materials process design and control laboratory the stefan problem: a stochastic analysis using the...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

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/




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




1. Motivation

2. Stochastic Preliminaries

3. Smolyak theorem

4. Stochastic Stefan problem

5. Solution methodology and Implementation issues

6. Results

7. Conclusions

OUTLINE




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?




MOTIVATION

Interested in control.

Non-linear process

How do small variations in the conditions affect evolution

Boundary conditions

Initial conditions

Material Properties

Interfacial kinetics




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.




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




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




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




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




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




EXTENDED FINITE ELEMENT METHOD (X-FEM) APPROXIMATION

SOLUTION METHODOLOGY: TEMPERATURE

The Standard FE Approximation The X-FEM Approximation




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.




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





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





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.





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




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




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




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




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




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 collocation points is 801




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




NUMERICAL EXAMPLE:3

Uncertainty in the thermal property

Due to the presence of impurities

Can be a control mechanism

Variation of 10%


Variation in the y direction

Can be due to flow




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.




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

materials process design and control laboratory the stefan problem: a stochastic analysis using the...

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