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Solving stationary inverse heat conduction in a thin plate
Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering. Department of Applied Mathematics, University of Nairobi, Nairobi, Kenya.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-2681-8965
Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
2023 (English)In: Partial Differential Equations and Applications, ISSN 2662-2971, Vol. 4, no 6Article in journal (Refereed) Published
Abstract [en]

We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence.

Place, publisher, year, edition, pages
2023. Vol. 4, no 6
Keywords [en]
Cauchy problem, Stationary heat equation, Degenerate elliptic equation, Landweber iterative method
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-201113DOI: 10.1007/s42985-023-00267-7OAI: oai:DiVA.org:liu-201113DiVA, id: diva2:1839768
Available from: 2024-02-22 Created: 2024-02-22 Last updated: 2024-02-22Bibliographically approved
In thesis
1. Regularization methods for solving Cauchy problems for elliptic and degenerate elliptic equations
Open this publication in new window or tab >>Regularization methods for solving Cauchy problems for elliptic and degenerate elliptic equations
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we study Cauchy problems for the elliptic and degenerate elliptic equations. These problems are ill-posed. We split the boundary of the domain into two parts. On one of them, say Γ0, we have available Cauchy data and on remaining part Γ1 we introduce unknown Robin data. To construct the operator equation which replaces our Cauchy problem we use two boundary value problems (BVP). The first one is the mixed BVP with Robin condition on Γ1 and with Dirichlet condition on Γ0 and the second BVP with Dirichlet Data on Γ1 and with Robin data on Γ0. The well–posedness of these problems is achieved by an appropriate choice of parameters in Robin boundary conditions. The first Dirichlet–Robin BVP is used to construct the operator equation replacing the Cauchy problem and the second Robin–Dirichlet problem for adjoint operator. Using these problems we can apply various regularization methods for stable reconstruction of the solution. In Paper I, the Cauchy problem for the elliptic equation with variable coefficients, which includes Helmholtz type equations, is analyzed. A proof showing that the Dirichlet–Robin alternating algorithm is convergent is given, provided that the parameters in the Robin conditions are chosen appropriately. Numerical experiments that shows the behaviour of the algorithm are given. In particular, we show how the speed of convergence depends on the choice of Robin parameters. 

In Paper II, the Cauchy problem for the Helmholtz equation, for moderate wave numbers k2, is considered. The Cauchy problem is reformulated as an operator equation and iterative method based on Krylov subspaces are implemented. The aim is to achieve faster convergence in comparison to the Alternating algorithm from the previous paper. Methods such as the Landweber iteration, the Conjugate gradient method and the generalized minimal residual method are considered. We also discuss how the algorithms can be adapted to also cover the case of non–symmetric differential operators. 

In Paper III, we look at a steady state heat conduction problem in a thin plate. The plate connects two cylindrical containers and fix their relative positions. A two dimensional mathematical model of heat conduction in the plate is derived. Since the plate has sharp edges on the sides we obtained a degenerate elliptic equation. We seek to find the temperature on the interior cylinder by using data on the exterior cylinder. We reformulate the Cauchy problem as an operator equation, with a compact operator. The operator equation is solved using the Landweber method and the convergence is investigated. 

In Paper IV, the Cauchy problem for a more general degenerate elliptic equation is considered. We stabilize the computations using Tikhonov regularization. The normal equation, in the Tikhonov algorithm, is solved using the Conjugate gradient method. The regularization parameter is picked using either the L–curve or the Discrepancy principle. 

In all papers, numerical examples are given where we solve the various boundary value problems using a finite difference scheme. The results show that the suggested methods work quite well. 

Abstract [sv]

I denna avhandling studerar vi Cauchy-problem för elliptiska och degenererade elliptiska ekvationer. Dessa problem är illa ställda. Vi delar upp randen till området i två delar. På en av dem, säg Γ0, har vi Cauchy–data tillgängligt, och på den återstående delen, Γ1 introducerar vi okända Robin-villkor.

För att konstruera operatorekvationen som ersätter vårt Cauchy-problem använder vi två randvärdesproblem (BVP). Det första problemet är ett BVP med Robin–villkor på Γ1 och Dirichlet–villkor på Γ0. Det andra problemet är ett BVP med Dirichlet–data på Γ1 och med Robin–data på Γ0. Dessa problem är välställda om parametrar i Robinvillkoren väljs lämpligt. Det första Dirichlet–Robin problemet används för att konstruera operatorekvationen som ersätter Cauchy problemet, och det andra Robin–Dirichlet-problemet används för att definiera den adjungerande operatorn. Vi kan sedan tillämpa olika regulariseringsmetoder och återskapa lösningen till problemet på ett stabilt sätt.

I Artikel I analyseras Cauchy-problemet för den elliptiska ekvationen med variabla koefficienter, vilket inkluderar ekvationer av Helmholtz-typ. Ett bevis som visar att den Dirichlet–Robin alternerande algoritmen är konvergent ges, förutsatt att parametrarna i Robin–villkoren väljs på lämpligt sätt. Numeriska experiment som illustrerar algoritmens beteende ges. I synnerhet visar vi hur konvergen-shastigheten beror på valet av Robin-parametrar.

I Artikel II behandlas Cauchy–problemet för Helmholtz ekvation, för medelstora vågtal k2. Cauchy–problemet omformuleras som en operatorekvation och iterativa metoder, baserade på Krylov rum, implementeras. Syftet är att uppnå snabbare konvergens jämfört med den ursprungliga alternerande algoritmen som studerades i den föregående artikeln. Vi diskuterar också hur algoritmerna kan anpassas till fallet med icke-symmetriska differentialoperatorer.

I Artikel III tittar vi på ett stationärt värmeledningsproblem i en tunn platta. Plattan sammanbinder två cylindriska behållare och fixerar deras relativa position. En tvådimensionell matematisk modell av värmeledning i plattan härleds. Eftersom plattan har vassa kanter på sidorna får vi en degenererad elliptisk ekvation. Vi försöker hitta temperaturen på den inre cylindern genom att använda data på den yttre cylindern. Vi omformulerar Cauchy–problemet som en operatorekvation, med en kompakt operator. Operatorekvationen löses med Landwebers metod och konvergensen undersöks.

I Artikel IV behandlas Cauchy problemet för en mer allmän degenererad elliptisk ekvation. Vi stabiliserar beräkningarna med hjälp av Tikhonov–regularisering, där normal ekvationen löses med Konjugerade gradientmetoden. Reguleringsparametern väljs med antingen L–kurva eller Diskrepansprincipen.

I alla artiklar ges numeriska exempel där vi löser de olika randvärdesproblemen med hjälp av finita differenser. Resultaten visar att de föreslagna metoderna fungerar ganska bra.  

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2024. p. 19
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2371
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-201114 (URN)10.3384/9789180755047 (DOI)9789180755030 (ISBN)9789180755047 (ISBN)
Public defence
2024-03-26, BL32 (Nobel), B-building, Campus Valla, Linköping, 13:15 (English)
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Supervisors
Available from: 2024-02-22 Created: 2024-02-22 Last updated: 2024-02-22Bibliographically approved

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Chepkorir, JenniferBerntsson, FredrikKozlov, Vladimir

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