:

            This new equation looks like a structure equation, except for the extra term ρ f MA
            The fluid-structure coupling can be condensed into an extra mass acting on the
            structure (hence the terminology “added mass effect”)




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
The Added Mass Effect

  The simplified coupled problem:
                                         8
                                         >
                                         >     −∆p = 0,      in    Ωf ,
                                         >
                                         <
                                           ∂p
                             Fluid:           = −ρf η ,
                                                    ¨        on        Γw ,
                                         > ∂n
                                         >
                                         >
                                         :                                                        (1)
                                               p = 0,        on        Γin−out .

                             Solid:     ρs hs η + Lη = p|Γw ,
                                              ¨                        in     Γw ,
  Condensed solid problem: we have p|Γw = MA (−ρf η ), thus
                                                  ¨
                         ` s s       f
                                            ´
                          ρ h +     ρ MA     η + Lη = 0, in
                                              ¨                                  Γw .             (2)
                                    | {z }
                                 added mass

        Remarks:

            This new equation looks like a structure equation, except for the extra term ρ f MA
            The fluid-structure coupling can be condensed into an extra mass acting on the
            structure (hence the terminology “added mass effect”)


        Major question:
        What kind of time integration scheme of (2) arises from the a explicit coupling
        of (1)?
                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Time Discretization: Explicit Coupling (I)

  Fluid:    8    n+1 − un
            > fu
            >ρ
            >              + p n+1 = 0,       8
            >
            >
            <      ∆t                         >
                                              <                 −∆p n+1 = 0,
                                  n+1
                            div u     = 0, =⇒              n − 2η n−1 + η n−2
            >
            >                                 > ∂p = −ρf η
                                              :                               .
            >
            >                 η n − η n−1       ∂n              (∆t)2
                  un+1 · n =
            >
            :                             .
                                   ∆t




                                          a
                            Miguel A. Fern´ndez   Mini-Workshop I
Time Discretization: Explicit Coupling (I)

  Fluid:    8    n+1 − un
            > fu
            >ρ
            >              + p n+1 = 0,       8
            >
            >
            <      ∆t                         >
                                              <                 −∆p n+1 = 0,
                                  n+1
                            div u     = 0, =⇒              n − 2η n−1 + η n−2
            >
            >                                 > ∂p = −ρf η
                                              :                               .
            >
            >                 η n − η n−1       ∂n              (∆t)2
                  un+1 · n =
            >
            :                             .
                                   ∆t
  Solid:
                                  η n+1 − 2η n + η n−1             n+1
                         ρs h s                        + Lη n+1 = p|Γw .
                                         (∆t)2




                                          a
                            Miguel A. Fern´ndez   Mini-Workshop I
Time Discretization: Explicit Coupling (I)

  Fluid:     8    n+1 − un
             > fu
             >ρ
             >              + p n+1 = 0,       8
             >
             >
             <      ∆t                         >
                                               <                 −∆p n+1 = 0,
                                   n+1
                             div u     = 0, =⇒              n − 2η n−1 + η n−2
             >
             >                                 > ∂p = −ρf η
                                               :                               .
             >
             >                 η n − η n−1       ∂n              (∆t)2
                   un+1 · n =
             >
             :                             .
                                    ∆t
  Solid:
                                        η n+1 − 2η n + η n−1             n+1
                               ρs h s                        + Lη n+1 = p|Γw .
                                               (∆t)2
  Condensed solid problem:

                       η n+1 − 2η n + η n−1          η n − 2η n−1 + η n−2
              ρs h s                        + ρf M A                      + Lη n+1 = 0.
                              (∆t)2                         (∆t)2




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
Time Discretization: Explicit Coupling (I)

  Fluid:     8    n+1 − un
             > fu
             >ρ
             >              + p n+1 = 0,       8
             >
             >
             <      ∆t                         >
                                               <                 −∆p n+1 = 0,
                                   n+1
                             div u     = 0, =⇒              n − 2η n−1 + η n−2
             >
             >                                 > ∂p = −ρf η
                                               :                               .
             >
             >                 η n − η n−1       ∂n              (∆t)2
                   un+1 · n =
             >
             :                             .
                                    ∆t
  Solid:
                                       η n+1 − 2η n + η n−1             n+1
                              ρs h s                        + Lη n+1 = p|Γw .
                                              (∆t)2
  Condensed solid problem:

                       η n+1 − 2η n + η n−1        η n − 2η n−1 + η n−2
              ρs h s                        +ρf MA                      + Lη n+1 = 0.
                              (∆t)2                       (∆t)2
                       |        {z        }
                             implicit




                                               a
                                 Miguel A. Fern´ndez   Mini-Workshop I
Time Discretization: Explicit Coupling (I)

  Fluid:     8    n+1 − un
             > fu
             >ρ
             >              + p n+1 = 0,       8
             >
             >
             <      ∆t                         >
                                               <                 −∆p n+1 = 0,
                                   n+1
                             div u     = 0, =⇒              n − 2η n−1 + η n−2
             >
             >                                 > ∂p = −ρf η
                                               :                               .
             >
             >                 η n − η n−1       ∂n              (∆t)2
                   un+1 · n =
             >
             :                             .
                                    ∆t
  Solid:
                                       η n+1 − 2η n + η n−1             n+1
                              ρs h s                        + Lη n+1 = p|Γw .
                                              (∆t)2
  Condensed solid problem:

                       η n+1 − 2η n + η n−1         η n − 2η n−1 + η n−2
              ρs h s                         +ρf MA                      +Lη n+1 = 0.
                              (∆t)2                         (∆t)2
                       |        {z         }        |         {z       }
                             implicit                      explicit




                                               a
                                 Miguel A. Fern´ndez   Mini-Workshop I
Time Discretization: Explicit Coupling (I)

  Fluid:        8    n+1 − un
                > fu
                >ρ
                >              + p n+1 = 0,       8
                >
                >
                <      ∆t                         >
                                                  <                 −∆p n+1 = 0,
                                      n+1
                                div u     = 0, =⇒              n − 2η n−1 + η n−2
                >
                >                                 > ∂p = −ρf η
                                                  :                               .
                >
                >                 η n − η n−1       ∂n              (∆t)2
                      un+1 · n =
                >
                :                             .
                                       ∆t
  Solid:
                                          η n+1 − 2η n + η n−1             n+1
                                 ρs h s                        + Lη n+1 = p|Γw .
                                                 (∆t)2
  Condensed solid problem:

                          η n+1 − 2η n + η n−1         η n − 2η n−1 + η n−2
                 ρs h s                         +ρf MA                      +Lη n+1 = 0.
                                 (∆t)2                         (∆t)2
                          |        {z         }        |         {z       }
                                implicit                      explicit


           Observation:
           Explicit coupling leads to a explicit discretization of the added mass term.




