Problems from Industry: Case Studies

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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
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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.

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Stress Reduction during Crystal Growth

• Growth Process

• Simulation

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Problem and Objective

• Problem

• Objective model and reduce thermal stress

Thermal Stress
Dislocations
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Full Problem

• Temperature flow equations phase change

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Basic Thermal Elasticity

• Thermal elasticity
• Equilibrium equation
• von Mises stress
• Resolved stress (in the slip directions)

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A Simplified Model for Thermal Stress

• Temperature
• Growth (of moving interface)
• Meniscus and corner
• Other boundary conditions

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Non-dimensionalisation

• Temperature
• Boundary conditions
• Interface

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Approximate Solution

• Asymptotic expansion
• Equations up-to 1st order
• Lateral boundary condition
• Interface
• Top boundary

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0th Order Solution

• Reduced to 1D!
• Pseudo-steady state
• Cylindrical crystals
• Conic crystals

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1st Order Solution

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

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Thermal Stress

• Plain stress assumption
• Stress components
• von Mises stress
• Maximum von Mises stress

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Size and Shape Effects
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Shape Effect II
Convex Modification
Concave Modification
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Stress Control and Reduction

• Examples from the Nature taken from Design in
  Nature, 1998

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Other Examples
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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.

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Stress Reduction by Thermal Flux Control

• Problem setup
• Alternative (optimal control) formulation
• Constraint

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Method of Lagrange Multiplier

• Modified objective functional
• Euler-Lagrange equations

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Stress Reduction by Shape Control

• Optimal control setup
• Euler-Lagrange equations

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Results I Conic Crystals
History of Max Stress
Three Flux Variations
Stress at Final Length
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Results II Linear Thermal Flux
Max Stress
Growth Angle
Crystal Shape
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Results III Optimal Thermal Flux
Growth Angle
Max Stress
Crystal Shape
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Parametric Studies Effect of Penalty Parameters
Growth Angle
Crystal Shape
Max Stress
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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).

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Temperature Control in Hot-Rolling Mills

• Cooling by laminar flow
• Q1 Bao Steels rule of thumb
• Q2 Is full numerical solution necessary for the
  control problem?

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Model

• Temperature equation and boundary conditions

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Non-dimensionalization

• Scaling
• Equations and BCs
• Simplified equation

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Discussion

• Exact solution
• Leading order approximation
• Temperature via optimal control

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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.

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Model

• Continuous-time optimal consumption model due to
  Merton (1969, 1971)
• Stochastic processes for market and stock
• Maximize expected utility

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Model (cont.)

• Dynamics of the option
• Dynamics of the total wealth
• Proportions of wealth

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Hamilton-Jacobi-Bellman Equation

• A 2nd order, 3D, highly nonlinear PDE.

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Solution of HJB

• First order conditions
• HJB
• Terminal condition (zero bequest)
• Two-period Approach

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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).

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Vesting Period (stock only)

• Incomplete similarity reduction
• Simplified HJB (1D)
• Numerical issues
• Explicit or implicit?
• Boundary conditions loss of positivity, etc.

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Vesting Period (stock-option)

• Incomplete similarity reduction
• Reduced HJB (2D)
• Numerical method ADI.

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Results value function
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Results optimal weight and consumption
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Option or stock?