1. Problems from Industry: Case Studies Huaxiong Huang Department of Mathematics and Statistics York University Toronto, Ontario, Canada M3J 1P3 http://www.math.yorku.ca/~hhuang Supported by: NSERC, MITACS, Firebird, BCASI

2. Outline Stress Reduction for Semiconductor Crystal Growth. –Collaborators: S. Bohun, I. Frigaard, S. Liang. Temperature Control in Hot Rolling Steel Plant. –Collaborators: J. Ockendon, Y. Tan. Optimal Consumption in Personal Finance. –Collaborators: M. Cao, M. Milevsky, J. Wei, J. Wang.

3. Stress Reduction during Crystal Growth Growth Process: Simulation:

4. Problem and Objective Problem:Objective: model and reduce thermal stress Dislocations Thermal Stress

5. Full Problem Temperature + flow equations + phase change:

6. Basic Thermal Elasticity Thermal elasticity Equilibrium equation von Mises stress Resolved stress (in the slip directions)

7. A Simplified Model for Thermal Stress Temperature Growth (of moving interface) Meniscus and corner Other boundary conditions

8. Non-dimensionalisation Temperature Boundary conditions Interface

9. Approximate Solution Asymptotic expansion Equations up-to 1 st order Lateral boundary condition Interface Top boundary

10. 0 th Order Solution Reduced to 1D! Pseudo-steady state Cylindrical crystals Conic crystals

11. 1 st Order Solution Also reduced to 1D! Cylindrical crystals Conic crystals General shape Stress is determined by the first order solution (next slide).

12. Thermal Stress Plain stress assumption Stress components von Mises stress Maximum von Mises stress

13. Size and Shape Effects

14. Shape Effect II Convex ModificationConcave Modification

15. Stress Control and Reduction Examples from the Nature [taken from Design in Nature, 1998 ]

16. Other Examples

17. Stress Control and Reduction in Crystals Previous work –Capillary control: controls crystal radius by pulling rate; –Bulk control: controls pulling rate, interface stability, temperature, thermal stress, etc. by heater power, melt flow; –Feedback control: controls radial motion stability; –Optimal control: using reduced model (Bornaide et al, 1991; Irizarry- Rivera and Seider, 1997; Metzger and Backofen, 2000; Metzger 2002); –Optimal control: using full numerical simulation (Gunzburg et al, 2002; Muller, 2002, etc.) ; –All assume cylindrical shape (reasonable for silicon); no shape optimization was attempted. Our approach –Optimal control: using semi-analytical solution (Huang and Liang, 2005); –Both shape and thermal flux are used as control functions.

18. Stress Reduction by Thermal Flux Control Problem setup Alternative (optimal control) formulation Constraint

19. Method of Lagrange Multiplier Modified objective functional Euler-Lagrange equations

20. Stress Reduction by Shape Control Optimal control setup Euler-Lagrange equations

21. Results I: Conic Crystals Three Flux VariationsStress at Final Length History of Max Stress

22. Results II: Linear Thermal Flux Crystal Shape Max Stress Growth Angle

23. Results III: Optimal Thermal Flux Crystal Shape Max Stress Growth Angle

24. Parametric Studies: Effect of Penalty Parameters Crystal ShapeMax Stress Growth Angle

25. Conclusion and Future Work Stress can be reduced significantly by control thermal flux or crystal shape or both; Efficient solution procedure for optimal control is developed using asymptotic solution; Sensitivity and parametric study show that the solution is robust; Improvements can be made by – incorporating the effect of melt flow (numerical simulation is currently under way); –incorporating effect of gas flow (fluent simulation shows temporary effect may be important); –Incorporating anisotropic effect (nearly done).

26. Temperature Control in Hot-Rolling Mills Cooling by laminar flow Q1: Bao Steel’s rule of thumb Q2: Is full numerical solution necessary for the control problem?

27. Model Temperature equation and boundary conditions

28. Non-dimensionalization Scaling Equations and BCs Simplified equation

29. Discussion Exact solution Leading order approximation Temperature via optimal control

30. Optimal Consumption with Restricted Assets Examples of illiquid assets: –Lockup restrictions imposed as part of IPOs; –Selling restrictions as part of stock or stock-option compensation packages for executives and other employees; –SEC Rule 144. Reasons for selling restriction: –Retaining key employees; –Encouraging long term performance. Financial implications for holding restricted stocks: –Cost of restricted stocks can be high (30-80%) [KLL, 2003]; Purpose of present study: –Generalizing KLL (2003) to the stock-option case.; –Validate (or invalidate) current practice of favoring stocks.

31. Model Continuous-time optimal consumption model due to Merton (1969, 1971): –Stochastic processes for market and stock –Maximize expected utility

32. Model (cont.) –Dynamics of the option –Dynamics of the total wealth –Proportions of wealth

33. Hamilton-Jacobi-Bellman Equation A 2 nd order, 3D, highly nonlinear PDE.

34. Solution of HJB First order conditions HJB Terminal condition (zero bequest) Two-period Approach

35. Post-Vesting (Merton) Similarity solution Key features of the Merton solution –Holing on market only; –Constant portfolio distribution; –Proportional consumption rate (w.r..t. total wealth).

36. Vesting Period (stock only) Incomplete similarity reduction Simplified HJB (1D) Numerical issues –Explicit or implicit? –Boundary conditions; loss of positivity, etc.

37. Vesting Period (stock-option) Incomplete similarity reduction Reduced HJB (2D) Numerical method: ADI.

38. Results: value function

39. Results: optimal weight and consumption

40. Option or stock?