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
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.
Stress Reduction during Crystal Growth
• Growth Process:
•
Simulation:
Problem and Objective
• Problem:
• Objective: model and reduce
thermal stress
Thermal Stress
Dislocations
Full Problem
• Temperature + flow equations + phase change:
Basic Thermal Elasticity
• Thermal elasticity
• Equilibrium equation
• von Mises stress
• Resolved stress (in the slip directions)
A Simplified Model for Thermal Stress
• Temperature
• Growth (of moving interface)
• Meniscus and corner
• Other boundary conditions
Non-dimensionalisation
• Temperature
• Boundary conditions
• Interface
Approximate Solution
• Asymptotic expansion
• Equations up-to 1st order
• Lateral boundary condition
• Interface
• Top boundary
0th Order Solution
• Reduced to 1D!
• Pseudo-steady state
• Cylindrical crystals
• Conic crystals
1st Order Solution
• Also reduced to 1D!
• Cylindrical crystals
• Conic crystals
• General shape
• Stress is determined by the first order solution (next slide).
Thermal Stress
• Plain stress assumption
• Stress components
• von Mises stress
• Maximum von Mises stress
Size and Shape Effects
Shape Effect II
Convex Modification
Concave Modification
Stress Control and Reduction
• Examples from the Nature [taken from Design in Nature, 1998 ]
Other Examples
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; IrizarryRivera 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.
Stress Reduction by Thermal Flux Control
• Problem setup
• Alternative (optimal control) formulation
• Constraint
Method of Lagrange Multiplier
• Modified objective functional
• Euler-Lagrange equations
Stress Reduction by Shape Control
• Optimal control setup
• Euler-Lagrange equations
Results I: Conic Crystals
Three Flux Variations
Stress at Final Length
History of Max Stress
Results II: Linear Thermal Flux
Crystal Shape
Max Stress
Growth Angle
Results III: Optimal Thermal Flux
Crystal Shape
Max Stress
Growth Angle
Parametric Studies: Effect of Penalty Parameters
Crystal Shape
Max Stress
Growth Angle
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).
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?
Model
• Temperature equation and boundary conditions
Non-dimensionalization
• Scaling
• Equations and BCs
• Simplified equation
Discussion
• Exact solution
• Leading order approximation
• Temperature via optimal control
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.
Model
• Continuous-time optimal consumption model due to
Merton (1969, 1971):
– Stochastic processes for market and stock
– Maximize expected utility
Model (cont.)
– Dynamics of the option
– Dynamics of the total wealth
– Proportions of wealth
Hamilton-Jacobi-Bellman Equation
• A 2nd order, 3D, highly nonlinear PDE.
Solution of HJB
• First order conditions
• HJB
• Terminal condition (zero bequest)
• Two-period Approach
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).
Vesting Period (stock only)
• Incomplete similarity reduction
• Simplified HJB (1D)
• Numerical issues
– Explicit or implicit?
– Boundary conditions; loss of positivity, etc.
Vesting Period (stock-option)
• Incomplete similarity reduction
• Reduced HJB (2D)
• Numerical method: ADI.
Results: value function
Results: optimal weight and consumption
Option or stock?