Problems from Industry: Case Studies

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 Thermal Stress Dislocations

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 1st order • Lateral boundary condition • Interface • Top boundary

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

11. 1st 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 Modification Concave 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 History of Max Stress Three Flux Variations Stress at Final Length

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

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

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

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 2nd 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?