Regularization of the inverse heat conduction problem by the discrete Fourier transform

Wróblewska, Agnieszka; Frąckowiak, Andrzej; Ciałkowski, Michał

doi:10.1080/17415977.2015.1017480

Cited by 25 publications

(10 citation statements)

References 10 publications

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“…To solve the inverse problem, the method described in Wróblewska et al (2015) and Frąckowiak and Ciałkowski (2018) will be adapted. In those papers, an unknown distribution of temperature or heat flux on the boundary of the region was sought in the form of linear combinations of functions of known shape.…”

Section: Solution Of the Inverse Problemmentioning

confidence: 99%

Determination of heat transfer coefficient in a T-shaped cavity by means of solving the inverse heat conduction problem

Frąckowiak

Spura

Gampe

et al. 2019

HFF

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Purpose T-shaped cavities occur by design in many technical applications. An example of such a stator cavity is the side space between the guide vane carriers and the outer casing of a steam turbine. Thermal conditions inside it have a significant impact on the deformation of the turbine casing. In order to improve its prediction, the purpose of this paper is to provide a methodology to gain better knowledge of the local heat transfer at the cavity boundaries based on experimental results. Design/methodology/approach To determine the heat transfer coefficient distribution inside a model cavity with the help of a scaled generic test rig, an inverse heat conduction problem is posed and a method for solving such type of problems in the form of linear combinations of Trefftz functions is presented. Findings The results of the calculations are compared with another inverse method using first-order gradient optimization technique as well as with estimated values obtained with an analytic two-dimensional thermal network model, and they show an excellent agreement. The calculation procedure is proved to be numerically stable for different degrees of complexity of the sought boundary conditions. Originality/value This paper provides a universal and robust methodology for the fast direct determination of an arbitrary distribution of heat transfer coefficients based on material temperature measurements spread over the confining wall.

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Section: Solution Of the Inverse Problemmentioning

confidence: 99%

Determination of heat transfer coefficient in a T-shaped cavity by means of solving the inverse heat conduction problem

Frąckowiak

Spura

Gampe

et al. 2019

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“…Due to its importance the literature on this problem is vast. Recent results on this particular problem includes meshless methods [12,13], an optimal regularization method [14], a fitting algorithm [15], singular boundary method [16], discrete Fourier transform method [17], a method based on proper solution space [18], and an energy regularization method [19].…”

Section: Introductionmentioning

confidence: 99%

Non-Iterative Solution Methods for Cauchy Problems for Laplace and Helmholtz Equation in Annulus Domain

Tadi

Radenković

2021

Mathematics

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This note is concerned with two new methods for the solution of a Cauchy problem. The first method is based on homotopy-perturbation approach which leads to solving a series of well-posed boundary value problems. No regularization is needed in this method. Laplace and Helmholtz equations are considered in an annular region. It is also proved that the homotopy solution for the Laplace operator converges to the actual exact solution. The second method is also non-iterative. It is based on the application of the Green’s second identity which leads to a moment problem for the unknown boundary condition. Tikhonov regularization is used to obtain a stable and close approximation of the missing boundary condition. A number of examples are used to study the applicability of the methods with the presence of noise.

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“…There are many methods used to regularize inverse problems. Among them, there is the Tikhonov regularization (Beck and Woodbury, 2016; Chen et al , 2019; Djerrar et al , 2017; Frąckowiak et al , 2019a; Laneev, 2018; Marin, 2010, 2016; Niu et al , 2014; Sun, 2016; Tikhonov and Arsenin, 1977; Yaparova, 2016), the Tikhonov–Philips regularization (Joachimiak et al , 2019a), the discrete Fourier transform (Frąckowiak and Ciałkowski, 2018; Wróblewska et al , 2015) and SVD algorithm (Hasanov and Mukanova, 2015). In her article, Cheruvu (2017) applied the wavelet regularization of Laplace’s equation in the arbitrarily shaped domain.…”

Section: Introductionmentioning

confidence: 99%

Choice of the regularization parameter for the Cauchy problem for the Laplace equation

Joachimiak

2020

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Purpose In this paper, the Cauchy-type problem for the Laplace equation was solved in the rectangular domain with the use of the Chebyshev polynomials. The purpose of this paper is to present an optimal choice of the regularization parameter for the inverse problem, which allows determining the stable distribution of temperature on one of the boundaries of the rectangle domain with the required accuracy. Design/methodology/approach The Cauchy-type problem is ill-posed numerically, therefore, it has been regularized with the use of the modified Tikhonov and Tikhonov–Philips regularization. The influence of the regularization parameter choice on the solution was investigated. To choose the regularization parameter, the Morozov principle, the minimum of energy integral criterion and the L-curve method were applied. Findings Numerical examples for the function with singularities outside the domain were solved in this paper. The values of results change significantly within the calculation domain. Next, results of the sought temperature distributions, obtained with the use of different methods of choosing the regularization parameter, were compared. Methods of choosing the regularization parameter were evaluated by the norm Nmax. Practical implications Calculation model described in this paper can be applied to determine temperature distribution on the boundary of the heated wall of, for instance, a boiler or a body of the turbine, that is, everywhere the temperature measurement is impossible to be performed on a part of the boundary. Originality/value The paper presents a new method for solving the inverse Cauchy problem with the use of the Chebyshev polynomials. The choice of the regularization parameter was analyzed to obtain a solution with the lowest possible sensitivity to input data disturbances.

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Regularization of the inverse heat conduction problem by the discrete Fourier transform

Cited by 25 publications

References 10 publications

Determination of heat transfer coefficient in a T-shaped cavity by means of solving the inverse heat conduction problem

Determination of heat transfer coefficient in a T-shaped cavity by means of solving the inverse heat conduction problem

Non-Iterative Solution Methods for Cauchy Problems for Laplace and Helmholtz Equation in Annulus Domain

Choice of the regularization parameter for the Cauchy problem for the Laplace equation

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