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Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
A robust stabilised immersed finite element
framework for complex fluid-structure interaction
Dr. Chennakesava Kadapa∗
Dr. Wulf G Dettmer and Prof. Djordje Peri´c
Zienkiewicz center for Computational Engineering
Swansea University, Swansea, UK.
07-April-2017, FEF 2017, Rome, Italy.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Outline
1 Motivation
2 Formulation
3 Fluid-rigid body interaction
4 Fluid-flexible body interaction
5 Summary and Conclusions
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Motivation
Valve plate
Valveseat
Requirements:
Industrial requirement for FSI in check valves
Need to capture large displacements
Need to capture topological changes
Ability to include generic geometries
Efficiency
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Motivation
Valve plate
Valveseat
Requirements:
Industrial requirement for FSI in check valves
Need to capture large displacements
Need to capture topological changes
Ability to include generic geometries
Efficiency
Bottlenecks:
Serious limitations of commercial software
tools for complex FSI
Mainly based on body-fitted meshes
Coupling and the order of accuracy?
Difficulties associated with body-fitted meshes
Requires mesh generation
Difficulty increases for complex geometries
Requires re-meshing for large deformations
Extremely difficult to capture topological
changes
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Motivation (contd..)
1 Immersed boundary method (Peskin, 1972)
− Diffuse interfaces
No added-mass instabilities
× Explicitness
× Mostly O(∆t) → small ∆t
× Coupling via body-force on the fluid using artificial springs
× Very weak coupling → even smaller ∆t
2 Immersed interface method (LeVeque, 1994)
Clean interfaces
× Correction terms - too complicated in 3D
× Force terms - matrix inverses/SVD
× Explicit schemes for fluid → smaller ∆t
3 Fictitious Domain Method (Glowinski, 1994)
− Diffuse interfaces
Implicit schemes, O(∆t2
) possible
Strong or weak coupling via Lagrange multipliers
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Motivation (contd..)
Fictitious Domain/Distributed Lagrange multiplier method on b-splines
C. Kadapa, W. G. Dettmer, and D. Peri´c. A fictitious domain/distributed Lagrange
multiplier based fluid-structure interaction scheme with hierarchical B-Spline grids.
CMAME, 301:1-27, 2016.
pres pres
0.2
0.4
0.6
0.8
2.879e-04
9.365e-01
vel Magnitude
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Motivation (contd..)
Fictitious Domain/Distributed Lagrange multiplier method on b-splines
Advantages
Implicit and O(∆t2
)
Equal-order interpolation for velocity and
pressure
Local refinement
Works very well for thin structures
Issues
Zeroes on the matrix diagonal due to
Lagrange multipliers
PSPG + Lagrange multipliers → conservation
issues → drop PSPG
Unnecessary DOF when bulky solids are
present
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
A stabilised immersed framework - ingredients
Fluid:
Incompressible Navier-Stokes
Hierarchical b-splines
Mixed-Galerkin formulation
SUPG/PSPG stabilisation
Cut-cell approach
Nitsche’s method
Ghost-penalty stabilisation
Sub-triangulation and adaptive
integration
Implicit O(∆t) and O(∆t2
)
Solid:
Rigid solids
Solid-solid contact
Flexible solids (continuum)
Lagrange elements
Finite strains
F-bar for ν → 0.5
Implicit O(∆t) and O(∆t2
)
Coupling:
Dirichlet-Neumann coupling
Staggered solution schemes - O(∆t) and O(∆t2
)
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Governing equations
Fluid
ρf ∂vf
∂t
+ ρf
(vf
· ∇)vf
− µf
∆vf
+ ∇p = ff
and
∇ · vf
= 0
Solid
ρs ∂2
ds
∂t2
+ ∇ · σs
= fs
Interface conditions
vf
= vs
σf
· nf
+ σs
· ns
= 0
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Cut-cell approach
Ωf
Ωs
(a) Typical scenario (b) Subtriangulation (c) Adaptive integration
Advantages
Clean interfaces.
Stabilised formulation for fluids.
B-Splines of any order, along
with hierarchical refinement.
Fewer DOF.
Issues
Integration of cut cells.
