Void Growth and Coalescence by vacancy diffusion using level sets

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Void Growth and Coalescence by vacancy diffusion
usinglevel sets

• Kinjal Dhruva, A.M. Cuitiño
• Michael Ortiz and Marisol Koslowski
• Updated Presentation
• July 24th 2004

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Void growth coalescence Level set

• Void-size distribution coarsens as a result of
  differential void growth, coalescence
• Outlook
• Level set calculations of void coarsening in
  plastic single-crystal matrix
• Fitting of Becker-Doring coeffients as a function
  of pressure, temperature and rate of deformation
• EoS ? macroscopic model ? VTF

Level set calculations of void coalescence
(Cuitiño 02)
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Level Set Representation(Overview)
This method represents an interface using a
level set function f(X,t) defined at every
grid point having the following
properties f(X,t)lt0 Inside the
interface f(X,t)gt0 Outside the interface
f(X,t)0 Interface
Subsequent interface motion under the velocity
field v is governed by the equation
(Sethian, Aslam)
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Advantages of Level Set Representation

• Grid for interface tracking is same as the
  computational grid.
• Uses standard Hamilton-Jacobi solvers for front
  propagation.
• Interface motion under a velocity field can be
  formulated in a straightforward way.
• Interfaces can develop sharp corners,break apart,
  
• and merge together.

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Interface velocity dependence

• Velocity Field for subsequent motion depends on
• any of the following factors
• Position
• Time
• Geometry of interface(e.g. its normal or its mean
  curvature)
• External Physics

In our case for void growth
Velocity Field
fvacancies (Velocity component due to vacancy
diffusion)
fdislocations (Velocity component due to
dislocation motion)
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Void velocity for growth coalescence
fdislocations
fvacancies

• Vacancy Diffusion
• Surface Bulk diffusion
• Diffusivity of material
• Curvature and gradients

• Phase Field Method
• Phase field
• Dislocation flow
• Dislocation densities

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Vacancy Diffusion
Diffusion Based Motion

• Bulk Diffusion
• -Diffusivity
• -Curvature gradient
• Surface Diffusion
• -Diffusivity
• -Curvature

Solve the diffusion problem Cinterfacef(k)
Flux (dC/dn)
fvacancies
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Dislocation Dynamics
Phase Field Method -Dislocation Densities
-Dislocation Geometries -Loop Nucleation
Dislocation loop
Solve the Mechanics Problem se sd ? s s(e)

Elasticity(No dislocation) ( se )
Phase Field ( sd )
Update Dislocation field
Dislocation Motion?Flux
fd
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Superposition of two components
Linear Elasticity allows us to superpose separate
components from diffusion equations and
dislocation dynamics


Diffusion based motion (fvacancies )
Dislocation based motion (fd )
Resultant velocity component (f)
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Methodology (Diffusion Problem)
Level set for initial void interface
Calculate Curvature(k)
B.C. ? Cinterfacef(k)

• Ghost Fluid Method
• Immersed Interface Method

Solution of interface problem
Flux b(dC/dn)

• Image Analysis of void
• Area
• Effective Radius
• Growth Rate

Flux?Velocity Field
Update level set with velocity field to locate
new interface position
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Different BCs applied on solution domain
Neumann BC
Periodic BC
Dirichlet BC
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Implicit Periodic BC
Implicit Boundary Conditions ensure that the
solution domain can be used as a Representative
Cell
Representative cell
Representative cell in a continuous domain
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Time Dependent Parabolic Equation
(For a given void location)
(Click to see movies)
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Results A
Flux dependent velocity growth
Free Growth
Voids2
Voids2
Voids3
Voids3
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Results-B
3 VOID GROWTH
Concentration Plot
Vector Plot
Contour Plot
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Results C
Vector Plots of Concentration
Close to interface
Overview
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Current Status

• Level set representation of complicated geometry

• Applied Ghost Fluid Method (Fedkiw) based
  algorithm for interface problem.

• Analyzed the growing void geometry using Image
  Analysis capabilities of Matlab

• Solved curvature and flux dependent void growth
  problem in 2-D 3-D

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Future Outlook

• Implement GFM based code on VTF for 3-D case
• Develop an alternate algorithm based on Immersed
  Interface method(Z Li) for inclusions problem.
• Solve the elasticity equation for 2-D and 3-D
  case
• Superimpose diffusion problem with elasticity
  equation.