Reduced Formulation of Steady Fluid-structure Interaction with - PowerPoint PPT Presentation

Reduced Formulation of Steady Fluid-structure Interaction with Parametric Coupling Toni Lassila ∗ , ◦ , Alfio Quarteroni † , × , Gianluigi Rozza † ∗ Department of Mathematics and Systems Analysis × MOX - Modellistica e Calcolo Scientifico School of Science and Technology Dipartimento di Matematica “F . Brioschi” Aalto University Politecnico di Milano † Chair of Modelling and Scientific Computing Mathematics Institute of Computational Science and Engineering Ecole Polytechnique F´ ´ ed´ erale de Lausanne ◦ Supported by the Emil Aaltonen Foundation IV International Symposium on Modelling of Physiological Flows, Chia Laguna, Sardinia, June 2-5, 2010 Fluid-structure Interaction with Parametric Coupling 1 / 21

Outline “Towards reducing the geometric complexity of FSI problems” Motivation for fluid-structure interaction Previous three days of this conference... Steady fluid-structure interaction problem Incompressible Stokes equations for fluid Generalized 1-d string model for the structure Coupling between traction applied by fluid and structural displacement Parametric flow geometry Parametric free-form deformation of geometry Fluid equations on fixed domain with parametric coefficients Model reduction Fluid-structure coupling variables are the geometric parameters Iterative scheme in parameter space Reduced basis method for approximation of parametric Stokes Fluid-structure Interaction with Parametric Coupling 2 / 21

Approaches to Reduced Modelling of Fluid-Structure Interaction Classical model reduction applied to linear systems of ODEs and PDEs ⇒ not very useful for FSI with strong geometric nonlinearity. Some new approaches have been proposed: Proper Orthogonal Decomposition (review in [DH01]) Eigendecomposition-based method for approximating an ensemble of trajectories of a given dynamical system Widely used in aeroelasticity simulations (not so much in hemodynamics) Cons: Computationally expensive, error of reduced model difficult to estimate Geometrical multiscale (review in [FQV09]) Different fidelity models (0D vs. 1D vs. 3D-models) used in different parts of the cardiovascular system, coupled together with suitable boundary conditions Combines modelling scales ranging from peripheral circulation all the way to the major arteries Cons: Physically meaningful boundary conditions between 0D-1D-3D models are challenging (talk of C. Malossi) Reduced basis element method [LMR06] Decomposition of complex flow network to a small collection of “simple elements” like T-junctions and straight pipes, combined with reduced basis method (talk by L. Iapichino) Fluid-structure Interaction with Parametric Coupling 3 / 21

Our Model Reduction Strategy for Fluid-Structure Interaction Standard Fluid-Structure Interaction Reduced Fluid-Structure Interaction Fluid-structure Interaction with Parametric Coupling 4 / 21

Steady Fluid-Structure Interaction Model Problem Σ( η ) 0 η ( x 1 ) L −△ u + ∇ p = 0 Γ in Γ out Ω( η ) ∇ · u = 0 Fluid: Structure:   ε d 4 η − K d 2 η  − ν △ u + ∇ p = f , in Ω( η )    + η = τ Σ , for x 1 ∈ ( 0 , L )   dx 4 dx 2   in Ω( η ) ∇ · u = 0 , 1 1   η ( 0 ) = η ′ ( 0 ) = η ( L ) = η ′ ( L ) = 0  u = u 0 , on Γ in ∪ Γ out      Coupling condition: u = 0 , on Σ( η ) � � � � ∇ u + ∇ u t � � 0 τ Σ = p n − ν n · , on Σ( η ) 1 Existence proved in [G98] with fixed point argument + some additional regularity assumptions. Fluid-structure Interaction with Parametric Coupling 5 / 21

Choice of Structural Model and Treatment of Boundary Conditions Generalized 1-d string model for arterial wall [QTV00] in the steady case: − kGh ∂ 2 η Eh η R 0 ( x 1 ) 2 = τ Σ , + on x 1 ∈ ( 0 , L ) , ∂ x 2 1 − ν 2 1 P where h = wall thickness, k = Timoshenko shear correction factor, G = shear modulus, E = Young modulus, ν P = Poisson ratio, and R 0 ( x 1 ) = radius of the reference configuration at distance x 1 from inflow. ε d 4 η We choose to include a fourth order singular perturbation term � , which after dx 4 1 nondimensionalizing the equations gives ε d 4 η − K d 2 η + η = τ Σ , for x 1 ∈ ( 0 , L ) dx 4 dx 2 1 1 with the boundary conditions η ( 0 ) = η ′ ( 0 ) = η ( L ) = η ′ ( L ) = 0 . Fluid-structure Interaction with Parametric Coupling 6 / 21

Steady Fluid-Structure Interaction Problem (weak form) Incompressible Stokes fluid + 1-d membrane structure Find ( u , p , η ) ∈ V (Ω( η )) × Q (Ω( η )) × S ( 0 , L ) s.t.   A ( u , v )+ B ( p , v ) = � F , v � for all v ∈ V (Ω( η ))  , B ( q , u ) = 0 for all q ∈ Q (Ω( η ))   C ( η , φ ) = � R ( u , p ) , φ � for all φ ∈ S ( 0 , L ) . where we have the bilinear forms for the incompressible Stokes equations � � A ( u , v ) = ν Ω( η ) ∇ u · ∇ v d Ω , B ( q , v ) = − Ω( η ) q ∇ · v d Ω , and the linear form � � F , v � = Ω( η ) f · v d Ω and the structural bilinear form � � � d 2 η d 2 φ d η d φ C ( η , φ ) = ε dx 1 + K dx 1 + ηφ dx 1 . dx 2 dx 2 dx 1 dx 1 Σ 0 Σ 0 Σ 0 1 1 The fluid residual is the normal component of the normal Cauchy stress of the fluid. Fluid-structure Interaction with Parametric Coupling 7 / 21