                                                  a
                                    Miguel A. Fern´ndez   Mini-Workshop I
Times discretization: Explicit Coupling (II)


Theorem (Causin-Gerbeau-Nobile 04)
Let λmax be the for largest eigenvalue of MA and assume that Lη = aη. Then, the previous
explicit coupling scheme is unconditionally unstable whenever

                                           ρf λmax
                                                   ≥ 1.                                    (3)
                                            ρs h s




                                            a
                              Miguel A. Fern´ndez   Mini-Workshop I
Times discretization: Explicit Coupling (II)


Theorem (Causin-Gerbeau-Nobile 04)
Let λmax be the for largest eigenvalue of MA and assume that Lη = aη. Then, the previous
explicit coupling scheme is unconditionally unstable whenever

                                                  ρf λmax
                                                          ≥ 1.                                             (3)
                                                   ρs h s


Remarks:
    The stability of the coupling scheme does not depend on the time step ∆t.
    The “instability” condition (3) confirms empirical observations:
           instabilities may occur when the structure is light, as in haemodynamic applications ρ f ≈ ρs
           in aeroelasticity ρf    ρs , hence explicit coupling is stable




                                                   a
                                     Miguel A. Fern´ndez   Mini-Workshop I
 Times discretization: Explicit Coupling (II)


 Theorem (Causin-Gerbeau-Nobile 04)
 Let λmax be the for largest eigenvalue of MA and assume that Lη = aη. Then, the previous
 explicit coupling scheme is unconditionally unstable whenever

                                                   ρf λmax
                                                           ≥ 1.                                             (3)
                                                    ρs h s


 Remarks:
     The stability of the coupling scheme does not depend on the time step ∆t.
     The “instability” condition (3) confirms empirical observations:
            instabilities may occur when the structure is light, as in haemodynamic applications ρ f ≈ ρs
            in aeroelasticity ρf    ρs , hence explicit coupling is stable

Idea of the proof.
    Note that MA is a compact, postive and self-adjoint operator on L2 (Γw ).
    Expand η n+1 , η n , η n−1 and η n−2 on a orthonormal basis made of eigen-vectors of MA .
    Conclude by proving that characteristic polynomial of the resulting difference equation, has a
    root with magnitude larger than 1 under condition (3).


                                                    a
                                      Miguel A. Fern´ndez   Mini-Workshop I
Time Discretization: Implicit Coupling

  Fluid:

                                                    df,n+1 − df,n
              df,n+1 = Ext(ds,n+1 |Γw ), wn+1 =                      , Ωf,n+1 = (I + df,n+1 )(Ωf ),
                                                                                                  0
                  Z
                                       0
                                             Z              ∆t Z
               1                           1
                           ρf un+1 · vf −          ρf u n · v f +         ρf (div wn+1 )un+1 · vf
              ∆t Ωf,n+1                    ∆t Ωf,n                  f,n+1
                 Z                                               ZΩ
              +           ρf (wn+1 − un+1 ) · un+1 · vf + 2µ             ε(un+1 ) : ε(vf )
                   Ωf,n+1                                         Ωf,n+1
                 Z                       Z
              −           p n+1 div vf =         gn+1 · vf , ∀vf ∈ V f (tn+1 ),
                    Ωf,n+1                 Γin−out

              un+1
               |Γw,n+1
                            n+1
                         = w|Γw,n+1

  Solid:
     Z                                                Z
                   ds,n+1 − 2ds,n + ds,n−1
              ρs                           · vs +              F (ds,n+1 )S(ds,n+1 ) :   0d
                                                                                              s
         Ωs
          0
                           (∆t)2                          Ωs
                                                           0
                                                                    `                              ´
                                                               = −Rf un+1 , p n+1 , df,n+1 ; L(vs ) ,   ∀vs ∈ V s .




                                                    a
                                      Miguel A. Fern´ndez         Mini-Workshop I
Time Discretization: Implicit Coupling

  Fluid:

                                                    df,n+1 − df,n
              df,n+1 = Ext(ds,n+1 |Γw ), wn+1 =                      , Ωf,n+1 = (I + df,n+1 )(Ωf ),
                                                                                                  0
                  Z
                                       0
                                             Z              ∆t Z
               1                           1
                           ρf un+1 · vf −          ρf u n · v f +         ρf (div wn+1 )un+1 · vf
              ∆t Ωf,n+1                    ∆t Ωf,n                  f,n+1
                 Z                                               ZΩ
              +           ρf (wn+1 − un+1 ) · un+1 · vf + 2µ             ε(un+1 ) : ε(vf )
                   Ωf,n+1                                         Ωf,n+1
                 Z                       Z
              −           p n+1 div vf =         gn+1 · vf , ∀vf ∈ V f (tn+1 ),
                    Ωf,n+1                 Γin−out

              un+1
               |Γw,n+1
                            n+1
                         = w|Γw,n+1

  We formally note this problem as (df,n+1 , un+1 , p n+1 ) = F (ds,n+1 ).
  Solid:
     Z                                                Z
                   ds,n+1 − 2ds,n + ds,n−1
              ρs                           · vs +              F (ds,n+1 )S(ds,n+1 ) :   0d
                                                                                              s
         Ωs
          0
                           (∆t)2                          Ωs
                                                           0
                                                                    `                              ´
                                                               = −Rf un+1 , p n+1 , df,n+1 ; L(vs ) ,   ∀vs ∈ V s .




                                                    a
                                      Miguel A. Fern´ndez         Mini-Workshop I
Time Discretization: Implicit Coupling

  Fluid:

                                                    df,n+1 − df,n
              df,n+1 = Ext(ds,n+1 |Γw ), wn+1 =                      , Ωf,n+1 = (I + df,n+1 )(Ωf ),
                                                                                                  0
                  Z
                                       0
                                             Z              ∆t Z
               1                           1
                           ρf un+1 · vf −          ρf u n · v f +         ρf (div wn+1 )un+1 · vf
              ∆t Ωf,n+1                    ∆t Ωf,n                  f,n+1
                 Z                                               ZΩ
              +           ρf (wn+1 − un+1 ) · un+1 · vf + 2µ             ε(un+1 ) : ε(vf )
                   Ωf,n+1                                         Ωf,n+1
                 Z                       Z
              −           p n+1 div vf =         gn+1 · vf , ∀vf ∈ V f (tn+1 ),
                    Ωf,n+1                 Γin−out

              un+1
               |Γw,n+1
                            n+1
                         = w|Γw,n+1

  We formally note this problem as (df,n+1 , un+1 , p n+1 ) = F (ds,n+1 ).
  Solid:
     Z                                                Z
                   ds,n+1 − 2ds,n + ds,n−1
              ρs                           · vs +              F (ds,n+1 )S(ds,n+1 ) :   0d
                                                                                              s
         Ωs
          0
                           (∆t)2                          Ωs
                                                           0
                                                                    `                              ´
                                                               = −Rf un+1 , p n+1 , df,n+1 ; L(vs ) ,   ∀vs ∈ V s .

  We formally note this problem as ds,n+1 = S(df,n+1 , un+1 , p n+1 ).


                                                    a
                                      Miguel A. Fern´ndez         Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:
    Implicit coupling leads to stable schemes. Velocity continuity is exactly enforced

                                                          ds,n+1 − ds,n
                                                           |Γw      Γw
                                          un+1
                                           |Γw
                                                 =           0          0
                                             0                  ∆t
                                                          |     {z     }
                                                        interface velocity

    We then recover an energy estimate without artificial power at the interface.