Ill-conditioned matrices due to
small cut cells.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Formulation - staggered scheme
Coupling: Dirichlet-Neumann
For every time step:
1 predict force on the solid: FP
n+1
2 solve the solid problem for ds
n+1 and vs
n+1 using FP
n+1
3 reposition immersed solid(s) and update the fluid mesh
4 solve the fluid problem and obtain the force Ff
n+1
5 average the force: Fn+1 = −β Ff
n+1 + (1 − β) FP
n+1
6 proceed to next time step
FP
n+1 = FP1
= Fn, O(∆t)
FP
n+1 = FP2
= 2Fn − Fn−1, O(∆t2
)
W. G. Dettmer and D. Peri´c. A new staggered scheme for fluid-structure
interaction, IJNME, 93:1-22, 2013.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Variational formulation for fluid problem
Bf
Gal({wf
, q}, {vf
, p}) + Bf
Stab({wf
, q}, {vf
, p}) + Bf
Nitsche({wf
, q}, {vf
, p})
+ Bf
GP({wf
, q}, {vf
, p}) = F f
Gal({wf
, q}) (1)
Standard Galerkin terms
Bf
Gal({wf
, q}, {vf
, p}) =
Ωf
wf
· ρf ∂vf
∂t
+ vf
· ∇vf
dΩf
+
Ωf
µ ∇wf
: ∇vf
dΩf
−
Ωf
(∇ · wf
) p dΩf
+
Ωf
q (∇ · vf
) dΩf
(2)
F f
Gal({wf
, q}) =
Ωf
wf
· gf
dΩf
+
Γ
f
N
wf
· tf
dΓ (3)
C. Kadapa, W. G. Dettmer, and D. Peri´c. A stabilised immersed boundary method on
hierarchical b-spline grids for fluid-rigid body interaction with solid-solid contact.
CMAME, 318:242-269, 2017.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Variational formulation for fluid problem (contd..)
Stabilisation for the fluid
Bf
Stab({wf
, q}, {vf
, p}) =
nel
e=1 Ωfe
1
ρf
[τSUPG ρf
vf
· ∇wf
+ τPSPG ∇q] · rM dΩf
+
nel
e=1 Ωfe
τLSIC ρf
(∇ · wf
) (∇ · vf
) dΩf
(4)
where, rM is the residual of the momentum equation,
rM = ρf ∂vf
∂t
+ ρf
(vf
· ∇vf
) − µf
∆vf
+ ∇p − gf
(5)
Nitsche’s method
Bf
Nitsche({wf
, q}, {vf
, p}) = γN1
ΓD
wf
· (vf
− vs
) dΓ −
ΓD
wf
· (σ({vf
, p}) · nf
) dΓ
− γN2
ΓD
(σ({wf
, q}) · nf
) · (vf
− vs
) dΓ (6)
We use γN1
= 0 and γN2
= −1.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Variational formulation for fluid problem (contd..)
Ghost penalty
Bf
GP({wf
, q)}, {(v, p)}) = γu
GP µ G1(wf
, vf
) + γp
GP
1
µ
g3(q, p)
where,
G1(w, v) :=
d
i=1 F ∈F
h2a−1
F
[Da
wi][Da
vi] ds
g3(w, v) :=
F ∈F
h2a+1
F
[Da
w][Da
v] ds
[Djz] normal derivative of z, of order j, on face F
and a is degree of b-splines. Figure: Ghost-penalty operators
are applied on blue coloured
edges.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Fluid-rigid body interaction
Fluid: Generalised-α, O(∆t2
)
Solid: Generalised-α, O(∆t2
)
Coupling: FP
= FP2
= 2Fn − Fn−1, O(∆t2
)
C. Kadapa, W. G. Dettmer, and D. Peri´c. A stabilised immersed boundary method on
hierarchical b-spline grids for fluid-rigid body interaction with solid-solid contact.
CMAME, 318:242-269, 2017.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Rotational galloping of a rectangular body - Re=250
Figure: Hierarchical mesh.
