An Arbitrary LagrangianEulerian Finite Element Formulation for Dynamics and Finite Strain Plasticity

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An Arbitrary Lagrangian-Eulerian Finite Element
Formulation for Dynamics andFinite Strain
Plasticity Models

• Gang Cao
• Dept. of Civil Engineering
• Oregon State University

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Outline

• Basics of Continuum Mechanics
• Constitutive Model Finite Strain Plasticity
• Initial Boundary Value Problem
• Uncoupled ALE Formulation
• Numerical simulations

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1. Basics of Continuum Mechanics
is a material body
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The motion of a material body
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1. Basics of Continuum Mechanics
Picking out two neighboring configurations, we
get
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1. Basics of Continuum Mechanics
Reference configuration M is fixed and
independent of any placement of the material body
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Displacement Field
Physical displacement in Reference configuration
Material displacement
Spatial displacement
Displacement of mesh point m
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Velocity Fields
Using chain rule
Mesh velocity
Convective velocity
Material velocity
Spatial velocity
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Acceleration Fields
In reference configuration
Material Acceleration
Spatial Acceleration
Mesh Acceleration
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General material time derivative
The physical acceleration can also be written as
For any function f(x), the material time
derivative can be written as
Local accel.
convective accel.
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Deformation gradient
Similarly, we can define
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Concept of Stress
Cut out a part of the body and denote the
remaining part as and in the material and
the spatial configuration. is
the Cauchy traction vector.
is the First piola-Kirchhoff (or nominal)
traction vector

• Eulers cut principle

Cauchy stress tensor
First Piola Kirchhoff stress tensor
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Concept of Stress
Second Piola Kirchhoff stress tensor
Kirchhoff stress tensor
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2. Constitutive Model Finite Strain Plasticity
Representation of the multiplicative
decomposition of the deformation gradient into
its plastic and elastic contribution
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Finite Strain Plasticity in Tensor Notation
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Numerical Implementation

• Integration of the Flow Rules----Implicit Scheme

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The Elastic Predictor
The elastic predictor leads to the elastic trial
state
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The Elastic Predictor
Total Lagrange Formulation
Updated Lagrange Formulation
The Kuhn-Tucker complementary conditions
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The Elastic Predictor

• Elastic step

If lt0, then the trial elastic state with
satisfies the Kuhn-Tucker conditions.
Then the trial elastic step is the solution at
time

• Plastic step

When gt0, it means that is non
admissible and cannot be the solution at time
. This should be corrected by plastic corrector
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The Plastic Corrector
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Spectral Decomposition
And define principal elastic logarithmic
stretches by
Substitute these formulas into previous
equations, we get
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Finite strain plasticity in spectral
Decomposition
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3. Initial Boundary Value Problem (IBVP)
Problem Definition in the referential
configuration
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Strong Form of the IBVP
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Weak Form of the IBVP
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Finite Element Discretization
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Finite Element Discretization

• Spatial velocity

Convective velocity
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Finite Element Discretization

• Approximation for the internal variables

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Finite Element Discretization
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4. Uncoupled ALE Formulation
----The Operator Split

• Linear advection equation

Split
Eulerian Equation
Lagrangian Equation
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The Operator Split

• By using Taylor-series expansion for variable
  , we get

• Lagrangian phase

Lagrangian Equation

• Eulerian phase

Eulerian Equation
Where,
and
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The Operator Split

• The features of Smoothing Phase

• Boundary nodes are required to remain on the
  boundary, since this is the main advantage of the
  ALE formulation compared to the Eulerian one.
  This can be obtained by allowing only a
  tangential movement to the boundary of these
  nodes
• Mesh distortion is controlled by moving inner
  nodes in an appropriate way

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Application of Operator Split Technique to the
ALE Formulation
Not the same point any more
Keep the physical motion constant
Material Body and mesh move together
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Application of Operator Split Technique to the
ALE Formulation
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Application of Operator Split Technique to the
ALE Formulation
Illustration of the Updated Lagrange Formulation
used in the Lagrangian Phase of the ALE
formulation
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Flowchart of the Uncoupled ALE formulation
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The Smoothing Phase

• Laplacian Based Approach

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The Smoothing Phase

• Element Area Based Approach

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Convective Velocity
after smooth
before smooth

• Mesh velocity

• Convective velocity

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Eulerian Phase----Final solution of the internal
variables

• Velocities are computed at mesh nodes
• The stress-related internal variables normally
  lie inside the elements
• This yields a discontinuous stress field, because
  we need we need compute spatial gradient of these
  internal variables

Two method to compute the derivatives of the
Lagrangian solution of theinternal variable with
respect to time

• Lax-Wendroff Scheme
• Godunov-scheme

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Lax-Wendroff Scheme
We need a smoothed field of spatial gradient of
the internal variable
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Lax-Wendroff Scheme
classical least squares projection
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Godunov Scheme

• Without computing the spatial gradient
• Directly compute the derivatives of the
  Lagrangian solution of the internal variable with
  respect to time

Define
Conservative form
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Godunov Scheme ----One-point-quadruatu
re
Assume piecewise constant field of the solution
of the internal variable after the Lagrangian
phase.
Take the weak form
Where is the internal variable component
along the side s of the element under
consideration, stands for the total number
of sides of the element and is the
volume of the element
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Godunov Scheme ----One-point-quadruatu
re
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Godunov Scheme ----One-point-quadruatu
re
where
is the flux of convective velocity
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Godunov Scheme ----multiple-point-quad
ruature
Illustration of the Godunov scheme for
Multiple-point-quadrature
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5. Numerical Simulations
----Impact of a Circular Bar
Smoothing strategy
Equidistantly distributed in radial direction
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Impact of a Circular Bar
Non-uniform distribution in radial direction
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Impact of a Circular Bar
Equidistantly distributed in radial direction
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Impact of a Circular Bar

• Theres not much difference between the
  Lagrangian and the ALE formulation
• The time step in ALE formulation is four times
  larger than one in Lagrangian formulation, which
  means ALE formulation can be much faster than the
  Lagrangian formulation although in every time
  step ALE need extra smooth phase and Eulerian
  phase after the Lagrangian phase

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Necking of a Circular Bar
For ALE and Lagrangian computation
For the reference solution
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Smooth Strategy
Equidistantly distributed
Fixed the vertical position
Equidistantly distributed in radial direction
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Comparison of Spatial Meshes
Large deformation
Uniform distributed
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Necking of a Circular Bar
constant in horizontal direction
not constant in horizontal direction
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Conclusion

• The results from uncoupled ALE formulation are
  reliable
• The ALE formulation can be much faster than
  Lagrangian formulation because it can use larger
  time step without large element deformation

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Reference
1 Christian Linder, An Arbitrary
Lagrangian-Eulerian Finite Element Formulation
for Dynamics and Finite Strain Plasticity Models,
Master thesis, University of Stuttgart,
German,2003
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• Thank you, everyone!!!