A fictitious domain approach for the finite element discretization - PowerPoint PPT Presentation

A fictitious domain approach for the finite element discretization of FSI Lucia Gastaldi Universit` a di Brescia http://lucia-gastaldi.unibs.it MWNDEA 2020

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Outline Fluid-Structure Interaction 1 FSI with Lagrange multiplier 2 Computational aspects 3 Time marching schemes 4 Main collaborators : Daniele Boffi, Luca Heltai, Nicola Cavallini, Sebastian Wolf, Miguel A. Fern´ andez, Michele Annese, Simone Scacchi page 2

Outline 1 Fluid-Structure Interaction FSI with Lagrange multiplier 2 Computational aspects 3 Time marching schemes 4

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Fluid-structure interaction Ω ⊂ R d , d = 2 , 3 x Eulerian variable in Ω B t B t deformable structure domain B t ⊂ R m , m = d , d − 1 Ω s Lagrangian variable in B X X ( · , t ) : B → B t position of the solid F = ∂ X ∂ s deformation gradient B page 1

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Fluid-structure interaction Ω ⊂ R d , d = 2 , 3 x Eulerian variable in Ω B t B t deformable structure domain B t ⊂ R m , m = d , d − 1 Ω s Lagrangian variable in B X X ( · , t ) : B → B t position of the solid F = ∂ X ∂ s deformation gradient B u ( x , t ) material velocity u ( x , t ) = ∂ X ∂ t ( s , t ) where x = X ( s , t ) page 1

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Numerical approaches to FSI Boundary fitted approaches The fluid problem is solved on a mesh that deforms around a Lagrangian structure mesh, using arbitary Lagrangian–Eulerian (ALE) coordinate system. In case of large deformation the boundary fitted fluid mesh can become severely distorted. Non boundary fitted approaches A separate structural discretization is superimposed onto a background fluid mesh ◮ fictitious domain < Glowinski-Pan-P´ eriaux ’94, Yu ’05 > ◮ level set method < Chang-Hou-Merriman-Osher ’96 > ◮ immersed boundary method (IBM) < Peskin ’02 > ◮ Nitsche-XFEM method < Burman-Fern´ andez ’14, Alauzet-Fabr` eges-Fern´ andez-Landajuela ’16 > ◮ immersogeometric FSI (thin structures) < Kamensky-Hsu-Schillinger-Evans-Aggarwal-Bazilevs-Sacks-Hughes ’15 > ◮ divergence conforming B-splines < Casquero-Zhang-Bona-Casas-Dalcin-Gomez ’18 > page 2

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Numerical approaches to FSI Boundary fitted approaches The fluid problem is solved on a mesh that deforms around a Lagrangian structure mesh, using arbitary Lagrangian–Eulerian (ALE) coordinate system. In case of large deformation the boundary fitted fluid mesh can become severely distorted. Non boundary fitted approaches A separate structural discretization is superimposed onto a background fluid mesh ◮ fictitious domain < Glowinski-Pan-P´ eriaux ’94, Yu ’05 > ◮ level set method < Chang-Hou-Merriman-Osher ’96 > ◮ immersed boundary method (IBM) < Peskin ’02 > ◮ Nitsche-XFEM method < Burman-Fern´ andez ’14, Alauzet-Fabr` eges-Fern´ andez-Landajuela ’16 > ◮ immersogeometric FSI (thin structures) < Kamensky-Hsu-Schillinger-Evans-Aggarwal-Bazilevs-Sacks-Hughes ’15 > ◮ divergence conforming B-splines < Casquero-Zhang-Bona-Casas-Dalcin-Gomez ’18 > Our approach originates from the immersed boundary method IBM and moved towards a fictitious domain method FDM. page 2

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes FSI problem (thick incompressible solid) � ∂ u f � Ω ρ f ∂ t + u f · ∇ u f = div σ f in Ω \ B t Ω f t div u f = 0 in Ω \ B t B t ∂ 2 X s F −⊤ + P ( F )) ∂ t 2 = div s ( | F | σ f ρ s in B div s u s = 0 in B ∂ B t u f = ∂ X on ∂ B t ∂ t σ f n f = − ( σ f s + | F | − 1 PF ⊤ ) n s on ∂ B t u s = ∂ X σ f σ f = − p f I + ν f ∇ sym u f s = − p s I + ν s ∇ sym u s ∂ t s F −⊤ and P ( F ) Piola–Kirchhoff stress tensor such that P = | F | σ e P ( F ) = ∂ W ∂ F where W is the potential energy density + initial and boundary conditions page 3

Outline 1 Fluid-Structure Interaction FSI with Lagrange multiplier 2 Computational aspects 3 Time marching schemes 4

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Fictitious domain approach < Boffi–Cavallini–G. ’15 > ◮ Fluid velocity and pressure are extended into the solid domain � u f � p f in Ω \ B t in Ω \ B t u = p = u s in B t in B t p s page 4

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Fictitious domain approach < Boffi–Cavallini–G. ’15 > ◮ Fluid velocity and pressure are extended into the solid domain � u f � p f in Ω \ B t in Ω \ B t u = p = u s in B t in B t p s ◮ Body motion u ( x , t ) = ∂ X ∂ t ( s , t ) for x = X ( s , t ) page 4