DOF max(θ) fo
Robertson et al. [3] - 0.2620 0.762 fn
Dettmer and Peri´c [4] - 0.2670 0.780 fn
Present - Level-3, Q1 61560 0.2233 0.832 fn
Present - Level-4, Q1 109713 0.2262 0.800 fn
Present - Level-3, Q2 61491 0.2833 0.800 fn
Present - Level-4, Q2 109395 0.2688 0.768 fn
Table: ρs
∞ = ρf
∞ = 0.8, β = 0.9, ∆t = 0.05.
-3.00
0.00
3.00
-5.00
5.00
vortz
-8.00
-4.00
0.00
-11.88
3.81
pres
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Sedimentation of multiple particles
Mesh: 180 × 300, Q1.
ρf
= 0.998, µf
= 0.0101, D = 0.2, ρs
= 1.002ρf
, g = 981
ρf
∞ = ρs
∞ = 0.0, β = 0.1, ∆t = 0.005
3
5 0.7 0.4
0.4
0.4
vf
=0
vf
=0
vf
= 0
tf
= 0
0.15
0.30
0.45
0.00
0.49
vel Magn
0.20
0.40
0.60
0.00
0.73
vel Magn
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Model turbines
-0.5
0
0.5
-1.000
1.000
vortz
8.0
16.0
24.0
0.0
30.0
vel Magn
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Swing check valve
ρf
= 1.0, µf
= 0.1, Iθθ = 5.71e6, Re ≈ 500, Q1, ρs
∞ = ρf
∞ = 0.5, β =
0.5, ∆t = 0.2
250
50
115
85
r=20
65
5
5
0 50 100 150 200
Time
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
Angleofrotation(deg)
pin
=5
pin
=10
0 50 100 150 200
Time
−120
−100
−80
−60
−40
−20
0
20
Flowrate
pin
=5
pin
=10
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Pressure relief valve - 3D
vel Magn vel Magn
25.00
50.00
0.00
60.00
vel Magn
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Fluid-flexible body interaction
1 Type A
Fluid: Generalised-α, O(∆t2
)
Solid: Generalised-α, O(∆t2
)
Coupling: FP
= FP2
= 2Fn − Fn−1, O(∆t2
)
Added-mass instabilities
2 Type B
Fluid: Backward-Euler, O(∆t)
Solid: Backward-Euler, O(∆t)
Coupling: FP
= FP1
= Fn, O(∆t)
Can deal with significant added-mass
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Turek benchmark
0.2 2.3
0.20.21
0.05
0.35
0.02A
vf
x = vin
vf
y = 0
no-slip
no-slip
tractionfree
Figure: Geometry and BCs, fluid mesh, and solid mesh (200 × 10).
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Turek benchmark - FSI2
0 2 4 6 8 10 12 14
Time
−0.10
−0.05
0.00
0.05
0.10
Y-displacement
∆t =0.002
∆t =0.001
DOF dy fo
Turek and Hron - Level-3, ∆t = 0.002 [2] 76672 1.20 ± 79.2 2.0
Turek and Hron - Level-4, ∆t = 0.002 [2] 304128 1.25 ± 80.7 2.0
Present - Level-3, Q1, ∆t = 0.005 ≈ 70000 1.39 ± 79.6 2.0
Present - Level-3, Q1, ∆t = 0.002 ≈ 70000 1.41 ± 80.9 2.0
Present - Level-3, Q1, ∆t = 0.001 ≈ 70000 1.25 ± 79.9 2.0
Table: Y-displacement of A. Type A scheme.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Leaf in cross flow
ρf
= 1, µf
= 0.01, vin = 1.5 y(2 − y) sin(2πt)
ρs
= 1, Es
= 3000, νs
= 0.4
Type B. Relaxation parameter, β = 0.02.
