materials process design and control laboratory the stefan problem: a stochastic analysis using the...
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
THE STEFAN PROBLEM: A STOCHASTIC ANALYSIS USING THE EXTENDED FINITE ELEMENT METHOD
Baskar Ganapathysubramanian, Nicholas ZabarasMaterials Process Design and Control Laboratory
Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall
Cornell University Ithaca, NY 14853-3801
Email: [email protected]: http://mpdc.mae.cornell.edu/
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
FUNDING SOURCES:
Air Force Research Laboratory
Air Force Office of Scientific Research
National Science Foundation (NSF)
ALCOA
Army Research Office
COMPUTING SUPPORT:
Cornell Theory Center (CTC)
ACKNOWLEDGEMENTSACKNOWLEDGEMENTS
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
1. Motivation
2. Stochastic Preliminaries
3. Smolyak theorem
4. Stochastic Stefan problem
5. Solution methodology and Implementation issues
6. Results
7. Conclusions
OUTLINE
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
All physical systems have an inherent associated randomness
SOURCES OF UNCERTAINTIES
•Multiscale material information – inherently statistical in nature.
•Uncertainties in process conditions
•Input data
•Model formulation – approximations, assumptions.
Why uncertainty modeling ?
Assess product and process reliability.
Estimate confidence level in model predictions.
Identify relative sources of randomness.
Provide robust design solutions.
Engineering component
Heterogeneous random
Microstructural features
MOTIVATION
Process
Control?
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MOTIVATION
Interested in control.
Non-linear process
How do small variations in the conditions affect evolution
Boundary conditions
Initial conditions
Material Properties
Interfacial kinetics
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S
Sample space of elementary events
Real line
Random
variable
MAP
Collection of all possible outcomes
Each outcome is mapped to a
corresponding real value
Interpreting random variables as functions
A general stochastic process is a random field with variations along space and time – A
function with domain (Ω, Τ, S)
REPRESENTING RANDOMNESS:1
1. Interpreting random variables
2. Distribution of the random variable
Ex. Inlet velocity, Inlet temperature
1 0.1o
3. Correlated data
Ex. Presence of impurities, porosity
Usually represented with a correlation function
We specifically concentrate on this.
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REPRESENTING RANDOMNESS:2
1. Representation of random process
- Karhunen-Loeve, Polynomial Chaos expansions
2. Infinite dimensions to finite dimensions
- depends on the covarience
Karhunen-Loèvè expansion
Based on the spectral decomposition of the covariance kernel of the stochastic process
Random process Mean
Set of random variables to
be found
Eigenpairs of covariance
kernel
• Need to know covariance
• Converges uniformly to any second order process
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Index
Eig
enva
lue
5 10 15 200
5
10
15
Set the number of stochastic dimensions, N
Dependence of variables
Pose the (N+d) dimensional problem
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SOLUTION TECHNIQUES FOR STOCHASTIC PDE’s
Monte Carlo:
- Sample stochastic space
- Easy to implement
- Embarrassingly parallel
- Large number of realizations necessary for convergence
- Impractical as number of dimensions increases
Spectral Stochastic Method:
- Dependant variables projected onto a stochastic space spanned by a
set of complete orthogonal polynomials
- Use the Galerkin projection
- Good convergence
- But coupled set of equations
- Substantial changes to deterministic code
Curse of Dimensionality
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COLLOCATION STRATEGIES
Decoupled system
Convergence proofs
For larger stochastic dimensions: Need to combine the decoupled nature of Monte Carlo with the fast convergence of the spectral stochastic methods.
- Use sampling- Construct interpolating functions
Collocation
How is this different from MC?
Use Galerkin projection
Given a set of points
A smooth function
Find the interpolating function
All variables can be represented in terms of the Lagrange polynomials and values at the points
Optimal choice of points
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SMOLYAK ALGORITHM
Extensively used in statistical mechanics
Provides a way to construct interpolation functions based on minimal number of points
Univariate interpolations to multivariate interpolations
( ) ( )i
i i
i i
xx X
U f a f x
1
0 11
, 1,
0, ,
( ) ( ) ( )( )d
i i id
iiq d q d
i q
U U U i i i
A f A f f
Uni-variate interpolation
Multi-variate interpolation
Smolyak interpolation
ORDER CC FE
3 1581 1000
4 8801 10000
5 41625 100000
D = 10
Some degradation in accuracy
Maximal reduction when the function is assumed to be smooth
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Temperature
i is the thermal diffusivity
( , ) c vt v Interface Kinetics
Growth rate: Stefan condition
1 s ls l
H
v k kL n n
SOLUTION PROCEDURE
Set Stochastic dimensions
Choose collocation points
Perform deterministic simulation at each stochastic collocation point
Use the sparse grid interpolation functions to compute moments and other statistics
Boundary conditions
Initial conditions
Material Properties
Interfacial kinetics
2( , , )( ) ( , , )i
x t ww x t w
t
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EXTENDED FINITE ELEMENT METHOD (X-FEM) APPROXIMATION
SOLUTION METHODOLOGY: TEMPERATURE
The Standard FE Approximation The X-FEM Approximation
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SOLUTION METHODOLOGY : TEMPERATURE
LEVEL SET FORMULATION
Interface tracked explicitly using level sets
Enrichment function also defined using nodal values of the level sets
Level set evolution: calculation of the extension velocity
Equation moves with the correct velocity V at the interface.
