An update on the Eulerian formulation for the

Introduction

Governing equations for fluids and solids

Numerical method and implementation

An update on the Eulerian formulation for the simulation of soft solids in fluids Suhas S Jain and Ali Mani Flow Physics and Computational Engineering, Stanford University

20th November 2017

Suhas S Jain and Ali Mani

APS DFD 2017

1 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Introduction and Motivation

System of soft solids in a fluid are ubiquitous Animal tissues, cell membrane.

Studied for decades - arbitrary Lagrangian-Eulerian (ALE) method - Stiff solids. Need for a fully Eulerian approach with “true solid constitutive laws”. Eulerian Godunov method (Miller & Colella 2001) - unbounded domains Reference Map Technique (RMT) (Kamrin et al. 2012)

Suhas S Jain and Ali Mani

APS DFD 2017

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Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Reference Map Technique

Reference configuration

Deformed configuration

Reference map ~ x, t) = X ~ ξ(~ ~ x, t) Dξ(~ =0 Dt

⇒

~ x, t) ∂ ξ(~ ~ x, t) = 0 ~ ξ(~ +~ u.∇ ∂t

Deformation gradient → →− ~ t) = ∂~ ~ = (− F(X, x/∂ X ∇ ξ (~ x, t))−1 [Valkov et al., J. Appl. Mech., 82, 2015]. Suhas S Jain and Ali Mani

APS DFD 2017

3 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Governing equations for solids and fluids Momentum balance ∂(ρ~ u) + ∇.(ρ~ u⊗~ u) = ∇.σ ∂t Mass balance Fluid:

∂ρ ~ uρ) = 0 + ∇.(~ ∂t

Solid:

⇒

−1

⇒

ρ = ρo (det(F))

~ u=0 ∇.~ det(F) = 1

Cauchy stress Newtonian fluid: σ

f

=µ



  T    ~ u + ∇~ ~u ~ ∇~ − 1 ∇P

Neo-Hookean solid: −1

s

σ = 2(detF) ˆ ψ(C) = µ[trC − 3]

⇒

F

ˆ ∂ ψ(C) T F ∂C s

s

T

~ (∇ ~ ~ ξ) ~ ξ)] σ = 2µ [(∇

−1

[Valkov et al., J. Appl. Mech., 82, 2015]. Suhas S Jain and Ali Mani

APS DFD 2017

4 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Basic methodology

ξ~ defined only within the solid.

[Valkov et al., J. Appl. Mech., 82, 2015].

Suhas S Jain and Ali Mani

APS DFD 2017

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Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Basic methodology

Procedure: Solve for ξ~n+1 in ΩS .

Suhas S Jain and Ali Mani

APS DFD 2017

6 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Basic methodology

Procedure: Solve for ξ~n+1 in ΩS . ξ~n+1 → Extrapolate outside ΩS .

Suhas S Jain and Ali Mani

APS DFD 2017

7 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Basic methodology

Procedure: Solve for ξ~n+1 in ΩS . ξ~n+1 → Extrapolate outside ΩS . ξ~n+1 → Construct φ → Construct H(φ).

Suhas S Jain and Ali Mani

APS DFD 2017

8 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Basic methodology

Procedure: Solve for ξ~n+1 in ΩS . ξ~n+1 → Extrapolate outside ΩS . ξ~n+1 → Construct φ → Construct H(φ). Compute σ s , σ f .

Suhas S Jain and Ali Mani

APS DFD 2017

9 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Basic methodology

Procedure: Solve for ξ~n+1 in ΩS . ξ~n+1 → Extrapolate outside ΩS . ξ~n+1 → Construct φ → Construct H(φ). Compute σ s , σ f . σ ← Blend f (H(φ), σ s , σ f ) ~ u = 0. σ → Compute ~ u → Project ∇.~

Suhas S Jain and Ali Mani

APS DFD 2017

10 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Closure model

Fluid-Solid coupling is based on the “one-fluid formulation”. Smoothed heaviside function  0   1 x 1 πx (1 + + sin( )) H(x) =  2 wT π wT  

x ≤ −wT |x| < wT x ≥ wT

1

constructed based on the reinitialized level-set field. Mixture model σ = H(φ(~ x, t))σ f + (1 − H(φ(~ x, t)))σ s ρ = H(φ(~ x, t))ρf + (1 − H(φ(~ x, t)))ρs Capability to handle Solid-solid and solid-wall contact.

Suhas S Jain and Ali Mani

APS DFD 2017

11 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Updates and major issues fixed

Momentum conservation - conservative form of equations. X ∂ρ~ u ~ ~ + ∇.(ρ~ u~ u) = ∇.( σ ) i ∂t i Non-dissipative schemes - stress evaluation, convective flux evaluation. Central difference schemes No viscous damping in the solid

Modified reference map equation - momentum consistent solid advection. Improved robustness

~ x, t) ∂ ξ(~ ~ x, t) = 0 ~ ξ(~ + H(ψ)~ u.∇ ∂t Least-squares based extrapolation method. Collocated grid.