0.8
8.0
1.0
0.0212vx = vin
vy = 0.0
no-slip
slip
tx=ty=0
Figure: Fluid mesh ≈ 70000. Solid mesh: 4 × 100, Q1.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Leaf in cross flow
0 5 10 15 20 25 30 35 40
Time
−0.8
−0.4
0.0
0.4
0.8
1.2
X-displacement
Ref
∆t =0.005
∆t =0.002
∆t =0.001
Figure: X-displacement of the midpoint of the leaf’s tip. Ref - [1], ∆t = 0.02.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Leaf in cross flow
0.25
0.50
0.75
1.00
0.00
1.21
vel Magn
0.20
0.40
0.60
0.00
0.76
vel Magn
1.40
1.75
2.10
1.10
2.44
pres
2.10
2.45
2.80
3.15
1.82
3.49
pres
Figure: Velocity and pressure contour plots obtained with ∆t = 0.005.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Squeezing a flexible membrane through a constriction
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Plate in cross flow
Domain: [0, 2] × [0.0, 0.6] × [−0.6, 0.6]
ρf
= 1.0, µf
= 0.01, vin = 10000
324 (y)(0.6 − y)(z + 0.6)(0.6 − z)
ρs
= 1.0, Es
= 200, νs
= 0.3
Type B, β = 0.02.
Figure: Fluid mesh: Q1, DOF ≈ 130,000. Solid mesh: Q1, 4 × 20 × 40.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Plate in cross flow
0 1 2 3 4 5
Time
0.00
0.05
0.10
0.15
0.20
0.25
0.30X-displacement
∆t =0.05
∆t =0.02
∆t =0.01
Figure: X-displacement of mid point of top surface.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Plate in cross flow
0.01
0.03
0.04
0.06
-0.00
0.07
disp X
0.25
0.50
0.75
-0.06
1.00
vel X
0.05
0.10
0.15
0.00
0.20
disp X
0.25
0.50
0.75
0.00
1.00
vel X
0.01
0.03
0.04
0.06
-0.00
0.07
disp X
0.15
0.30
0.45
0.01
0.61
pres
0.05
0.10
0.15
0.00
0.20
disp X
0.00
0.25
0.50
0.75
-0.10
0.94
pres
Figure: Plots of X-velocity and pressure obtained with ∆t = 0.02.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Summary
Summary and Conclusions
A robust and efficient stabilised immersed framework.
Higher-order spatial discretisations for the background grid.
Implicit O(∆t) and O(∆t2
) time integration schemes.
O(∆t) and O(∆t2
) staggered solution schemes.
Can capture large deformations and topological changes effectively.
Can deal with significant added-mass.
Applicable to wide variety of complex FSI problems.
On-going work
Parallelisation.
Fine tuning ghost-penalty parameters.
Completely closed scenarios.
Flexible solids with contact.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Acknowledgements
We thank Schaeffler Technologies AG & Co. KG, Germany, for funding
this project.
Thank you
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
References
C. Kadapa, W. G. Dettmer, and D. Peri´c.
A fictitious domain/distributed Lagrange multiplier based
fluid-structure interaction scheme with hierarchical B-Spline grids.
Computer Methods in Applied Mechanics and Engineering,
301:1–27, 2016.
W. G. Dettmer, C. Kadapa, and D. Peri´c.
A stabilised immersed boundary method on hierarchical b-spline
grids.
Computer Methods in Applied Mechanics and Engineering,
311:415–437, 2016.
C. Kadapa, W. G. Dettmer, and D. Peri´c.
A stabilised immersed boundary method on hierarchical b-spline
grids for fluid-rigid body interaction with solid-solid contact.
Computer Methods in Applied Mechanics and Engineering,
318:242–269, 2017.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
References (contd..)
E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing.
CutFEM: Discretizing geometry and partial differential equations.
International Journal for Numerical Methods in Engineering,
104:472–501, 2014.
S. Turek, and J. Hron.
Proposal for Numerical Benchmarking of Fluid-Structure Interaction
between an Elastic Object and Laminar Incompressible Flow.
In: Fluid-Structure Interaction, Springer, Berlin, 371–385, 2006.
I. Robertson, S. J. Sherwin, and P. W. Bearman.
A numerical study of rotational and transverse galloping rectangular
bodies.
Journal of Fluids and Structures, 17:681–699, 2003.
W. G. Dettmer, and D. Peri´c.
A computational framework for fluid-rigid body interaction: Finite
element formulation and applications.
Computer Methods in Applied Mechanics and Engineering,
195:1633–1666, 2006.
