Reconstruction of an unknown boundary portion from Cauchy data in
                    <i>n</i>
                    dimensions

Bryan, Kurt; Caudill, Lester

doi:10.1088/0266-5611/21/1/015

Cited by 39 publications

(26 citation statements)

References 11 publications

(20 reference statements)

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“…Recently, Banks et al [3,4] proposed some reconstruction methods for the boundary determination problem of a parabolic equation defined in a rectangular domain ⊂ 2 . More recently, Bryan and Caudill [5,6] investigated these kinds of boundary determination problems for more general domains. We note here that the unknown boundary to be determined in these works does not depend on time, whereas the unknown free boundary to be determined in this paper moves with respect to time.…”

Section: Y C Hon and M Limentioning

confidence: 99%

A computational method for inverse free boundary determination problem

Hon

2007

Numerical Meth Engineering

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SUMMARYBased on the method of fundamental solutions and discrepancy principle for the choice of location for source points, we extend in this paper the application of the computational method to determine an unknown free boundary of a Cauchy problem of parabolic-type equation from measured Dirichlet and Neumann data with noises. The standard Tikhonov regularization technique with the L-curve method for an optimal regularized parameter is adopted for solving the resultant highly ill-conditioned system of linear equations. Both one-dimensional and two-dimensional numerical examples are given to verify the efficiency and accuracy of the proposed computational method.

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Section: Y C Hon and M Limentioning

confidence: 99%

A computational method for inverse free boundary determination problem

Hon

2007

Numerical Meth Engineering

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“…Reconstruction methods for the heat equation have been proposed by Banks, Kojima and Winfree [2,3] in the case where is a rectangle, Chapko, Kress and Yoon [8] where domain is the unit disk, Bryan and Caudill [4,5] where is a strip. Recently, Bryan and Caudill also considered the inverse Cauchy problem in n dimensions and develop a Newton-like algorithm [7].…”

Section: Introductionmentioning

confidence: 99%

Numerical method for the inverse heat transfer problem in composite materials with Stefan-Boltzmann conditions

Chen

2009

Adv Comput Math

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In this paper, one specific kind of heat transfer problem with nonlinear Stefan-Boltzmann conditions are considered in a three dimensional multi-layer domain. Theoretical results for forward and inverse problems are presented. Numerical simulations of specific models from applications are provided to demonstrate the heat transfer process in the composite materials of forward problem. One reconstruction method is proposed to find the corrosion part, and the numerical examples show that the reconstruction algorithm is effective.

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“…Ω 1 The section of Ω where the material has not been corroded. Ω 2 The section of Ω where the material has been corroded. u 0 The temperature profile in the uncorroded plate Ω made of material (1), using the same input flux used to compute u 1 , u 2 .…”

Section: Terms and Definitionsmentioning

confidence: 99%

“…u 1 The temperature profile in Ω 1 , function of (x, y, t). u 2 The temperature profile in Ω 2 , function of (x, y, t). C(x) The corrosion profile; the function of the curve that separates Ω 1 from Ω 2 .…”

Section: Terms and Definitionsmentioning

confidence: 99%

Thermal Detection of Inaccessible Corrosion

2013

SIURO

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In this paper, we explore the mathematical inverse problem of detecting corroded material on the reverse side of a partially accessible metal plate. We provide a novel formulation of the two-dimensional problem using a heat source as the detection method, developing a numerical method for performing these reconstructions. The reconstruction is performed via integration against test functions, and we will show how a linearization can be used to simplify the initial problem and explain a regularization method used to obtain acceptable results for the corrosion profile. Results will be shown for a variety of corrosion profiles and system thermal parameters and error will be quantified for each case. It is shown that the reconstruction of small corrosion profiles is within a reasonable amount of error for real-world applications (20%), and while larger corrosion is less accurately reconstructed, this method will allow for detection of corrosion of any size. Possibilities for future work are also outlined, including the definition of regularization parameters and extending the procedure to different domains.

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Reconstruction of an unknown boundary portion from Cauchy data in n dimensions

Cited by 39 publications

References 11 publications

A computational method for inverse free boundary determination problem

A computational method for inverse free boundary determination problem

Numerical method for the inverse heat transfer problem in composite materials with Stefan-Boltzmann conditions

Thermal Detection of Inaccessible Corrosion

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