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Comparison of four different versions of the variable metric method for solving inverse heat conduction problems

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Abstract

In this study, four different versions of the variable metric method (VMM) are investigated in solving standard one-dimensional inverse heat conduction problems in order to evaluate their efficiency and accuracy. These versions include Davidon–Fletcher–Powell (DFP), Broydon–Fletcher–Goldfarb–Shanno (BFGS), Symmetric Rank-one (SR1), and Biggs formula of the VMM. These investigations are carried out using temperature data obtained from numerical simulations.

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Abbreviations

e RMS :

relative root mean square error, Eq. 23

f :

function of sum of the squares of errors, Eq. 5

K :

number of sensors

M :

total number of time steps (equal to total number of unknowns that should be estimated.)

q :

heat flux (W/m2)

\(\vec{q}\) :

unknowns vector (design vector with dimension M × 1), Eq. 2

S :

vector of descent direction, Eq. 6

T :

temperature calculated by the model (K)

t :

time (s)

X :

pulse sensitivity coefficient (K m2 /W) defined by Eq. 3

Y :

temperature measured by sensors

(x, y ):

Cartesian coordinate system

α:

thermal diffusivity (m2/s)

ɛ:

tolerance error

λ:

search step length

λ* :

optimal step length

∇:

gradient

T:

matrix transpose symbol

→:

vector

′:

derivative symbol

k :

sensor index number

M :

total number of time steps

\(m, \tilde{m}\) :

indices of time step, each between 1,2,..., M

i :

iteration number of VMM cycles

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

    A. Pourshaghaghy & F. Kowsary

  2. Department of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran

    A. Behbahaninia

Authors
  1. A. Pourshaghaghy
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  2. F. Kowsary
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  3. A. Behbahaninia
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Corresponding author

Correspondence to F. Kowsary.

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Pourshaghaghy, A., Kowsary, F. & Behbahaninia, A. Comparison of four different versions of the variable metric method for solving inverse heat conduction problems. Heat Mass Transfer 43, 285–294 (2007). https://doi.org/10.1007/s00231-006-0107-9

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  • DOI: https://doi.org/10.1007/s00231-006-0107-9

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