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Appendix
Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions
Covering/uncovering
Without mapping
0.06
0.12
0.18
0.00
0.20
velo Magnitude
With mapping - solid nodal velocities → fluid grid
0.06
0.12
0.18
0.00
0.20
velo Magnitude

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A robust stabilised immersed finite element framework for complex fluid-structure interaction

  • 1. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions A robust stabilised immersed finite element framework for complex fluid-structure interaction Dr. Chennakesava Kadapa∗ Dr. Wulf G Dettmer and Prof. Djordje Peri´c Zienkiewicz center for Computational Engineering Swansea University, Swansea, UK. 07-April-2017, FEF 2017, Rome, Italy.
  • 2. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Outline 1 Motivation 2 Formulation 3 Fluid-rigid body interaction 4 Fluid-flexible body interaction 5 Summary and Conclusions
  • 3. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Motivation Valve plate Valveseat Requirements: Industrial requirement for FSI in check valves Need to capture large displacements Need to capture topological changes Ability to include generic geometries Efficiency
  • 4. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Motivation Valve plate Valveseat Requirements: Industrial requirement for FSI in check valves Need to capture large displacements Need to capture topological changes Ability to include generic geometries Efficiency Bottlenecks: Serious limitations of commercial software tools for complex FSI Mainly based on body-fitted meshes Coupling and the order of accuracy? Difficulties associated with body-fitted meshes Requires mesh generation Difficulty increases for complex geometries Requires re-meshing for large deformations Extremely difficult to capture topological changes
  • 5. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Motivation (contd..) 1 Immersed boundary method (Peskin, 1972) − Diffuse interfaces No added-mass instabilities × Explicitness × Mostly O(∆t) → small ∆t × Coupling via body-force on the fluid using artificial springs × Very weak coupling → even smaller ∆t 2 Immersed interface method (LeVeque, 1994) Clean interfaces × Correction terms - too complicated in 3D × Force terms - matrix inverses/SVD × Explicit schemes for fluid → smaller ∆t 3 Fictitious Domain Method (Glowinski, 1994) − Diffuse interfaces Implicit schemes, O(∆t2 ) possible Strong or weak coupling via Lagrange multipliers
  • 6. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Motivation (contd..) Fictitious Domain/Distributed Lagrange multiplier method on b-splines C. Kadapa, W. G. Dettmer, and D. Peri´c. A fictitious domain/distributed Lagrange multiplier based fluid-structure interaction scheme with hierarchical B-Spline grids. CMAME, 301:1-27, 2016. pres pres 0.2 0.4 0.6 0.8 2.879e-04 9.365e-01 vel Magnitude
  • 7. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Motivation (contd..) Fictitious Domain/Distributed Lagrange multiplier method on b-splines Advantages Implicit and O(∆t2 ) Equal-order interpolation for velocity and pressure Local refinement Works very well for thin structures Issues Zeroes on the matrix diagonal due to Lagrange multipliers PSPG + Lagrange multipliers → conservation issues → drop PSPG Unnecessary DOF when bulky solids are present
  • 8. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions A stabilised immersed framework - ingredients Fluid: Incompressible Navier-Stokes Hierarchical b-splines Mixed-Galerkin formulation SUPG/PSPG stabilisation Cut-cell approach Nitsche’s method Ghost-penalty stabilisation Sub-triangulation and adaptive integration Implicit O(∆t) and O(∆t2 ) Solid: Rigid solids Solid-solid contact Flexible solids (continuum) Lagrange elements Finite strains F-bar for ν → 0.5 Implicit O(∆t) and O(∆t2 ) Coupling: Dirichlet-Neumann coupling Staggered solution schemes - O(∆t) and O(∆t2 )
  • 9. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Governing equations Fluid ρf ∂vf ∂t + ρf (vf · ∇)vf − µf ∆vf + ∇p = ff and ∇ · vf = 0 Solid ρs ∂2 ds ∂t2 + ∇ · σs = fs Interface conditions vf = vs σf · nf + σs · ns = 0
  • 10. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Cut-cell approach Ωf Ωs (a) Typical scenario (b) Subtriangulation (c) Adaptive integration Advantages Clean interfaces. Stabilised formulation for fluids. B-Splines of any order, along with hierarchical refinement. Fewer DOF. Issues Integration of cut cells. Ill-conditioned matrices due to small cut cells.
