CutFEM on hierarchical B-Spline Cartesian grids with applications to fluid-structure interaction

Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions
CutFEM on hierarchical B-Spline Cartesian grids
with applications to ﬂuid-structure interaction
Dr. C. Kadapa, Dr. W. G. Dettmer and Prof. D. Peri´c
Zienkiewicz center for Computational Engineering
Swansea University, Swansea, UK.
06-June-2016, ECCOMAS 2016, Crete Island, Greece.

Outline
1 Motivation
2 Formulation
3 Adaptive integration
4 Numerical examples
5 Summary and Conclusions

Motivation
Large deformations
Topological changes
Generic geometries
Added-mass instabilities

Fictitious Domain/Distributed Lagrange multiplier method
Figure: Hierarchical B-Spline mesh.
200.00
250.00
300.00
148.39
326.94
pres
-60.00
-45.00
-30.00
-76.71
-20.91
pres
-320.00
-280.00
-240.00
-200.00
-355.18
-166.04
pres
30.00
45.00
60.00
75.00
21.46
80.45
pres
Figure: Pressure contour plots

0.2
0.4
0.6
0.8
2.879e-04
9.365e-01
vel Magnitude

Advantages
Equal-order interpolation for velocity and
pressure
Works very well for thin structures
Issues
Zeroes on the matrix diagonal
Zeroes on the matrix diagonal due to
Lagrange multipliers
Unnecessary DOF when bulky solids are
present

What is CutFEM?

What is CutFEM?
Advantages
Clean interfaces.
B-Splines of any order, along
with hierarchical reﬁnement.
Stabilised formulation for ﬂuids.
Fewer DOF.

What is CutFEM?
Advantages
Clean interfaces.
B-Splines of any order, along
with hierarchical reﬁnement.
Stabilised formulation for ﬂuids.
Fewer DOF.
Issues
Integration of cut cells.
Ill-conditioned matrices due to
small cut cells.

Formulation
Incompressible Navier-Stokes
Find velocity, v = (v1, v2, v3), and pressure, p, such that
ρf ∂vf
∂t
+ ρf
(vf
· ∇)vf
− µf
∆vf
+ ∇p = gf
in Ωf
(1a)
∇ · vf
= 0 in Ωf
(1b)
vf
= vs
on Γf
D (1c)
σf
· nf
= tf
on Γf
N (1d)
where,
σf
= µ∇vf
− p I (2)

Formulation (contd..)
Variational formulation
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}) (3)
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
(4)
F f
Gal({wf
, q}) =
Ωf
wf
· gf
dΩf
+
Γ
f
N
wf
· tf
dΓ (5)

Stabilisation
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
(6)
where, rM is the residual of the momentum equation,
rM = ρf ∂vf
∂t
+ ρf
(vf
· ∇vf
) − µf
∆vf
+ ∇p − gf
(7)
Nitsche’s method
Bf
N({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 (8)

Ghost penalty (Burman et al. [1])
Jump operator for a scalar valued problem:
gs(w, v) :=
k
j=1 F ∈F
h2(j−1)+s
F
[Dj
w][Dj
v] ds
[Djz] normal derivative of z, of order j, on face F
and k is degree of B-Splines.
Jump operator for a vector valued problem:
Gs(w, v) :=
d
i=1
k
j=1 F ∈F
h2(j−1)+s
F
[Dj
wi][Dj
vi] ds
Figure: Ghost-penalty
operators are applied on blue
coloured edges.
The ghost-penalty operator is given as,
Bf
GP({wf
, q)}, {(v, p)}) = γu
GP µ G1(wf
, vf
) + γp
GP
1
µ
g3(q, p) (9)
In this work, only the non-zero derivative of highest-order is considered.

Integration of cut-cells - Subtriangulation
Advantages
Exact integration for boundaries with straight
edges.
Easy when boundaries are discretised with
straight edges.
Fewer Gauss points.
Disadvantages
Diﬃcult for generic geometries in 3D. Need to
use constrained tetrahedralisation.
Diﬃcult when boundaries are discretised with
curved elements or NURBS.

