Numerically Solving the Coupled Motion of Fluid and Contained - PowerPoint PPT Presentation

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects Elijah Newren December 7, 2004 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 1/16

Outline • Example Biological Problems Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

Outline • Example Biological Problems • Equations for Fluid Motion Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

Outline • Example Biological Problems • Equations for Fluid Motion • Immersed Boundary Method Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

Outline • Example Biological Problems • Equations for Fluid Motion • Immersed Boundary Method • Immersed Interface Method Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

Outline • Example Biological Problems • Equations for Fluid Motion • Immersed Boundary Method • Immersed Interface Method • Incoherent Ramblings Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

Outline • Example Biological Problems • Equations for Fluid Motion • Immersed Boundary Method • Immersed Interface Method • (Even More) Incoherent Ramblings Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

Biological Problems Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

Biological Problems • Beating Heart Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

Biological Problems • Beating Heart • Platelet Aggregation Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

Biological Problems • Beating Heart • Platelet Aggregation • Insect Flight Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

Biological Problems • Beating Heart • Platelet Aggregation • Insect Flight • Cochlear Dynamics Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

Biological Problems • Beating Heart • Platelet Aggregation • Insect Flight • Cochlear Dynamics • Mechanical Properties of Cells Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

Biological Problems • Beating Heart • Platelet Aggregation • Insect Flight • Cochlear Dynamics • Mechanical Properties of Cells • Swimming of Organisms Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

Fluid Motion ρ ( u t + ( u · ∇ ) u ) = −∇ p + µ ∆ u + f ∇ · u = 0 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

Fluid Motion ρ ( u t + ( u · ∇ ) u ) = −∇ p + µ ∆ u + f Momentum ∇ · u = 0 • Change in Momentum (“Mass times acceleration”) Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

Fluid Motion ρ ( u t + ( u · ∇ ) u ) = −∇ p + µ ∆ u + f Pressure Gradient ∇ · u = 0 • Change in Momentum (“Mass times acceleration”) • Pressure Gradient (normal force between volumes) Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

Fluid Motion ρ ( u t + ( u · ∇ ) u ) = −∇ p + µ ∆ u + f Viscosity ∇ · u = 0 • Change in Momentum (“Mass times acceleration”) • Pressure Gradient (normal force between volumes) • Viscosity (tangential force between volumes) Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

Fluid Motion ρ ( u t + ( u · ∇ ) u ) = −∇ p + µ ∆ u + f Other Forces ∇ · u = 0 • Change in Momentum (“Mass times acceleration”) • Pressure Gradient (normal force between volumes) • Viscosity (tangential force between volumes) • Other forces (gravity, psychokinesis, etc.) Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

Fluid Motion ρ ( u t + ( u · ∇ ) u ) = −∇ p + µ ∆ u + f Incompressibility Constraint ∇ · u = 0 • Change in Momentum (“Mass times acceleration”) • Pressure Gradient (normal force between volumes) • Viscosity (tangential force between volumes) • Other forces (gravity, psychokinesis, etc.) • Incompressibility Constraint (Volume doesn’t change) Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

Navier Stokes Equations u t + ( u · ∇ ) u = −∇ p + ν ∆ u + f ∇ · u = 0 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 5/16

Navier Stokes Equations u t + ( u · ∇ ) u = −∇ p + ν ∆ u + f ∇ · u = 0 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 5/16

Navier Stokes Equations u t + ∇ p = ( u · ∇ ) u + ν ∆ u + f ∇ · u = 0 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 5/16

Navier Stokes Equations u t + ∇ p = ( u · ∇ ) u + ν ∆ u + f ∇ · u = 0 ⇒ u t = P ( − ( u · ∇ ) u + ν ∆ u + f ) Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 5/16

Hodge Decomposition Given periodic ω , ∃ ! periodic u & ∇ φ such that ω = u + ∇ φ ∇ · u = 0 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

Hodge Decomposition Given periodic ω , ∃ ! periodic u & ∇ φ such that ω = u + ∇ φ ∇ · u = 0 Proof (of existence): Taking the divergence: ∇ · ω = ∇ · u + ∇ · ( ∇ φ ) = ∆ φ Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

