Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method

1. Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method MSc Thesis presentation – ThijsBosma – December 4th 2013 Supervisors: MatthijsLangelaar(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: Levelsetbased 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: Levelsetbased 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: Levelsetbased 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 • 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 • 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 • 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 The monolithic and the staggered approach Staggered Monolithic Fluid and structure solved simultaneously Complete Jacobian is used Residual error complete • Fluid and structure are solved separately • Complex FSI coupling terms in Jacobian are ignored • Residual error complete

23. The Solver The monolithic and the staggered approach Staggered Monolithic Efficient Suitable for optimization Difficult to find steady state solution • Inefficient • Unsuitable for optimization • Guarantees a 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 discretizationJacobian is a matrix

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

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

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 Actual Presumed

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 Jacobianis 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?

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. ‘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 ThijsBosma

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. 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