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Optimization and Engineering

, Volume 10, Issue 1, pp 19–42 | Cite as

A parameter optimization heuristic for a temperature estimation model

  • J. Cole Smith
  • Dale L. Henderson
  • Alfonso Ortega
  • Jason DeVoe
Article

Abstract

We present a heuristic technique for solving a parameter estimation problem that arises in modeling the thermal behavior of electronic chip packages. Compact Thermal Models (CTMs) are network models of steady state thermal behavior, which show promise in augmenting the use of more detailed and computationally expensive models. The CTM parameter optimization problem that we examine is a nonconvex optimization problem in which we seek a set of CTM parameters that best predicts, under general conditions, the thermal response of a particular chip package geometry that has been tested under a small number of conditions. We begin by developing a nonlinear programming formulation for this parameter optimization problem, and then develop an algorithm that uses special characteristics of the optimization problem to quickly generate heuristic solutions. Our algorithm descends along a series of solutions to one-dimensional nonconvex optimization problems, obtaining a locally optimal set of model parameters at modest computational cost. Finally, we provide some experimental results and recommendations for extending this research.

Keywords

Heuristic Nonconvex optimization Parameter estimation Compact thermal models 

Nomenclature

n

The number of CTM boundary nodes, not including the junction node;

m

The number boundary condition observations under which the electronic chip package is evaluated;

q

The input power parameter at the junction node under all observations;

hik

The heat transfer coefficient at boundary node i=1,…,n for observation k=1,…,m;

Si

The surface area represented by boundary node i=1,…,n;

\(\hat{T}_{i}^{k}\)

The reference temperature for node i=0,…,n and observation k=1,…,m;

TA

The ambient environmental air temperature;

wik

The relative importance weight of correctly estimating the temperature of node i=0,…,n under observation k=1,…,m;

h(δ)

The objective function of the CTM model with respect to the perturbation δ of a single thermal resistance value;

g(δ)

A single term of the objective function of the CTM model with respect to the perturbation δ of a single thermal resistance value;

\(\mathcal{V}\)

A parameter used in the weighting scheme for junction nodes;

\(\mathcal{W}\)

A parameter used in the weighting scheme for boundary nodes;

Rij

The resistance parameter value for link (i,j), for 0≤i<jn;

rij

The inverse of the resistance parameter R ij ;

Tik

The predicted temperature for node i=0,…,n under observation k=1,…,m;

z

The current objective function value

Ak

A square symmetric matrix composed of given values for r ij , h i k , and S i ;

α

A variable used in the heuristic to substitute for the difference in certain terms of A k −1 ;

Fk

A nonsingular transformation matrix of the same dimensions as A k used in the heuristic;

Δ

A perturbation matrix of the same dimensions as A;

Δik

A variable used to translate T i k into a positive range, T i k =T i k i k ;

δij or δ

A perturbation of a variable r ij ;

eij

A variable used in the heuristic representing the current error between T i k and \(\hat{T}_{i}^{k}\)

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • J. Cole Smith
    • 1
  • Dale L. Henderson
    • 2
  • Alfonso Ortega
    • 3
  • Jason DeVoe
    • 4
  1. 1.Department of Industrial and Systems EngineeringThe University of FloridaGainesvilleUSA
  2. 2.Department of Systems EngineeringUnited States Military AcademyWest PointUSA
  3. 3.Department of Mechanical EngineeringVillanova UniversityVillanovaUSA
  4. 4.ASE TechnologiesPhoenixUSA

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