Advances in the mathematical theory of the finite element immersed boundary method Daniele Boffi Dipartimento di Matematica “F. Casorati”, Universit` a di Pavia http://www-dimat.unipv.it/boffi May 12, 2014
Outline Immersed boundary method 1 Mass conservation 2 IBM with Lagrange multiplier 3 Main collaborators: Lucia Gastaldi , Nicola Cavallini, Francesca Gardini
Outline Immersed boundary method 1 The model FE approximation CFL condition Numerical results Mass conservation 2 IBM with Lagrange multiplier 3
Immersed boundary method Mass conservation IBM with Lagrange multiplier IBM – Immersed boundary method Introduced by Peskin for the simulation of the blood flow in the heart. � Peskin ’72–’77 � � McQueen–Peskin ’83– � � Peskin ’02 � Successfully applied to many biological problems, where a fluid interacts with a flexible structure. Daniele Boffi IBM and Finite Elements May 12, 2014 page 3
Immersed boundary method Mass conservation IBM with Lagrange multiplier IBM – Immersed boundary method Introduced by Peskin for the simulation of the blood flow in the heart. � Peskin ’72–’77 � � McQueen–Peskin ’83– � � Peskin ’02 � Successfully applied to many biological problems, where a fluid interacts with a flexible structure. The main feature is that the structure is considered as a part of the fluid by introducing suitable additional forces and masses. The Navier–Stokes equations are solved in the whole domain (fluid + solid) by finite differences and the interaction with the structure is obtained by means of singular force and mass terms defined by a Dirac delta function localized in the solid domain. Daniele Boffi IBM and Finite Elements May 12, 2014 page 3
Immersed boundary method Mass conservation IBM with Lagrange multiplier Finite elements for IBM At the beginning, we used finite elements mainly because we thought this would simplify the mathematical analysis. Indeed, it turned out that this was a good choice also from the practical point of view. � B.–Gastaldi ’03 � � B.–Gastaldi–Heltai ’04–’07 � � B.–Gastaldi–Heltai–Peskin ’08 � No need to approximating the Dirac delta functions, since the variational formulation takes care of it in a natural way Better interface approximation (less diffusion, sharp pressure jump) The fluid equations can be approximated with standard mixed schemes ( Q 2 − P 1 , Hood–Taylor, P 1 iso P 2 − P c 1 , . . . ) Daniele Boffi IBM and Finite Elements May 12, 2014 page 4
Immersed boundary method Mass conservation IBM with Lagrange multiplier Immersed elastic bodies Ω Elastic body Ω Elastic boundary B t B t Fluid Fluid Immersed body of codimension 0 Immersed body of codimension 1 the fluid domain and the immersed the immersed body can be either a body have the same dimension curve in 2D or a surface in 3D Daniele Boffi IBM and Finite Elements May 12, 2014 page 5
Immersed boundary method Mass conservation IBM with Lagrange multiplier Notation X(t) ω Ω B B t Ω fluid + solid B t deformable structure domain Ω ⊂ R d , d = 2 , 3 B t ⊂ R m , m = d , d − 1 x Euler. var. in Ω s Lagrangian var. in B B reference domain u ( x , t ) fluid velocity X ( · , t ) : B → B t position of the solid F = ∂ X p ( x , t ) fluid pressure ∂ s deformation grad. (det F > 0) u ( x , t ) = ∂ X ∂ t ( s , t ) where x = X ( s , t ) Daniele Boffi IBM and Finite Elements May 12, 2014 page 6
Immersed boundary method Mass conservation IBM with Lagrange multiplier From conservation of momenta, in absence of external forces, it holds � ∂ u � ρ ˙ u = ρ ∂ t + u · ∇ u = ∇ · σ in Ω In our case the Cauchy stress tensor has the following form � in Ω \ B t σ f σ = σ f + σ s in B t Daniele Boffi IBM and Finite Elements May 12, 2014 page 7
Immersed boundary method Mass conservation IBM with Lagrange multiplier From conservation of momenta, in absence of external forces, it holds � ∂ u � ρ ˙ u = ρ ∂ t + u · ∇ u = ∇ · σ in Ω In our case the Cauchy stress tensor has the following form � in Ω \ B t σ f σ = σ f + σ s in B t Incompressible fluid: σ = σ f = − p I + µ ( ∇ u + ( ∇ u ) T ) Visco-elastic material: σ = σ f + σ s with σ s elastic part of the stress Daniele Boffi IBM and Finite Elements May 12, 2014 page 7
