a stabilized and coupled meshfree/meshbased method for fsi

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


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


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 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 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


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


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


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


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


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.


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.


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.


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 *


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 \ *


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:


Numerical Results Vortex excited beam (no connectivity change)


Numerical Results


Numerical Results Rotating

object


Numerical Results Connectivity changes due to large nodal

displacements.


Numerical Results


Numerical Results Moving flap


Numerical Results The flap is considered as a discontinuity. “Visibility criterion” is used for the construction of the

MLS-functions.


Numerical Results Connectivity changes due to “visibility criterion”.


Numerical Results


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.

a stabilized and coupled meshfree/meshbased method for fsi

Documents

stabilization method

squares mls functions

efficient meshbased

coupled shape functions

galerkin setting

fem influence

fem functionssupports

mesh causes problems