1. A stabilized and coupled meshfree/meshbased method for FSI Hermann G. Matthies Thomas-Peter Fries Technical University Braunschweig, Germany Institute of Scientific Computing

2. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 2 Contents Motivation Meshfree/meshbased flow solver Meshfree Method (EFG) Stabilization Coupling Fluid-Structure-Interaction (FSI) Numerical Results Summary and Outlook

3. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 3 Motivation In geometrically complex FSI problems, a mesh may be difficult to be maintained meshbased methods like the FEM may fail without remeshing. For example: large geometry deformations, moving and rotating objects

4. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 4 Motivation Use meshfree methods (MMs): no mesh needed, but more time-consuming. Coupling: Use MMs only where mesh makes problems, otherwise employ FEM. Fluid: Coupled MM/FEM Structure: Standard FEM

5. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 5 Meshfree Method Meshfree method for the fluid part: Approx. incompressible Navier-Stokes equations. Moving least-squares (MLS) functions in a Galerkin setting. Nodes are not material points: allows Eulerian or ALE- formulation. Closely related to the element-free Galerkin (EFG) method. For the success of this method, Stabilization and Coupling with meshbased methods is needed.

6. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 6 Meshfree Method Element-free Galerkin method (EFG) MLS functions in a Galerkin setting. Moving Least Squares (MLS) Discretize the domain by a set of nodes. Define local weighting functions around each node.

7. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 7 Sufficient overlap of the weighting fcts. required Meshfree Method Local approximation around an arbitrary point. : Complete basis vector, e.g.. : Unknown coefficients of the approximation. Minimize a weighted error functional. For the approximation follows: meshfree MLS functions

8. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 8 MLS functionslin. FEM functions Supports in MMs are ”adjustable”. Supports in FEM are pre-defined by mesh. Meshfree Method Meshfree. No Kronecker- property. Non-polynomial functions. 1 0 1 0

9. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 9 Incompressible Navier-Stokes equations: Momentum equation: Continuity equation: Newtonian fluid: Stabilization is necessary for Advections terms: NS-eqs. in Eulerian or ALE formulation. “Equal-order-interpolation“: incompressible NS-eqs. with the same shape functions for velocity and pressure. Stabilization

10. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 10 Stabilization Stabilization is realized by SUPG/PSPG. SUPG/PSPG-stabilized weak form (Petrov-Galerkin): SUPGPSPG

11. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 11 Stabilization The stucture of the stabilization method may be applied to meshbased and meshfree methods. SUPG/PSPG-stabilization turned out to be slightly less diffusive than GLS. The stabilization parameter requires special attention: MLS functions are not interpolating different character. Standard formulas are only valid for small supports.

12. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 12 Coupling Our aim is a fluid solver for geometrically complex situations. The coupling is realized on shape function level. Meshfree area Meshbased area Transition area MMs shall be used where a mesh causes problems, in the rest of the domain the efficient meshbased FEM is used.

13. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 13 Coupling Coupling approach of Huerta et al.: Coupling with modified MLS technique, which considers the FEM influence in the transition area. Modifications are required in order to obtain coupled shape functions that may be stabilized reliably.

14. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 14 Coupling Original approach of Huerta et al. Modification: Add MLS nodes along and superimpose shape functions there.

15. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 15 Coupling Superimpose FEM and MLS nodes along smaller supports reliable stabilization. The resulting stabilized and coupled flow solver has been validated by a number of test cases. modified:original:

16. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 16 Fluid-Structure Interaction fluid domain structure domain displacement of Strongly coupled, partitioned strategy: Fluid: coupled MLS/FEM Structure: FEM Mesh movement traction along fluid domain +mesh velocity Situation:

17. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 17 Numerical Results Vortex excited beam (no connectivity change)

18. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 18 Numerical Results

19. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 19 Numerical Results Rotating object

20. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 20 Numerical Results Connectivity changes due to large nodal displacements.

21. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 21 Numerical Results

22. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 22 Numerical Results

23. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 23 Numerical Results Moving flap

24. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 24 Numerical Results The flap is considered as a discontinuity. “Visibility criterion” is used for the construction of the MLS- functions.

25. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 25 Numerical Results Connectivity changes due to “visibility criterion”.

26. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 26 Numerical Results

27. TU BraunschweigMeshfree/Meshbased Method for FSISlide: 27 Summary Meshfree Method: ALE-formulation, MLS functions in a Galerkin setting similar to EFG. SUPG/PSPG-Stabilization in extension of meshbased approach. Approach of Huerta et al. modified and extended by stabilization for coupling with FEM-fluid and FEM-solid. Method shows good behaviour, avoids remeshing. No conceptual difficulties seen for extension into 3-D and for higher order shape functions.