Reduction Step #1: Free-form Deformations of the Fluid Domain fixed reference domain deformed parametric domain T ( · , µ ) Ω 0 Ω( µ ) FFD map Ψ − 1 affine map Ψ P 0 P 0 ℓ, m + µ ℓ, m ℓ, m � T ( · , µ ) parameter matrix µ FFD control points parameters = displacements of control points Recalling from the talk of A. Manzoni... Fluid-structure Interaction with Parametric Coupling 8 / 21

Parametric Fluid Equations on the Fixed Reference Domain Parametric FFD deformation map T ( · ; µ ) : Ω 0 → Ω( µ ) defined as T = Ψ − 1 ◦ � T ◦ Ψ where � � L M � b L , M P 0 T ( � ∑ ∑ ℓ, m ( � x ; µ ) = x ) ℓ, m + µ ℓ, m ℓ = 0 m = 0 and its Jacobian matrix J T ( x ; µ ) := ∇ x T define the transformation tensors [RV07] ν T ( x ; µ ) := J − t T J − 1 χ T ( x ; µ ) := J − 1 T det ( J T ) , T det ( J T ) used to map fluid problem back to reference domain: find ( � u , � p ) ∈ V (Ω 0 ) × Q (Ω 0 ) s.t. � �  � � ν ∂ � ∂ � ∂ � u k v k v k  det ( J T η )[ f F + f lift ] k d Ω 0 , + �  [ ν T η ] i , j p [ χ T η ] k , j d Ω 0 =   ∂ � ∂ � ∂ � x i x j x j  Ω 0 Ω 0      for all � v ∈ V (Ω 0 ) �  ∂ � u k   �  q [ χ T η ] k , j d Ω 0 = 0 ,   ∂ � x j  Ω 0    for all � q ∈ Q (Ω 0 ) Recalling from the talk of A. Manzoni... Fluid-structure Interaction with Parametric Coupling 9 / 21

Parametric Coupling of Fluid and Structure Standard iterative scheme for fluid-structure coupling Ω( η k ) ( u k , p k ) − → Stokes update ↑ ↓ fluid residual η k + 1 R ( u k , p k ) geometry ← − structural equation Our parametric coupling approach u k , � µ k Ω( µ k ) p k ) − → − → ( � parametric domain Stokes update ↑ ↓ µ k + 1 R ( µ k ) � η parameters ← − ← − least squares fit structure Fluid-structure Interaction with Parametric Coupling 10 / 21

Parametric Coupling Algorithm Fixed-point algorithm for weak parametric coupling Start with initial guess for the parameter µ 0 and set k = 0. u h ( µ k ) , � p h ( µ k )) [ Fluid substep ] Solve the discretized fluid problem in Ω 0 to obtain ( � 1 Form the discrete fluid residual 2 � � u k R ( µ k ) := G p k F − A � h − B � h where G ( � v h ) takes the boundary normal trace of any � v h on Σ 0 [ Structure substep ] Solve for assumed structural displacement ˆ η ( µ k ) ∈ S s.t. 3 η h , φ ) = � R ( µ k ) , φ � C (ˆ for all φ ∈ S [ Parametric projection substep ] Solve “inverse problem” of finding parameter value that gives 4 best fit to the assumed displacement ˆ η ( µ k ) � µ k + 1 := argmin η h ( µ k ) | 2 d Γ Σ | η h (¯ µ ) − ˆ ¯ µ to obtain next parameter value. Displacement η h ( x ; ¯ µ ) = T ( x ; ¯ µ ) − T ( x ; 0 ) is given by the FFD and requires no structural equation solutions. Iterate until | µ k + 1 − µ k | < TOL. 5 Fluid-structure Interaction with Parametric Coupling 11 / 21

Reduction Step #2: Reduced Basis Methods for Parametric PDEs Problem: FE solution ( u h ( µ ) , p h ( µ )) ∈ V h × Q h too expensive to compute for many different values of µ . Observation: Dependence of the bilinear forms A ( · , · ; µ ) and B ( · , · ; µ ) on µ is smooth ⇒ parametric manifold of solutions in V h × Q h is smooth Solution: Choose a representative set of parameter values µ 1 ,..., µ N with N ≪ N Snapshot solutions u h ( µ 1 ) ,..., u h ( µ N ) span a subspace V N h for the velocity and p ( µ 1 ) ,..., p ( µ N ) span a subspace Q N h for the pressure Galerkin reduced basis formulation For given parameter vector µ ∈ D find approximate solution u N h ( µ ) ∈ V N h and p N h ( µ ) ∈ Q N h in reduced spaces such that A ( u N h ( µ ) , v ; µ )+ B ( p N for all v ∈ V N h ( µ ) , v ; µ ) = � F h ( µ ) , v � h B ( q , u N for all q ∈ Q N h ( µ ); µ ) = 0 h Recalling from the talk of A. Manzoni... Fluid-structure Interaction with Parametric Coupling 12 / 21