A stable scheme for simulation of incompressible flows in - PowerPoint PPT Presentation

A stable scheme for simulation of incompressible flows in time-dependent domains and hemodynamic applications Yuri Vassilevski 1 , 2 , 3 Maxim Olshanskii 4 Alexander Danilov 1 , 2 , 3 Alexander Lozovskiy 1 Victoria Salamatova 1 , 2 , 3 1 Marchuk Institute of Numerical Mathematics RAS 2 Moscow Institute of Physics and Technology 3 Sechenov University 4 University of Houston Modeling, Simulation and Optimization of the Cardiovascular system October 23, 2018, Magdeburg The work was supported by the Russian Science Foundation

Workshop Announcement, 6-8 November 2018 Marchuk Institute of Numerical Mathematics RAS, Moscow, Russia The 10th Workshop on Numerical Methods and Mathematical Modelling in Biology and Medicine www.dodo.inm.ras.ru/biomath

Workshop Announcement, 7-11 October 2019 Far East Federal University, island Russky, Vladivostok, Russia The Fourth German-Russian Workshop on Numerical Methods and Mathematical Modelling in Geophysical and Biomedical Sciences

Acknowledge the talk by Boris Muha: Stability and convergence analysis of the kinematically coupled scheme for the fluid-structure interaction

Fluid-Structure Interaction

Fluid-Structure Interaction problem Prerequisites for FSI ◮ reference subdomains Ω f , Ω s ◮ transformation ξ maps Ω f , Ω s to Ω f ( t ), Ω s ( t ) ◮ v and u denote velocities and displacements in � Ω := Ω f ∪ Ω s ◮ ξ ( x ) := x + u ( x ), F := ∇ ξ = I + ∇ u , J := det( F ) ◮ Cauchy stress tensors σ f , σ s ◮ pressures p f , p s ◮ density ρ f is constant

Fluid-Structure Interaction problem Universal equations in reference subdomains Dynamic equations  ρ − 1 s div ( J σ s F − T )  in Ω s ,  ∂ v � � �� ∂ t = v − ∂ u  ( J ρ f ) − 1 div ( J σ f F − T ) − ∇ v F − 1 in Ω f  ∂ t

Fluid-Structure Interaction problem Universal equations in reference subdomains Dynamic equations  ρ − 1 s div ( J σ s F − T )  in Ω s ,  ∂ v � � �� ∂ t = v − ∂ u  ( J ρ f ) − 1 div ( J σ f F − T ) − ∇ v F − 1 in Ω f  ∂ t Kinematic equation ∂ u ∂ t = v in Ω s

Fluid-Structure Interaction problem Universal equations in reference subdomains Dynamic equations  ρ − 1 s div ( J σ s F − T )  in Ω s ,  ∂ v � � �� ∂ t = v − ∂ u  ( J ρ f ) − 1 div ( J σ f F − T ) − ∇ v F − 1 in Ω f  ∂ t Kinematic equation ∂ u ∂ t = v in Ω s Fluid incompressibility J ∇ v : F − T = 0 div ( J F − 1 v ) = 0 in Ω f or in Ω f

Fluid-Structure Interaction problem Universal equations in reference subdomains Dynamic equations  ρ − 1 s div ( J σ s F − T )  in Ω s ,  ∂ v � � �� ∂ t = v − ∂ u  ( J ρ f ) − 1 div ( J σ f F − T ) − ∇ v F − 1 in Ω f  ∂ t Kinematic equation ∂ u ∂ t = v in Ω s Fluid incompressibility J ∇ v : F − T = 0 div ( J F − 1 v ) = 0 in Ω f or in Ω f Constitutive relation for the fluid stress tensor σ f = − p f I + µ f (( ∇ v ) F − 1 + F − T ( ∇ v ) T ) in Ω f

FSI problem User-dependent equations in reference subdomains Constitutive relation for the solid stress tensor σ s = σ s ( J , F , p s , λ s , µ s , . . . ) in Ω s 1 Michler et al (2004), Hubner et al (2004), Hron&Turek (2006),...

FSI problem User-dependent equations in reference subdomains Constitutive relation for the solid stress tensor σ s = σ s ( J , F , p s , λ s , µ s , . . . ) in Ω s Monolithic approach 1 : Extension of the displacement field to the fluid domain G ( u ) = 0 in Ω f , u = u ∗ on ∂ Ω f 1 Michler et al (2004), Hubner et al (2004), Hron&Turek (2006),...

FSI problem User-dependent equations in reference subdomains Constitutive relation for the solid stress tensor σ s = σ s ( J , F , p s , λ s , µ s , . . . ) in Ω s Monolithic approach 1 : Extension of the displacement field to the fluid domain G ( u ) = 0 in Ω f , u = u ∗ on ∂ Ω f for example, vector Laplace equation or elasticity equation 1 Michler et al (2004), Hubner et al (2004), Hron&Turek (2006),...

FSI problem User-dependent equations in reference subdomains Constitutive relation for the solid stress tensor σ s = σ s ( J , F , p s , λ s , µ s , . . . ) in Ω s Monolithic approach 1 : Extension of the displacement field to the fluid domain G ( u ) = 0 in Ω f , u = u ∗ on ∂ Ω f for example, vector Laplace equation or elasticity equation + Initial, boundary, interface conditions ( σ f F − T n = σ s F − T n ) 1 Michler et al (2004), Hubner et al (2004), Hron&Turek (2006),...

Numerical scheme ◮ Conformal triangular or tetrahedral mesh Ω h in � Ω ◮ LBB-stable pair for velocity and pressure P 2 / P 1 , P 2 for displacements ◮ Fortran open source software Ani2D , Ani3D (Advanced numerical instruments 2D/3D, K.Lipnikov, Yu.Vassilevski et al.) http://sf.net/p/ani3d/ : http://sf.net/p/ani2d/ ◮ mesh generation ◮ FEM systems ◮ algebraic solvers

Numerical scheme Find { u k +1 , v k +1 , p k +1 } ∈ V 0 h × V h × Q h s.t. � ∂ u � v k +1 = g h ( · , ( k + 1)∆ t ) on Γ f 0 , = v k +1 on Γ fs ∂ t k +1

Numerical scheme Find { u k +1 , v k +1 , p k +1 } ∈ V 0 h × V h × Q h s.t. � ∂ u � v k +1 = g h ( · , ( k + 1)∆ t ) on Γ f 0 , = v k +1 on Γ fs ∂ t k +1 where V h ⊂ H 1 ( � Ω) 3 , Q h ⊂ L 2 ( � Ω) , V 0 h = { v ∈ V h : v | Γ s 0 ∪ Γ f 0 = 0 } , V 00 h = { v ∈ V 0 h : v | Γ fs = 0 } � ∂ f � := 3 f k +1 − 4 f k + f k − 1 ∂ t 2∆ t k +1

Numerical scheme � � ∂ v � � u k ) S ( u k +1 , � u k ) : ∇ ψ d Ω + J k F ( � ρ s ψ d Ω + ∂ t Ω s Ω s k +1 � � � ∂ v � � ∂ u � � � � v k − ρ f J k ∇ v k +1 F − 1 ( � u k ) ρ f J k ψ d Ω + � ψ d Ω + ∂ t ∂ t Ω f Ω f k +1 k � � u k v k +1 : D � p k +1 J k F − T ( � ∀ ψ ∈ V 0 u k ) : ∇ ψ d Ω = 0 2 µ f J k D � u k ψ d Ω − h Ω f Ω { A } s := 1 f k := 2 f k − f k − 1 , 2 ( A + A T ) u k ) , � D u v := {∇ vF − 1 ( u ) } s , J k := J ( �

