1. MSc Thesis presentation – Thijs Bosma – December 4th 2013
Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method
MSc Thesis presentation – Thijs Bosma – December 4th 2013
Supervisors:
Matthijs Langelaar(DUT)
Fred van Keulen(DUT)
Kurt Maute(CU)

2. Introduction to Fluid-Structure Interaction (FSI)

3. Introduction to Fluid-Structure Interaction (FSI)
Ultimate goal is Topology Optimization
ALE-method, computationally expensive (re-meshing)
Density-based methods, unclear interface
[James, 2012]

4. Introduction to my work
Goals of the research
Model: Levelset based based geometry description for fluid-structure interaction (FSI) problems with eXtended Finite Element Method (XFEM) approximation
Goal: Develop an efficient solver scheme that finds the steady state solution of the FSI problem (simultaneously for fluid and structure), such that it can be used in an optimization framework

5. Introduction to my work
Goals of the research
Model: Levelset based based geometry description for fluid-structure interaction (FSI) problems with eXtended Finite Element Method (XFEM) approximation
Goal: Develop an efficient solver scheme that finds the steady state solution of the FSI problem (simultaneously for fluid and structure), such that it can be used in an optimization framework

6. Introduction to my work
Goals of the research
Model: Levelset based geometry description for fluid-structure interaction (FSI) problems with eXtended Finite Element Method (XFEM) approximation
Goal: Develop an efficient solver scheme that finds the steady state solution of the FSI problem (simultaneously for fluid and structure), such that it can be used in an optimization framework
Does the approximated solution describe the physics of the system?
How can the problem be solved efficiently?
What makes this approach suitable for optimization?

7. Content The model The solvers Results staggered Results monolithic
Monolithic and staggered solver
Results staggered
1) Does the approximated solution describe the physics of the system?
Results monolithic
2) How can the problem be solved efficiently?
Outlook
3) What makes this approach suitable for optimization?
Conclusions/Recommendations

8. The model
An overview of the modeling process

9. The modeled problem Length tunnel: 300 μm Height tunnel: 100 μm
Height structure: 50 μm
Width: 5 μm

10. The model
An overview of the process

11. Fluid-structure interaction

12. The model
An overview of the process

13. Levelset Method LSF zero contour
Zero contour of signed distance function φ(x) describes the interface
Shortest distance from a point in the domain to the interface determines levelset field (LSF)
LSF zero contour

14. Levelset Method 6000 ft. contour φ(x)<0 φ(x)=0 φ(x)>0
Divides the domain in 3 parts:
Fluid (φ(x)<0)
Zero contour (φ(x)=0)
Structure (φ(x)>0)
Concept similar to elevation map of Boulder, CO, USA
φ(x)<0
φ(x)=0
φ(x)>0

15. Levelset Method
If the structure deforms/displaces the levelset field changes
The levelset field depends on the structural displacements
Structural
displacement u

16. The model
An overview of the process

17. eXtended Finite Element Method
Discontinuity turns off
part of the element
Approximation/discretization technique, based on FEM
Only find solution at discrete points in domain (nodes)
Assume solution and allow discontinuous solution between nodes
Discontinuity is transition from fluid to structure

18. eXtended Finite Element Method
LSF zero contour determines location of discontinuity
Two meshes
Approximation introduces Residual error
Residual is function of solution and LSF
If error is zero, approximated solution is found + =

19. The model
An overview of the process

20. The Solver The Newton-Raphson method for non-linear problems
R(un) is residual error function
u0 is initial solution
How to get to solution from initial solution?

21. The Solver The Newton-Raphson method for non-linear problems
Iteratively using the ‘slope’ is an efficient and accurate way
Slope can be found analytically, but is difficult
J is the slope of function R, called Jacobian
Principle holds for N dimensions

22. The Solver Staggered Monolithic
The monolithic and the staggered approach
Staggered
Monolithic
Fluid and structure are solved separately
Complex FSI coupling terms in Jacobian are ignored
Residual error complete
Fluid and structure solved simultaneously
Complete Jacobian is used
Residual error complete

23. The Solver Staggered Monolithic Staggered: check the Residual function
The monolithic and the staggered approach
Staggered
Monolithic
Inefficient
Unsuitable for optimization
Guarantees a steady state solution
Efficient
Suitable for optimization
Difficult to find steady state solution
Staggered: check the Residual function
Monolithic: check the Jacobian

24. The model
An overview of the process

25. Results – Staggered scheme
Velocity and displacement field – Steady state
XFEM-staggered:
[-]
[-]
COMSOL-ALE:
[m/s]
[μm]

26. Results – Staggered scheme
Velocity and displacement field – Steady state
XFEM-staggered:
[-]
[-] ≈
COMSOL-ALE:
[m/s]
[μm]

27. Results – Staggered scheme
Velocity and displacement field – Steady state
XFEM-staggered:
[-]
[-] ≈ ≈
COMSOL-ALE:
Staggered: Residual function is ok
[m/s]
[μm]

28. Goal
Goal: Develop an efficient solver scheme that finds the steady state solution of the FSI problem (simultaneously for fluid and structure), such that it can be used in an optimization framework
Does the approximated solution describe the physics of the system? Yes, based on qualitative check!
How can we efficiently solve the system?
What makes this approach suitable for optimization?

