Numerical identification of source terms for a two dimensional heat conduction problem in polar coordinate system

Yang, Liu; Jiang, Yu; Luo, Guoxi; Deng, Zui‐Cha

doi:10.1016/j.apm.2012.03.024

Cited by 20 publications

(10 citation statements)

References 27 publications

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“…The above fundamental governing bio-heat equation (1) represents a balance between the accumulation of energy (in the left-hand side of (1)) and the superposition of heat conduction (diffusion), heat transfer effect due to the blood flowing through the capillary network and heat generation due to the cell metabolism. The inverse linear problem of finding the metabolic heat source f has been considered elsewhere, [20,21,22], herein we address the more difficult nonlinear problem of finding the blood perfusion coefficient q(x). The importance of the blood perfusion contribution to the heat generation in tissue has been stressed in carcinogenic skin and brest tumours because of the increased nutrition and oxygen demand [23].…”

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confidence: 99%

Simultaneous reconstruction of the perfusion coefficient and initial temperature from time-average integral temperature measurements

Cao

Lesnic

2019

Applied Mathematical Modelling

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Simultaneous reconstruction of the perfusion coefficient and initial temperature from time-average integral temperature measurements. Applied Mathematical Modelling, 68. pp. 523-539. AbstractInverse coefficient identification formulations give rise to some of the most important mathematical problems because they tell us how to determine the unknown physical properties of a given medium under inspection from appropriate extra measurements. Such an example occurs in bioheat transfer where the knowledge of the blood perfusion is of critical importance for calculating the temperature of the blood flowing through the tissue. Furthermore, in many related applications the initial temperature of the diffusion process is also unknown. Therefore, in this framework the simultaneous reconstruction of the space-dependent perfusion coefficient and initial temperature from two linearly independent weighted time-integral observations of temperature is investigated. The quasi-solution of the inverse problem is obtained by minimizing the least-squares objective functional, and the Fréchet gradients with respect to both of the two unknown space-dependent quantities are derived. The stabilisation of the conjugate gradient method (CGM) is established by regularising the algorithm with the discrepancy principle. Three numerical tests for one-and two-dimensional examples are illustrated to reveal the accuracy and stability of the numerical results. IntroductionThe inverse problem of identifying the space-dependent perfusion/radiative coefficient from integral observation was previously studied in [1,2,3]. This unknown coefficient was numerically determined in the one-dimensional bio-heat equation with heat flux or time-average temperature measurement by minimising the Tikhonov regularisation functional using the NAG routine E04FCF together with the finite-difference method (FDM), [4]. Recently, the space-dependent perfusion coefficient was recovered by the CGM from the final or time-average temperature measurement in [5]. Also, the inverse problem of determining the initial temperature from temperature measurements at a later time was extensively studied, e.g. [6,7]. Besides, there are many numerical techniques that had been developed to reconstruct the unknown initial temperature, including the iterative CGM [8,9], the boundary element method (BEM) with regularisation [10], the elliptic approximation together with the BEM [11], the Tikhonov regularisation approach [12], the Fourier regularisation method [13] and the self-adaptive Lie-group adaptive method [14].In [15], the space-dependent radiative coefficient and the initial temperature were simultaneously reconstructed from temperature measurements at a fixed time θ > 0 and in ω × (0, T ), where ω is a subregion of the space domain Ω; the stability of the inverse problem was established, the existence of the minimizer of Tikhonov's first-order regularisation functional was proved, and the numerical results were obtained by using a nonlinear gradient multigrid technique. Similarly, t...

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confidence: 99%

Simultaneous reconstruction of the perfusion coefficient and initial temperature from time-average integral temperature measurements

Cao

Lesnic

2019

Applied Mathematical Modelling

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“…The error bound (28) does not provide the convergence as → 0 obviously. Hence, we need to choose a proper parameter .…”

Section: Remarkmentioning

confidence: 98%

“…It is worth pointing out that ill posed problems of a large number of diffusion equations, both fractionalorder as well as integral order, have been discussed by many authors. Yang et al [24][25][26][27][28] discuss the identification of source terms for some integral-order diffusion equations using some regularization strategies. Hon et al [29,30] apply some meshless methods to the ill posed problems of heat conduction equations.…”

Section: Introductionmentioning

confidence: 99%

A Mollification Regularization Method for a Fractional-Diffusion Inverse Heat Conduction Problem

Deng

Yang

Feng

2013

Mathematical Problems in Engineering

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The ill-posed problem of attempting to recover the temperature functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor when the governing linear diffusion equation is of fractional type is discussed. A simple regularization method based on Dirichlet kernel mollification techniques is introduced. We also proposea priorianda posterioriparameter choice rules and get the corresponding error estimate between the exact solution and its regularized approximation. Moreover, a numerical example is provided to verify our theoretical results.

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“…Although ( ) and ℎ ( ) belong to the space 2 (0, ), the Fourier coefficients { } and {ℎ } no longer satisfy inequalities (11) and (12) since the random noise does not decay and will be dramatically amplified by 2 and 2 1 . Therefore, the inverse problem is ill-posed for determining the heat source and the initial distribution simultaneously in 2 (0, ).…”

Section: Ill-posednessmentioning

confidence: 99%

“…In the past decades, various classes of inverse heat conduction equation problems have been studied by many scholars including recovery of the initial temperature [1][2][3][4][5], reconstruction of the heat source [6][7][8][9][10][11][12], and identification of thermal diffusion coefficients [13,14]. The inverse problems of heat equations such as the backward problems and the source reconstruction problems arise from various scientific and engineering fields, including heat conduction, hydrology, environmental controlling.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Determination of the Space-Dependent Source and the Initial Distribution in a Heat Equation by Regularizing Fourier Coefficients of the Given Measurements

Qiu

Zhang

Peng

2018

Advances in Mathematical Physics

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We consider an inverse problem for simultaneously determining the space-dependent source and the initial distribution in heat conduction equation. First, we study the ill-posedness of the inverse problem. Then, we construct a regularization problem to approximate the originally inverse problem and obtain the regularization solutions with their stability and convergence results. Furthermore, convergence rates of the regularized solutions are presented under a prior and a posteriori strategies for selecting regularization parameters. Results of numerical examples show that the proposed regularization method is stable and effective for the considered inverse problem.

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Numerical identification of source terms for a two dimensional heat conduction problem in polar coordinate system

Cited by 20 publications

References 27 publications

Simultaneous reconstruction of the perfusion coefficient and initial temperature from time-average integral temperature measurements

Simultaneous reconstruction of the perfusion coefficient and initial temperature from time-average integral temperature measurements

A Mollification Regularization Method for a Fractional-Diffusion Inverse Heat Conduction Problem

Simultaneous Determination of the Space-Dependent Source and the Initial Distribution in a Heat Equation by Regularizing Fourier Coefficients of the Given Measurements

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