Stability and convergence analysis of the kinematically coupled - - PowerPoint PPT Presentation

Fluid-structure interaction Kinematically coupled scheme. Applications.

Stability and convergence analysis of the kinematically coupled scheme for the fluid-structure interaction

Boris Muha

Department of Mathematics, Faculty of Science, University of Zagreb

2018 Modeling, Simulation and Optimization of the Cardiovascular System Magdeburg, 22-24 October 2018 Joined work with M. Bukaˇ c (Notre Dame) and S. ˇ Cani´ c (UC Berkeley)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Talk summary

• 1. Fluid-structure interaction.
• 2. Kinematically coupled scheme.
• 3. Applications.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Motivation

• Fluid-structure interaction (FSI) problems describe the dynamics of a

multiphysics system involving fluid and solid components.

• They are everyday phenomena in nature and arise in various applications

ranging from biomedicine to engineering.

• Examples: blood flow in vessels, artificial heart valves, vocal cords,

valveless pumping, airway closure in lungs, geophysics (underground flows, hydraulic fracturing), classical industrial applications (aeroelasticity, offshore structures), artificial micro-swimmers in body liquids, micro-(and nano-)electro-mechanical systems (MEMS), various sports equipment

Fluid-structure interaction Kinematically coupled scheme. Applications.

Motivation II

• Main motivation for our work comes from biofluidic applications.
• Main example in this talk will be blood flow through compliant vessel.
• Densities of the structure and the fluid are comparable (unlike in e.g.

aeroelasticity) - highly nonlinear coupling.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Formulation of an example of FSI problem

• 3D fluid is coupled with 3D nonlinear elasticity.
• ΩF - the fluid reference domain.
• ΩS - the structure reference domain.
• ΩF - the fluid-structure interface .

Fluid-structure interaction Kinematically coupled scheme. Applications.

The structure equations

• Mathematically, FSI systems are described in terms of continuum

mechanics, which gives rise to a system of partial differential equations (PDEs).

• More precisely, a non-linear system of partial differential equations of

mixed parabolic-hyperbolic type with a moving boundary, i.e. part of the domain is also an unknown of the system.

• Unknowns: u - the fluid velocity and η - the structure displacement.
• The elastodynamics equations:

̺s ∂2 ∂t2 η = ∇ · T(∇η) in (0, T) × ΩS,

• Constitutive relation:

First Piola-Kirchhoff stress tensor: T(F) =

∂ ∂F W (F), where

W : M3(R) → R is a stored energy function.

Fluid-structure interaction Kinematically coupled scheme. Applications.

The fluid equations

• The physical fluid domain is given by

Ωf (t) = ϕf (t, Ωf ), t ∈ (0, T), where φf the fluid domain displacement (i.e. an arbitrary extension of η to ΩF).

• The fluid equations:

ρf (∂tu + u · ∇u) = ∇ · σ(∇u, p), ∇ · u = 0,

• in Ωf (t) = ϕf (t, Ωf ), t ∈ (0, T).

Constitutive relation: Cauchy stress tensor is given by relation σ(∇u, p) = −pI + 2µD(u)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Coupling and boundary condition

• Dynamic coupling condition (balance of forces):

Tn = (σ ◦ ϕ)∇ϕ−τn

• n (0, T) × Γ
• Kinematic coupling condition (no-slip):

∂tη(t, .) = u(t, .)|Γ ◦ ϕf ,

• n Γ, t ∈ (0, T).
• In some physical situations different kinematic boundary condition

might be more appropriate (slip, Signorini type BC).

• Boundary data: dynamic pressure/stress free or periodic.

Fluid-structure interaction Kinematically coupled scheme. Applications.

An FSI problem - summary

find (u, η) such that ̺s ∂2

∂t2 η = ∇ · T(∇η) in

(0, T) × Ωs, ρf (∂tu + u · ∇u) = ∇ · σ(∇u, p), ∇ · u = 0,

• in Ωf (t) = ϕf (t, Ωf ), t ∈ (0, T),

t

0 u|Γ ◦ ϕ = ϕΓ, Tn = (σ ◦ ϕ)∇ϕ−τn

• n Γ,

(1) where u is the fluid velocity, η the structure deformation, ϕf the fluid domain displacement

Fluid-structure interaction Kinematically coupled scheme. Applications.

Energy inequality

• We formally multiply the structure equations by ∂tη and integrate over

ΩS. Then formally multiply the fluid equations by u and integrate over ΩF.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Energy inequality

• We formally multiply the structure equations by ∂tη and integrate over

ΩS. Then formally multiply the fluid equations by u and integrate over ΩF.

• By adding resulting equalities, integrating by parts and using the

coupling conditions we obtain formal energy inequality: d dt

• ∂tη2

L2(Ωs) + u2 L2(Ωf (t) +

• Ωs

W (∇η)

• + µD(u)2

L2(Ωf (t))

≤ C(data).

Fluid-structure interaction Kinematically coupled scheme. Applications.

Energy inequality

• We formally multiply the structure equations by ∂tη and integrate over

ΩS. Then formally multiply the fluid equations by u and integrate over ΩF.

• By adding resulting equalities, integrating by parts and using the

coupling conditions we obtain formal energy inequality: d dt

• ∂tη2

L2(Ωs) + u2 L2(Ωf (t) +

• Ωs

W (∇η)

• + µD(u)2

L2(Ωf (t))

≤ C(data).