                                                a
                                  Miguel A. Fern´ndez       Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:
    Implicit coupling leads to stable schemes. Velocity continuity is exactly enforced

                                                          ds,n+1 − ds,n
                                                           |Γw      Γw
                                          un+1
                                           |Γw
                                                 =           0          0
                                             0                  ∆t
                                                          |     {z     }
                                                        interface velocity

    We then recover an energy estimate without artificial power at the interface.
    Highly non-linear coupled problem (geometrical non-linearities):

                                    (df,n+1 , un+1 , p n+1 ) = F (ds,n+1 )
                                       ds,n+1 = S(df,n+1 , un+1 , p n+1 )




                                                a
                                  Miguel A. Fern´ndez       Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:
    Implicit coupling leads to stable schemes. Velocity continuity is exactly enforced

                                                          ds,n+1 − ds,n
                                                           |Γw      Γw
                                          un+1
                                           |Γw
                                                 =           0          0
                                             0                  ∆t
                                                          |     {z     }
                                                        interface velocity

    We then recover an energy estimate without artificial power at the interface.
    Highly non-linear coupled problem (geometrical non-linearities):

                                    (df,n+1 , un+1 , p n+1 ) = F (ds,n+1 )
                                       ds,n+1 = S(df,n+1 , un+1 , p n+1 )

    Thus, by composition:

                               Fixed-point problem:

                                          ds,n+1 = S ◦ F (ds,n+1 )




                                                a
                                  Miguel A. Fern´ndez       Mini-Workshop I
 Explicit Coupling (II)

Some observations are in order:
    Implicit coupling leads to stable schemes. Velocity continuity is exactly enforced

                                                           ds,n+1 − ds,n
                                                            |Γw      Γw
                                           un+1
                                            |Γw
                                                    =         0          0
                                              0
                                                           |     ∆t
                                                                 {z     }
                                                         interface velocity

    We then recover an energy estimate without artificial power at the interface.
    Highly non-linear coupled problem (geometrical non-linearities):

                                     (df,n+1 , un+1 , p n+1 ) = F (ds,n+1 )
                                        ds,n+1 = S(df,n+1 , un+1 , p n+1 )

    Thus, by composition:

                                  Fixed-point problem:

                                            ds,n+1 = S ◦ F (ds,n+1 )

                   Root-finding problem:
                                              def
                              R(ds,n+1 ) = ds,n+1 − S ◦ F (ds,n+1 ) = 0

                                                 a
                                   Miguel A. Fern´ndez       Mini-Workshop I
Solution Methods: Fixed-Point Based Methods

                         def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid: ds = S ◦ F (ds )
                              k+1         k
  (b) Relaxation step: ds ←− ωk ds + (1 − ωk )ds
                         k+1      k+1          k




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Solution Methods: Fixed-Point Based Methods

                          def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid: ds = S ◦ F (ds )
                              k+1         k
  (b) Relaxation step: ds ←− ωk ds + (1 − ωk )ds
                         k+1      k+1          k

  Acceleration via dynamic relaxation: Aitken’s method
  (Mok-Wall-Ramm 01, Gerbeau-Vidrascu 03)

                         (ds − ds ) · (ds − S ◦ F (ds ) − ds
                           k    k−1     k           k
                                                                         s
                                                           k−1 + S ◦ F (dk−1 ))
                  ωk =
                                |ds − S ◦ F (ds ) − ds
                                  k           k
                                                                   s
                                                     k−1 + S ◦ F (dk−1 )|
                                                                          2




                                              a
                                Miguel A. Fern´ndez   Mini-Workshop I
Solution Methods: Fixed-Point Based Methods

                          def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid: ds = S ◦ F (ds )
                              k+1         k
  (b) Relaxation step: ds ←− ωk ds + (1 − ωk )ds
                         k+1      k+1          k

  Acceleration via dynamic relaxation: Aitken’s method
  (Mok-Wall-Ramm 01, Gerbeau-Vidrascu 03)

                         (ds − ds ) · (ds − S ◦ F (ds ) − ds
                           k    k−1     k           k
                                                                         s
                                                           k−1 + S ◦ F (dk−1 ))
                  ωk =
                                |ds − S ◦ F (ds ) − ds
                                  k           k
                                                                   s
                                                     k−1 + S ◦ F (dk−1 )|
                                                                          2


  Reducing the fluid computational cost via transpiration conditions
  (Deparis-MF-Formaggia 03)




                                              a
                                Miguel A. Fern´ndez   Mini-Workshop I
Solution Methods: Fixed-Point Based Methods

                            def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid: ds = S ◦ F (ds )
                              k+1         k
  (b) Relaxation step: ds ←− ωk ds + (1 − ωk )ds
                         k+1      k+1          k

  Acceleration via dynamic relaxation: Aitken’s method
  (Mok-Wall-Ramm 01, Gerbeau-Vidrascu 03)

                         (ds − ds ) · (ds − S ◦ F (ds ) − ds
                           k    k−1     k           k
                                                                         s
                                                           k−1 + S ◦ F (dk−1 ))
                  ωk =
                                  |ds − S ◦ F (ds ) − ds
                                    k           k
                                                                     s
                                                       k−1 + S ◦ F (dk−1 )|
                                                                            2


  Reducing the fluid computational cost via transpiration conditions
  (Deparis-MF-Formaggia 03)


              Observation
              Too expensive and may fail to converge !




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
Solution Methods: Newton Based Methods

                         def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid:  ds = S ◦ F (ds )
                                 k+1         k
                              s def s
  (b) Evaluate residual:                          = d s − ds
                           R(dk ) = dk − S ◦ F (ds )
                                                 k    k    k+1
  (c) Solve tangent problem
                         ˆ        s ˜   s      s
                           Dds R(dk ) δd = −R(dk ) (solved via GMRES)
  (d) Update rule:   ds ←− ds + δds
                      k+1   k




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Solution Methods: Newton Based Methods

                         def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid:  ds = S ◦ F (ds )
                                 k+1         k
                              s def s
  (b) Evaluate residual:                          = d s − ds
                           R(dk ) = dk − S ◦ F (ds )
                                                 k    k    k+1
  (c) Solve tangent problem
                         ˆ        s ˜   s      s
                           Dds R(dk ) δd = −R(dk ) (solved via GMRES)
  (d) Update rule:   ds ←− ds + δds
                      k+1   k



      The bottleneck of Newton’s method
                                                          ˆ             ˜
      For a given solid variation z, how to compute           Dds R(ds ) z ?
                                                                     k




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Solution Methods: Newton Based Methods

                            def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid:    ds = S ◦ F (ds )
                                   k+1         k
                                s def s
  (b) Evaluate residual:                          = d s − ds
                             R(dk ) = dk − S ◦ F (ds )
                                                   k  k    k+1
  (c) Solve tangent problem
                         ˆ        s ˜   s      s
                           Dds R(dk ) δd = −R(dk ) (solved via GMRES)
  (d) Update rule:   ds ←− ds + δds
                      k+1   k



      The bottleneck of Newton’s method
                                                             ˆ             ˜
      For a given solid variation z, how to compute              Dds R(ds ) z ?
                                                                        k

  Finite difference Jacobian approximations
  (Matthies-Steindorf 01, Heil 03),

                      ˆ             ˜    R(ds + z) − R(ds )
                                            k           k
                          Dds R(ds ) z ≈
                                 k                          ,              0<     1




                                                a
                                  Miguel A. Fern´ndez   Mini-Workshop I
Solution Methods: Newton Based Methods

                         def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid:  ds = S ◦ F (ds )
                                 k+1         k
                              s def s
  (b) Evaluate residual:                          = d s − ds
                           R(dk ) = dk − S ◦ F (ds )
                                                 k    k    k+1
  (c) Solve tangent problem
                         ˆ        s ˜   s      s
                           Dds R(dk ) δd = −R(dk ) (solved via GMRES)
  (d) Update rule:   ds ←− ds + δds
                      k+1   k



      The bottleneck of Newton’s method
                                                          ˆ             ˜
      For a given solid variation z, how to compute           Dds R(ds ) z ?
                                                                     k

  Finite difference Jacobian approximations
  (Matthies-Steindorf 01, Heil 03),
  too expensive and may fail to converge !




