problems from industry: case studies huaxiong huang department of mathematics and statistics york...
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Problems from Industry: Case Studies
Huaxiong HuangDepartment of Mathematics and Statistics
York UniversityToronto, Ontario, Canada M3J 1P3http://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
Dislocations
Thermal Stress
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; 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.
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?