Dual-phase lag equation. Stability conditions of a numerical algorithm based on the explicit scheme of the finite difference method

• Autorzy: Majchrzak E. , Mochnacki B.

Identyfikatory

DOI

10.17512/jamcm.2016.3.09

• Języki publikacji: EN

Abstrakty

EN

The dual-phase lag equation (DPLE) is considered. This equation belongs to the group of hyperbolic PDE, contains a second order time derivative and higher order mixed derivative in both time and space. From the engineer’s point of view, the DPLE results from the generalized form of the Fourier law. It is applied as a mathematical model of thermal processes proceeding in the micro-scale and also in the case of bio-heat transfer problem analysis. At the stage of numerical computations the different approximate methods of the PDE solving can be used. In this paper, the authors present the considerations concerning the stability conditions of the explicit scheme of finite difference method (FDM). The appropriate conditions have been found using the von Neumann analysis. In the final part of the paper, the results of testing computations are shown.

Słowa kluczowe

EN

heat conduction; dual-phase lag equation; finite difference method; stability conditions of FDM explicit scheme

PL

przewodzenie ciepła; metoda różnic skończonych

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2016
• Tom: Vol. 15, nr 3
• Strony: 89--96
• Opis fizyczny: Bibliogr. 12 poz., rys.

Twórcy

autor

Majchrzak E.

• Silesian University of Technology Gliwice, Poland

autor

Mochnacki B.

• University of Occupational Safety Management in Katowice Katowice, Poland

Bibliografia

• [1] Cattaneo M.C., A form of heat conduction equation which eliminates the paradox of instantaneous propagation, C.R. Acad. Sci. I - Math. 1958, 247, 431-433.
• [2] Tzou D.Y., Macro- to Microscale Heat Transfer: The Lagging Behavior, John Wiley & Sons, Ltd. 2015.
• [3] Zhang Z.M., Nano/Microscale Heat Transfer, McGraw-Hill, New York 2007.
• [4] Dai W., Nassar R., A domain decomposition method for solving three-dimensional heat transport equations in a double-layered thin film with microscale thickness, Numerical Heat Transfer 2000, Part A, 38, 243-255.
• [5] Antaki P.J., New interpretation of non-Fourier heat conduction in processed meat, ASME J. Heat Transfer 2005,127, 189-193.
• [6] Roetzel W., Putra N., Das S.K., Experiment and analysis for non-Fourier conduction in materials with non-homogeneous inner structure, Int. J. Therm. Sci. 2003, 42, 541-552.
• [7] McDonough J.M., Kunadian I., Kumar R.R., Yang T., An alternative discretization and solution procedure for the dual phase-lag equation, Journal of Computational Physics 2006, 219, 163-171.
• [8] Young D.M., Iterative Solution of Large Linear Systems, Academic Press, New York 1971.
• [9] Majchrzak E., Mochnacki B., Sensitivity analysis of transient temperature field in microdomains with respect to the dual-phase-lag model parameters, Journal for Multiscale Computational Engineering 2014, 12(1), 65-77.
• [10] Mochnacki B., Ciesielski M., Micro-scale heat transfer. Algorithm basing on the Control Volume Method and the identification af relaxation and thermalization times using the search method, Computer Methods in Material Science 2015, 15, 2, 353-361.
• [11] Ciesielski M., Mochnacki B., Application of the control volume method using the Voronoi polygons for numerical modeling of bio-heat transfer problems, Journal of Theoretical and Applied Mechanics 2014, 52, 4, 927-935.
• [12] Mochnacki B., Majchrzak E., Numerical modeling of casting solidification using generalized finite difference method, THERMEC 2009, PTS 1-4 Book Series: Materials Science Forum 2010, 638-642, 2676-2681.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.