• From the analysis point of view such a FSI problem is still out of reach

(some results Coutand, Shkoller (’06), Grandmont (’02), Galdi, Kyed ’09, Boulakia, Guerrero (’16), ˇ Cani´ c, BM ’16) - various simplified models in the literature.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Main challenges

• Nonlinear elastodynamics.
• Navier-Stokes equations.
• Nonlinear coupling - geometrical nonlinearity.
• Fluid domain deformation (injectivity, regularity).
• Hyperbolic-parabolic coupling.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Main challenges

• Nonlinear elastodynamics.
• Navier-Stokes equations.
• Nonlinear coupling - geometrical nonlinearity.
• Fluid domain deformation (injectivity, regularity).
• Hyperbolic-parabolic coupling.

It is natural to consider various simplifications of the general FSI model. Which simplifications are physically relevant? Simplifications are usually

• btained by neglecting some terms (physically - some small parameters)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Introduction

• Monolithic and partitioned approach.
• Kinematically coupled scheme is a partitioned scheme introduced by

Guidoboni, Glowinski, Cavallini, ˇ Cani´ c (JCP 2009).

• This lecture will be mostly based on convergence analysis in Bukaˇ

c, BM (SINUM 2016).

• The scheme is based on the Lie operator splitting, where the fluid and

the structure subproblems are fully decoupled and communicate only via the interface conditions.

• Advantages are modularity, stability, and easy implementation.
• Several extensions and implementations by different groups.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Lie-Trotter formula

• u′(t) = Au(t) → u(x) = eAtu0, A ∈ Mn
• Let us decompose A = A1 + A2?
• eA+B = eAeB ⇔ AB = BA.
• However:

eA+B = lim

N→∞(eA/NeB/N)N.

• This can be generalized to certain unbounded operators.
• However it is not directly applicable to FSI problems.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Lie (Marchuk-Yanenko) operator splitting scheme

• We consider initial value problem d

dt φ + A(φ) = 0, φ(0) = φ0.

• We suppose that A = A1 + A2.
• Let k = T/N be time-dicretization step and tn = nk. Then we define:

d dt φn+ i

2 + Ai(φn+ i 2 ) = 0

in (tn, tn+1), φn+ i

2 (tn) = φn+ i−1 2 , n = 0, . . . , N − 1, i = 1, 2,

where φn+ i

2 = φn+ i 2 (tn+1).

Fluid-structure interaction Kinematically coupled scheme. Applications.

Simplified linear model

ρf ∂tu = ∇ · σ(u, p), ∇ · u = 0 in (0, T) × Ω, σ(u, p)n = −pin/out(t)n

• n (0, T) × Σ,

ρsǫ∂2

t η + Lsη = −σ(u, p)n

• n (0, T) × Γ,

∂tη = u

• n (0, T) × Γ,

η(., 0) = η0, ∂tη(., 0) = v0

• n Γ,

u(., 0) = u0 in Ω.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Monolithic formulation

Given t ∈ (0, T) find (✉, η, p) ∈ V f × V s × Qf with ✉ = ∂tη on Γ, such that for all (ϕ, ξ, q) ∈ V fsi × Qf ρf

• Ω

∂t✉ · ϕd① + 2µ

• Ω

❉(✉) : ❉(ϕ)d① −

• Ω

p∇ · ϕd① +

• Ω

(∇ · ✉)qd① + ρsǫ

• Γ

∂ttη · ξdx +

• Γ

Lsη · ξdS =

• Σ

pin/out(t)ϕ · ♥dS. V f = (H1(Ω))d, Qf = L2(Ω), V s = (H1

0(Γ))d,

V fsi = {(ϕ, ξ) ∈ V f × V s| ϕ|Γ = ξ},

Fluid-structure interaction Kinematically coupled scheme. Applications.

Fluid and structure sub-problems

• Step 1: The structure sub-problem. Find ˜

✈ n+1, and ηn+1 such that ρsǫ ˜ ✈ n+1 − ✈ n ∆t + LSηn+1 = −βσ(✉n, pn)♥

• n Γ,

(2) dtηn+1 = ˜ ✈ n+1

• n Γ,

(3)

• Step 2. The fluid sub-problem. Find ✉n+1, pn+1 and ✈ n+1 such that

ρf dt✉n+1 = ∇ · σ(✉n+1, pn+1) in Ω, (4) ∇ · ✉n+1 = 0 in Ω, (5) ρsǫ✈ n+1 − ˜ ✈ n+1 ∆t = −σ(✉n+1, pn+1)♥ + βσ(✉n, pn)♥

• n Γ,

(6) ✉n+1 = ✈ n+1

• n Γ,

(7)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Improving the accuracy

• Original kiniematically couple scheme β = 0
• Splitting error: ˜

✈ n+1 − ✈ n+1 - accuracy O((∆t)1/2). ✈ ✈ ♥ ♥ ♥ ♥ ♥

Fluid-structure interaction Kinematically coupled scheme. Applications.

Improving the accuracy

• Original kiniematically couple scheme β = 0
• Splitting error: ˜

✈ n+1 − ✈ n+1 - accuracy O((∆t)1/2).

