Computational Mechanics

, Volume 46, Issue 4, pp 597–607 | Cite as

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

  • Majid Siavashi
  • Farshad Kowsary
  • Ehsan Abbasi-Shavazi
Original Paper


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.


Inverse heat conduction problem (IHCP) Conjugate gradients method Parameter estimation Flaw identification Crack detection Non-destructive test 

List of symbols


Mean squared error (target function)


Heat convection coefficient (Wm−2 K−1)


Heat conduction coefficient (Wm−1 K−1)


Length of crack (m)


Number of measurements


Number of unknown parameters


Heat flux (Wm−2)


Radius of flaw (m)


Surrounding temperature (K)


x coordinate position of flaw (m)


y coordinate position of flaw (m)


Vector of search direction


Vector of estimated temperatures (K)


Vector of measured temperatures (K)


Operator to convert vector to diagonal matrix


Matrix of sensitivity coefficients

Greek symbols


Angle of elliptical flaw by horizontal (°)


Vector of unknown parameters


Vector of optimal step size



Number of iteration


Optimal value



Sensor index


Unknown parameter index



Inverse heat conduction problem


Conjugate gradient method


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Majid Siavashi
    • 1
  • Farshad Kowsary
    • 1
  • Ehsan Abbasi-Shavazi
    • 2
  1. 1.School of Mechanical EngineeringUniversity of TehranTehranIran
  2. 2.Faculty of Engineering and Built EnvironmentUniversity of NewcastleCallaghanAustralia

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