  • 11. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Formulation - staggered scheme Coupling: Dirichlet-Neumann For every time step: 1 predict force on the solid: FP n+1 2 solve the solid problem for ds n+1 and vs n+1 using FP n+1 3 reposition immersed solid(s) and update the fluid mesh 4 solve the fluid problem and obtain the force Ff n+1 5 average the force: Fn+1 = −β Ff n+1 + (1 − β) FP n+1 6 proceed to next time step FP n+1 = FP1 = Fn, O(∆t) FP n+1 = FP2 = 2Fn − Fn−1, O(∆t2 ) W. G. Dettmer and D. Peri´c. A new staggered scheme for fluid-structure interaction, IJNME, 93:1-22, 2013.
  • 12. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Variational formulation for fluid problem Bf Gal({wf , q}, {vf , p}) + Bf Stab({wf , q}, {vf , p}) + Bf Nitsche({wf , q}, {vf , p}) + Bf GP({wf , q}, {vf , p}) = F f Gal({wf , q}) (1) Standard Galerkin terms Bf Gal({wf , q}, {vf , p}) = Ωf wf · ρf ∂vf ∂t + vf · ∇vf dΩf + Ωf µ ∇wf : ∇vf dΩf − Ωf (∇ · wf ) p dΩf + Ωf q (∇ · vf ) dΩf (2) F f Gal({wf , q}) = Ωf wf · gf dΩf + Γ f N wf · tf dΓ (3) C. Kadapa, W. G. Dettmer, and D. Peri´c. A stabilised immersed boundary method on hierarchical b-spline grids for fluid-rigid body interaction with solid-solid contact. CMAME, 318:242-269, 2017.
  • 13. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Variational formulation for fluid problem (contd..) Stabilisation for the fluid Bf Stab({wf , q}, {vf , p}) = nel e=1 Ωfe 1 ρf [τSUPG ρf vf · ∇wf + τPSPG ∇q] · rM dΩf + nel e=1 Ωfe τLSIC ρf (∇ · wf ) (∇ · vf ) dΩf (4) where, rM is the residual of the momentum equation, rM = ρf ∂vf ∂t + ρf (vf · ∇vf ) − µf ∆vf + ∇p − gf (5) Nitsche’s method Bf Nitsche({wf , q}, {vf , p}) = γN1 ΓD wf · (vf − vs ) dΓ − ΓD wf · (σ({vf , p}) · nf ) dΓ − γN2 ΓD (σ({wf , q}) · nf ) · (vf − vs ) dΓ (6) We use γN1 = 0 and γN2 = −1.
  • 14. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Variational formulation for fluid problem (contd..) Ghost penalty Bf GP({wf , q)}, {(v, p)}) = γu GP µ G1(wf , vf ) + γp GP 1 µ g3(q, p) where, G1(w, v) := d i=1 F ∈F h2a−1 F [Da wi][Da vi] ds g3(w, v) := F ∈F h2a+1 F [Da w][Da v] ds [Djz] normal derivative of z, of order j, on face F and a is degree of b-splines. Figure: Ghost-penalty operators are applied on blue coloured edges.
  • 15. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Fluid-rigid body interaction Fluid: Generalised-α, O(∆t2 ) Solid: Generalised-α, O(∆t2 ) Coupling: FP = FP2 = 2Fn − Fn−1, O(∆t2 ) C. Kadapa, W. G. Dettmer, and D. Peri´c. A stabilised immersed boundary method on hierarchical b-spline grids for fluid-rigid body interaction with solid-solid contact. CMAME, 318:242-269, 2017.