• We improve the accuracy by taking into account fluid stress in the

structure sub-problem (β = 1): ˜ ✈ n+1 − ✈ n+1 = ∆t ̺sǫ

• σn+1♥ − βσn♥
• = ∆t

̺sǫ

• β
• σn+1♥ − σn♥
• + (1 − β)σn+1♥
• n Γ.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Improving the accuracy

• Original kiniematically couple scheme β = 0
• Splitting error: ˜

✈ n+1 − ✈ n+1 - accuracy O((∆t)1/2).

• We improve the accuracy by taking into account fluid stress in the

structure sub-problem (β = 1): ˜ ✈ n+1 − ✈ n+1 = ∆t ̺sǫ

• σn+1♥ − βσn♥
• = ∆t

̺sǫ

• β
• σn+1♥ − σn♥
• + (1 − β)σn+1♥
• n Γ.
• Optimal first order accuracy for β = 1.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Improving the accuracy

• Original kiniematically couple scheme β = 0
• Splitting error: ˜

✈ n+1 − ✈ n+1 - accuracy O((∆t)1/2).

• We improve the accuracy by taking into account fluid stress in the

structure sub-problem (β = 1): ˜ ✈ n+1 − ✈ n+1 = ∆t ̺sǫ

• σn+1♥ − βσn♥
• = ∆t

̺sǫ

• β
• σn+1♥ − σn♥
• + (1 − β)σn+1♥
• n Γ.
• Optimal first order accuracy for β = 1.
• Extension to second order accuracy - Oyekole, Trenchea, Bukaˇ

c (SINUM 2018)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Stability estimate

• We obtain stability estimate in the standard way by taking ˜

✈ n+1 and (✉n+1, ✈ n+1) and using identity (a − b)a = 1 2a2 − 1 2b2 + 1 2(a − b)2. ✉ ♥ ✈ ✈ ✉ ♥ ✉ ♥ ✉ ♥ ✉ ♥ ✉ ♥ ✈ ✈

Fluid-structure interaction Kinematically coupled scheme. Applications.

Stability estimate

• We obtain stability estimate in the standard way by taking ˜

✈ n+1 and (✉n+1, ✈ n+1) and using identity (a − b)a = 1 2a2 − 1 2b2 + 1 2(a − b)2.

• Only splitting term is non-standard:

I = ∆t

• Γ

σ(✉n

h, pn h)♥ ·

• ✈ n+1

h

− ˜ ✈ n+1

h

• dS

✉ ♥ ✉ ♥ ✉ ♥ ✉ ♥ ✉ ♥ ✈ ✈

Fluid-structure interaction Kinematically coupled scheme. Applications.

Stability estimate

• We obtain stability estimate in the standard way by taking ˜

✈ n+1 and (✉n+1, ✈ n+1) and using identity (a − b)a = 1 2a2 − 1 2b2 + 1 2(a − b)2.

• Only splitting term is non-standard:

I = ∆t

• Γ

σ(✉n

h, pn h)♥ ·

• ✈ n+1

h

− ˜ ✈ n+1

h

• dS
• ∆t2

ρsǫ

• Γ

σ(✉n

h, pn h)♥ ·

• σ(✉n

h, pn h)♥ − σ(✉n+1 h

, pn+1

h

)♥

• dS

= ∆t2 2ρsǫ

• σ(✉n

h, pn h)♥2 L2(Γ) − σ(✉n+1 h

, pn+1

h

)♥2

L2(Γ)

• +ρsǫ

2 ✈ n+1

h

− ˜ ✈ n+1

h

2

L2(Γ).

Fluid-structure interaction Kinematically coupled scheme. Applications.

Stability analysis

We define discrete energies: Ef (✉n

h) = ρf

2 ✉n

h2 L2(Ω),

Ev(✈ n

h) = ρsǫ

2 ✈ n

h2 L2(Γ),

Es(ηn

h) = 1

2ηn

h2

• S. (8)

Theorem (Bukaˇ c, BM, SINUM ’16)

Let {(✉n

h, pn h, ˜

✈ n

h, ✈ n h, ηn h}0≤n≤N obtained by the numerical scheme.

Ef (✉N

h ) + Ev(✈ N h ) + Es(ηN h ) + ∆t2

2ρsǫσ(✉N

h , pN h )♥2 L2(Γ) + ρf ∆t2

2

N−1

• n=0

dt✉n+1

h

2

L

+∆t2 2

N−1

• n=0

dtηn+1

h

2

S + µ∆t N−1

• n=0

✉n+1

h

2

F + ρsǫ

2

N−1

• n=0

˜ ✈ n+1

h

− ✈ n

h2 L2(Γ)

Ef (✉0

h) + Ev(✈ 0 h) + Es(η0 h) + ∆t2

2ρsǫσ(✉0

h, p0 h)♥2 L2(Γ) + ∆t N−1

• n=0

pin/out(tn+1)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Error analysis

Theorem

Consider the numerical solution (✉h, ph, ˜ ✈ h, ✈ h, ηh) with discrete initial data (✉0

h, p0 h, ˜

✈ 0

h, ✈ 0 h, η0 h) = (Sh✉0, Πhp0, Ih˜

✈ 0, Ih✈ 0, Rhη0). Assume that β = 1 and the exact solution satisfies additional regularity assumptions. Furthermore, we assume that γ∆t < 1, γ1 < ρsǫ 8∆t , γ2 < 1 4, where γ > 0, γ1 > 0, γ2 > 0. Let ˜ γ = max{γ, γ2, γ3}. ✉N − ✉N

h L2(Ω) + ✉N − ✉N h L2(0,T;F) + ✈ N − ✈ N h L2(Γ)

+ηN − ηN

h S + σ(✉N, pN)♥ − σ(✉N h , pN h )♥L2(Γ)

e˜

γT

• ∆tA1 + ∆t2

∆t1/2 + 1 γ2 + 1 γ1 + γ1∆t

• A2

+hkB1 + hk+1B2 + hs+1B3

Fluid-structure interaction Kinematically coupled scheme. Applications.