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Fictitious domain approach < Boffi–Cavallini–G. ’15 > ◮ Fluid velocity and pressure are extended into the solid domain � u f � p f in Ω \ B t in Ω \ B t u = p = u s in B t in B t p s ◮ Body motion u ( x , t ) = ∂ X ∂ t ( s , t ) for x = X ( s , t ) ◮ We introduce two functional spaces Λ and Z and a bilinear form c : Λ × Z → R such that c ( µ, z ) = 0 ∀ µ ∈ Λ ⇒ z = 0 page 4

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Notation: � ν f in Ω \ B t a ( u , v ) = ( ν ∇ sym u , ∇ sym v ) with ν = ν s in B t b ( u , v , w ) = ρ f 2 (( u · ∇ v , w ) − ( u · ∇ w , v )) � � ( u , v ) = uv d x , ( X , z ) B = Xz d s Ω B δ ρ = ρ s − ρ f page 5

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Variational form with Lagrange multiplier Problem For t ∈ [0 , T ], find u ( t ) ∈ H 1 0 (Ω) d , p ( t ) ∈ L 2 0 (Ω), X ( t ) ∈ W 1 , ∞ ( B ) d , and λ ( t ) ∈ Λ such that ρ d dt ( u ( t ) , v ) + a ( u ( t ) , v ) + b ( u ( t ) , u ( t ) , v ) ∀ v ∈ H 1 0 (Ω) d − ( div v , p ( t )) + c ( λ ( t ) , v ( X ( · , t ))) = 0 ∀ q ∈ L 2 ( div u ( t ) , q ) = 0 0 (Ω) � ∂ 2 X � ∀ z ∈ H 1 ( B ) d δ ρ ∂ t 2 ( t ) , z + ( P ( F ( t )) , ∇ s z ) B − c ( λ ( t ) , z ) = 0 B � µ , u ( X ( · , t ) , t ) − ∂ X ( t ) � c = 0 ∀ µ ∈ Λ ∂ t page 6

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Definition of c The fact that X ∈ W 1 , ∞ ( B ) d implies v ( X ( · )) ∈ H 1 ( B ) d Case 1 Z = H 1 ( B ) d , Λ dual space of H 1 ( B ) d , �· , ·� B duality pairing λ ∈ Λ = ( H 1 ( B ) d ) ′ , z ∈ H 1 ( B ) d c ( λ , z ) = � λ , z � B Case 2 Z = H 1 ( B ) d , Λ = H 1 ( B ) d � λ ∈ Λ , z ∈ H 1 ( B ) d c ( λ , z ) = ( ∇ s λ · ∇ s z + λ · z ) ds B page 7

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Energy estimate Stability estimate If ρ s > ρ f , then the following bound holds true 2 � � ρ f d 0 + d dt E ( X ( t )) + 1 d ∂ X dt || u ( t ) || 2 0 + µ || ∇ u ( t ) || 2 � � 2 δ ρ = 0 � � 2 ∂ t dt � � B � where E ( X ( t )) = W ( F ( s , t )) ds B Remark Similar bound holds true if the condition ρ s > ρ f is not satisfied. page 8

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Time advancing scheme - Backward Euler BE Problem 0 (Ω) d and X 0 ∈ W 1 , ∞ ( B ) d , for n = 1 , . . . , N , find Given u 0 ∈ H 1 0 (Ω), X n ∈ W 1 , ∞ ( B ) d , and λ n ∈ Λ , such that 0 (Ω) d × L 2 ( u n , p n ) ∈ H 1 � u n +1 − u n � + a ( u n +1 , v ) + b ( u n +1 , u n +1 , v ) ρ f , v ∆ t − ( div v , p n +1 ) + c ( λ n +1 , v ( X n +1 ( · ))) = 0 ∀ v ∈ H 1 0 (Ω) d ( div u n +1 , q ) = 0 ∀ q ∈ L 2 0 (Ω) � X n +1 − 2 X n + X n − 1 � + ( P ( F n +1 ) , ∇ s z ) B δ ρ , z ∆ t 2 B − c ( λ n +1 , z ) = 0 ∀ z ∈ H 1 ( B ) d µ , u n +1 ( X n +1 ( · )) − X n +1 − X n � � c = 0 ∀ µ ∈ Λ ∆ t page 9

Fluid-Structure Interaction FSI with Lagrange multiplier Computational aspects Time marching schemes Time advancing scheme - Mofified backward Euler MBE Problem 0 (Ω) d and X 0 ∈ W 1 , ∞ ( B ) d , for n = 1 , . . . , N , find Given u 0 ∈ H 1 0 (Ω), X n ∈ W 1 , ∞ ( B ) d , and λ n ∈ Λ , such that 0 (Ω) d × L 2 ( u n , p n ) ∈ H 1 � u n +1 − u n � + a ( u n +1 , v ) + b ( u n , u n +1 , v ) ρ f , v ∆ t − ( div v , p n +1 ) + c ( λ n +1 , v ( X n ( · ))) = 0 ∀ v ∈ H 1 0 (Ω) d ( div u n +1 , q ) = 0 ∀ q ∈ L 2 0 (Ω) � X n +1 − 2 X n + X n − 1 � + ( P ( F n +1 ) , ∇ s z ) B δ ρ , z ∆ t 2 B − c ( λ n +1 , z ) = 0 ∀ z ∈ H 1 ( B ) d µ , u n +1 ( X n ( · )) − X n +1 − X n � � c = 0 ∀ µ ∈ Λ ∆ t page 10