NUMERICAL RESOLUTION OF AN EXACT HEAT CONDUCTION MODEL WITH A DELAY TERM

Marco Campo; José R. Fernández; Ramón Quintanilla

doi:10.11948/2019.332

2019 Volume 9 Issue 1

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Marco Campo, José R. Fernández, Ramón Quintanilla. NUMERICAL RESOLUTION OF AN EXACT HEAT CONDUCTION MODEL WITH A DELAY TERM[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 332-344. doi: 10.11948/2019.332

Citation:

Marco Campo, José R. Fernández, Ramón Quintanilla. NUMERICAL RESOLUTION OF AN EXACT HEAT CONDUCTION MODEL WITH A DELAY TERM[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 332-344. doi: 10.11948/2019.332

NUMERICAL RESOLUTION OF AN EXACT HEAT CONDUCTION MODEL WITH A DELAY TERM

1.
Departamento de Matemáticas, ETS de Ingenieros de Caminos, Canales y Puertos, Universidade da Coruña, Campus de Elviña, 15071 A Coruña, Spain
2.
Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
3.
Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, 08222 Terrassa, Barcelona, Spain

Corresponding author: Email address: jose.fernandez@uvigo.es(J.R. Fernández)
Fund Project: The work of M. Campo and J. R. Fernández has been supported by the Ministerio de Economía y Competitividad under the research project MTM2015-66640-P (with the participation of FEDER). The work of R. Quintanilla has been supported by the Ministerio de Economía y Competitividad under the research project MTM2016-74934-P (AEI/FEDER, UE)

Abstract

Abstract

In this paper we analyze, from the numerical point of view, a dynamic thermoelastic problem. Here, the so-called exact heat conduction model with a delay term is used to obtain the heat evolution. Thus, the thermomechanical problem is written as a coupled system of partial differential equations, and its variational formulation leads to a system written in terms of the velocity and the temperature fields. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, from which the linear convergence of the algorithm could be derived under suitable additional regularity conditions. Finally, a two-dimensional numerical example is solved to show the accuracy of the approximation and the decay of the discrete energy.
- Thermoelasticity /
- exact heat condution /
- delay parameter /
- finite elements /
- a priori error estimates
MSC: 74K10, 35G50, 37N15, 74F05, 65M60, 65M12

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References

[1]	M. Campo, J. R. Fernández, K. L. Kuttler, M. Shillor and J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Engrg., 2006, 196(1-3), 476-488. doi: 10.1016/j.cma.2006.05.006 CrossRef Google Scholar
[2]	D. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, ASME. Appl. Mech. Rev., 1998, 51(12), 705-729. doi: 10.1115/1.3098984 CrossRef Google Scholar
[3]	S. K. R. Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 2007, 30, 231-238. doi: 10.1080/01495730601130919 CrossRef Google Scholar
[4]	P. G. Ciarlet, Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds., 1993, vol I, 17-351. Google Scholar
[5]	M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Letters, 2009, 22, 1374-1379. doi: 10.1016/j.aml.2009.03.010 CrossRef Google Scholar
[6]	M. Fabrizio and F. Franchi, Delayed thermal models: stability and thermodynamics, J. Thermal Stresses, 2014, 37, 160-173. doi: 10.1080/01495739.2013.839619 CrossRef Google Scholar
[7]	A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 1992, 15, 253-264. doi: 10.1080/01495739208946136 CrossRef Google Scholar
[8]	A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 1993, 31, 189-208. doi: 10.1007/BF00044969 CrossRef Google Scholar
[9]	F. Hecht, New development in FreeFem++, J. Numer. Math., 2012, 20(3-4), 251-265. Google Scholar
[10]	R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 1999, 22, 451-470. doi: 10.1080/014957399280832 CrossRef Google Scholar
[11]	R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, Int. J. Solids Struct. 2000, 37, 215-224. doi: 10.1016/S0020-7683(99)00089-X CrossRef Google Scholar
[12]	J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford Mathematical Monographs, Oxford, 2010. Google Scholar
[13]	S. Kant, M. Gupta, O. N. Shivay and S. Mukhopadhyay, An investigation on a two-dimensional problem of Mode-I crack in a thermoelastic medium, Z. Ang. Math. Phys., 2018, 69, 21. doi: 10.1007/s00033-018-0914-0 CrossRef Google Scholar
[14]	A. Kumar and S. Mukhopadhyay, Investigation on the effects of temperature dependency of material parameters on a thermoelastic loading problem, Z. Ang. Math. Phys., 2017, 68, 98. doi: 10.1007/s00033-017-0843-3 CrossRef Google Scholar
[15]	B. Kumari, A. Kumar and S. Mukhopadhyay, Investigation of harmonic plane waves: detailed analysis of recent thermoplastic model with single delay term, Math. Mech. Solids, 2018. DOI: 10.1177/1081286518755232. CrossRef Google Scholar
[16]	B. Kumari and S. Mukhopadhyay, Fundamental solutions of thermoelasticity with a recent heat conduction model with a single delay term, J. Thermal Stresses, 2017, 40, 866-878. doi: 10.1080/01495739.2017.1281715 CrossRef Google Scholar
[17]	B. Kumari, and S. Mukhopadhyay, Some theorems on linear theory of thermoelasticity for an anisotropic medium under an exact heat conduction model with a delay, Math. Mech. Solids, 2017, 22, 1177-1189. doi: 10.1177/1081286515620263 CrossRef Google Scholar
[18]	M. C. Leseduarte and R. Quintanilla, Phragmen-Lindelof alternative for an exact heat conduction equation with delay, Comm. Pure Appl. Anal., 2013, 12, 1221-1235. Google Scholar
[19]	R. Quintanilla, A well-posed problem for the Dual-Phase-Lag heat conduction, J. Thermal Stresses, 2008, 31, 260-269. doi: 10.1080/01495730701738272 CrossRef Google Scholar
[20]	R. Quintanilla, A well-posed problem for the Three-Dual-Phase-Lag heat conduction, J. Thermal Stresses, 2009, 32, 1270-1278. doi: 10.1080/01495730903310599 CrossRef Google Scholar
[21]	R. Quintanilla, Some solutions for a family of exact phase-lag heat conduction problems, Mech. Res. Commun., 2011, 38, 355-360. doi: 10.1016/j.mechrescom.2011.04.008 CrossRef Google Scholar
[22]	R. Quintanilla, On uniqueness and stability for a thermoelastic theory, Math. Mech. Solids, 2017, 22(6), 1387-396. doi: 10.1177/1081286516634154 CrossRef Google Scholar
[23]	O. N. Shivay and S. Mukhopadhyay, Some basic theorems on a recent model of linear thermoelasticity for a homogeneous and isotropic medium, Math. Mech. Solids, 2018. DOI: 10.1177/1081286518762612. CrossRef Google Scholar
[24]	D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 1995, 117, 8-16. doi: 10.1115/1.2822329 CrossRef Google Scholar
[25]	D. Y. Tzou, The generalized lagging response in small-scale and high-rate heating, Int. J. Heat Mass Transfer, 1995, 38, 3231-3240. doi: 10.1016/0017-9310(95)00052-B CrossRef Google Scholar
[26]	L. Wang, X. Zhou and X. Wei, Heat Conduction: Mathematical Models and Analyticial solutions, Springer, 2008. Google Scholar