Hodge Decomposition Given periodic ω , ∃ ! periodic u & ∇ φ such that ω = u + ∇ φ ∇ · u = 0 Proof (of existence): Taking the divergence: ∇ · ω = ∇ · u + ∇ · ( ∇ φ ) = ∆ φ Thus we merely need to solve ∆ φ = ∇ · ω to find φ . Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

Hodge Decomposition Given periodic ω , ∃ ! periodic u & ∇ φ such that ω = u + ∇ φ ∇ · u = 0 Proof (of existence): Taking the divergence: ∇ · ω = ∇ · u + ∇ · ( ∇ φ ) = ∆ φ Thus we merely need to solve ∆ φ = ∇ · ω to find φ . A solution exists since (using the Fredholm alternative theorem): � � �∇ · ω , c � = c ( ∇ · ω ) = c ( ω · n ) = 0 . Ω ∂ Ω Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

Hodge Decomposition Given periodic ω , ∃ ! periodic u & ∇ φ such that ω = u + ∇ φ ∇ · u = 0 Proof (of existence): Taking the divergence: ∇ · ω = ∇ · u + ∇ · ( ∇ φ ) = ∆ φ Thus we merely need to solve ∆ φ = ∇ · ω to find φ . A solution exists since (using the Fredholm alternative theorem): � � �∇ · ω , c � = c ( ∇ · ω ) = c ( ω · n ) = 0 . Ω ∂ Ω Finally, we just set u = ω − ∇ φ . Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

Navier Stokes Solver u t + ∇ p = − ( u · ∇ ) u + ν ∆ u + f ∇ · u = 0 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

Navier Stokes Solver u t + ∇ p = − ( u · ∇ ) u + ν ∆ u + f ∇ · u = 0 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

Navier Stokes Solver u t + ∇ p = − ( u · ∇ ) u + ν ∆ u + f ∇ · u = 0 y ′ = f ( y ) Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

Navier Stokes Solver u t + ∇ p = − ( u · ∇ ) u + ν ∆ u + f ∇ · u = 0 y ′ = f ( y ) � t 2 y 2 = y 1 + f ( y ( t )) dt t 1 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

Navier Stokes Solver u t + ∇ p = − ( u · ∇ ) u + ν ∆ u + f ∇ · u = 0 y ′ = f ( y ) � t 2 y 2 = y 1 + f ( y ( t )) dt t 1 y 2 = y 1 + 1 2∆ t ( f ( y 2 ) + f ( y 1 )) Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

Navier Stokes Solver u t + ∇ p = − ( u · ∇ ) u + ν ∆ u + f ∇ · u = 0 y ′ = f ( y ) � t 2 y 2 = y 1 + f ( y ( t )) dt t 1 y 2 = y 1 + 1 2∆ t ( f ( y 2 ) + f ( y 1 )) y n +1 − y n = 1 2( f ( y n +1 ) + f ( y n )) ∆ t Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

Navier Stokes Solver u n +1 − u n 2 + ν 2∆( u n +1 + u n ) + f n + 1 + ∇ p n + 1 2 = − [( u · ∇ ) u ] n + 1 2 ∆ t ∇ · u n +1 = 0 y ′ = f ( y ) � t 2 y 2 = y 1 + f ( y ( t )) dt t 1 y 2 = y 1 + 1 2∆ t ( f ( y 2 ) + f ( y 1 )) y n +1 − y n = 1 2( f ( y n +1 ) + f ( y n )) ∆ t Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

Navier Stokes Solver u n +1 − u n 2 + ν 2∆( u n +1 + u n ) + f n + 1 + ∇ p n + 1 2 = − [( u · ∇ ) u ] n + 1 2 ∆ t ∇ · u n +1 = 0 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

Navier Stokes Solver u ∗ − u n 2 + ν 2∆( u ∗ + u n ) + f n + 1 + 0 = − [( u · ∇ ) u ] n + 1 2 ∆ t ∇ · u n +1 = 0 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16