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?