Integration of cut-cells - adaptive integration
Subtriangularion: nGP = 420; % error = 1.55e-6.
(a) Level-3 adaptive integration
102
103
104
105
106
Number of Gauss points in cut-cells
10-5
10-4
10-3
10-2
10-1
100
abs(%errorinarea)
(b) nGP for diﬀerent levels

Integration of cut-cells - adaptive integration - merging
Subtriangularion: nGP = 420; % error = 1.55e-6.
102
103
104
105
106
10-5
10-4
10-3
10-2
10-1
100
abs(%errorinarea)
without merging
with merging

Integration of cut-cells - binary tree
(a) quad-tree (b) binary-tree
Figure: Adaptive integration - quad-tree Vs binary-tree.

Integration of cut-cells - adaptive integration
102
103
104
105
106
10-5
10-4
10-3
10-2
10-1
100
abs(%errorinarea)
quad-tree without merging
binary-tree without merging
quad-tree with merging
binary-tree with merging
Binary subdivision helps to reduce number of Gauss points in cut-cells by
20%. (In 3D, for sphere, the reduction is about 30%.)

Numerical examples

CutFEM - Basic scheme
W. G. Dettmer, C. Kadapa, and D. Peri´c. Ingredients for an
immersed interface methods for FSI. COUPLED Problems, Venice,
Italy, May-2015.
W. G. Dettmer, C. Kadapa, and D. Peri´c. A stabilised immersed
boundary method on hierarchical b-spline grids. Submitted for
publication.

Flow past a cylinder
10 20
15
15
1
vx = 1.0
vy = 0
vy = 0, tx = 0
vy = 0, tx = 0
tx=ty=0

Flow past a cylinder - Re = 20
1 2 3 4 5 6 7 8 9
Number of Gauss points
1.0
1.5
2.0
2.5
3.0
Dragcoefficient(CD)
Level-0
Level-1
Level-2
Level-3
Level-4
(a) Q1
1 2 3 4 5 6 7 8 9
1.0
1.5
2.0
2.5
3.0
Dragcoefficient(CD)
Level-0
Level-1
Level-2
Level-3
Level-4
(b) Q2
1 2 3 4 5 6 7 8 9
1.0
1.5
2.0
2.5
3.0
Dragcoefficient(CD)
Level-0
Level-1
Level-2
Level-3
Level-4
(c) Q3
0 50 100 150 200 250 300 350
Number of edges
1.0
1.5
2.0
2.5
3.0
Dragcoefficient(CD)
Q1
Q2
Q3
(d) nIE
0 2 4 6 8 10 12 14
1.0
1.5
2.0
2.5
3.0
Dragcoefficient(CD)
Q1
Q2
Q3
(e) nGPtria
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Ghost penalty parameter (γGP)
1.0
1.5
2.0
2.5
3.0
Dragcoefficient(CD)
Q1
Q2
Q3
(f) γGP

(a) Q2, Level-1 (b) Q2, Level-2
-0.300
0.000
0.300
-0.500
0.700
pres
(c) Q2, Level-3
(d) Q1, γGP = 100 (e) Q1, γGP = 10−2
-0.300
0.000
0.300
-0.500
0.700
pres
(f) Q1, γGP = 10−4
(g) Q2, nIE = 20 (h) Q2, nIE = 80
-0.300
0.000
0.300
-0.500
0.700
pres
(i) Q2, nIE = 320

0 50 100 150 200
Time
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Dragcoefficient(CD)
0 50 100 150 200
Time
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
Liftcoefficient(CL)
Figure: CD and CL for Re = 100 with Q2 B-Splines with ∆t = 0.1.
Data CD CL St
Liu et al. [4] 1.35 ±0.339 0.165
Le et al. [3] 1.37 ±0.323 0.160
Kadapa et al. [2] 1.39 ±0.339 0.166
Present 1.37 ±0.331 0.167
Table: Comparison of CD, CL and St.

