CutFEM on hierarchical B-Spline Cartesian grids with applications to fluid-structure interaction
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The document presents CutFEM, a fictitious domain/distributed Lagrange multiplier method on hierarchical B-Spline Cartesian grids. It discusses the formulation of CutFEM for incompressible Navier-Stokes equations, including stabilization techniques. It also describes adaptive integration techniques for cut cells. Numerical examples applying CutFEM to flow past a cylinder and a pressure relief valve problem are presented to demonstrate the method's accuracy and ability to reduce degrees of freedom compared to traditional finite difference methods.
![Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions
Formulation (contd..)
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)](https://image.slidesharecdn.com/eccomas2016-171213013345/85/CutFEM-on-hierarchical-B-Spline-Cartesian-grids-with-applications-to-fluid-structure-interaction-12-320.jpg)
![Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions
Formulation (contd..)
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.](https://image.slidesharecdn.com/eccomas2016-171213013345/85/CutFEM-on-hierarchical-B-Spline-Cartesian-grids-with-applications-to-fluid-structure-interaction-13-320.jpg)
![Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions
Flow past a cylinder - Re = 100
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.](https://image.slidesharecdn.com/eccomas2016-171213013345/85/CutFEM-on-hierarchical-B-Spline-Cartesian-grids-with-applications-to-fluid-structure-interaction-24-320.jpg)
CutFEM on hierarchical B-Spline Cartesian grids with applications to fluid-structure interaction
- 1. 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.
- 2. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Outline 1 Motivation 2 Formulation 3 Adaptive integration 4 Numerical examples 5 Summary and Conclusions
- 3. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Motivation Large deformations Topological changes Generic geometries Added-mass instabilities
- 4. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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
- 5. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Fictitious Domain/Distributed Lagrange multiplier method 0.2 0.4 0.6 0.8 2.879e-04 9.365e-01 vel Magnitude
- 6. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Fictitious Domain/Distributed Lagrange multiplier method 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
- 7. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions What is CutFEM?
- 8. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions What is CutFEM? Advantages Clean interfaces. B-Splines of any order, along with hierarchical refinement. Stabilised formulation for fluids. Fewer DOF.
- 9. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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.
- 10. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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)
- 11. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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)
- 12. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Formulation (contd..) 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)
- 13. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Formulation (contd..) 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.
- 14. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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.
- 15. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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
- 16. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Integration of cut-cells - adaptive integration - merging 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) without merging with merging (b) nGP for different levels
- 17. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Integration of cut-cells - binary tree (a) quad-tree (b) binary-tree Figure: Adaptive integration - quad-tree Vs binary-tree.
- 18. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Integration of cut-cells - adaptive integration (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) quad-tree without merging binary-tree without merging quad-tree with merging binary-tree with merging (b) nGP for different levels Binary subdivision helps to reduce number of Gauss points in cut-cells by 20%. (In 3D, for sphere, the reduction is about 30%.)
- 19. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Numerical examples
- 20. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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.
- 21. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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
- 22. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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 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 (b) Q2 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 (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 Number of Gauss points 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
- 23. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Flow past a cylinder - Re = 20 (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
- 24. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Flow past a cylinder - Re = 100 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.
- 25. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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
- 26. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Pressure relief valve 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.
- 27. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Pressure relief valve 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.
- 28. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Pressure relief valve 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
- 29. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Ball check valve 401248 7.5 3 17 7.4 15.2 4.6 pin pout no-slip no-slip
- 30. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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
- 31. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Ball check valve 4.00 8.00 12.00 0.00 16.00 vel Magn Figure: Velocity magnitude.
- 32. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Bearing rotor (a) Domains (b) Discretisation Figure: Animation - velocity magnitude.
- 33. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Flow past a sphere in 3D Figure: Triangles = 776. DOF = 178084 for Q1.
- 34. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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).
- 35. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Pressure relief valve - 3D (a) Geometry (b) Discretisation (c) Streamlines
- 36. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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.
- 37. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions Acknowledgements We thank Schaeffler Technologies AG & Co. KG, Germany, for funding this project. Thank you
- 38. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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. Peri´c. 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.
- 39. Motivation Formulation Adaptive integration Numerical examples Summary and Conclusions 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.