1. Materials Process Design and Control Laboratory THE STEFAN PROBLEM: A STOCHASTIC ANALYSIS USING THE EXTENDED FINITE ELEMENT METHOD Baskar Ganapathysubramanian, Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/

2. Materials Process Design and Control Laboratory 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) ACKNOWLEDGEMENTS

3. Materials Process Design and Control Laboratory 1. Motivation 2. Stochastic Preliminaries 3. Smolyak theorem 4. Stochastic Stefan problem 5. Solution methodology and Implementation issues 6. Results 7. Conclusions OUTLINE

4. Materials Process Design and Control Laboratory 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 ProcessControl?

5. Materials Process Design and Control Laboratory MOTIVATION Interested in control. Non-linear process How do small variations in the conditions affect evolution Boundary conditions Initial conditions Material Properties Interfacial kinetics

6. Materials Process Design and Control Laboratory 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 3. Correlated data Ex. Presence of impurities, porosity Usually represented with a correlation function We specifically concentrate on this.

7. Materials Process Design and Control Laboratory 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 Set the number of stochastic dimensions, N Dependence of variables Pose the (N+d) dimensional problem

8. Materials Process Design and Control Laboratory 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

9. Materials Process Design and Control Laboratory 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

10. Materials Process Design and Control Laboratory 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 Uni-variate interpolation Multi-variate interpolation Smolyak interpolation ORDERCCFE 315811000 4880110000 541625100000 D = 10 Some degradation in accuracy Maximal reduction when the function is assumed to be smooth

11. Materials Process Design and Control Laboratory Temperature is the thermal diffusivity Interface Kinetics Growth rate: Stefan condition 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

12. Materials Process Design and Control Laboratory EXTENDED FINITE ELEMENT METHOD (X-FEM) APPROXIMATION SOLUTION METHODOLOGY: TEMPERATURE The Standard FE ApproximationThe X-FEM Approximation

13. Materials Process Design and Control Laboratory 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.

14. Materials Process Design and Control Laboratory 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

15. Materials Process Design and Control Laboratory 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 r n smaller quadrilaterals - r = 10 in 2D, r = 6 in 3D

16. Materials Process Design and Control Laboratory 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.

17. Materials Process Design and Control Laboratory IMPLEMENTATION ISSUES Temperature evolution complete. Evaluating the propagating velocity of the interface Requires estimation of the heat flux jump across the interface xdxd Consider a point x d on the interface: Find temperature at the two new points x s and x l. Finding points x s and x l is non-trivial. Search through points. Neighbor list

18. Materials Process Design and Control Laboratory IMPLEMENTATION ISSUES Improve 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

19. Materials Process Design and Control Laboratory 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

20. Materials Process Design and Control Laboratory 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

21. Materials Process Design and Control Laboratory 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

22. Materials Process Design and Control Laboratory 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

23. Materials Process Design and Control Laboratory 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

24. Materials Process Design and Control Laboratory 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

25. Materials Process Design and Control Laboratory 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.

26. Materials Process Design and Control Laboratory 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