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Numerical methods for inverse heat conduction problems
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
2001 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In many industrial applications one wishes to determine the temperature history on the surface of a body, where the surface itself is inaccessible for measurements. The sideways heat equation is a model of this situation. In a one-dimensional setting this is formulated mathematically as a Cauchy problem for the heat equation, where temperature and heat--flux data are available along the line x=1, and a solution is sought for 0 ≤ x< 1. This problem is ill-posed in the sense that the solution does not depend continuously on the data. Stability can be restored by replacing the time derivative in the heat equation by a bounded approximation. We consider both spectral and wavelet approximations of the derivative. The resulting problem is a system of ordinary differential equations in the space variable, that can be solved using standard methods, e.g. a Runge-Kutta method. The methods are analyzed theoretically, and error estimates are derived, that can be used for selecting the appropriate level of regularization. The numerical implementation of the proposed methods is discussed. Numerical experiments demonstrate that the proposed methods work well, and can be implemented efficiently. Furthermore, the numerical methods can easily be adapted to solve problems with variable coefficients, and also non-linear equations. As test problems we take model equations, with constant and variable coefficients. Also, we solve problems from applications, with actual measured data.

Inverse problems for the stationary heat equation are also discussed. Suppose that the Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are given on the outer boundary, and we wish to compute the steady state temperature on the inner boundary. A standard approach is to discretize the equation by finite differences, and use Tikhonov's method for stabilizing the discrete problem, which leads to a large sparse least squares problem. Alternatively, we propose to use a conformal mapping to transform the domain into an annulus, where the equivalent problem can be solved using separation of variables. The ill-posedness is dealt with by filtering away high frequencies from the solution. Numerical results using both methods are presented. A closely related problem is that of determining the stationary temperature inside a body, from temperature and heat-flux measurements on a part of the boundary. In practical applications it is sometimes the case that the domain, where the differential equation is valid, is partly unknown. In such cases we want to determine not only the temperature, but also the shape of the boundary of the domain. This problem arises, for instance, in iron production, where the walls of a melting furnace is subject to both physical and chemical wear. In order to avoid a situation where molten metal breaks out through the walls the thickness of the walls should be constantly monitored. This is done by solving an inverse problem for the stationary heat equation, where temperature and heat-flux data are available at certain locations inside the walls of the furnace. Numerical results are presented also for this problem.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet , 2001. , 14 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 723
Keyword [en]
Ill-posed, Heat Conduction, Regularization
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-34894Local ID: 23830ISBN: 91-7373-132-3 (print)OAI: oai:DiVA.org:liu-34894DiVA: diva2:255742
Public defence
2001-12-14, Sal Key 1, Key-huset, Linköpings universitet, Linköping, 13:15 (Swedish)
Opponent
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2013-02-15
List of papers
1. Wavelet and Fourier methods for solving the sideways heat equation
Open this publication in new window or tab >>Wavelet and Fourier methods for solving the sideways heat equation
2000 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 21, no 6, 2187-2205 p.Article in journal (Refereed) Published
Abstract [en]

We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 ≤ x < 1.

The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge–Kutta method.

We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-47660 (URN)10.1137/S1064827597331394 (DOI)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
2. A spectral method for solving the sideways heat equation
Open this publication in new window or tab >>A spectral method for solving the sideways heat equation
1999 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 15, no 4, 891-906 p.Article in journal (Refereed) Published
Abstract [en]

We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data are given along the line x = 1, and a solution is sought in the interval 0 ≤ x < 1.

The problem is ill-posed, in the sense that the solution does not depend continuously on the data. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded spectral-based approximation. The cut-off level in the spectral approximation acts as a regularization parameter. Error estimates for the regularized solution are derived and a procedure for selecting an appropriate regularization parameter is given. The discretized problem is an initial value problem for an ordinary differential equation in the space variable, which can be solved using standard numerical methods, for example a Runge-Kutta method. As test problems we take equations with constant and variable coefficients.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-68233 (URN)10.1088/0266-5611/15/4/305 (DOI)
Available from: 2011-05-13 Created: 2011-05-13 Last updated: 2017-12-11
3. An inverse heat conduction problem and an application to heat treatment of aluminium
Open this publication in new window or tab >>An inverse heat conduction problem and an application to heat treatment of aluminium
2000 (English)In: Inverse Problems in Engineering Mechanics II / [ed] Masataka Tanaka, G.S. Dulikravich, 2000, 99-106 p.Conference paper, Published paper (Refereed)
Abstract [en]

We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem where one wants to determine the temperature on the surface of a body using internal measurements. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We discuss the nature of the ill-posedness as well as methods for restoring stability with respect to measurement errors.

Successful heat treatment requires good control of the temperature and cooling rates during the process. In an experiment a aluminium block, of the alloy AA7010, was cooled rapidly by spraying water on one surface. Thermocouples inside the block recorded the temperature, and we demonstrate that it is possible to find the temperature distribution in the region between the thermocouple and the surface, by solving numerically the sideways heat equation.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-68235 (URN)978-0-08-043693-7 (ISBN)008053516X (ISBN)
Conference
International Symposium on Inverse Problems in Engineering Mechanics, Nagano, Japan, March 2000
Available from: 2011-05-13 Created: 2011-05-13 Last updated: 2014-12-15
4. Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation
Open this publication in new window or tab >>Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation
2001 (English)Report (Other academic)
Abstract [en]

Numerical procedures for solving a non-Characteristic Cauchy problem for the heat equation are discussed. More precisely we consider a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted 0 ≤ x <1. The problem is often referred to as the sideways heat equation.

The problem is analyzed, using both Fourier analysis and the singular value decomposition, and is found to be severely ill-posed. The literature is vast, and many authors have proposed numerical methods that regularize the IHCP. In this paper we attempt to give an overview that covers the most popular methods that have been considered.

Numerical examples that illustrate the numerical algorithms are given.

Publisher
33 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 17
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-88733 (URN)
Available from: 2013-02-15 Created: 2013-02-15 Last updated: 2013-02-15
5. Numerical solution of a Cauchy problem for the Laplace equation
Open this publication in new window or tab >>Numerical solution of a Cauchy problem for the Laplace equation
2001 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 17, no 4, 839-853 p.Article in journal (Refereed) Published
Abstract [en]

We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.

National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-47296 (URN)10.1088/0266-5611/17/4/316 (DOI)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
6. Boundary identification for an elliptic equation
Open this publication in new window or tab >>Boundary identification for an elliptic equation
2001 (English)Report (Other academic)
Abstract [en]

We consider an inverse problem for the two dimensional steady state heat equation. More precisely, the heat equation is valid in a domain Ω, that is a subset of the unit square, temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and x = 1 are assumed to be insulated. From these we wish to determine the temperature in the domain Ω. Furthermore, a part of the boundry ∂Ω is considered to be unknown, and must also be determined.

The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equation, which can be solved using standard numerical methods.

The inverse problem, that we consider in this paper, arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an invers heat conduction problem.

Publisher
16 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 23
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-88734 (URN)
Available from: 2013-02-15 Created: 2013-02-15 Last updated: 2013-02-15

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