[Jain & Mani, CTR Annual Research Briefs, 2017]

Suhas S Jain and Ali Mani

APS DFD 2017

12 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Results

Least-square method for extrapolation

Locally linear approximation ξ = ax + by + c Solid

Solid 1

PDE

2

3

4

1

least-squares 10−4

Error (L2 )

8.32 ×

Cost

≈ 1550ms

6.72 × 10−9 ≈ 100ms

Zalesak disk

Suhas S Jain and Ali Mani

APS DFD 2017

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2

3

4

5

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Results

Lid-driven cavity

Re = 1000 100x100 grid.

1.0

0.4

Present work Ghia et al. (1982)

0.2 x coordinate

0.8 u velocity

Present work Ghia et al. (1982)

0.6 0.4 0.2

0.0 −0.2 −0.4

0.0 −0.25

0.00

0.25 0.50 y coordinate

Suhas S Jain and Ali Mani

0.75

1.00

0.0

0.2

0.4 0.6 v velocity

APS DFD 2017

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1.0

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Problems with advecting level-set field

[Valkov et al., 2015] Suhas S Jain and Ali Mani

APS DFD 2017

15 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Two solids in a Taylor-Green Vortex

Initial field

After collision

Smooth interface

Modified reference map advection equation ~ x, t) ∂ ξ(~ ~ x, t) = 0 ~ ξ(~ + H(ψ)~ u.∇ ∂t Suhas S Jain and Ali Mani

APS DFD 2017

16 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Simulations of a solid in a fluid with wall contact Collision in microgravity:

Collision in gravity - Stiff solid:

Collision in gravity - Soft solid:

Suhas S Jain and Ali Mani

APS DFD 2017

17 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Conservative vs Non-conservative formulation

Initial state Non-conservative formulation:

Conservative formulation:

Suhas S Jain and Ali Mani

APS DFD 2017

18 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Conclusion

We here presented: An incompressible fully Eulerian formulation for soft solids in fluids using an approximate Projection method. Improvements to the previously proposed Reference Map Technique: Momentum conserving formulation. Non-dissipative schemes for flux computation - better KE conservation. Momentum consistent solid advection. An accurate and cost effective extrapolation procedure. Use of collocated grids.

Future work: Analytical results for a quantitative comparison of fully coupled systems. An implicit formulation. Multigrid solver to speed up the poisson solution.

Suhas S Jain and Ali Mani

APS DFD 2017

19 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

THANK YOU Acknowledgements:

Franklin P. and Caroline M. Johnson Fellowship

Ken Kamrin, MIT

Reference: Jain, S, S, & Mani, A, ’An incompressible Eulerian formulation for soft solids in fluids’, CTR Annual Research briefs, 2017.

Suhas S Jain and Ali Mani

APS DFD 2017

20 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Reference map advection and Level-set reconstruction Analytical expression for φ(~ x, t = 0) is known. Level-set reconstruction ~ t = 0) φ(~ x, t) = φ(ξ, Reinitialize φ using Fast-marching method. Modified advection equation ~ x, t) ∂ ξ(~ ~ x, t) = 0 ~ ξ(~ + H(ψ)~ u.∇ ∂t and

( H(ψ) =

1

ΩS

0

else

Advantages of this modification: removes high frequency content in ξ~ field - simple central schemes. robust solver. [Jain & Mani, CTR Annual Research Briefs, 2017] Suhas S Jain and Ali Mani

APS DFD 2017

21 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Comparison and updates on the recent approach

RMT15 (Valkov et al. 2015)

Present approach

Grid

Staggered

Collocated

Nature of the solver

Compressible

Incompressible

Reference map Extrapolation

PDE based

Least-square based

Level-set construction

Advected in time

Reconstructed from t = 0 (Analytical - exact)

Smoothing routines

Required

No

Discretization stencil

one-sided

central

Global damping

Yes

No

Suhas S Jain and Ali Mani

APS DFD 2017

22 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Robustness due to modified advection equation

2 1

1 2

Suhas S Jain and Ali Mani

APS DFD 2017

23 / 24

Results

Introduction

Governing equations for fluids and solids

Numerical method and implementation

Solid-Solid contact Define, φ(1) − φ(2) 2 = 0 represents a midsurface between two solids. φ12 =

where φ12

Repulsive force ( f~i,j =

γi,j n ˆ 12i,j

φ(1) < 0 or φ(2) < 0

0

otherwise γi,j = krep δs (φ12i,j )

where n ˆ 12i,j is the unit vector normal to contours of φ12 and pointing away from the midsurface, krep is a prefactor and δs (x) is a compactly supported influence function,  πx  1 + cos w   T |x| < −wT δs (x) = 2wT    0 |x| ≥ wT Suhas S Jain and Ali Mani

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Results