  • 16. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Rotational galloping of a rectangular body - Re=250 Figure: Hierarchical mesh. DOF max(θ) fo Robertson et al. [3] - 0.2620 0.762 fn Dettmer and Peri´c [4] - 0.2670 0.780 fn Present - Level-3, Q1 61560 0.2233 0.832 fn Present - Level-4, Q1 109713 0.2262 0.800 fn Present - Level-3, Q2 61491 0.2833 0.800 fn Present - Level-4, Q2 109395 0.2688 0.768 fn Table: ρs ∞ = ρf ∞ = 0.8, β = 0.9, ∆t = 0.05. -3.00 0.00 3.00 -5.00 5.00 vortz -8.00 -4.00 0.00 -11.88 3.81 pres
  • 17. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Sedimentation of multiple particles Mesh: 180 × 300, Q1. ρf = 0.998, µf = 0.0101, D = 0.2, ρs = 1.002ρf , g = 981 ρf ∞ = ρs ∞ = 0.0, β = 0.1, ∆t = 0.005 3 5 0.7 0.4 0.4 0.4 vf =0 vf =0 vf = 0 tf = 0 0.15 0.30 0.45 0.00 0.49 vel Magn 0.20 0.40 0.60 0.00 0.73 vel Magn
  • 18. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Model turbines -0.5 0 0.5 -1.000 1.000 vortz 8.0 16.0 24.0 0.0 30.0 vel Magn
  • 19. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Swing check valve ρf = 1.0, µf = 0.1, Iθθ = 5.71e6, Re ≈ 500, Q1, ρs ∞ = ρf ∞ = 0.5, β = 0.5, ∆t = 0.2 250 50 115 85 r=20 65 5 5 0 50 100 150 200 Time −45 −40 −35 −30 −25 −20 −15 −10 −5 0 Angleofrotation(deg) pin =5 pin =10 0 50 100 150 200 Time −120 −100 −80 −60 −40 −20 0 20 Flowrate pin =5 pin =10
  • 20. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Pressure relief valve - 3D vel Magn vel Magn 25.00 50.00 0.00 60.00 vel Magn
  • 21. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Fluid-flexible body interaction 1 Type A Fluid: Generalised-α, O(∆t2 ) Solid: Generalised-α, O(∆t2 ) Coupling: FP = FP2 = 2Fn − Fn−1, O(∆t2 ) Added-mass instabilities 2 Type B Fluid: Backward-Euler, O(∆t) Solid: Backward-Euler, O(∆t) Coupling: FP = FP1 = Fn, O(∆t) Can deal with significant added-mass
  • 22. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Turek benchmark 0.2 2.3 0.20.21 0.05 0.35 0.02A vf x = vin vf y = 0 no-slip no-slip tractionfree Figure: Geometry and BCs, fluid mesh, and solid mesh (200 × 10).
  • 23. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Turek benchmark - FSI2 0 2 4 6 8 10 12 14 Time −0.10 −0.05 0.00 0.05 0.10 Y-displacement ∆t =0.002 ∆t =0.001 DOF dy fo Turek and Hron - Level-3, ∆t = 0.002 [2] 76672 1.20 ± 79.2 2.0 Turek and Hron - Level-4, ∆t = 0.002 [2] 304128 1.25 ± 80.7 2.0 Present - Level-3, Q1, ∆t = 0.005 ≈ 70000 1.39 ± 79.6 2.0 Present - Level-3, Q1, ∆t = 0.002 ≈ 70000 1.41 ± 80.9 2.0 Present - Level-3, Q1, ∆t = 0.001 ≈ 70000 1.25 ± 79.9 2.0 Table: Y-displacement of A. Type A scheme.
  • 24. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Leaf in cross flow ρf = 1, µf = 0.01, vin = 1.5 y(2 − y) sin(2πt) ρs = 1, Es = 3000, νs = 0.4 Type B. Relaxation parameter, β = 0.02. 0.8 8.0 1.0 0.0212vx = vin vy = 0.0 no-slip slip tx=ty=0 Figure: Fluid mesh ≈ 70000. Solid mesh: 4 × 100, Q1.
  • 25. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Leaf in cross flow 0 5 10 15 20 25 30 35 40 Time −0.8 −0.4 0.0 0.4 0.8 1.2 X-displacement Ref ∆t =0.005 ∆t =0.002 ∆t =0.001 Figure: X-displacement of the midpoint of the leaf’s tip. Ref - [1], ∆t = 0.02.