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Solution Methods: Newton Based Methods

                         def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid:  ds = S ◦ F (ds )
                                 k+1         k
                              s def s
  (b) Evaluate residual:                          = d s − ds
                           R(dk ) = dk − S ◦ F (ds )
                                                 k    k    k+1
  (c) Solve tangent problem
                         ˆ        s ˜   s      s
                           Dds R(dk ) δd = −R(dk ) (solved via GMRES)
  (d) Update rule:   ds ←− ds + δds
                      k+1   k



      The bottleneck of Newton’s method
                                                          ˆ             ˜
      For a given solid variation z, how to compute           Dds R(ds ) z ?
                                                                     k

  Finite difference Jacobian approximations
  (Matthies-Steindorf 01, Heil 03),
  too expensive and may fail to converge !
  Physically based Jacobian approximations
  (Tezduyar 01, Gerbeau-Vidrascu 03, Deparis-Gerbeau-Vaseur 04),




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Solution Methods: Newton Based Methods

                         def
1 Initial guess ds , ds = ds,n
                 0    0
2 For k = 0, 1, . . . until convergence
  (a) Solve fluid and solid:  ds = S ◦ F (ds )
                                 k+1         k
                              s def s
  (b) Evaluate residual:                          = d s − ds
                           R(dk ) = dk − S ◦ F (ds )
                                                 k    k    k+1
  (c) Solve tangent problem
                         ˆ        s ˜   s      s
                           Dds R(dk ) δd = −R(dk ) (solved via GMRES)
  (d) Update rule:   ds ←− ds + δds
                      k+1   k



      The bottleneck of Newton’s method
                                                          ˆ             ˜
      For a given solid variation z, how to compute           Dds R(ds ) z ?
                                                                     k

  Finite difference Jacobian approximations
  (Matthies-Steindorf 01, Heil 03),
  too expensive and may fail to converge !
  Physically based Jacobian approximations
  (Tezduyar 01, Gerbeau-Vidrascu 03, Deparis-Gerbeau-Vaseur 04),
  cheaper but still may fail to converge !




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
 Exact Jacobian Evaluation
                                                      ˆ             ˜
Given a solid displacement z we need to evaluate                ¯
                                                          Dds R(ds ) z with

                                             def
                                   R(ds ) = ds − S ◦ F (ds ).




                                              a
                                Miguel A. Fern´ndez       Mini-Workshop I
 Exact Jacobian Evaluation
                                                       ˆ             ˜
Given a solid displacement z we need to evaluate                 ¯
                                                           Dds R(ds ) z with

                                              def
                                    R(ds ) = ds − S ◦ F (ds ).

One application of the chain rule yields
                     ˆ           ˜       ˆ                     ˜ˆ          ˜
                              ¯                           ¯            ¯
                       Dds R(ds ) z = z − D(ds ,u,p) S(F (ds )) Dds F (ds ) z.




                                               a
                                 Miguel A. Fern´ndez       Mini-Workshop I
 Exact Jacobian Evaluation
                                                        ˆ             ˜
Given a solid displacement z we need to evaluate                  ¯
                                                            Dds R(ds ) z with

                                               def
                                     R(ds ) = ds − S ◦ F (ds ).

One application of the chain rule yields
                  ˆ           ˜          ˆ                   ˜                ˆ           ˜
                           ¯                           ¯
                    Dds R(ds ) z = z − D(df ,u,p) S(F (ds ))                          ¯
                                                                             D ds F ( ds ) z .
                                                                           |      {z        }
                                                                        δu f = (δu, δp, δdf )




                                                a
                                  Miguel A. Fern´ndez       Mini-Workshop I
 Exact Jacobian Evaluation
                                                        ˆ             ˜
Given a solid displacement z we need to evaluate                  ¯
                                                            Dds R(ds ) z with

                                               def
                                     R(ds ) = ds − S ◦ F (ds ).

One application of the chain rule yields
                  ˆ           ˜          ˆ                   ˜                ˆ          ˜
                           ¯                           ¯
                    Dds R(ds ) z = z − D(df ,u,p) S(F (ds ))                         ¯
                                                                            D ds F ( ds ) z .
                                                                          |      {z        }
                                                                       δu f = (δu, δp, δdf )
                                          |                         {z                       }
                                                                    δz




                                                a
                                  Miguel A. Fern´ndez       Mini-Workshop I
 Fluid Tangent Problem

Using shape-derivative calculus (Sokolowski-Zolesio 91, Delfour-Zolesio 01), (δu, δp)
solves the following linear problem:
         8 ρ               h                              i     ρ
                                   n         ¯f                         f
         > ∆t δu + ρ div δu ⊗ (u − wg (d )) − σ(δu, δp) = ∆t (div δd )¯      u
         >
         >
         >
         >
         >
         >              nh                             ih                      io
         >
         >                                 ¯
                 − div ρ¯ ⊗ (un − wg (df )) − σ(¯ , p ) I div δdf − ( δdf )T
                            u                      u ¯
         >
         >
         >
         >                                  n h                           io
         >
         >            ρ                                                           ¯
         >
         >
         <       +      div(¯ ⊗ δdf ) − div µ
                             u                   u δdf + ( δdf )T ( u)T , in Ωf (d),
                                                 ¯                   ¯
                    ∆t
                            n h                    io
         > div δu = − div u I div δdf − ( δdf )T , in Ωf (df ),
         >                    ¯                                   ¯
         >
         >
         >
         >
         >          f
         > δu = δd , on Γw (df ),
         >
         >
         >                        ¯
         >
         >
         >
         >        ∆t
         >
         >                   h                         i
         : σ(δu, δp)n = µ       u δdf + ( δdf )T ( u)T n, on Γ
                                ¯                   ¯                      ,
                                                                      in−out

                ¯
with δdf = Ext (ds )z (MF-Moubachir 03,04).




                                            a
                              Miguel A. Fern´ndez   Mini-Workshop I
 Fluid Tangent Problem Approximations

An approximate fluid tangent problem can be derived by neglecting the shape derivative terms,
yielding                     h                                i
            8 ρ
            >                                ¯                               ¯
                   δu + ρ div δu ⊗ (un − wg (df )) − σ(δu, δp) = 0, in Ωf (df ),
            >
            > ∆t
            >
            >
            >                        ¯
            < div δu = 0, in Ωf (df ),
            >
                       f
             > δu = δd , on Γw (df ),
             >
             >                    ¯
             >
             >
             >
             >       ∆t
             :
               σ(δu, δp)n = 0, on Γin−out .