Error analysis II

+∆thk ∆t + 1 γ2 + 1 γ1 + γ1∆t2 C1 + ∆ths+1 ∆t + 1 γ2 + 1 γ1 + γ1∆t2 C2,

• A1 = ∂tt✉L2

t (L2(Ω)) + 1

γ ∂tt✈L2

t (L2(Γ)) + 1

γ ∂ttηL2

t (H1(Γ)) + 1

γ1 ∂tσ♥L2

t (L2 x)

A2 = ∂tσ♥L2(0,T;L2(Γ)), B1 = 1 γ ✈L2

t (Hk+1(Γ)) + ∂t✉L2 t (Hk+1(Ω)) + ✉L2 t (Hk+1(Ω))

+ 1 γ1 ✉L2

t (Hk+1(Γ)) + ✉L∞ t (Hk+1(Ω)) + ✉L∞ t (Hk+1(Γ)) + ηL∞ t (Hk+1(Γ)),

B2 =

• 1 + 1

γ1

• ∂t✈L2

t (Hk+1(Γ)) + ✈L∞ t (Hk+1(Γ)),

B3 = p2

L2

t (Hs+1(Ω)) + 1

γ1 p2

L2

t (Hs+1(Γ)) + pL∞ t (Hs+1(Γ)),

C1 = ∂t✉2

2 k+1

, C2 = ∂tp2

2 s+1

.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Thick structure - simplified problem

• ΩF = (0, 1)2 × (−1, 0), ΩS = (0, 1)2 × (0, 1), Γ = (0, 1)2 × {0}.
• S(η) = 2µsD(η) + λs(∇ · η)I.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Thick structure - simplified problem

• ΩF = (0, 1)2 × (−1, 0), ΩS = (0, 1)2 × (0, 1), Γ = (0, 1)2 × {0}.
• S(η) = 2µsD(η) + λs(∇ · η)I.

ρf ∂tu = ∇ · σ(u, p), ∇ · u = 0 in (0, T) × Ω, ̺s∂2

t η = ∇ · S(η)

in (0, T) × ΩS, S(η)n = σ(u, p)n

• n (0, T) × Γ,

∂tη = u

• n (0, T) × Γ,

Fluid-structure interaction Kinematically coupled scheme. Applications.

Thick structure - simplified problem

• ΩF = (0, 1)2 × (−1, 0), ΩS = (0, 1)2 × (0, 1), Γ = (0, 1)2 × {0}.
• S(η) = 2µsD(η) + λs(∇ · η)I.

ρf ∂tu = ∇ · σ(u, p), ∇ · u = 0 in (0, T) × Ω, ̺s∂2

t η = ∇ · S(η)

in (0, T) × ΩS, S(η)n = σ(u, p)n

• n (0, T) × Γ,

∂tη = u

• n (0, T) × Γ,

If we consider coupling via elastic interface, coupling condition reads: ρsǫ∂2

t η + Lsη = S(η)n − σ(u, p)n

Fluid-structure interaction Kinematically coupled scheme. Applications.

Splitting for the thick structure

• Function spaces:

V f = H1(ΩF)3, Q = L2(ΩF), V s = H1(ΩS)3, V fsi = {(ϕ, ξ) ∈ V f ×V s|

• Step 1: Find (˜

✉n+1

h

, ˜ ✈ n+1

h

) ∈ V fsi

h , ηn+1 h

∈ V s

h such that for every

(ϕh, ξh) ∈ V fsi

h

the following equality holds: ρs

• ΩS

˜ ✈ n+1

h

− ✈ n

h

∆t · ξh + ats(ηn+1

h

, ξh) +ρf

• ΩF

˜ ✉n+1

h

− ✉n

h

∆t · ϕh = −

• Γ

σn

h♥ · ξ,

˜ ✈ n+1

h

= ηn+1

h

− ηn

h

∆t , (˜ ✈ n+1

h

)|Γ = (˜ ✉n+1

h

)|Γ, (9) where σn

h = σ(✉n h, pn h).

Fluid-structure interaction Kinematically coupled scheme. Applications.

