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

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 Perić
Zienkiewicz center for Computational Engineering
Swansea University, Swansea, UK.
07-April-2017, FEF 2017, Rome, Italy.

Outline
1 Motivation
2 Formulation
3 Fluid-rigid body interaction
4 Fluid-ﬂexible body interaction
5 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
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 (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

Fictitious Domain/Distributed Lagrange multiplier method on b-splines
C. Kadapa, W. G. Dettmer, and D. Perić. 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

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

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
)
Coupling:
Dirichlet-Neumann coupling
Staggered solution schemes - O(∆t) and O(∆t2
)

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

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.

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. Perić. A new staggered scheme for fluid-structure
interaction, IJNME, 93:1-22, 2013.

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. Perić. A stabilised immersed boundary method on
hierarchical b-spline grids for fluid-rigid body interaction with solid-solid contact.
CMAME, 318:242-269, 2017.

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.

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.

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.

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
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

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

Model turbines
-0.5
0
0.5
-1.000
1.000
vortz
8.0
16.0
24.0
0.0
30.0
vel Magn

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

Pressure relief valve - 3D
vel Magn vel Magn
25.00
50.00
0.00
60.00
vel Magn

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

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).

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.

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.

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.

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.

Squeezing a ﬂexible membrane through a constriction

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.

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.

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.

Summary
Summary and Conclusions
A robust and efficient stabilised immersed framework.
Higher-order spatial discretisations for the background grid.
) 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.

Acknowledgements
We thank Schaeﬄer Technologies AG & Co. KG, Germany, for funding
this project.
Thank you

References
C. Kadapa, W. G. Dettmer, and D. Perić.
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. Perić.
A stabilised immersed boundary method on hierarchical b-spline
grids.
311:415–437, 2016.
C. Kadapa, W. G. Dettmer, and D. Perić.
A stabilised immersed boundary method on hierarchical b-spline
grids for fluid-rigid body interaction with solid-solid contact.
318:242–269, 2017.

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. Perić.
A computational framework for fluid-rigid body interaction: Finite
element formulation and applications.
195:1633–1666, 2006.

Appendix

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

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

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