29. Results – Monolithic scheme
Velocity and displacement field – Exploded
XFEM: ≠
[-]
[-]
COMSOL:
Monolithic: Jacobian is not ok
[m/s]
[μm]

30. Results – Monolithic scheme
Jacobian check – Test Case Finite differences (FD)
FD is expensive, but reliable
Four element problem, all elements intersected
3 problems discovered – 1 discussed
After discretization Jacobian is a matrix

31. Results – Monolithic scheme
Jacobian check – Overview of the matrice entries
dus
duf
dRs
dRf
Analytic - Desired
Finite Difference - Comparison

32. Results – Monolithic scheme
Jacobian check – Overview of the matrices
dus
duf
dRs
dRf
Analytic - Desired
Finite Difference - Comparison

33. Results – Monolithic scheme
Jacobian check – Schematic of 2 element problem
Location zero contour structure depends on displacements
Zero contour fluid depends on orthogonal distance to zero contour
Zero contours determine what part is deleted from solution

34. Results – Monolithic scheme
Jacobian check – Schematic of 2 element problem with displacements

35. Results – Monolithic scheme
Jacobian check – Schematic of 2 element problem with displacements

36. Results – Monolithic scheme
Jacobian check – Schematic of 2 element problem with displacements

37. Results – Monolithic scheme
Jacobian check – Schematic of 2 element problem with displacements
Displacements of structural element 1 affect zero contour in both fluid elements
Presumed
Actual

38. Results – Monolithic scheme
Jacobian check – Schematic of 2 element problem with displacements
Displacements of element 1 affect zero contour in both elements
Secondary coupling introduced between intersected elements through LSM
Secondary coupling not incorporated in
analytic Jacobian

39. Goal
Goal: Develop an efficient solver scheme that finds the steady state solution of the FSI problem (simultaneously for fluid and structure), such that it can be used in an optimization framework
Does the approximated solution describe the physics of the system? Yes, based on qualitative check!
How can we efficiently solve the system? Monolithically, but analytic Jacobian is not numerically consistent
What makes this approach suitable for optimization?

40. Outlook
What makes this approach suitable for optimization?

41. Outlook What makes this approach suitable for optimization?
Move the extra beam to the top wall

42. Outlook
What makes this approach suitable for optimization?

43. Goal
Goal: Develop an efficient solver scheme that finds the steady state solution of the FSI problem (simultaneously for fluid and structure), such that it can be used in an optimization framework
Does the approximated solution describe the physics of the system? Yes, based on qualitative check!
How can we efficiently solve the system? Monolithically, but Jacobian is not numerically consistent
What makes this approach suitable for optimization? Flexible geometry description, accurate physical behavior at interface

44. Conclusions
The staggered setup has qualitatively shown that the steady state solution is comparable with the solution from ALE-based method
The FSI problem can not be solved with a monolithic setup yet
Jacobian is not numerically consistent
Flexible geometry description with physically relevant results

45. Recommendations
More elaborate and quantitative validation of the results should performed
The analytic Jacobian needs to be improved
Secondary coupling
Two other issues
Topology Optimization

46. * Loosely translated by Thijs Bosma
‘The primary product of science is failure, but failure teaches us where not to go in the future’ – Vincent Icke, physics professor University of Leiden in DWDD 27/11/2013* Thanks for the attention!
* Loosely translated by Thijs Bosma

47. References
James, K.A. and Martins, J.R. (2012). An isoparametric approach to level set topology optimization using a body fitted finite element mesh. Computers & Structures, 90-91:97-106

48. Backup slides

49. How to describe the behavior of the system?
The modeled problem
The physical configuration
Abstract blood vessel with valve
2D horizontal tunnel with structure fixed at bottom
Fluid flows from left to right
Steady state
Fluid applies force on structure
Structure changes flow path
Dimensions in μm
How to describe the behavior of the system?

50. Discontinuous shape functions

51. Results – Staggered scheme
The process

52. Results – Staggered scheme
The process

53. Results – Staggered scheme
Residual development

54. Levelset update - Changing DOFs

55. Results – Staggered scheme
Pressure and displacement field – XFEM model and COMSOL
XFEM:
BACKUP SLIDE PRESSURE
COMSOL:

56. Finite Differences

57. Non-dimensional numbers
BACKUP

58. Mesh mismatch

59. Residual equations
BACKUP

60. 3-field setup

61. Projection onto fluid mesh