Probing for inclusions in heat conductive bodies

Gaitan, Patricia; Isozaki, Hiroshi; Poisson, Olivier; Siltanen, Samuli; Tamminen, Janne P.

doi:10.3934/ipi.2012.6.423

Cited by 12 publications

(12 citation statements)

References 15 publications

(20 reference statements)

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“…The idea is based on the complex spherical wave given by Ide-Isozaki-Nakata-Siltanen-Uhlmann [11] for the elliptic case. Other computational approaches to static inclusion detection include [5,8].…”

Section: 3mentioning

confidence: 99%

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

Gaitan

Isozaki

Poisson

et al. 2015

Communications in Partial Differential Equations

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Abstract. We consider an inverse boundary value problem for the heat equation ∂tu = div (γ∇xu) in (0, T ) × Ω, u = f on (0, T ) × ∂Ω, u t=0 = u 0 , in a bounded domain Ω ⊂ R n , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time :. Fix a direction e * ∈ S n−1 arbitrarily. Assuming that ∂D(t) is strictly convex for 0 ≤ t ≤ T , we show that k and sup{e * ·x ; x ∈ D(t)} (0 ≤ t ≤ T ), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ∂ν u(t, x) (0,T )×∂Ω . The knowledge of the initial data u 0 is not used in the proof. If we know min 0≤t≤T sup x∈D(t) x · e * , we have the same conclusion from the local Dirichlet-to-Neumann map. The results have applications to nondestructive testing. Consider a physical body consisiting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.

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Section: 3mentioning

confidence: 99%

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

Gaitan

Isozaki

Poisson

et al. 2015

Communications in Partial Differential Equations

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“…Ikehata [9], and Ikehata and Kawashita [10] developed the probe method for the heat equation with time-independent inclusions. In [6], the case of time-independent inclusions was treated and a numerical computation result was given. The idea is based on the complex spherical wave given by Ide et al [8] for the elliptic case.…”

Section: Inverse Heat Conductivity Problemmentioning

confidence: 99%

“…The second case is more complicated: We can determine the value (1 − 1 k )s(t) and we show that k (and so, s(t)) can take two values at most.Theoretically, the infinite-precision measurement needs to be repeated infinitely many times to recover D(t) and γ(t, x) perfectly. However, approximate recovery should be possible from a finite number of finite-precision measurements similarly to [8,6,7], but this is outside the scope of the present paper.…”

mentioning

confidence: 99%

Inverse Problems for Time-Dependent Singular Heat Conductivities---One-Dimensional Case

Gaitan¹,

Isozaki²,

Poisson³

et al. 2013

SIAM J. Math. Anal.

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International audienceWe consider an inverse boundary value problem for the heat equation on the interval (0, 1), where the heat conductivity γ(t, x) is piecewise constant and the point of discontinuity depends on time : γ(t, x) = k 2 (0 < x < s(t)), γ(t, x) = 1 (s(t) < x < 1). Firstly we show that k and s(t) on the time interval [0, T ] are determined from a partial Dirichlet-to-Neumann map : u(t, 1) → ∂xu(t, 1), 0 < t < T , u(t, x) being the solution to the heat equation such that u(t, 0) = 0, independently of the initial data u(0, x). Secondly we show that another partial Dirichlet-to-Neumann map u(t, 0) → ∂xu(t, 1), 0 < t < T , u(t, x) being the solution to the heat equation such that u(t, 1) = 0, restricts the pair (k, s(t)) to at most two cases on the time interval [0, T ], independently of the initial data u(0, x)

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“…These particular solutions are constructed by a control method as it has been done in the original work [37]. The technics used to prove our main results are different from that involved in [2,10,16,19]. By means of specific test functions our main result can be read as an approximation to the Fourier transformation of the delta distributions at the centers of the inhomogeneities and this is suggested as an idea for a numerical reconstruction algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…There are lots of works on inverse problem of heat conductivity [5,8,14,16,18,19,21,24,31,35]. Generally, the determination of conductivity profiles from knowledge of boundary measurements has received a great deal of attention (see for example [1,9,15,33]) the reconstruction of imperfections within dynamics is much less investigated.…”

Section: Introductionmentioning

confidence: 99%

On an inverse boundary problem for the heat equation when small heat conductivity defects are present in a material

2015

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For the heat equation in a bounded domain we consider the inverse problem of identifying locations and certain properties of the shapes of small heat-conducting inhomogeneities from dynamic boundary measurements on part of the boundary and for finite interval in time. The key ingredient is an asymptotic method based on appropriate averaging of the partial dynamic boundary measurements. Our approach is expected to lead to very effective computational identification algorithms.

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Probing for inclusions in heat conductive bodies

Cited by 12 publications

References 15 publications

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

Inverse Problems for Time-Dependent Singular Heat Conductivities---One-Dimensional Case

On an inverse boundary problem for the heat equation when small heat conductivity defects are present in a material

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