                                              a
                                Miguel A. Fern´ndez   Mini-Workshop I
 Fluid Tangent Problem Approximations

An approximate fluid tangent problem can be derived by neglecting the shape derivative terms,
yielding                     h                                i
            8 ρ
            >                                ¯                               ¯
                   δu + ρ div δu ⊗ (un − wg (df )) − σ(δu, δp) = 0, in Ωf (df ),
            >
            > ∆t
            >
            >
            >                        ¯
            < div δu = 0, in Ωf (df ),
            >
                       f
             > δu = δd , on Γw (df ),
             >
             >                    ¯
             >
             >
             >
             >       ∆t
             :
               σ(δu, δp)n = 0, on Γin−out .
On the other hand, neglecting here the convective and diffusive terms (Gerbeau-Vidrascu
2003), we get the following problem
                              8 ρ                        f ¯f
                              > ∆t δu + δp = 0, in Ω (d ),
                              >
                              >
                              >
                              >
                              >                    ¯
                              > div δu = 0, in Ωf (df ),
                              <
                                       f
                              > δu = δd , on Γw (df ),
                              >
                              >                  ¯
                              >
                              >
                              >
                              >      ∆t
                              :                        ¯
                                δpn = 0, on Γin−out (df ).




                                              a
                                Miguel A. Fern´ndez   Mini-Workshop I
  Iterations History (∆t = 10−4 s)


                       50
                                                                                 Newton: exact
                                                                              Newton: not exact
                       45                                                           FP-Aitken


                       40


                       35
number of iterations




                       30


                       25


                       20


                       15


                       10


                        5


                        0
                            0   0.005   0.01           0.015         0.02          0.025          0.03
                                                         time




                                                             a
                                               Miguel A. Fern´ndez   Mini-Workshop I
Computational Cost (∆t = 10−4 s)

                    Algorithm          CPU time (dimensionless)
                  Picard-Aitken                 1.00
                  Quasi-Newton                  0.55
                     Newton                     0.60




                                     a
                       Miguel A. Fern´ndez   Mini-Workshop I
Computational Cost (∆t = 10−4 s)

                        Algorithm          CPU time (dimensionless)
                      Picard-Aitken                 1.00
                      Quasi-Newton                  0.55
                         Newton                     0.60



           Question
           What about increasing the time step ∆t ?




                                         a
                           Miguel A. Fern´ndez   Mini-Workshop I
 Residuals History (∆t = 10−3 s)


                                                                        Newton: exact
             0.01                                                    Newton: not exact




            0.001



           0.0001
residual




            1e-05



            1e-06



            1e-07



            1e-08
                    0   20      40                    60                80               100
                                     number of iterations




                                           a
                             Miguel A. Fern´ndez            Mini-Workshop I
Some Remarks on Implicit Schemes




  Newton based methods perform better than fixed-point based methods

  Approximated Jacobians may lead to divergent quasi-Newton methods

  Exact Jacobians may improve the convergence of the Newton’s loop (moderate time steps)
  but are too expensive

  Realistic simulations with this kind of methods are very expensive




                                           a
                             Miguel A. Fern´ndez   Mini-Workshop I
               Part III

 Semi-Implicit Coupling




              a
Miguel A. Fern´ndez   Mini-Workshop I
Let’s Summarize

  Explicit coupling is cheap but unstable




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Let’s Summarize

  Explicit coupling is cheap but unstable
  Implicit coupling is stable but too expensive




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Let’s Summarize

  Explicit coupling is cheap but unstable
  Implicit coupling is stable but too expensive
  Added-mass effect seems to be the source of numerical instabilities in explicit coupling




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Let’s Summarize

  Explicit coupling is cheap but unstable
  Implicit coupling is stable but too expensive
  Added-mass effect seems to be the source of numerical instabilities in explicit coupling
  Geometrical non-linearities (moving domains), convective and viscous effects do not seem to
  affect the stability of a coupling algorithm. However, they are implicitly treated in fully
  implicit schemes (very expensive!)




                                             a
                               Miguel A. Fern´ndez   Mini-Workshop I
Let’s Summarize

  Explicit coupling is cheap but unstable
  Implicit coupling is stable but too expensive
  Added-mass effect seems to be the source of numerical instabilities in explicit coupling
  Geometrical non-linearities (moving domains), convective and viscous effects do not seem to
  affect the stability of a coupling algorithm. However, they are implicitly treated in fully
  implicit schemes (very expensive!)


              Major question:
              How to balance stability and computational cost?




                                              a
                                Miguel A. Fern´ndez   Mini-Workshop I
Let’s Summarize

  Explicit coupling is cheap but unstable
  Implicit coupling is stable but too expensive
  Added-mass effect seems to be the source of numerical instabilities in explicit coupling
  Geometrical non-linearities (moving domains), convective and viscous effects do not seem to
  affect the stability of a coupling algorithm. However, they are implicitly treated in fully
  implicit schemes (very expensive!)


              Major question:
              How to balance stability and computational cost?

      First idea: semi-implicit coupling
           Treat implicitly the added-mass effect
           Treat explicitly the geometrical non-linearities and the convective and
           viscous effects




                                              a
                                Miguel A. Fern´ndez   Mini-Workshop I
Let’s Summarize

  Explicit coupling is cheap but unstable
  Implicit coupling is stable but too expensive
  Added-mass effect seems to be the source of numerical instabilities in explicit coupling
  Geometrical non-linearities (moving domains), convective and viscous effects do not seem to
  affect the stability of a coupling algorithm. However, they are implicitly treated in fully
  implicit schemes (very expensive!)


              Major question:
              How to balance stability and computational cost?

      First idea: semi-implicit coupling
           Treat implicitly the added-mass effect
           Treat explicitly the geometrical non-linearities and the convective and
           viscous effects

      Second idea: semi-implicit coupling
      Perform this using a projection scheme (Chorin-Teman) within the fluid



                                              a
                                Miguel A. Fern´ndez   Mini-Workshop I
 The Chorin-Teman Projection Scheme

Main feature
Incompressibility and viscous/convective effects are decoupled




Set u0 = u0 , for n ≥ 0 compute (˜n+1 , un+1 , p n+1 )
                                 u
     Elliptic step:
                        8 „ n+1                                 «
                             ˜
                        > ρf u  − un
                        <            + un+1 ·
                                       ˜                 un+1
                                                         ˜          − 2µ div ε(˜n+1 ) = f n+1 ,
                                                                               u
                               ∆t
                        >
                        :                                                         un+1 = 0.
                                                                                  ˜|Γ

     Projection step:
                                     8      n+1 − un+1
                                     > ρf u       ˜
                                     >
                                     >                 +             p n+1 = 0,
                                     <         ∆t
                                     >
                                     >                              · un+1 = 0,
                                     >
                                     :
                                                           un+1 · n|Γ = 0.
(see Guermond-Minev-Shen 04, Rannacher 91, Shen 92)



                                                 a
                                   Miguel A. Fern´ndez    Mini-Workshop I
 The Chorin-Teman Projection Scheme

Main feature
Incompressibility and viscous/convective effects are decoupled



Set u0 = u0 , for n ≥ 0 compute (˜n+1 , un+1 , p n+1 )
                                 u
     Elliptic step:
                          8 „ n+1                                  «
                               ˜
                          > ρf u  − un
                          <            + un+1 ·
                                         ˜                  un+1
                                                            ˜          − 2µ div ε(˜n+1 ) = f n+1 ,
                                                                                  u
                                 ∆t
                          >
                          :                                                        un+1 = 0.
                                                                                   ˜|Γ