Step 2

Find (✉n+1

h

, ✈ n+1

h

, pn+1) ∈ V fsi

h

× Qh, such that for (ϕh, ξh, qh) ∈ V fsi

h

× Qh, the following equality holds: ρf

• ΩF

✉n+1

h

− ˜ ✉n+1

h

∆t · ϕh + ρs

• ΩS

✈ n+1

h

− ˜ ✈ n+1

h

∆t · ξh +af (✉n+1

h

, ϕh)−b(pn+1

h

, ϕh) + b(qh, ✉n+1

h

) =

• Γ

σn

h♥ · ϕh,

(✈ n+1

h

)|Γ = (✉n+1

h

)|Γ. (10)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Thick structure - stability theorem

Theorem

Let {(✉n

h, ˜

✈ n

h, ✈ n h, ηn h}0≤n≤N be the numerical solution. Then, the following

estimate holds: Ef (✉N

h ) + Ev(✈ N h ) + Es(ηN h ) + ∆t2

ρsh σ(✉N

h , pN h )♥2 L2(Γ)

+ρf ∆t2 2

N−1

• n=0

dt✉n+1

h

2

L2(ΩF ) + ∆t2

2

N−1

• n=0

as(dtηn+1

h

, dtηn+1

h

) +µ∆t

N−1

• n=0

✉n+1

h

2

F + ρs

2

N−1

• n=0

˜ ✈ n+1

h

− ✈ n

h2 L2(ΩS)

Ef (✉0

h) + Ev(✈ 0 h) + Es(η0 h) + ∆t2

ρsh σ(✉0

h, p0 h)♥2 L2(Γ) + ∆t N−1

• n=0

pin/out(tn+1)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Numerical convergence - thin structure

Fluid-structure interaction Kinematically coupled scheme. Applications.

Numerical convergence - thick structure

Fluid-structure interaction Kinematically coupled scheme. Applications.

Nonlinear moving boundary problem

• η denote the vertical displacement of the deformable boundary.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Nonlinear moving boundary problem

• η denote the vertical displacement of the deformable boundary.
• Since domain is symmetric we consider only upper half of the domain.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Nonlinear moving boundary problem

• η denote the vertical displacement of the deformable boundary.
• Since domain is symmetric we consider only upper half of the domain.
• Fluid domain at time t:

Ωη(t) = {(z, r) : 0 < z < L, 0 < r < R + η(t, z)}

Fluid-structure interaction Kinematically coupled scheme. Applications.

Nonlinear moving boundary problem

• η denote the vertical displacement of the deformable boundary.
• Since domain is symmetric we consider only upper half of the domain.
• Fluid domain at time t:

Ωη(t) = {(z, r) : 0 < z < L, 0 < r < R + η(t, z)}

• Γ(t) = {(z, R + η(t, z)) : 0 < z < 1} is deformable boundary.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Nonlinear moving boundary problem

• η denote the vertical displacement of the deformable boundary.
• Since domain is symmetric we consider only upper half of the domain.
• Fluid domain at time t:

Ωη(t) = {(z, r) : 0 < z < L, 0 < r < R + η(t, z)}

• Γ(t) = {(z, R + η(t, z)) : 0 < z < 1} is deformable boundary.
• Longitudinal displacement is neglected.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Full FSI problem

Find u = (uz(t, z, r), ur(t, z, r)), p(t, z, r), and η(t, z) such that ρf

• ∂tu + (u · ∇)u
• =

∇ · σ ∇ · u =

• in Ωη(t), t ∈ (0, T),

u = ∂tηer ρsh∂2

t η + C0η − C1∂2 z η + C2∂4 z η

= −Jσn · er

• n (0, T) × (0, L),

ur = 0, ∂ruz =

• n (0, T) × Γb,

p + ρf

2 |u|2

= Pin/out(t), ur = 0,

• n (0, T) × Γin/out,

u(0, .) = u0, η(0, .) = η0, ∂tη(0, .) = v0.    at t = 0.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Energy inequality

By formally taking solution (u, ∂tη) as test function in the weak formulation we get following energy estimate: d dt E + D ≤ C(Pin/out), where

Fluid-structure interaction Kinematically coupled scheme. Applications.

Energy inequality

By formally taking solution (u, ∂tη) as test function in the weak formulation we get following energy estimate: d dt E + D ≤ C(Pin/out), where E = ̺f 2 u2

L2(Ω) + ̺sh

2 ηt2

L2(Γ) + 1

2

• C0η2

L2 + C1∂zη2 L2 + C2∂2 z η2 L2

• ,

D = µD(u)2

L2(Ω).

Fluid-structure interaction Kinematically coupled scheme. Applications.

ALE formulation on reference domain

• We want to rewrite problem in the reference configuration

Ω = (0, L) × (0, 1).

Fluid-structure interaction Kinematically coupled scheme. Applications.

ALE formulation on reference domain

• We want to rewrite problem in the reference configuration

Ω = (0, L) × (0, 1).

• Since we consider control domain, we can not use Lagrangian

coordinates.

Fluid-structure interaction Kinematically coupled scheme. Applications.

ALE formulation on reference domain

• We want to rewrite problem in the reference configuration

Ω = (0, L) × (0, 1).

• Since we consider control domain, we can not use Lagrangian

coordinates.

• We use ALE mapping Aη(t) : Ω → Ωη(t),

Aη(t)(˜ x, ˜ z) =

• ˜

x (R + η(t, x))˜ z

• ,

(˜ x, ˜ z) ∈ Ω.

Fluid-structure interaction Kinematically coupled scheme. Applications.

ALE formulation on reference domain

• We want to rewrite problem in the reference configuration

Ω = (0, L) × (0, 1).

• Since we consider control domain, we can not use Lagrangian

coordinates.

• We use ALE mapping Aη(t) : Ω → Ωη(t),

Aη(t)(˜ x, ˜ z) =

• ˜

x (R + η(t, x))˜ z

• ,

(˜ x, ˜ z) ∈ Ω.

