liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Boundary identification for an elliptic equation
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
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.

Place, publisher, year, edition, pages
2001. , 16 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 23
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-88734OAI: oai:DiVA.org:liu-88734DiVA: diva2:605787
Available from: 2013-02-15 Created: 2013-02-15 Last updated: 2013-02-15
In thesis
1. Numerical methods for inverse heat conduction problems
Open this publication in new window or tab >>Numerical methods for inverse heat conduction problems
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
Ill-posed, Heat Conduction, Regularization
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-34894 (URN)23830 (Local ID)91-7373-132-3 (ISBN)23830 (Archive number)23830 (OAI)
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

Open Access in DiVA

No full text

Authority records BETA

Berntsson, Fredrik

Search in DiVA

By author/editor
Berntsson, Fredrik
By organisation
Department of MathematicsThe Institute of Technology
Engineering and Technology

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 124 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf