Numerical Simulations of 3D Fluid-Structure interacion Martin M - PowerPoint PPT Presentation

Numerical Simulations of 3D Fluid-Structure interacion Martin M´ adl´ ık Mathematical Institute of Charles University Computational Methods with Applications Harrachov August 21 2007 Supervisor: Frantiˇ sek Marˇ s´ ık Thermo-mechanical Institute CAS

Physics Math Numerical method Results Conclusions Outline 1 Physical problem 2 Mathematical formulation 3 Numerical method 4 Results 5 Conclusions Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model Motivation Biological problem: Blood flow in vessels Solid and Fluid parts (2 domains) Interaction of materials Various kind of materials (Easy to change Constitutive relations) Full 3D setting Usual assumptions Incompressibility, Isotermicity Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The ALE V 0 Material volume initial configuration V ( t ) Material volume actual state V 0 x i = � x ( X i , t ) deform. � X V ( t ) Control volume y i = � y ( X i , t ) deform. Position and velocity in coordinate system y i y ( � x ( � w ( � X , t ) − � � X , t ) = � X , t ) � � ∂� y v − ∂� w � � = � v V ( t ) = � � � ∂ t ∂ t � � X X Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The ALE V 0 Material volume � x initial configuration V ( t ) � v V ( t ) Material volume actual state V 0 x i = � x ( X i , t ) deform. � X V ( t ) Control volume y i = � y ( X i , t ) deform. Position and velocity in coordinate system y i y ( � x ( � w ( � X , t ) − � � X , t ) = � X , t ) � � ∂� y v − ∂� w � � = � v V ( t ) = � � � ∂ t ∂ t � � X X Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The ALE V 0 Material volume � x initial configuration V ( t ) v � x 3 V ( t ) Material volume actual state V 0 x i = � x ( X i , t ) deform. � X V ( t ) Control volume x 2 y i = � y ( X i , t ) deform. x 1 Position and velocity in coordinate system y i y ( � x ( � w ( � X , t ) − � � X , t ) = � X , t ) � � ∂� y v − ∂� w � � = � v V ( t ) = � � � ∂ t ∂ t � � X X Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The ALE V 0 Material volume � x initial configuration V ( t ) v � x 3 , y 3 V ( t ) Material volume actual state V 0 x i = � x ( X i , t ) deform. � X V ( t ) Control volume x 2 , y 2 V ( t ) y � y i = � y ( X i , t ) deform. x 1 , y 1 Position and velocity in coordinate system y i y ( � x ( � w ( � X , t ) − � � X , t ) = � X , t ) � � ∂� y v − ∂� w � � = � v V ( t ) = � � � ∂ t ∂ t � � X X Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The ALE V 0 Material volume � x initial configuration V ( t ) v � x 3 , y 3 V ( t ) Material volume actual state ∂ � w ∂t V 0 x i = � x ( X i , t ) deform. � X v V t � V ( t ) Control volume x 2 , y 2 V ( t ) y � y i = � y ( X i , t ) deform. x 1 , y 1 Position and velocity in coordinate system y i y ( � x ( � w ( � X , t ) − � � X , t ) = � X , t ) � � ∂� y v − ∂� w � � = � v V ( t ) = � � � ∂ t ∂ t � � X X Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model Balance laws in the ALE Extensive quantity � � � Φ( � X , t ) d V = Φ( t ) = ϕ ( � x , t ) dv = ϕ ( � y , t ) dv y V 0 V ( t ) V ( t ) General balance d Φ dt = ˙ Φ = L (Φ) + P (Φ) L Total flux P Total production Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model Balance laws in the ALE Extensive quantity � � � Φ( � X , t ) d V = Φ( t ) = ϕ ( � x , t ) dv = ϕ ( � y , t ) dv y