• uη(t, .) = u(t, .) ◦ Aη(t),

pη(t, .) = p(t, .) ◦ Aη(t).

Fluid-structure interaction Kinematically coupled scheme. Applications.

ALE formulation on reference domain

• We want to rewrite problem in the reference configuration

Ω = (0, L) × (0, 1).

• Since we consider control domain, we can not use Lagrangian

coordinates.

• We use ALE mapping Aη(t) : Ω → Ωη(t),

Aη(t)(˜ x, ˜ z) =

• ˜

x (R + η(t, x))˜ z

• ,

(˜ x, ˜ z) ∈ Ω.

• uη(t, .) = u(t, .) ◦ Aη(t),

pη(t, .) = p(t, .) ◦ Aη(t).

• We have problem on a fixed domain, but with coefficients that depend
• n the solution.

Fluid-structure interaction Kinematically coupled scheme. Applications.

ALE formulation on reference domain

• We want to rewrite problem in the reference configuration

Ω = (0, L) × (0, 1).

• Since we consider control domain, we can not use Lagrangian

coordinates.

• We use ALE mapping Aη(t) : Ω → Ωη(t),

Aη(t)(˜ x, ˜ z) =

• ˜

x (R + η(t, x))˜ z

• ,

(˜ x, ˜ z) ∈ Ω.

• uη(t, .) = u(t, .) ◦ Aη(t),

pη(t, .) = p(t, .) ◦ Aη(t).

• We have problem on a fixed domain, but with coefficients that depend
• n the solution.
• Test functions still depend on the solution (because of divergence-free

condition).

Fluid-structure interaction Kinematically coupled scheme. Applications.

Few remarks

• ∇η is gradient in ALE coordinates:

∇η =    ∂˜

z − ˜

r ∂zη R + η∂˜

r

1 R + η∂˜

r

   .

Fluid-structure interaction Kinematically coupled scheme. Applications.

Few remarks

• ∇η is gradient in ALE coordinates:

∇η =    ∂˜

z − ˜

r ∂zη R + η∂˜

r

1 R + η∂˜

r

   .

• wη = ∂tη˜

rer is domain (ALE) velocity.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Few remarks

• ∇η is gradient in ALE coordinates:

∇η =    ∂˜

z − ˜

r ∂zη R + η∂˜

r

1 R + η∂˜

r

   .

• wη = ∂tη˜

rer is domain (ALE) velocity.

• In numerical simulations, one can use the ALE transformation Aηn to

“transform” the problem back to domain Ωηn and solve it there, thereby avoiding the un-necessary calculation of the transformed gradient. The ALE velocity is the only extra term that needs to be included with that approach.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Weak solution of the reference domain

We say that (u, η) is weak solution of

• 1. u ∈ L∞(L2) ∩ L2(H1), η ∈ L∞(H2) ∩ W 1,∞(L2),
• 2. ∇η · u = 0, u(t, z, 1) = ∂tη(t, z)er,
• 3. for every (q, ψ) such that q(t, z, 1) = ψ(t, z)er, ∇η · v = 0 following

equality holds: −ρf T

• Ω

(R + η)u · ∂tq + 1 2 T

• Ω

(R + η)

• ((u − wη) · ∇η)u · q

−((u − wη) · ∇η)q · u

• + 2µ

T

• Ω

(R + η)Dη(u) : Dη(q) −ρf 2 T

• Ω

(∂tη)u · q − ρsh T L ∂tη∂tψ + T L

• C0ηψ + C1∂zη∂zψ

+C2∂2

z η∂2 z ψ

• = ±R

T Pin/out

• Γin/out

qz + ρf

• Ωη0

u0 · q(0) + ρsh L v0ψ(0)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Weak formulation of Step 1

un+ 1

2 = un.

Then we define (vn+ 1

2 , ηn+ 1 2 ) ∈ H2

0(0, L) × H2 0(0, L) as a solution of the

following problem, written in weak form: L ηn+ 1

2 − ηn

∆t φ = L vn+ 1

2 φ,

φ ∈ L2(0, L), ρsh L vn+ 1

2 − vn

∆t ψ + T L

• C0ηn+ 1

2 ψ + C1∂zηn+ 1 2 ∂zψ + C2∂2

z ηn+ 1

2 ∂2

z ψ

• = 0,

ψ ∈ H2

0(0, L).

Fluid-structure interaction Kinematically coupled scheme. Applications.

Weak formulation of Step 2

for every (q, ψ), such that q(t, z, 1) = ψ(t, z), ∇ηn · q = 0 ρf

• Ω

(R + ηn)

• un+1 − un+ 1

2

∆t · q + 1 2

• (un − vn+ 1

2 rer) · ∇ηn

un+1 · q −1 2

• (un − vn+ 1

2 rer) · ∇ηn

q · un+1

• + ρf

2

• Ω

vn+ 1

2 un+1 · q

+2µ

• Ω

(R + ηn)Dηn(u) : Dηn(q) +ρshs L vn+1 − vn+ 1

2

∆t ψ = R

• Pn

in

• Γin

qz − Pn

• ut
• Γout

qz

• ,

with ∇ηn · un+1 = 0, un+1

|Γ

= vn+1er, ηn+1 = ηn+ 1

2 .

Fluid-structure interaction Kinematically coupled scheme. Applications.

