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A parameter optimization heuristic for a temperature estimation model

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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.

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Abbreviations

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;

h k i :

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

S i :

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;

T A :

The ambient environmental air temperature;

w k i :

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;

R ij :

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

r ij :

The inverse of the resistance parameter R ij ;

T k i :

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

z :

The current objective function value

A k :

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

α :

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

F k :

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;

Δ k i :

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

δ ij or δ :

A perturbation of a variable r ij ;

e j i :

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

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, The University of Florida, 303 Weil Hall, Gainesville, FL, 32611, USA

    J. Cole Smith

  2. Department of Systems Engineering, United States Military Academy, West Point, NY, 10996, USA

    Dale L. Henderson

  3. Department of Mechanical Engineering, Villanova University, Villanova, PA, 19085, USA

    Alfonso Ortega

  4. ASE Technologies, Phoenix, AZ, 85021, USA

    Jason DeVoe

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  1. J. Cole Smith
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  2. Dale L. Henderson
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  3. Alfonso Ortega
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  4. Jason DeVoe
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Correspondence to J. Cole Smith.

Additional information

The authors are indebted to four anonymous referees for their help in improving the contribution and presentation of this paper.

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Smith, J.C., Henderson, D.L., Ortega, A. et al. A parameter optimization heuristic for a temperature estimation model. Optim Eng 10, 19–42 (2009). https://doi.org/10.1007/s11081-008-9039-1

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