V 0 V ( t ) V ( t ) General balance d Φ dt = ˙ Φ = L (Φ) + P (Φ) L Total flux P Total production Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model Balance laws in the ALE The balance law in the integral form ∂ X K � ∂ � Φ( v k − v k ∂ t Φ j y d V + V t ) j y ∂ y k dA K = V 0 ∂ V 0 ∂ X K � � l k (Φ) j y σ (Φ) j y d V . ∂ y k dA K + ∂ V 0 V 0 Local balance in the ALE ∂ X K �� � ∂ ∂ φ ( v k − v k � V t ) − l k (Φ) − j y σ (Φ) = 0 ∂ t ( j y φ ) + j y ∂ X K ∂ y k Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model Balance laws in the ALE The balance law in the integral form ∂ X K � ∂ � Φ( v k − v k ∂ t Φ j y d V + V t ) j y ∂ y k dA K = V 0 ∂ V 0 ∂ X K � � l k (Φ) j y σ (Φ) j y d V . ∂ y k dA K + ∂ V 0 V 0 Local balance in the ALE ∂ X K �� � ∂ ∂ φ ( v k − v k � V t ) − l k (Φ) − j y σ (Φ) = 0 ∂ t ( j y φ ) + j y ∂ X K ∂ y k Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The mass balance V 0 ̺ ( � � � The total mass: m ( t ) = X , t ) d V = V ( t ) ̺ ( � x , t ) dv The balance ∂ X K �� � ∂ ∂ ̺ ( v k − v k � ∂ t ( j y ̺ ) + V t ) j y = 0 ∂ X K ∂ y k Question Why is the ALE a generalization of the Lagrangian, Eulerian view? Let V ( t ) be static Let V ( t ) be a flow y = � j y = j , � v V ( t ) = � v , remains: � X , j y = 1, remains: ∂ ∂ t + ∂ ( ̺ v k ) ∂̺ ∂ t ( j ̺ ) = 0 = 0 . ∂ x k Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The mass balance V 0 ̺ ( � � � The total mass: m ( t ) = X , t ) d V = V ( t ) ̺ ( � x , t ) dv The balance ∂ X K �� � ∂ ∂ ̺ ( v k − v k � ∂ t ( j y ̺ ) + V t ) j y = 0 ∂ X K ∂ y k Question Why is the ALE a generalization of the Lagrangian, Eulerian view? Let V ( t ) be static Let V ( t ) be a flow y = � j y = j , � v V ( t ) = � v , remains: � X , j y = 1, remains: ∂ ∂ t + ∂ ( ̺ v k ) ∂̺ ∂ t ( j ̺ ) = 0 = 0 . ∂ x k Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The mass balance V 0 ̺ ( � � � The total mass: m ( t ) = X , t ) d V = V ( t ) ̺ ( � x , t ) dv The balance ∂ X K �� � ∂ ∂ ̺ ( v k − v k � ∂ t ( j y ̺ ) + V t ) j y = 0 ∂ X K ∂ y k Question Why is the ALE a generalization of the Lagrangian, Eulerian view? Let V ( t ) be static Let V ( t ) be a flow y = � j y = j , � v V ( t ) = � v , remains: � X , j y = 1, remains: ∂ ∂ t + ∂ ( ̺ v k ) ∂̺ ∂ t ( j ̺ ) = 0 = 0 . ∂ x k Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The mass balance V 0 ̺ ( � � � The total mass: m ( t ) = X , t ) d V = V ( t ) ̺ ( � x , t ) dv The balance ∂ X K �� � ∂ ∂ ̺ ( v k − v k � ∂ t ( j y ̺ ) + V t ) j y = 0 ∂ X K ∂ y k Question Why is the ALE a generalization of the Lagrangian, Eulerian view? Let V ( t ) be static Let V ( t ) be a flow y = � j y = j , � v V ( t ) = � v , remains: � X , j y = 1, remains: ∂ ∂ t + ∂ ( ̺ v k ) ∂̺ ∂ t ( j ̺ ) = 0 = 0 . ∂ x k Martin M´ adl´ ık Numerical simulations of FSI problems

Physics Math Numerical method Results Conclusions Motivation Balance laws in the ALE The Model The mass balance V 0 ̺ ( � � � The total mass: m ( t ) = X , t ) d V = V ( t ) ̺ ( � x , t ) dv The balance ∂ X K �� � ∂ ∂ ̺ ( v k − v k � ∂ t ( j y ̺ ) + V t ) j y = 0 ∂ X K ∂ y k Question Why is the ALE a generalization of the Lagrangian, Eulerian view? Let V ( t ) be static Let V ( t ) be a flow y = � j y = j , � v V ( t ) = � v , remains: � X , j y = 1, remains: ∂ ∂ t + ∂ ( ̺ v k ) ∂̺ ∂ t ( j ̺ ) = 0 = 0 . ∂ x k Martin M´ adl´ ık Numerical simulations of FSI problems