Discrete energy inequality

Lemma

(The Uniform Energy Bounds) Let ∆t > 0 and N = T/∆t > 0. There exists a constant C > 0 independent of ∆t (and N), such that the following estimates hold:

• 1. E

n+ 1

2

N

≤ C, E n+1

N

≤ C, for all n = 0, ..., N − 1,

• 2. N

j=1 Dj N ≤ C,

3.

N−1

• n=0
• Ω

(R + ηn)|un+1 − un|2 + vn+1 − vn+ 1

2 2

L2(0,L)

+vn+ 1

2 − vn2

L2(0,L)

• ≤ C,

4.

N−1

• n=0
• (C0ηn+1 − ηn2

L2(0,L) + C1∂z(ηn+1 − ηn)2 L2(0,L)

+C2∂2

z (ηn+1 − ηn)2 L2(0,L)

• ≤ C.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Discrete energy inequality II

• E

n+ 1

2

N

, E n+1

N

, and Dj

N are discrete kinetic energy and dissipation:

E

n+ i

2

N

= 1 2

• ρf
• Ω

(R + ηn)|u

n+ i

2

N

|2 + ρshsv

n+ i

2

N

2

L2(0,L)

+C0η

n+ i

2

N

2

L2(0,L) + C1∂zη n+ i

2

N

2

L2(0,L) + C2∂2 z η n+ i

2

N

2

L2(0,L)

• ,

Dn+1

N

= ∆tµ

• Ω

(R + ηn)|Dηn(un+1

N

)|2, n = 0, . . . , N, i = 0, 1.

• C depends only on the parameters in the problem, on the kinetic energy
• f the initial data E0, and on the energy norm of the inlet and outlet

data Pin/out2

L2(0,T)

Fluid-structure interaction Kinematically coupled scheme. Applications.

Approximate solutions

• We define approximate solutions as piece-wise constant in time:

uN(t, .) = un

N, ηN(t, .) = ηn N, vN(t, .) = vn N, v∗ N(t, .) = v n− 1

2

N

, t ∈ ((n − 1)∆t, n∆t], n = 1 . . . N.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Approximate solutions

• We define approximate solutions as piece-wise constant in time:

uN(t, .) = un

N, ηN(t, .) = ηn N, vN(t, .) = vn N, v∗ N(t, .) = v n− 1

2

N

, t ∈ ((n − 1)∆t, n∆t], n = 1 . . . N.

• We show that approximate solutions are well defined for small T, i.e.

R + ηN > 0.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Approximate solutions

• We define approximate solutions as piece-wise constant in time:

uN(t, .) = un

N, ηN(t, .) = ηn N, vN(t, .) = vn N, v∗ N(t, .) = v n− 1

2

N

, t ∈ ((n − 1)∆t, n∆t], n = 1 . . . N.

• We show that approximate solutions are well defined for small T, i.e.

R + ηN > 0.

• Discrete energy inequality implies:
• 1. Sequence (ηN)n∈N is uniformly bounded in L∞(0, T; H2

0(0, L)).

• 2. Sequence (vN)n∈N is uniformly bounded in L∞(0, T; L2(0, L)).
• 3. Sequence (v ∗

N)n∈N is uniformly bounded in L∞(0, T; L2(0, L)).

• 4. Sequence (uN)n∈N is uniformly bounded in

L∞(0, T; L2(Ω)) ∩ L2(0, T; H1(Ω)).

Fluid-structure interaction Kinematically coupled scheme. Applications.

Convergence of approximate solution

Let us now summarize obtained convergence results: ηN → η weakly∗ in L∞(0, T; H2

0(0, L))

vn → v weakly∗ in L∞(0, T; L2(0, L)) uN → u weakly∗ in L∞(0, T; L2(Ω)) uN → u weakly in L2(0, T; H1(Ω)) uN → u in L2(0, T; L2(Ω)), vN → v in L2(0, T; L2(0, L)), ηN → η in L∞(0, T; Hs(0, L)), 0 ≤ s < 2.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Convergence(existence) theorem

Theorem

Let ̺f , ̺s, µ, hs, Ci > 0, Di ≥ 0, i = 1, 2, 3. Suppose that the initial data v0 ∈ L2(0, L), u0 ∈ L2(Ωη0), and η0 ∈ H2

0(0, L) is such that (R + η0(z)) > 0,

z ∈ [0, L]. Furthermore, let Pin, Pout ∈ L2

loc(0, ∞).

Then there exist T > 0 and a weak solution of (u, η) of Considered FSI problem has at least one weak solution on (0, T), which satisfies the following energy estimate: E(t) + t D(τ)dτ ≤ E0 + C(Pin2

L2(0,T) + Pout2 L2(0,T)),

t ∈ [0, T], where C depends only on the coefficients, and E(t) and D(t) are given by

Fluid-structure interaction Kinematically coupled scheme. Applications.

Convergence(existence) II

E(t) = ρf 2 u2

L2(Ωη(t)) + ρsh

2 ∂tη2

L2(0,L)

+1 2

• C0η2

L2(0,L) + C1∂zη2 L2(0,L) + C2∂2 z η2 L2(0,L)

• ,

D(t) = µD(u)2

L2(Ωη(t)))

Furthermore, one of the following is true:

• 1. T = ∞,
• 2. lim

t→T min z∈[0,L](R + η(z)) = 0.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Regularization by a thin structure

• The interface is linearly elastic Koiter shell:

ρs1h∂ttη + Lsη = η − J(t, z)(σn)|(t,z,R+η(t,z)) · er + S(t, z, R)er · er.