     Projection step:
                      8      n+1 − un+1                          8
                      > ρf u       ˜                                             f
                      >
                      >                 +        p n+1 = 0,      > −∆p n+1 = − ρ
                                                                 >
                                                                 >                             · un+1 ,
                                                                                                 ˜
                      <         ∆t                               <
                                                                                ∆t
                                                              =⇒
                      >
                      >                      · un+1      = 0,    > ∂p n+1
                                                                 >
                      >
                      :                    n+1
                                                                 >
                                                                 :         = 0.
                                       u         · n|Γ   = 0.       ∂n |Γ

(see Guermond-Minev-Shen 04, Rannacher 91, Shen 92)



                                                  a
                                    Miguel A. Fern´ndez       Mini-Workshop I
Semi-Implicit Coupling Algorithm: Explicit Part

  Update fluid domain (mesh) displacement and velocity:

                                                   df,n+1 − df,n
         df,n+1 = Ext(ds,n |Γw ),      wn+1 =                    ,        Ωf,n+1 = (I + df,n+1 )(Ωf ),
                                                                                                  0
                             0                          ∆t




                                               a
                                 Miguel A. Fern´ndez    Mini-Workshop I
Semi-Implicit Coupling Algorithm: Explicit Part

  Update fluid domain (mesh) displacement and velocity:

                                                    df,n+1 − df,n
          df,n+1 = Ext(ds,n |Γw ),      wn+1 =                    ,        Ωf,n+1 = (I + df,n+1 )(Ωf ),
                                                                                                   0
                              0                          ∆t

  Fluid: Elliptic step in the new domain Ωf,n+1 :
                     „ n+1                                           «
                       ˜
                       u     − un
                  ρf              + (˜n+1 − wn+1 ) ·
                                     u                        un+1
                                                              ˜            − 2µ div ε(˜n+1 ) = 0,
                                                                                      u
                           ∆t

                                        un+1 |Γw,n+1 = wn+1 |Γw,n+1 .
                                        ˜




                                                a
                                  Miguel A. Fern´ndez    Mini-Workshop I
Semi-Implicit Coupling Algorithm: Explicit Part

  Update fluid domain (mesh) displacement and velocity:

                                                            df,n+1 − df,n
                df,n+1 = Ext(ds,n |Γw ),           wn+1 =                 ,         Ωf,n+1 = (I + df,n+1 )(Ωf ),
                                                                                                            0
                                      0                          ∆t

  Fluid: Elliptic step in the new domain Ωf,n+1 :
            Z                                  Z                     Z
        1                                 1
                         ρf un+1 · v1 −
                            ˜       f
                                                    ρf u n · v 1 +
                                                               f
                                                                                  ρf (div wn+1 )˜n+1 · v1
                                                                                                u       f
       ∆t       Ωf,n+1                    ∆t  Ωf,n                       Ωf,n+1
       Z                                                       Z
   +              ρf (wn+1 − un+1 ) ·
                             ˜              un+1 · vf + 2µ
                                            ˜                              ε(˜n+1 ) : ε(v1 ) = 0,
                                                                             u           f              f
                                                                                                      ∀v1 ∈ V f (tn+1 ),
       Ωf,n+1                                                   Ωf,n+1

                                                   un+1 |Γw,n+1 = wn+1 |Γw,n+1 .
                                                   ˜




                                                        a
                                          Miguel A. Fern´ndez    Mini-Workshop I
Semi-Implicit Coupling Algorithm: Explicit Part

  Update fluid domain (mesh) displacement and velocity:

                                                            df,n+1 − df,n
                df,n+1 = Ext(ds,n |Γw ),           wn+1 =                 ,         Ωf,n+1 = (I + df,n+1 )(Ωf ),
                                                                                                            0
                                      0                          ∆t

  Fluid: Elliptic step in the new domain Ωf,n+1 :
            Z                                  Z                     Z
        1                                 1
                         ρf un+1 · v1 −
                            ˜       f
                                                    ρf u n · v 1 +
                                                               f
                                                                                  ρf (div wn+1 )˜n+1 · v1
                                                                                                u       f
       ∆t       Ωf,n+1                    ∆t  Ωf,n                       Ωf,n+1
       Z                                                       Z
   +              ρf (wn+1 − un+1 ) ·
                             ˜              un+1 · vf + 2µ
                                            ˜                              ε(˜n+1 ) : ε(v1 ) = 0,
                                                                             u           f              f
                                                                                                      ∀v1 ∈ V f (tn+1 ),
       Ωf,n+1                                                   Ωf,n+1

                                                   un+1 |Γw,n+1 = wn+1 |Γw,n+1 .
                                                   ˜



        Observations
        Geometrical non-linearities, viscous and convective effects are out of inner
        iterations




                                                        a
                                          Miguel A. Fern´ndez    Mini-Workshop I
Semi-Implicit Coupling Algorithm: Implicit Part

  Fluid: Projection step in Ωf,n+1 (known domain)
                          8
                          > f un+1 − un+1
                                       ˜
                          <ρ               + p n+1 = 0,                    in    Ωf,n+1
                                    ∆t
                          >
                          :                  · un+1 = 0,                   in    Ωf,n+1 ,

                                              ds,n+1 − ds,n
                               un+1 · n =                   · n,     on         Γw,n+1 .
                                                   ∆t
  Solid:
           8
                        ds,n+1 − 2ds,n + ds,n−1       `                     ´
           >
           <       ρs                 2
                                                − div0 F (ds,n+1 )S(ds,n+1 ) = 0,                  in   Ωs ,
                                                                                                         0
                                (∆t)
           >
           :                                                                           −T
               F (ds,n+1 )S(ds,n+1 )n0 = J(df,n+1 )σ(˜n+1 , p n+1 )F (df,n+1 )
                                                     u                                      n0 ,   on   Γw .
                                                                                                         0




                                                 a
                                   Miguel A. Fern´ndez   Mini-Workshop I
Semi-Implicit Coupling Algorithm: Implicit Part

  Fluid: Projection step in Ωf,n+1 (known domain)
           Z                                  Z                             Z                             Z
     1                                   1
                        ρf un+1 · v2 −
                                   f
                                                           ρf un+1 · v2 −
                                                              ˜       f
                                                                                         p n+1 div v2 +
                                                                                                    f
                                                                                                                       q div un+1 = 0,
     ∆t        Ωf,n+1                    ∆t       Ωf,n+1                        Ωf,n+1                        Ωf,n+1
                                                                    f
                                                                  ∀v2 ∈ HΓw,n+1 (div; Ωf,n+1 ),                 q ∈ L2 (Ωf,n+1 ),

                                                      ds,n+1 − ds,n
                                    un+1 · n =                      · n,            on     Γw,n+1 .
                                                           ∆t
  Solid:
           8
                             ds,n+1 − 2ds,n + ds,n−1       `                     ´
           >
           <            ρs                           − div0 F (ds,n+1 )S(ds,n+1 ) = 0,                             in    Ωs ,
                                                                                                                          0
                                     (∆t)2
           >
           :                                                                                       −T
                 F (ds,n+1 )S(ds,n+1 )n0 = J(df,n+1 )σ(˜n+1 , p n+1 )F (df,n+1 )
                                                       u                                                n0 ,      on     Γw .
                                                                                                                          0




