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 c (Notre Dame) and S. ˇ Joined work with M. Bukaˇ 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 • 3 D fluid is coupled with 3 D nonlinear elasticity. • Ω F - the fluid reference domain. • Ω S - the structure reference domain. • Ω F - 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: ∂ 2 ̺ s ∂ t 2 η = ∇ · T ( ∇ η ) in (0 , T ) × Ω S , • Constitutive relation: ∂ First Piola-Kirchhoff stress tensor: T ( F ) = ∂ F W ( F ) , where W : M 3 ( R ) → R is a stored energy function.

Fluid-structure interaction Kinematically coupled scheme. Applications. The fluid equations • The physical fluid domain is given by Ω f ( t ) = ϕ f ( t , Ω f ) , t ∈ (0 , T ) , where φ f the fluid domain displacement (i.e. an arbitrary extension of η to Ω F ). • The fluid equations: � ρ f ( ∂ t u + u · ∇ u ) = ∇ · σ ( ∇ u , p ) , in Ω f ( t ) = ϕ f ( t , Ω f ) , t ∈ (0 , T ) . ∇ · u = 0 , Constitutive relation: Cauchy stress tensor is given by relation σ ( ∇ u , p ) = − p I + 2 µ D ( u )

Fluid-structure interaction Kinematically coupled scheme. Applications. Coupling and boundary condition • Dynamic coupling condition (balance of forces): T n = ( σ ◦ ϕ ) ∇ ϕ − τ n on (0 , T ) × Γ • Kinematic coupling condition (no-slip): ∂ t η ( t , . ) = u ( t , . ) | Γ ◦ ϕ f , on 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 ∂ t 2 η = ∇ · T ( ∇ η ) in (0 , T ) × Ω s , � ρ f ( ∂ t u + u · ∇ u ) = ∇ · σ ( ∇ u , p ) , in Ω f ( t ) = ϕ f ( t , Ω f ) , t ∈ (0 , T ) , ∇ · u = 0 , � t 0 u | Γ ◦ ϕ = ϕ Γ , T n = ( σ ◦ ϕ ) ∇ ϕ − τ n on Γ , (1) where u is the fluid velocity, η the structure deformation, ϕ f the fluid domain displacement

Fluid-structure interaction Kinematically coupled scheme. Applications. Energy inequality • We formally multiply the structure equations by ∂ t η and integrate over Ω S . Then formally multiply the fluid equations by u and integrate over Ω F .

Fluid-structure interaction Kinematically coupled scheme. Applications. Energy inequality • We formally multiply the structure equations by ∂ t η and integrate over Ω S . Then formally multiply the fluid equations by u and integrate over Ω F . • By adding resulting equalities, integrating by parts and using the coupling conditions we obtain formal energy inequality: d � � ∂ t η � 2 L 2 (Ω s ) + � u � 2 + µ � D ( u ) � 2 � � L 2 (Ω f ( t ) + W ( ∇ η ) L 2 (Ω f ( t )) dt Ω s ≤ C ( data ) .

Fluid-structure interaction Kinematically coupled scheme. Applications. Energy inequality • We formally multiply the structure equations by ∂ t η and integrate over Ω S . Then formally multiply the fluid equations by u and integrate over Ω F . • By adding resulting equalities, integrating by parts and using the coupling conditions we obtain formal energy inequality: d � � ∂ t η � 2 L 2 (Ω s ) + � u � 2 + µ � D ( u ) � 2 � � L 2 (Ω f ( t ) + W ( ∇ η ) L 2 (Ω f ( t )) dt Ω s ≤ 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 obtained 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 ) = e At u 0 , A ∈ M n • Let us decompose A = A 1 + A 2 ? • e A + B = e A e B ⇔ AB = BA . • However: e A + B = lim N →∞ ( e A / N e B / 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) = φ 0 . • We suppose that A = A 1 + A 2 . • Let k = T / N be time-dicretization step and t n = nk . Then we define: d dt φ n + i 2 + A i ( φ n + i 2 ) = 0 in ( t n , t n +1 ) , 2 ( t n ) = φ n + i − 1 2 , n = 0 , . . . , N − 1 , i = 1 , 2 , φ n + i where φ n + i 2 = φ n + i 2 ( t n +1 ).

Fluid-structure interaction Kinematically coupled scheme. Applications. Simplified linear model ρ f ∂ t u = ∇ · σ ( u , p ) , ∇ · u = 0 in (0 , T ) × Ω , σ ( u , p ) n = − p in / out ( t ) n on (0 , T ) × Σ , ρ s ǫ∂ 2 t η + L s η = − σ ( u , p ) n on (0 , T ) × Γ , ∂ t η = u on (0 , T ) × Γ , η ( ., 0) = η 0 , ∂ t η ( ., 0) = v 0 on Γ , u ( ., 0) = u 0 in Ω .

Fluid-structure interaction Kinematically coupled scheme. Applications. Monolithic formulation Given t ∈ (0 , T ) find ( ✉ , η , p ) ∈ V f × V s × Q f with ✉ = ∂ t η on Γ, such that for all ( ϕ , ξ , q ) ∈ V fsi × Q f � � � ρ f ∂ t ✉ · ϕ d ① + 2 µ ❉ ( ✉ ) : ❉ ( ϕ ) d ① − p ∇ · ϕ d ① Ω Ω Ω � � � � + ( ∇ · ✉ ) qd ① + ρ s ǫ ∂ tt η · ξ dx + L s η · ξ dS = p in / out ( t ) ϕ · ♥ dS . Ω Γ Γ Σ V f = ( H 1 (Ω)) d , Q f = L 2 (Ω) , V s = ( H 1 0 (Γ)) d , V fsi = { ( ϕ , ξ ) ∈ V f × V s | ϕ | Γ = ξ } ,