Solution of inverse heat conduction equation with the use of Chebyshev polynomials

• Autorzy: Joachimiak M. , Frąckowiak A. , Ciałkowski M.

Identyfikatory

DOI

10.1515/aoter-2016-0028

• Języki publikacji: EN

Abstrakty

EN

A direct problem and an inverse problem for the Laplace’s equation was solved in this paper. Solution to the direct problem in a rectangle was sought in a form of finite linear combinations of Chebyshev polynomials. Calculations were made for a grid consisting of Chebyshev nodes, what allows us to use orthogonal properties of Chebyshev polynomials. Temperature distributions on the boundary for the inverse problem were determined using minimization of the functional being the measure of the difference between the measured and calculated values of temperature (boundary inverse problem). For the quasi-Cauchy problem, the distance between set values of temperature and heat flux on the boundary was minimized using the least square method. Influence of the value of random disturbance to the temperature measurement, of measurement points (distance from the boundary, where the temperature is not known) arrangement as well as of the thermocouple installation error on the stability of the inverse problem was analyzed.

Słowa kluczowe

EN

Laplace’s equation; boundary inverse problem; quasi-Cauchy problem; stability of the inverse problem

PL

równanie Laplace'a; brzegowe zagadnienie odwrotne; problem Quasi-Cauchy'ego

• Wydawca: Wydawnictwo Instytutu Maszyn Przepływowych PAN ; Komitet Termodynamiki i Spalania PAN
• Czasopismo: Archives of Thermodynamics
• Rocznik: 2016
• Tom: Vol. 37, no. 4
• Strony: 73--88
• Opis fizyczny: Bibliogr. 16 poz., rys., tab., wz.

Twórcy

autor

Joachimiak M.

• Poznan University of Technology, Chair of Thermal Engineering, Piotrowo 3, 60-965 Poznań, Poland

autor

Frąckowiak A.

• Poznan University of Technology, Chair of Thermal Engineering, Piotrowo 3, 60-965 Poznań, Poland

autor

Ciałkowski M.

• Poznan University of Technology, Chair of Thermal Engineering, Piotrowo 3, 60-965 Poznań, Poland

Bibliografia

• [1] Yaparova N.: Numerical methods for solving a boundary-value inverse heat conduction problem. Inverse Probl. Sci. En. 22(2014), 5, 832–847.
• [2] Solodusha S.V., Yaparova N.M.: Numerical solving an inverse boundary value problem of heat conduction using Volterra equations of the first kind. Num. Anal. Appl. 8(2015), 3, 267–274.
• [3] Xiong X.T., Hon Y.C.: Regularization error analysis on a one-dimensional inverse heat conduction problem in multilayer domain. Inverse Probl. Sci. En. 21(2013), 5, 865–887.
• [4] Frąckowiak A., Botkin N.D., Ciałkowski M., Hoffmann K.H.: Iterative algorithm for solving the inverse heat conduction problems with the unknown source function. Inverse Probl. Sci. En. 23(2015), 6, 1056–1071.
• [5] Liu C.S.: An iterative algorithm for identifying heat source by using a DQ and a Lie-group method. Inverse Probl. Sci. En. 23(2015), 1, 67–92.
• [6] Nguyen H.T., Luu V.C.H.: Two new regularization methods for solving sideways heat equation. J. Inequal. Appl. 65(2015), 1–17.
• [7] Shidfar A., Karamali G.R., Damirchi J.: An inverse heat conduction problem with a nonlinear source term. Nonlinear Analysis 65(2006), 615–621.
• [8] Wang F., Chen W., Qu W., Gu Y.: A BEM formulation in conjunction with parametric equation approach for three-dimensional Cauchy problems of steady heat conduction. Eng. Anal. Bound. Elem. 63(2016), 1–14.
• [9] Liu C.S., Kuo C.L., Chang J.R.: Recovering a heat source and initial value by a Lie-group differential algebraic equations method. Numer. Heat Tr. B-Fund. 67(2015), 231–254.
• [10] Ciałkowski M., Grysa K.: A sequential and global method of solving an inverse problem of heat conduction equation. J. Theor. App. Mech.-Pol. 48(2010), 1, 111–134.
• [11] Joachimiak M., Ciałkowski M.: Optimal choice of integral parameter in a process of solving the inverse problem for heat equation. Arch. Thermodyn. 35(2014), 3, 265–280.
• [12] Mierzwiczak M., Kołodziej J.A.: The determination temperature-dependent thermal conductivity as inverse steady heat conduction problem. Int. J. Heat . Mass Tran. 54(2011), 790–796.
• [13] Taler J., Taler D., Ludowski P.: Measurements of local heat flux to membrane water walls of combustion chambers. Fuel 115(2014), 70–83.
• [14] Taler D., Sury A.: Inverse heat transfer problem in digital temperature control in plate fin and tube heat exchangers. Arch. Thermodyn. 32(2011), 4, 17–31.
• [15] Marois M.A., M. Désilets M., Lacroix M.: What is the most suitable fixed grid solidification method for handling time-varying inverse Stefan problems in high temperature industrial furnaces?. Int. J. Heat Mass Tran. 55(2012), 5471–5478.
• [16] Paszkowski S.: Numerical application of multinomials and Chebyshev series. PWN, Warsaw 1975 (in Polish).