Ensure that the level set satisfies the signed distance property.
Reinitialize the level set; Fast marching.
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IMPLEMENTATION ISSUES
The level set evolution is solved using the Galerkin-Least square finite element method
The signed distance property is maintained through a choice of two techniques
- Fast marching technique
- Solving a pseudo-transient problem to steady state
Calculate the front velocity only at the zero level set.
Need to extend it into the computational domain
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IMPLEMENTATION ISSUES
Semi-discrete form of the temperature evolution derived from the weak form
Geometry of the interface is independent of the finite element mesh:
- necessary to modify the quadrature routines for the volume integrals
- elements intersected by the interface, this quadrature may not be accurate enough to capture the discontinuities and the change in material properties across the interface
- In n dimensions (n = 2; 3), divide the element that is cut by the interface into rn smaller quadrilaterals
- r = 10 in 2D , r = 6 in 3D
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IMPLEMENTATION ISSUES
Enforcing the temperature constraint at the interface
- Temperature is linearly distributed along any interface segment.
- Constraint enforced at the points where the interface intersects the element boundaries
Determining the points of intersection:
- In two dimensions, the interface intersects a quadrilateral grid at two and only two points
- the two-dimensional subdomain of intersection of a cubic element with the interface could have 3, 4 or 5 points of intersection with the element edges
- This calculation is implemented by looping over pairs of nodes and comparing the nodal level set values for a change in sign.
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IMPLEMENTATION ISSUES
Temperature evolution complete.
Evaluating the propagating velocity of the interface
Requires estimation of the heat flux jump across the interface
xd
Consider a point xd on the interface:
Find temperature at the two new points xs and xl. Finding points xs and xl is non-trivial.
Search through points.
Neighbor list
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IMPLEMENTATION ISSUESImprove computational efficiency:
- Reduce function calls in integrands
Utilize fact that computational grid is uniform grid
Precompute shape functions.
Parallelize solver:
- PETSC library, the matrix system can be easily parallelized. Parallelized KSPGMRES solver is used for solving the assembled linear systems.
Domain decomposition:
- Decompose the computational domain to reduce data storage and communication overheads.
Preconditioners
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NUMERICAL EXAMPLES
The growth of a circular disc, with a four-fold growth axis of symmetry is simulated for comparison with the prediction of solvability theory
Solvability theory admits a family of discrete solutions with one stable solution. This unique solution is also characterized by a unique tip shape and tip velocity. T
The growth of a circular disc,
four-fold growth axis of symmetry
Grid considered 800 x 800 quadrilateral.
The computational domain is a square region of side length 1200.
time step t = 50.
The undercooling is -0.55
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NUMERICAL EXAMPLE:1
Uncertainty in the boundary conditions.
Correlated noise in the boundary temperature.
Variation of 10%
Number of stochastic dimensions is 8
Number of collocation points is 3937
12 nodes in the cornell theory centre
The boundary conditions take a finite amount of time to influence the growth
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NUMERICAL EXAMPLE:1
Mean Temperature Mean shape
Deviation in Temperature Deviation in tip velocity
The effect of the boundary is not felt in the initial growth period.
The deviation of the temperature and velocity are negligible.
The formation of secondary dendrites proceeds
Notice the spots of high deviation along the arms of the crystal
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NUMERICAL EXAMPLE:2
Uncertainty in the initial conditions.
Assume radial correlation
Variation of 10%
This leads to changing undercooling as the solidification proceeds
Can expect richer structures
Number of stochastic dimensions is 8
Number of collocation points is 801
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NUMERICAL EXAMPLE:2
The deviation of the temperature is significant as soon as the solidification starts.
The mean structure has a set of nascent secondary dendrites
The variation in the tip velocity shows a cloud of possible dendrites growing
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NUMERICAL EXAMPLE:3
Uncertainty in the thermal property
Due to the presence of impurities
Can be a control mechanism
Variation of 10%
Number of stochastic dimensions is 8
Variation in the y direction
Can be due to flow
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NUMERICAL EXAMPLE:3
Can see clearly defined modes of growth in the standard deviation of the tip velocity.
Suggests multiple mode shifting takes place
Variation’s cause changes in tip velocity which changes the undercooling. This is seen in the steadily increasing deviation in front of the growing tip in the y direction.
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CONCLUSIONS/FUTURE WORK
N. Zabaras, B. Ganapathysubramanian and L. Tan, "Modeling dendritic solidification with melt convection using the extended finite element method (XFEM) and level set methods", Journal of Computational Physics, in press
B. Ganapathysubramanian and N. Zabaras, "Stochastic collocation methods for modeling thermal convection", Journal of Computational Physics, in preparation.
Changes in the initial condition cause maximal deviation, followed by changes in the thermal conditions. Perturbations to the boundary conditions take longer to affect growth.
Non-intrusive extension of the eXtended Finite Element method to solve stochastic stefan problems
Applied to effect of perturbation in boundary, initial and material properties
Computed ‘clouds’ of possible dendritic shapes due to these uncertainties.
FUTURE SCOPE
Provide bounds for different perturbations
Is it possible to control the structure using thermal and flow fields?
Couple with other scales