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Cavity detection in a heat conductor using linear sampling method

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Abstract

In this paper, solution of inverse problems in heat conduction transient fields is investigated. For this purpose, a new time-domain version of linear sampling method (TDLSM) is developed for cavity detection in a heat conductor. The linear sampling method (LSM) is an effective approach to image the geometrical features of unknown targets. Although this method has been used in the context of inverse scattering problems such as solid, acoustics, and electromagnetism, there is no specific attempt to apply this method to identification of cavities in heat conductors. This study emphasizes the implementation of the LSM in the time-domain fields using finite element method. A set of numerical simulations on two-dimensional transient heat conduction problems is presented to highlight many effective features of the proposed TDLSM fast qualitative identification method.

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Abbreviations

‖ ‖:

Norm’s vector

*:

Convolution operator

c p :

Specific heat

d :

Number of spatial dimensions

D:

Domain of the cavity

∂D :

Boundary of the cavity

g :

Density distribution

H():

Heaviside function

k :

Thermal conductivity

L 2 :

Hilbert space

N d :

Neumann-to-Dirichlet operator

N t :

Number of time steps

N r :

Number of receivers

N S :

Number of transmitters

n :

Outward unit normal vector

\(\bar{q}(\varvec{x},t)\) :

Prescribed inflow heat flux

r :

Distance

S :

External surface

S 1 :

Excitation surface

S 2 :

Observation surface

\(SL_{{S_{1} }}\) :

Single layer heat potential operator

t :

Time(s)

\(\bar{T}(\varvec{x},t)\) :

Prescribed temperature

\(T^{D} (\varvec{x},t)\) :

Total temperature field

\(T^{S} (\varvec{x},t)\) :

Differential temperature field

\(\widehat{T}\) :

Fundamental solution of the diffusion equation

U :

Left-singular vectors

\({v}(\varvec{x},t)\) :

Superposition temperature

V :

Right-singular vectors

\(w^{{S}} (\varvec{x},t)\) :

Superposition of differential temperature field

\(\varvec{x} = \left[ {x_{1} ,x_{2} } \right]^{T}\) :

Cartesian coordinates

z :

Sampling point

\(\phi_{\varvec{z}} (\varvec{x},t)\) :

Radiated temperature vector

ρ :

Mass density

Ω:

Homogeneous media

∇.(∇):

Laplace operator

δ :

Dirac delta function

\(\mathcal{L}\) :

Partial differential operator

τ :

Time

λ :

Tikhonov regularization parameter

ε :

Error

σ :

Singular value

D :

Dirichlet boundary condition

N :

Neumann boundary condition

n, m :

Time index

i, j :

Spatial index

FEM:

Finite element method

LSM:

Linear sampling method

MFS:

Method of fundamental solution

SVD:

Singular value decomposition

TDLSM:

Time domain linear sampling method

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Authors and Affiliations

  1. Faculty of Civil and Environmental Engineering, Tarbiat Modares University, P.O. Box 14115-397, Tehran, Iran

    S. H. Dehghan Manshadi & N. Khaji

Authors
  1. S. H. Dehghan Manshadi
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  2. N. Khaji
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Correspondence to N. Khaji.

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Dehghan Manshadi, S.H., Khaji, N. Cavity detection in a heat conductor using linear sampling method. Heat Mass Transfer 50, 973–984 (2014). https://doi.org/10.1007/s00231-014-1309-1

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