A stabilized and coupled meshfree/meshbased method for FSI
Hermann G. MatthiesThomas-Peter Fries
Technical University Braunschweig, Germany
Institute of Scientific Computing
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 2
Contents Motivation Meshfree/meshbased flow solver
Meshfree Method (EFG) Stabilization Coupling
Fluid-Structure-Interaction (FSI) Numerical Results Summary and Outlook
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 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
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 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
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 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.
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 6
Meshfree MethodElement-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. iw x x
ix
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 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.
Thu x p x a x
2T
1
N
i ii
J w u
M Ba x x p x a x x a = u
p x a x
For the approximation follows:
T 1
T :
hu M B x p u
N x meshfree MLS functions
1 x y
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 8
MLS functions lin. 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
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 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
, 0
0t
u u u f σ
u
I + 2 ,p p σ u, ε u
T12
ε u u+ u
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 10
Stabilization Stabilization is realized by SUPG/PSPG. SUPG/PSPG-stabilized weak form (Petrov-Galerkin):
,
,1
d : , d
d
, d
d
1eln
i
t
t
p
q
q
p
u u u
w u u u f ε w σ u
u
wh
σu f uwSUPG PSPG
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 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.
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 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 MLS
FEM
*
MMs shall be used where a mesh causes problems, in the rest of the domain the efficient meshbased FEM is used.
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 13
Coupling Coupling approach of Huerta et al.: Coupling with
modified MLS technique, which considers the FEM influence in the transition area.
FEM
FEM TMLS T
1
j
i
Ii
j
i
jNN
w
p
M
p
p
x xx x
x x x x
Modifications are required in order to obtain coupled shape functions that may be stabilized reliably.
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 14
Coupling Original approach of Huerta et al.
MLS*FEM
iw x x
2.0i x
MLS*FEM
iw x x
1.0i x
Modification: Add MLS nodes along and superimpose shape functions there.
FEM *
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 15
Coupling
MLS
*FEM
MLS
*FEM
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:
FEM \ *
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 16
Fluid-Structure Interaction
FS
Cfluid
domain
structure domain
displacement of
Strongly coupled, partitioned strategy:
Fluid: coupled MLS/FEM
Structure: FEM
Mesh movement
traction along
fluid domain +mesh velocity C
C
F
Situation:
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 17
Numerical Results Vortex excited beam (no connectivity change)
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 20
Numerical Results Connectivity changes due to large nodal
displacements.
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 24
Numerical Results The flap is considered as a discontinuity. “Visibility criterion” is used for the construction of the
MLS-functions.
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 25
Numerical Results Connectivity changes due to “visibility criterion”.
TU Braunschweig Meshfree/Meshbased Method for FSI Slide: 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.