Pressure relief valve
19111
5
3.2
pin
pout
noslip
noslip
1.0 0.50.5
2.1
0.4
g
r
Table: DOF comparison
Level FDM CutFEM % of FDM
2 99720 72030 72.23
3 149352 95289 63.80
4 327648 168360 51.38

Simulation time ≈ 2 hours.
Figure: Level-3
reﬁnement.
0 1 2 3 4 5 6 7 8
Time (ms)
−0.05
0.00
0.05
0.10
0.15
0.20
0.25
Valvelift(mm)
FDM-SC
CFM-subtrias
Figure: Valve lift with Q1 B-Splines. fP2
and β = 0.5.

0.6043
1.2086
1.8129
-0.235
2.182
pres
(a) 22 bar
0
0.6773
1.3546
2.0319
-0.319
2.391
pres
(b) 24 bar
0
0.7477
1.4953
2.243
-0.393
2.598
pres
(c) 26 bar
0
0.8154
1.6309
2.4463
-0.455
2.806
pres
(d) 28 bar
0
0.8808
1.7615
-0.508
3.015
pres
(e) 30 bar
0
0.9428
1.8857
-0.552
3.219
pres
(f) 32 bar
Figure: Pressure contour plots.

20 22 24 26 28 30
Inlet pressure (bar)
1.0
1.5
2.0
2.5
3.0
3.5
4.0Flowrate(l/min)
exp1
exp2
exp3
FDM
CFM
Figure: Flowrates - comparison with experiments

Ball check valve
401248
7.5
3 17
7.4
15.2
4.6
pin
pout
no-slip
no-slip

Ball check valve
ρf
= 800 kg/m3
. µf
= 800 cP. ms = 14.4 g. k = 0. Stop at 2mm.
pin = 0.5 sin(2πt/10)
−0.5
0.0
0.5
pin
0
1
2
disp
−2.5
0.0
2.0
velo
−50
100
250
Fbottom
0 5 10 15 20 25 30
Time (ms)
−50
100
250
Ftop

Ball check valve
4.00
8.00
12.00
0.00
16.00
vel Magn
Figure: Velocity magnitude.

Bearing rotor
(a) Domains (b) Discretisation
Figure: Animation - velocity magnitude.

Flow past a sphere in 3D
Figure: Triangles = 776. DOF = 178084 for Q1.

Flow past a sphere in 3D
0 20 40 60 80 100 120 140 160
Reynolds number (Re)
0
1
2
3
4
5
Dragcoefficient(CD)
Shirayama
Schlichting
Analytical
CutFEM-Q1
CutFEM-Q2
Figure: Drag coeﬃcient (CD) versus Reynolds number (Re).

Pressure relief valve - 3D
(a) Geometry
(b) Discretisation
(c) Streamlines

Summary and Conclusions and Future work
Summary and Conclusions
Implemented and tested CutFEM on hierarchical B-Spline grids.
Higher-order spatial discretisations for the background grid.
Demonstrated the performance with several examples.
The scheme is stable, accurate and robust.
Can capture large deformations and topological changes eﬀectively.
Future work
Fine tuning ghost-penalty parameters
Covering/uncovering
Completely closed scenarios
Parallelisation
Fluid-ﬂexible solid interactions.

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

References
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.
C. Kadapa, W. G. Dettmer, and D. Perić.
A fictitious domain/distributed Lagrange multiplier based
fluidstructure interaction scheme with hierarchical B-Spline grids.
Computer Methods in Applied Mechanics and Engineering,
301:1–27, 2016.
D. V. Le, B. C. Khoo, and J. Peraire.
An immersed interface method for viscous incompressible flows
involving rigid and flexible boundaries.
Journal of Computational Physics, 220:109–138, 2006.
C. Liu, X. Sheng, and C. H. Sung.
Preconditioned multigrid methods for unsteady incompressible flows.
Journal of Computational Physics, 139:35–57, 1998.

References (contd..)
H. Schlichting.
Boundary-Layer Theory
McGraw-Hill, USA, 1979.
S. Shirayama.
Flow past a shpere: Topological transitions of the vorticity ﬁeld
AIAA Journal, 30:349–358,1992.

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