A stabilized and coupled meshfree/meshbased method for FSI

1
A stabilized and coupled meshfree/meshbased
method for FSI
Technical University Braunschweig,
Germany Institute of Scientific Computing

• Hermann G. Matthies
• Thomas-Peter Fries

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Contents

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

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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

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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
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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.

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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.

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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
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Meshfree Method
MLS functions
lin. FEM functions
1
1
0
0
Supports in MMs are adjustable.
Supports in FEM are pre-defined by mesh.
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Stabilization

• 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.

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Stabilization

• Stabilization is realized by SUPG/PSPG.
• SUPG/PSPG-stabilized weak form (Petrov-Galerkin)

SUPG
PSPG
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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.

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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
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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.

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Coupling

• Original approach of Huerta et al.

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Coupling
modified
original

• 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.

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Fluid-Structure Interaction

• Situation

structure domain
fluid domain
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Numerical Results

• Vortex excited beam (no connectivity change)

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Numerical Results
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Numerical Results

• Rotating object

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Numerical Results

• Connectivity changes due to large nodal
  displacements.

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Numerical Results
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Numerical Results
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Numerical Results

• Moving flap

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Numerical Results

• The flap is considered as a discontinuity.
• Visibility criterion is used for the
  construction of the MLS-functions.

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Numerical Results

• Connectivity changes due to visibility
  criterion.

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Numerical Results
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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.