• S - the structure (linearized 3d elasticity) Piola-Kirchhoff stress tensor.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Regularization by a thin structure

• The interface is linearly elastic Koiter shell:

ρs1h∂ttη + Lsη = η − J(t, z)(σn)|(t,z,R+η(t,z)) · er + S(t, z, R)er · er.

• S - the structure (linearized 3d elasticity) Piola-Kirchhoff stress tensor.
• Numerical scheme is unconditionally stable and first order convergent.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Regularization by a thin structure

• The interface is linearly elastic Koiter shell:

ρs1h∂ttη + Lsη = η − J(t, z)(σn)|(t,z,R+η(t,z)) · er + S(t, z, R)er · er.

• S - the structure (linearized 3d elasticity) Piola-Kirchhoff stress tensor.
• Numerical scheme is unconditionally stable and first order convergent.
• Convergence is proven also in the nonlinear case (but not order of

convergence).

Fluid-structure interaction Kinematically coupled scheme. Applications.

Regularization by a thin structure

• The interface is linearly elastic Koiter shell:

ρs1h∂ttη + Lsη = η − J(t, z)(σn)|(t,z,R+η(t,z)) · er + S(t, z, R)er · er.

• S - the structure (linearized 3d elasticity) Piola-Kirchhoff stress tensor.
• Numerical scheme is unconditionally stable and first order convergent.
• Convergence is proven also in the nonlinear case (but not order of

convergence).

• Fluid dissipation is acts stronger on the structure when the interface is

shell - regularization mechanism.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Regularization by a thin structure

• The interface is linearly elastic Koiter shell:

ρs1h∂ttη + Lsη = η − J(t, z)(σn)|(t,z,R+η(t,z)) · er + S(t, z, R)er · er.

• S - the structure (linearized 3d elasticity) Piola-Kirchhoff stress tensor.
• Numerical scheme is unconditionally stable and first order convergent.
• Convergence is proven also in the nonlinear case (but not order of

convergence).

• Fluid dissipation is acts stronger on the structure when the interface is

shell - regularization mechanism.

• As thickness h → 0, the solution converges to the solution of FSI with

thick structure (numerical proof).

Fluid-structure interaction Kinematically coupled scheme. Applications.

FSI with stent

• Joined work with M. Bukaˇ

c, S. ˇ Cani´ c, M. Gali´ c, J. Tambaˇ ca, Y. Wang.

• Stents are metallic mesh-like biomedical devices used to prop the elastic

coronary arteries open.

• The idea is to model stent as a network of 1D inextensible rods

(Tambaˇ ca, Kosor, ˇ Cani´ c, Paniagua ’10).

• The resulting model gives good approximation of 3D displacement and

significantly reduces the computational cost.

• The 1D stent model is coupled with a 2D shell model and 3D fluid.

Fluid-structure interaction Kinematically coupled scheme. Applications.

FSI with stent

Fluid-structure interaction Kinematically coupled scheme. Applications.

FSI with stent

• Splitting strategy can be also applied to the FSI-stent problem.
• The approximate solutions constructed with the splitting scheme

converge to a weak solution in both linear and nonlinear moving boundary case.

• The scheme is unconditionally stable.
• Solution is not regular near the stent.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Numerical results

Fluid-structure interaction Kinematically coupled scheme. Applications.

Numerical results

Fluid-structure interaction Kinematically coupled scheme. Applications.

Numerical results

Fluid-structure interaction Kinematically coupled scheme. Applications.

Numerical results

Fluid-structure interaction Kinematically coupled scheme. Applications.

References

Martina Bukaˇ c, Sunˇ cica ˇ Cani´ c, and Boris Muha. A nonlinear fluid-structure interaction problem in compliant arteries treated with vascular stents.

• Appl. Math. Optim., 73(3):433–473, 2016.
• M. Bukac, S. Canic, and B. Muha.

A partitioned scheme for fluid-composite structure interaction problems. Journal of Computational Physics, 281(0):493 – 517, 2015. Martina Bukaˇ c and Boris Muha. Stability and Convergence Analysis of the Extensions of the Kinematically Coupled Scheme for the Fluid-Structure Interaction. SIAM J. Numer. Anal., 54(5):3032–3061, 2016. Giovanna Guidoboni, Roland Glowinski, Nicola Cavallini, and Sunˇ cica ˇ Cani´ c. Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow.

• J. Comput. Phys., 228(18):6916–6937, 2009.

Fluid-structure interaction Kinematically coupled scheme. Applications.

References II

• S. ˇ

Cani´ c, M. Gali´ c, B. Muha, J. Tambaˇ ca Analysis Of A Linear 3d Fluid-stent-shell Interaction Problem submitted Boris Muha and Sunˇ cica ˇ Cani´ c. Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls.

• Arch. Ration. Mech. Anal., 207(3):919–968, 2013.

Boris Muha and Sunˇ cica ˇ Cani´ c. Existence of a solution to a fluid-multi-layered-structure interaction problem.

• J. Differential Equations, 256(2):658–706, 2014.

Fluid-structure interaction Kinematically coupled scheme. Applications.

Thank you for your attention!