Numerical scheme � � ∂ v � � u k ) S ( u k +1 , � u k ) : ∇ ψ d Ω + J k F ( � ρ s ψ d Ω + ∂ t Ω s Ω s k +1 � � � ∂ v � � ∂ u � � � � v k − ρ f J k ∇ v k +1 F − 1 ( � u k ) ρ f J k ψ d Ω + � ψ d Ω + ∂ t ∂ t Ω f Ω f k +1 k � � u k v k +1 : D � p k +1 J k F − T ( � ∀ ψ ∈ V 0 u k ) : ∇ ψ d Ω = 0 2 µ f J k D � u k ψ d Ω − h Ω f Ω � � ∂ u � � � v k +1 φ d Ω + G ( u k +1 ) φ d Ω = 0 ∀ φ ∈ V 00 φ d Ω − h ∂ t Ω s Ω s Ω f k +1 { A } s := 1 f k := 2 f k − f k − 1 , 2 ( A + A T ) u k ) , � D u v := {∇ vF − 1 ( u ) } s , J k := J ( �

Numerical scheme � � ∂ v � � u k ) S ( u k +1 , � u k ) : ∇ ψ d Ω + J k F ( � ρ s ψ d Ω + ∂ t Ω s Ω s k +1 � � � ∂ v � � ∂ u � � � � v k − ρ f J k ∇ v k +1 F − 1 ( � u k ) ρ f J k ψ d Ω + � ψ d Ω + ∂ t ∂ t Ω f Ω f k +1 k � � u k v k +1 : D � p k +1 J k F − T ( � ∀ ψ ∈ V 0 u k ) : ∇ ψ d Ω = 0 2 µ f J k D � u k ψ d Ω − h Ω f Ω � � ∂ u � � � v k +1 φ d Ω + G ( u k +1 ) φ d Ω = 0 ∀ φ ∈ V 00 φ d Ω − h ∂ t Ω s Ω s Ω f k +1 � J k ∇ v k +1 : F − T ( � u k ) q d Ω = 0 ∀ q ∈ Q h Ω f { A } s := 1 f k := 2 f k − f k − 1 , 2 ( A + A T ) u k ) , � D u v := {∇ vF − 1 ( u ) } s , J k := J ( �

Numerical scheme � u k ) S ( u k +1 , � u k ) : ∇ ψ d Ω + . . . J k F ( � . . . + Ω s ◮ St. Venant–Kirchhoff model (geometrically nonlinear) : S ( u 1 , u 2 ) = λ s tr ( E ( u 1 , u 2 )) I + 2 µ s E ( u 1 , u 2 ); E ( u 1 , u 2 ) = { F ( u 1 ) T F ( u 2 ) − I } s ◮ inc. Blatz–Ko model: S ( u 1 , u 2 ) = µ s ( tr ( { F ( u 1 ) T F ( u 2 ) } s ) I − { F ( u 1 ) T F ( u 2 ) } s ) ◮ inc. Neo-Hookean model: u k ) → F ( u k +1 ) S ( u 1 , u 2 ) = µ s I ; F ( � { A } s := 1 2 ( A + A T )

Numerical scheme The scheme ◮ provides strong coupling on interface ◮ semi-implicit ◮ produces one linear system per time step ◮ second order in time

Numerical scheme The scheme ◮ provides strong coupling on interface ◮ semi-implicit ◮ produces one linear system per time step ◮ second order in time ◮ unconditionally stable (no CFL restriction), proved with assumptions: ◮ 1st order in time ◮ St. Venant–Kirchhoff inc./comp. (experiment: Neo-Hookean inc./comp.) ◮ extension of u to Ω f guarantees J k > 0 ◮ ∆ t is not large A.Lozovskiy, M.Olshanskii, V.Salamatova, Yu.Vassilevski. An unconditionally stable semi-implicit FSI finite element method. Comput.Methods Appl.Mech.Engrg., 297, 2015