Application of numerical methods for solving the non-Fourier equations. A review of our own and collaborators’ works

• Autorzy: Majchrzak E. , Mochnacki B.

Identyfikatory

DOI

10.17512/jamcm.2018.2.04

• Języki publikacji: EN

Abstrakty

EN

Thermal processes occuring in the solid bodies are, as a rule, described by the well-known Fourier equation (or the system of these equations) supplemented by the appropriate boundary and initial conditions. Such a mathematical model is sufficiently exact to describe the heat transfer processes in the macro scale for the typical materials. It turned out that the energy equation based on the Fourier law has the limitations and it should not be used in the case of the microscale heat transfer and also in the case of materials with a special inner structure (e.g. biological tissue). The better approximation of the real thermal processes assure the modifications of the energy equation, in particular the models in which the so-called lag times are introduced. The article presented is devoted to the numerical aspects of solving these types of equations (in the scope of the microscale heat transfer). The results published by the other authors can be found in the references posted in the works cited below.

Słowa kluczowe

EN

heat transfer; non-Fourier heat diffusion models; numerical methods; computational mechanics

PL

proces termiczny; wymiana ciepła; metody numeryczne; mechanika obliczeniowa; model dyfuzji cieplnej

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2018
• Tom: Vol. 17, nr 2
• Strony: 43--50
• Opis fizyczny: Bibliogr. 17 poz., rys.

Twórcy

autor

Majchrzak E.

• Institute of Computational Mechanics and Engineering, Silesian University of Technology Gliwice, Poland

autor

Mochnacki B.

• University of Occupational Safety Management in Katowice, Katowice, Poland

Bibliografia

• [1] Ciesielski, M. (2017). Analytical solution of the dual phase lag equation describing the laser heating of thin metal film. Journal of Applied Mathematics and Computational Mechanics, 16(1), 33-40.
• [2] Majchrzak, E., & Mochnacki, B. (2016). Dual-phase lag equation. Stability conditions of a numerical algorithm based on the explicit scheme of the finite difference method. Journal of Applied Mathematics and Computational Mechanics, 15(3), 89-96.
• [3] Mochnacki, B., & Tuzikiewicz, W. (2016). Cattaneo-Vernotte bioheat transfer equation. Stability conditions of numerical algorithm based on the explicit scheme of the finite difference method. Journal of Applied Mathematics and Computational Mechanics, 15(4), 137-144.
• [4] Majchrzak, E., & Mochnacki, B. (2017). Implicit scheme of the finite difference method for 1D dual-phase lag equation. Journal of Applied Mathematics and Computational Mechanics, 16(3), 37-46.
• [5] Majchrzak, E., & Mochnacki, B. (2018). Implicit scheme of the finite difference method for the second-order dual phase lag equation. Journal of Theoretical and Applied Mechanics, 56(2), 393-402.
• [6] Mochnacki, B., & Ciesielski, M. (2017). Numerical solution of the dual phase lag equation using the control volume method and Crank-Nicolson scheme. ECCOMAS MSF Thematic Conference, Ljubljana, Slovenia, 196-199.
• [7] Ciesielski, M. (2017). Application of the alternating direction implicit method for numerical solution od the dual phase lag equation. Journal of Theoretical and Applied Mechanics, 55(3), 839-852.
• [8] Majchrzak, E., Mochnacki, B., & Suchy, J.S. (2009). Numerical simulation of thermal processes proceeding in a multi-layered film subjected to ultrafast laser heating. Journal of Theoretical and Applied Mechanics, 47(2), 383-396.
• [9] Majchrzak, E., Mochnacki, B., Greer, A.L., & Suchy, J.S. (2009). Numerical modeling of short pulse laser interactions with multi-layered thin metal films. Computer Modeling in Engineering and Sciences, 41, 2, 131-146.
• [10] Majchrzak, E., & Kałuża, G. (2017). Analysis of thermal processes occurring in the heated multilayered metal films using the dual-phase lag model. Archives of Mechanics, 69, 4-5, 275-287.
• [11] Turchan, Ł. (2017). Solving the dual-phase lag bioheat transfer equation by the generalized finite difference method. Archives of Mechanics, 69, 4-5, 389-407.
• [12] Majchrzak, E., & Mochnacki, B. (2017). Numerical model of thin metal film heating using the boundary element method. Computer Methods in Materials Science, 17, 1, 12-17.
• [13] Mochnacki, B., & Ciesielski, M. (2016). Dual phase lag model of melting process in domain of metal film subjected to an external heat flux. Archives of Foundry Engineering, 16, 4, 85-90.
• [14] Ciesielski, M., Duda, M., & Mochnacki, B. (2016). Comparison of bio-heat transfer numerical models based on the Pennes and Cattaneo-Vernotte equations. Journal of Applied Mathematics and Computational Mechanics, 15, 4, 33-38.
• [15] Mochnacki, B., & Majchrzak, E. (2017). Numerical model of thermal interactions between cylindrical cryoprobe and biological tissue using the dual-phase lag equation. International Journal of Heat and Mass Transfer, 108, 1-10.
• [16] Kałuża, G., Majchrzak, E., & Turchan Ł. (2017). Sensitivity analysis of temperature field in the heated soft tissue with respect to the perturbations of porosity. Applied Mathematical Modelling, 49, 498-513.
• [17] Majchrzak, E., Jasiński, M., & Turchan Ł. (2017). Modeling of laser-soft tissue interactions using the dual-phase lag equation: sensitivity analysis with respect to selected tissue parameters. Defect and Diffusion Forum, 379, 108-123.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).