Immersed boundary method Mass conservation IBM with Lagrange multiplier From conservation of momenta, in absence of external forces, it holds � ∂ u � ρ ˙ u = ρ ∂ t + u · ∇ u = ∇ · σ in Ω In our case the Cauchy stress tensor has the following form � in Ω \ B t σ f σ = σ f + σ s in B t Incompressible fluid: σ = σ f = − p I + µ ( ∇ u + ( ∇ u ) T ) Visco-elastic material: σ = σ f + σ s with σ s elastic part of the stress Moreover, if the structural material has a density ρ s different from the fluid density ρ f , we have � ρ f in Ω \ B t ρ = ρ s in B t Daniele Boffi IBM and Finite Elements May 12, 2014 page 7
Immersed boundary method Mass conservation IBM with Lagrange multiplier We get the following variational formulation Navier–Stokes d ρ f dt ( u ( t ) , v ) + a ( u ( t ) , v ) + b ( u ( t ) , u ( t ) , v ) − ( div v , p ( t )) ∀ v ∈ H 1 0 (Ω) d = � d ( t ) , v � + � F ( t ) , v � ∀ q ∈ L 2 ( div u ( t ) , q ) = 0 0 (Ω) Excess Lagrangian mass density ∂ 2 X � � d ( t ) , v � = − δρ ∂ t 2 v ( X ( s , t )) ds B Load � P ( F ( s , t )) : ∇ s v ( X ( s , t )) ds ∀ v ∈ H 1 ˜ 0 (Ω) d � F ( t ) , v � = − B Body movement ∂ X ∂ t ( s , t ) = u ( X ( s , t ) , t ) ∀ s ∈ B Initial conditions u ( x , 0) = u 0 ( x ) ∀ x ∈ Ω , X ( s , 0) = X 0 ( s ) ∀ s ∈ B Daniele Boffi IBM and Finite Elements May 12, 2014 page 8
Immersed boundary method Mass conservation IBM with Lagrange multiplier The excess Lagrangian mass density is defined as follows � ρ s − ρ f codimension 0 δρ = t s ( ρ s − ρ f ) codimension 1 The following definition � codimension 0 P ˜ P = t s P codimension 1 makes use of the Piola–Kirchoff tensor and takes into account the change of variable P ( s , t ) = | F ( s , t ) | σ s ( X ( s , t ) , t ) F − T ( s , t ) in order to have � � σ s n da = P N dA ∀P t ∂ P t ∂ P Daniele Boffi IBM and Finite Elements May 12, 2014 page 9
Immersed boundary method Mass conservation IBM with Lagrange multiplier Stability � B.–Cavallini–Gastaldi ’11 � Recalling that ∂ X ∂ t ( s , t ) = u ( X ( s , t ) , t ) ∀ s ∈ B it holds ρ f d 0 + d dt || u ( t ) || 2 0 + µ || ∇ u ( t ) || 2 dt E ( X ( t )) 2 2 � � + 1 2( ρ s − ρ f ) d ∂ X � � = 0 � � dt ∂ t � � B where E is the total elastic potential energy � E ( X ( t )) = W ( F ( s , t )) ds B Daniele Boffi IBM and Finite Elements May 12, 2014 page 10
Immersed boundary method Mass conservation IBM with Lagrange multiplier Finite element approximation Uniform background grid T h for the domain Ω (meshsize h x ) Inf-sup stable finite element pair V h ⊂ H 1 0 (Ω) d Q h ⊂ L 2 0 (Ω) Grid S h for B (meshsize h s ) Piecewise linear finite element space for X S h = { Y ∈ C 0 ( B ; Ω) : Y ∈ P 1 } Notation T k , k = 1 , . . . , M e elements of S h s j , j = 1 , . . . , M vertices of S h E h set of the edges e of S h Daniele Boffi IBM and Finite Elements May 12, 2014 page 11
Immersed boundary method Mass conservation IBM with Lagrange multiplier Discrete source term � Source term: � F ( t ) , v � = − B P ( F h ( s , t )) : ∇ s v ( X h ( s , t )) ds ∀ v ∈ V h X h p.w. linear ⇒ F h , P h p.w. constant By integration by parts M e � � � F h ( t ) , v � h = − P h : ∇ s v ( X ( s , t )) ds T k k =1 M e � � = − P h Nv ( X ( s , t )) dA ∂ T k k =1 that is � � � F h ( t ) , v � h = − [ [ P h ] ] · v ( X ( s , t )) dA e e ∈E h ] = P + N + + P − N − jump of P across e for internal edges [ [ P ] [ [ P ] ] = P N jump when e ⊂ ∂ B Daniele Boffi IBM and Finite Elements May 12, 2014 page 12
Immersed boundary method Mass conservation IBM with Lagrange multiplier The semidiscrete problem becomes: find ( u h , p h ) : ]0 , T [ → V h × Q h and X h : [0 , T ] → S h such that d ρ f dt ( u h ( t ) , v ) + a ( u h ( t ) , v ) + b ( u h ( t ) , u h ( t ) , v ) ( ρ s − ρ f ) ∂ 2 X h � − ( div v , p h ( t )) = − ∂ t 2 v ( X h ( s , t )) ds B � � − [ [ P h ] ] · v ( X h ( s , t )) dA ∀ v ∈ V h e e ∈E h ( div u h ( t ) , q ) = 0 ∀ q ∈ Q h d X hi dt ( t ) = u h ( X hi ( t ) , t ) ∀ i = 1 , . . . , M u h (0) = u 0 h in Ω X hi (0) = X 0 ( s i ) ∀ i = 1 , . . . , M Daniele Boffi IBM and Finite Elements May 12, 2014 page 13
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