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Detection of flaws in a two-dimensional body through measurement of surface temperatures and use of conjugate gradient method

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Abstract

This paper aims to obtain parameters (i.e. location and dimensions) relevant to flaws in a two-dimensional body by measuring the temperature on its boundaries. In this endeavour, a steady-state heat conduction problem is formulated, and the geometry under study is subjected to a known heat load, resulting in a specific heat distribution in the body. By using a number of heat sensors, the temperature at selected points on the boundary of the body is obtained. Inverse heat conduction methods implement these temperature data, working toward estimating the flaw parameters. The objective function is optimized using conjugate gradients method, and in solving the direct problem, an FEM code is employed. To check the effectiveness of this method, sample cases with one or more circular, elliptical cavities or cracks in the body, and a case with unknown cavity shape is solved. Finally the ensuing results analyzed.

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Abbreviations

f :

Mean squared error (target function)

h :

Heat convection coefficient (Wm−2 K−1)

k :

Heat conduction coefficient (Wm−1 K−1)

l :

Length of crack (m)

M :

Number of measurements

N :

Number of unknown parameters

q′′:

Heat flux (Wm−2)

r :

Radius of flaw (m)

T ∞ :

Surrounding temperature (K)

x :

x coordinate position of flaw (m)

y :

y coordinate position of flaw (m)

\({\overline{S}}\) :

Vector of search direction

\({\bar{T}}\) :

Vector of estimated temperatures (K)

\({\bar{Y}}\) :

Vector of measured temperatures (K)

\({\overline{\overline{D}}(.)}\) :

Operator to convert vector to diagonal matrix

\({\overline{\overline{X}}}\) :

Matrix of sensitivity coefficients

θ :

Angle of elliptical flaw by horizontal (°)

\({\bar{\beta}}\) :

Vector of unknown parameters

\({\bar{\lambda}}\) :

Vector of optimal step size

(n):

Number of iteration

*:

Optimal value

i :

Sensor index

j :

Unknown parameter index

IHCP:

Inverse heat conduction problem

CGM:

Conjugate gradient method

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Authors and Affiliations

  1. School of Mechanical Engineering, University of Tehran, North Kargar St., Jala-Al-Ahmad Expressway, Tehran, Iran

    Majid Siavashi & Farshad Kowsary

  2. Faculty of Engineering and Built Environment, University of Newcastle, Callaghan, NSW, 2308, Australia

    Ehsan Abbasi-Shavazi

Authors
  1. Majid Siavashi
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  2. Farshad Kowsary
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  3. Ehsan Abbasi-Shavazi
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Corresponding author

Correspondence to Farshad Kowsary.

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Siavashi, M., Kowsary, F. & Abbasi-Shavazi, E. Detection of flaws in a two-dimensional body through measurement of surface temperatures and use of conjugate gradient method. Comput Mech 46, 597–607 (2010). https://doi.org/10.1007/s00466-010-0501-5

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  • DOI: https://doi.org/10.1007/s00466-010-0501-5

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