  • 26. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Leaf in cross flow 0.25 0.50 0.75 1.00 0.00 1.21 vel Magn 0.20 0.40 0.60 0.00 0.76 vel Magn 1.40 1.75 2.10 1.10 2.44 pres 2.10 2.45 2.80 3.15 1.82 3.49 pres Figure: Velocity and pressure contour plots obtained with ∆t = 0.005.
  • 27. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Squeezing a flexible membrane through a constriction
  • 28. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Plate in cross flow Domain: [0, 2] × [0.0, 0.6] × [−0.6, 0.6] ρf = 1.0, µf = 0.01, vin = 10000 324 (y)(0.6 − y)(z + 0.6)(0.6 − z) ρs = 1.0, Es = 200, νs = 0.3 Type B, β = 0.02. Figure: Fluid mesh: Q1, DOF ≈ 130,000. Solid mesh: Q1, 4 × 20 × 40.
  • 29. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Plate in cross flow 0 1 2 3 4 5 Time 0.00 0.05 0.10 0.15 0.20 0.25 0.30X-displacement ∆t =0.05 ∆t =0.02 ∆t =0.01 Figure: X-displacement of mid point of top surface.
  • 30. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Plate in cross flow 0.01 0.03 0.04 0.06 -0.00 0.07 disp X 0.25 0.50 0.75 -0.06 1.00 vel X 0.05 0.10 0.15 0.00 0.20 disp X 0.25 0.50 0.75 0.00 1.00 vel X 0.01 0.03 0.04 0.06 -0.00 0.07 disp X 0.15 0.30 0.45 0.01 0.61 pres 0.05 0.10 0.15 0.00 0.20 disp X 0.00 0.25 0.50 0.75 -0.10 0.94 pres Figure: Plots of X-velocity and pressure obtained with ∆t = 0.02.
  • 31. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Summary Summary and Conclusions A robust and efficient stabilised immersed framework. Higher-order spatial discretisations for the background grid. Implicit O(∆t) and O(∆t2 ) time integration schemes. O(∆t) and O(∆t2 ) staggered solution schemes. Can capture large deformations and topological changes effectively. Can deal with significant added-mass. Applicable to wide variety of complex FSI problems. On-going work Parallelisation. Fine tuning ghost-penalty parameters. Completely closed scenarios. Flexible solids with contact.
  • 32. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Acknowledgements We thank Schaeffler Technologies AG & Co. KG, Germany, for funding this project. Thank you
  • 33. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions References C. Kadapa, W. G. Dettmer, and D. Peri´c. A fictitious domain/distributed Lagrange multiplier based fluid-structure interaction scheme with hierarchical B-Spline grids. Computer Methods in Applied Mechanics and Engineering, 301:1–27, 2016. W. G. Dettmer, C. Kadapa, and D. Peri´c. A stabilised immersed boundary method on hierarchical b-spline grids. Computer Methods in Applied Mechanics and Engineering, 311:415–437, 2016. C. Kadapa, W. G. Dettmer, and D. Peri´c. A stabilised immersed boundary method on hierarchical b-spline grids for fluid-rigid body interaction with solid-solid contact. Computer Methods in Applied Mechanics and Engineering, 318:242–269, 2017.
  • 34. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions References (contd..) E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: Discretizing geometry and partial differential equations. International Journal for Numerical Methods in Engineering, 104:472–501, 2014. S. Turek, and J. Hron. Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow. In: Fluid-Structure Interaction, Springer, Berlin, 371–385, 2006. I. Robertson, S. J. Sherwin, and P. W. Bearman. A numerical study of rotational and transverse galloping rectangular bodies. Journal of Fluids and Structures, 17:681–699, 2003. W. G. Dettmer, and D. Peri´c. A computational framework for fluid-rigid body interaction: Finite element formulation and applications. Computer Methods in Applied Mechanics and Engineering, 195:1633–1666, 2006.
  • 35. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Appendix
  • 36. Motivation Formulation Fluid-rigid body interaction Fluid-flexible body interaction Summary and Conclusions Covering/uncovering Without mapping 0.06 0.12 0.18 0.00 0.20 velo Magnitude With mapping - solid nodal velocities → fluid grid 0.06 0.12 0.18 0.00 0.20 velo Magnitude