Xianwu Ling

1

A method for analyzing the stability of
non-iterative inverse heat conduction algorithms

• Xianwu Ling
• Russell Keanini
• Harish Cherukuri
• Department of Mechanical Engineering
• University of North Carolina at Charlotte
• Presented at the 2003 IPES Symposium

2
Acknowledgements

• NSF
• Alcoa Technical Center

3
Outline

• Objective
• Literature Review
• Inverse Problem Statement
• Direct Problem
• Inverse Algorithm Sequential Function
  Specification Method
• Derivation of Error Propagation Equation
• Stability Criterion Defined
• Application to 1-D Problem
• Summary and Conclusions

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Objectives

• Formulate a general, non-empirical approach for
  assessing the stability of Becks sequential
  function specification method

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Literature Review

• Maciag and Al-Khatih (2000). Int. J. Num. Meths.
  Heat Fluid Flow. Used integral (Greens
  function) solution and backward time differencing
  to obtain

Convergence, as determined by spectral radius of
B, determines stability.
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Literature Review, contd

• Liu (1996). J. Comp. Phys. Used Duhamels
  integral to obtain

where d is a response function that depends on
the measured data and where the set of
coefficients X are used to determine stability
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Overview of IHCP

• Inverse Heat Conduction Problem (IHCP)

Known Initial conditions
?1
Boundary conditions
Known temperatures at
?
? 1
Known temperature measurements
?2
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Inverse Problem Statement
? 1
?
?2
q
Known initial temperatures
Known temperatures on
? 1
Interior temperature measurements
I the total number of measurement sites (I6).
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Direct Heat Conduction Problem

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Inverse Algorithm

• Introduction to computational time steps
• Experimental time step, computational time
  step, and future time

,
R the number of future temperatures used.
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Inverse Algorithm

• Objective function

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Inverse Algorithm

• Minimization of

with respect to leads to
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Inverse Algorithm

• Introduction to function specification method
• Idea Assume a function form of the
  unknown, and convert IHCP into a problem in which
  the parameters for the function are solved for.

Piecewise constant function (1) q n1 are
solved for step by step (2) For each step
from n to n1, an unknown constant is assumed for
each future temperature time the final resultant
heat flux for the step is the average of the
unknown constants in the strict least squares
error sense.
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Inverse Algorithm

• A key observation

is a linear function of nodal heat fluxes at
? 2
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Inverse Algorithm

• Solve for computed temperatures at the
  measurement sites

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Inverse Algorithm

• Sensitivity coefficient matrix

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Inverse Algorithm

• Matrix normal equation

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Inverse Algorithm

• Inverse algorithm procedures

(1) Given the temperatures at n, and the
measured temperatures at some interior locations
at some future times, the heat fluxes from
n to n1 can be solved using the matrix normal
equation (together with the sensitivity
coefficient matrix equation)
(2) Given the heat fluxes from n to n1, the
temperature at the end of n1 can be updated
using
(3) Go to the next time step
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Numerical Tests
1. Step change in heat flux A
flat plate subjected to a constant heat flux qc
at x0 and insulated at xL.
Fictitious measurement site
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Numerical Tests

• Results

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Numerical Tests
2. Triangular heat flux

A flat plate subjected to a triangular heat flux
at x0 and insulated at xL. Noise input
temperatures data are simulated by (1) decimal
truncating, (2) adding a random error component
generated using a Gaussian probability
distribution .
Fictitious measurement site
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Numerical Tests

• Results

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Application to quenching

• Drayton Quenchalyer, Inconel 600 probe,
  Quenchant oil.
• Sampling Freq 8 Hz, Duration 60 S

Typical temperature history at the center of the
probe
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Application to quenching

• Burggrafs analytical solution

• Results

1. Excellent agreement 2. Influence of
small oscillations 3. Temperature comparison
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Error Propagation Equation
Global standard form equation

yields computed temperatures at measurement sites

where
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Error Propagation Equation
Matrix normal equation


and global force vector
then yield
where
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Error Propagation Equation
Substitution of
into standard form eqn. then gives

where

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Error Propagation Equation
Letting
be the computed global temperature
the measured temperature vector, the error
and
propagation equation is finally obtained

where
In linear problems
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Solution Stability to An Input Error

• One-dimensional axisymmetric problem

u Model
u Governing temperature equation (with no future
temperature regularization)
where
,
and
,
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Solution Stability to An Input Error
u Frobenius norm analysis
1. Assumption
and
2. Equations of temperature error propagations
,
3. Temperature error propagation factors
,
4. Convergence criterion
5. Frobenius norm
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Solution Stability to An Input Error
6. Results and discussions
1) Effect of measurement location and
computational time step
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Solution Stability to An Input Error
2) Effect of number of elements
a) Increasing the number of elements increases
the error suppression rates b) A choice of
20 element would be proper for the problem
under study, as observed as J. Beck.
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Questions