A stabilized and coupled meshfree/meshbased method for FSI

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

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

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 Meshfree/Meshbased Method for FSI

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 Meshfree/Meshbased Method for FSI

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. Meshfree/Meshbased Method for FSI

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. Meshfree/Meshbased Method for FSI

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 Meshfree/Meshbased Method for FSI

8. Meshfree. • No Kronecker- property. • Non-polynomial functions. Meshfree Method MLS functions lin. FEM functions 1 1 0 0 Supports in MMs are ”adjustable”. Supports in FEM are pre-defined by mesh. Meshfree/Meshbased Method for FSI

9. Stabilization • Incompressible Navier-Stokes equations: • Momentumequation: • Continuityequation: • Newtonianfluid: • Stabilization is necessary for • Advectionsterms: NS-eqs. in Eulerian or ALE formulation. • “Equal-order-interpolation“:incompressible NS-eqs. with the same shape functions for velocity and pressure. Meshfree/Meshbased Method for FSI

10. Stabilization • Stabilization is realizedby SUPG/PSPG. • SUPG/PSPG-stabilized weak form (Petrov-Galerkin): SUPG PSPG Meshfree/Meshbased Method for FSI

11. The stabilization parameter requires special attention: • MLS functions are not interpolating different character. • Standard formulas are only valid for small supports. 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. Meshfree/Meshbased Method for FSI

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

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. Meshfree/Meshbased Method for FSI

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

15. Coupling original: modified: • SuperimposeFEM and MLS nodes along smaller supports reliable stabilization. • The resulting stabilized and coupled flow solver has been validated by a number of test cases. Meshfree/Meshbased Method for FSI

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

17. Numerical Results • Vortex excited beam (no connectivity change) Meshfree/Meshbased Method for FSI

18. Numerical Results Meshfree/Meshbased Method for FSI

19. Numerical Results • Rotating object Meshfree/Meshbased Method for FSI

20. Numerical Results • Connectivity changes due to large nodal displacements. Meshfree/Meshbased Method for FSI

21. Numerical Results Meshfree/Meshbased Method for FSI

22. Numerical Results Meshfree/Meshbased Method for FSI

23. Numerical Results • Moving flap Meshfree/Meshbased Method for FSI

24. Numerical Results • The flap is considered as a discontinuity. • “Visibility criterion” is used for the construction of the MLS-functions. Meshfree/Meshbased Method for FSI

25. Numerical Results • Connectivity changes due to “visibility criterion”. Meshfree/Meshbased Method for FSI

26. Numerical Results Meshfree/Meshbased Method for FSI

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. Meshfree/Meshbased Method for FSI