                                                       a
                                         Miguel A. Fern´ndez        Mini-Workshop I
Semi-Implicit Coupling Algorithm: Implicit Part

  Fluid: Projection step in Ωf,n+1 (known domain)
              Z                                  Z                             Z                             Z
     1                                      1
                           ρf un+1 · v2 −
                                      f
                                                              ρf un+1 · v2 −
                                                                 ˜       f
                                                                                            p n+1 div v2 +
                                                                                                       f
                                                                                                                          q div un+1 = 0,
     ∆t           Ωf,n+1                    ∆t       Ωf,n+1                        Ωf,n+1                        Ωf,n+1
                                                                       f
                                                                     ∀v2 ∈ HΓw,n+1 (div; Ωf,n+1 ),                 q ∈ L2 (Ωf,n+1 ),

                                                         ds,n+1 − ds,n
                                       un+1 · n =                      · n,            on     Γw,n+1 .
                                                              ∆t
  Solid:
     Z                                            Z
                   ds,n+1 − 2ds,n + ds,n−1
              ρs                           · vs +      F (ds,n+1 )S(ds,n+1 ) : 0 ds
         Ωs
          0
                           (∆t)2                    Ωs
                                                     0
                                         Z                           Z
                                   = −           −p n+1 vs · n − 2µ           ε(˜n+1 )n · vs ,
                                                                                u                                           ∀vs ∈ V s .
                                                     Γw,n+1                                 Γw,n+1




                                                          a
                                            Miguel A. Fern´ndez        Mini-Workshop I
Semi-Implicit Coupling Algorithm: Implicit Part
  Fluid: Projection step in Ωf,n+1 (known domain)
              Z                                  Z                             Z                             Z
     1                                      1
                           ρf un+1 · v2 −
                                      f
                                                              ρf un+1 · v2 −
                                                                 ˜       f
                                                                                            p n+1 div v2 +
                                                                                                       f
                                                                                                                          q div un+1 = 0,
     ∆t           Ωf,n+1                    ∆t       Ωf,n+1                        Ωf,n+1                        Ωf,n+1
                                                                     ∀v2 ∈ HΓw,n+1 (div; Ωf,n+1 ),
                                                                       f
                                                                                                                   q ∈ L2 (Ωf,n+1 ),

                                                         ds,n+1 − ds,n
                                       un+1 · n =                      · n,            on     Γw,n+1 .
                                                              ∆t
  Solid:
     Z                                               Z
                   ds,n+1 − 2ds,n + ds,n−1
              ρs                 2
                                              · vs +      F (ds,n+1 )S(ds,n+1 ) : 0 ds
         Ωs
          0
                           (∆t)                        Ωs
                                                        0
                                       `                     ´      `                        ´
                            = − Rf p n+1 ; L2 (vs · n|Γw ) − Rf un+1 , df,n+1 ; L1 (v|Γw ) ,
                                     p                             u ˜
                                                                                         s
                                                                                                                            ∀vs ∈ V s .
                                                          0                                0
                                   |            {z           } |               {z            }
                                       Projection residual             Elliptic residual




                                                          a
                                            Miguel A. Fern´ndez        Mini-Workshop I
Semi-Implicit Coupling Algorithm: Implicit Part
  Fluid: Projection step in Ωf,n+1 (known domain)
              Z                                  Z                             Z                             Z
     1                                      1
                           ρf un+1 · v2 −
                                      f
                                                              ρf un+1 · v2 −
                                                                 ˜       f
                                                                                            p n+1 div v2 +
                                                                                                       f
                                                                                                                          q div un+1 = 0,
     ∆t           Ωf,n+1                    ∆t       Ωf,n+1                        Ωf,n+1                        Ωf,n+1
                                                                     ∀v2 ∈ HΓw,n+1 (div; Ωf,n+1 ),
                                                                       f
                                                                                                                   q ∈ L2 (Ωf,n+1 ),

                                                         ds,n+1 − ds,n
                                       un+1 · n =                      · n,            on     Γw,n+1 .
                                                              ∆t
  Solid:
     Z                                               Z
                   ds,n+1 − 2ds,n + ds,n−1
              ρs                 2
                                              · vs +      F (ds,n+1 )S(ds,n+1 ) : 0 ds
         Ωs
          0
                           (∆t)                        Ωs
                                                        0
                                       `                     ´      `                        ´
                            = − Rf p n+1 ; L2 (vs · n|Γw ) − Rf un+1 , df,n+1 ; L1 (v|Γw ) ,
                                     p                             u ˜
                                                                                         s
                                                                                                                            ∀vs ∈ V s .
                                                          0                                0
                                   |            {z           } |               {z            }
                                       Projection residual             Elliptic residual


      Observations
                   Projection sub-step in a fixed fluid domain (fixed matrix)
                   Linear coupled problem if linear solid
                   This coupled problem requires cheap inner iterations

                                                          a
                                            Miguel A. Fern´ndez        Mini-Workshop I
 Semi-Implicit Coupling: A Priori Energy Estimate (I)

Proposition
Assume that un+1 = 0 on Γin−out , the structure is an hyper-elastic material and
            ˜
                 »Z           Z       – Z
               1                                                        ∂q
                           q−        q =          div wn+1 q, ∀q with          = 0.
              ∆t    Ωf,n+1      Ωf,n       Ωf,n+1                       ∂t |x0

Then, the following energy inequality holds
        "Z                      Z                Z        ˛               ˛2 Z         ˛               ˛2 #
   1                ρf n+1 2           ρf n 2          ρs ˛ ds,n+1 − ds,n ˛         ρs ˛ ds,n − ds,n−1 ˛
                      |u   | −           |u | +           ˛
                                                          ˛
                                                                          ˛ −
                                                                          ˛
                                                                                       ˛
                                                                                       ˛
                                                                                                       ˛
                                                                                                       ˛
   ∆t        Ωf,n+1 2             Ωf,n 2           Ωs 2
                                                     0
                                                                 ∆t              Ωs 2
                                                                                  0
                                                                                              ∆t
                         "Z                      Z                  # Z
                      1
                   +           W (E (ds,n+1 )) −       W (E (ds,n )) +          2µ|ε(˜n+1 )|2
                                                                                     u
                      ∆t    Ωs
                             0                     Ωs
                                                    0                    Ωf,n+1

                                                          `                           ds,n+1 − ds,n ´
                                                      − Rf un+1 , df,n+1 ; L2 (˜n+1 −
                                                         u ˜                   u                   ) ≤0
                                                                                           ∆t




                                                    a
                                      Miguel A. Fern´ndez   Mini-Workshop I
 Semi-Implicit Coupling: A Priori Energy Estimate (I)

Proposition
Assume that un+1 = 0 on Γin−out , the structure is an hyper-elastic material and
            ˜
                 »Z           Z       – Z
               1                                                        ∂q
                           q−        q =          div wn+1 q, ∀q with          = 0.
              ∆t    Ωf,n+1      Ωf,n       Ωf,n+1                       ∂t |x0

Then, the following energy inequality holds
        "Z                      Z                Z        ˛               ˛2 Z         ˛               ˛2 #
   1                ρf n+1 2           ρf n 2          ρs ˛ ds,n+1 − ds,n ˛         ρs ˛ ds,n − ds,n−1 ˛
                      |u   | −           |u | +           ˛
                                                          ˛
                                                                          ˛ −
                                                                          ˛
                                                                                       ˛
                                                                                       ˛
                                                                                                       ˛
                                                                                                       ˛
   ∆t        Ωf,n+1 2             Ωf,n 2           Ωs 2
                                                     0
                                                                 ∆t              Ωs 2
                                                                                  0
                                                                                              ∆t
                         "Z                      Z                  # Z
                      1
                   +           W (E (ds,n+1 )) −       W (E (ds,n )) +          2µ|ε(˜n+1 )|2
                                                                                     u
                      ∆t    Ωs
                             0                     Ωs
                                                    0                    Ωf,n+1

                                                          `                             ds,n+1 − ds,n ´
                                                     − Rf un+1 , df,n+1 ; L1 (˜n+1 −
                                                         u ˜                   u                     ) ≤0
                                                                                             ∆t
                                                       |                        {z                     }
                                                          Z
                                                                       n+1         n+1   ds,n+1 − ds,n
                                                       2µ          ε(˜
                                                                     u     )n · (˜
                                                                                 u     −               )
                                                            Γw,n+1                             ∆t




                                                    a
                                      Miguel A. Fern´ndez   Mini-Workshop I
 Semi-Implicit Coupling: A Priori Energy Estimate (I)

Proposition
Assume that un+1 = 0 on Γin−out , the structure is an hyper-elastic material and
            ˜
                 »Z           Z       – Z
               1                                                        ∂q
                           q−        q =          div wn+1 q, ∀q with          = 0.
              ∆t    Ωf,n+1      Ωf,n       Ωf,n+1                       ∂t |x0

Then, the following energy inequality holds
        "Z                      Z                Z         ˛                ˛2 Z           ˛              ˛2 #
   1                ρf n+1 2           ρf n 2           ρs ˛ ds,n+1 − ds,n ˛           ρs ˛ ds,n − ds,n−1 ˛
                      |u   | −           |u | +            ˛
                                                           ˛
                                                                            ˛ −
                                                                            ˛
                                                                                           ˛
                                                                                           ˛
                                                                                                          ˛
                                                                                                          ˛
   ∆t        Ωf,n+1 2             Ωf,n 2           Ωs 2
                                                     0
                                                                  ∆t                Ωs 2
                                                                                     0
                                                                                                 ∆t
                         "Z                      Z                  # Z
                      1
                   +           W (E (ds,n+1 )) −       W (E (ds,n )) +             2µ|ε(˜n+1 )|2
                                                                                         u
                      ∆t    Ωs
                             0                     Ωs
                                                    0                      Ωf,n+1
                                               
                                  artificial              `                              ds,n+1 − ds,n ´
                                                  − Rf un+1 , df,n+1 ; L1 (˜n+1 −
                                                        u ˜                   u                       ) ≤0
                               viscous power                                                 ∆t
                                                      |                        {z                      }
                                                         Z
                                                                      n+1         n+1    ds,n+1 − ds,n
                                                     2µ           ε(˜
                                                                    u     )n · (˜
                                                                                u      −               )
                                                           Γw,n+1                              ∆t




                                                     a
                                       Miguel A. Fern´ndez   Mini-Workshop I
 Semi-Implicit Coupling: A Priori Energy Estimate (II)

Idea of the proof. We take appropriate test functions and then we add the fluid and the solid
variational formulations. In particular,
    for the solid:
                                                        ds,n+1 − ds,n
                                              vs =                    ∈ V s.
                                                             ∆t
    for the fluid (explicit part):

                           v1 = un+1 − L1 (v|Γw ) − L1 (˜n+1 − v|Γw ) ∈ V f (tn+1 ).
                            f
                                ˜           s
                                                        u|Γw    s
                                                        0              0           0


    for the fluid (implicit part):

                                v2 = un+1 − L2 (vs · n|Γw ) ∈ HΓw,n+1 (div; Ωf,n+1 ).
                                 f
                                                             0



                 Remarks:
                     1   v1|Γw = 0 and hence vf ∈ V f
                          f
                            0
                     2   v2 · n|Γw = 0 and hence v2 ∈ HΓw,n+1 (div; Ωf,n+1 )
                          f                       f
                                  0

                     3   Let us recall that un+1 = v|Γw
                                            ˜|Γw    s
                                                    0        0




                                                     a
                                       Miguel A. Fern´ndez       Mini-Workshop I
Stability Analysis: Simplified Case

  Convective and geometrical non-linearities are neglected (Stokes)

    Theorem (MF-Gerbeau-Grandmont 05)
    The scheme is stable, in the energy norm, under the following condition:
                                        „                    «
                                            hd        hd −2
                                ρs ≥ C ρf D + µ∆t D            ,
                                           H           H

    with h, H the fluid/solid mesh sizes and d, D the fluid/solid domains dimensions.




                                            a
                              Miguel A. Fern´ndez   Mini-Workshop I
Stability Analysis: Simplified Case

  Convective and geometrical non-linearities are neglected (Stokes)

    Theorem (MF-Gerbeau-Grandmont 05)
    The scheme is stable, in the energy norm, under the following condition:
                                        „                    «
                                            hd        hd −2
                                ρs ≥ C ρf D + µ∆t D            ,
                                           H           H

    with h, H the fluid/solid mesh sizes and d, D the fluid/solid domains dimensions.


    Remarks
         if d = D, we obtain a less restrictive condition than for the explicit coupling
         if D = d − 1, stability can be ensured keeping h = H




                                               a
                                 Miguel A. Fern´ndez   Mini-Workshop I
Stability Analysis: Simplified Case

  Convective and geometrical non-linearities are neglected (Stokes)

    Theorem (MF-Gerbeau-Grandmont 05)
    The scheme is stable, in the energy norm, under the following condition:
                                        „                    «
                                            hd        hd −2
                                ρs ≥ C ρf D + µ∆t D            ,
                                           H           H

    with h, H the fluid/solid mesh sizes and d, D the fluid/solid domains dimensions.


    Remarks
         if d = D, we obtain a less restrictive condition than for the explicit coupling
         if D = d − 1, stability can be ensured keeping h = H

  Idea of the proof:
      Start from the previous energy estimate.
      Use Cauchy-Schwartz and an inverse inequality in order to control the artificial viscous power

                                                                ds,n+1 − ds,n ´
                                                                 |Γw      |Γw
                                  Rf un+1 , df,n+1 ; L1 (˜n+1 −    0        0
                                    `
                                   u ˜                   u|Γw                 ) .
                                                            0        ∆t



                                               a
                                 Miguel A. Fern´ndez   Mini-Workshop I
Some open questions:


  Is incompressibility the only source of instabilities?
  Are there other kinds of semi-implicit coupling schemes?
  How to enforce physiological boundary conditions? Multi-scale modeling. . .



                                                                   3D models




                                                       1D models
                                Heart model




                                                                               OD models




                                                              OD models


                                                                                       OD models




                                               a
                                 Miguel A. Fern´ndez          Mini-Workshop I
        THANK YOU




              a
Miguel A. Fern´ndez   Mini-Workshop I

				
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