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April  2022, 27(4): 2221-2245. doi: 10.3934/dcdsb.2021130

On existence and numerical approximation in phase-lag thermoelasticity with two temperatures

1. 

Departamento de Matemáticas, ETS de Ingenieros de Caminos, Canales y Puertos, Universidade da Coruña, Campus de Elviña, 15071 A Coruña, Spain

2. 

Laboratório de Análise Numérica e Astrofísica, Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil

3. 

Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain

4. 

Departament de Matemàtiques, Universitat Politècnica de Catalunya, C. Colom 11, 08222 Terrassa, Barcelona, Spain

* Corresponding author: José R. Fernández

Received  August 2020 Revised  March 2021 Published  April 2022 Early access  April 2021

Fund Project: The work of M. Campo and J.R. Fernández has been partially supported by Ministerio de Ciencia, Innovación y Universidades under the research project PGC2018-096696-B-I00 (FEDER, UE). The work of M.I.M. Copetti has been partially supported by the Brazilian institution CNPq (grant 304709/2017-4). The work of R. Quintanilla has been supported by Ministerio de Economía y Competitividad under the research project "Análisis Matemático de Problemas de la Termomecánica" (MTM2016-74934-P), (AEI/FEDER, UE), and Ministerio de Ciencia, Innovación y Universidades under the research project "Análisis matemático aplicado a la termomecánica" (PID2019-105118GB-I00). The authors want to thank to the anonymous referees their useful comments which have allowed us to improve the paper

In this work we study from both variational and numerical points of view a thermoelastic problem which appears in the dual-phase-lag theory with two temperatures. An existence and uniqueness result is proved in the general case of different Taylor approximations for the heat flux and the inductive temperature. Then, in order to provide the numerical analysis, we restrict ourselves to the case of second-order approximations of the heat flux and first-order approximations for the inductive temperature. First, variational formulation of the corresponding problem is derived and an energy decay property is proved. Then, a fully discrete scheme is introduced by using the finite element method for the approximation of the spatial variable and the implicit Euler scheme for the discretization of the time derivatives. A discrete stability

Citation: Marco Campo, Maria I. M. Copetti, José R. Fernández, Ramón Quintanilla. On existence and numerical approximation in phase-lag thermoelasticity with two temperatures. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2221-2245. doi: 10.3934/dcdsb.2021130
References:
[1]

I. A. Abdallah, Dual phase lag heat conduction and thermoelastic properties of a semi-infinite medium induced by ultrashort pulsed laser, Prog. Phys., 3 (2009), 60-63. 

[2]

S. Banik and M. Kanoria, Effects of three-phase-lag on two temperatures generalized thermoelasticity for an infinite medium with a spherical cavity, Appl. Math. Mech., 33 (2012), 483-498.  doi: 10.1007/s10483-012-1565-8.

[3]

K. BorgmeyerR. Quintanilla and R. Racke, Phase-lag heat conduction: Decay rates for limit problems and well-posedness, J. Evol. Equ., 14 (2014), 863-884.  doi: 10.1007/s00028-014-0242-6.

[4]

M. CampoJ. R. FernándezK. L. KuttlerM. Shillor and J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Engrg., 196 (2006), 476-488.  doi: 10.1016/j.cma.2006.05.006.

[5]

C. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée, C. R. Acad. Sci. Paris, 247 (1958), 431-433. 

[6]

P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, ZAMP-Z. Angew. Math. Phys., 19 (1968), 614-627.  doi: 10.1007/BF01594969.

[7]

P. J. Chen and W. O. Williams, A note on non-simple heat conduction, ZAMP-Z. Angew. Math. Phys., 19 (1968), 969-970.  doi: 10.1007/BF01602278.

[8]

P. J. ChenM. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, ZAMP-Z. Angew. Math. Phys., 20 (1969), 107-112.  doi: 10.1007/BF01591120.

[9]

S. K. R. Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238.  doi: 10.1080/01495730601130919.

[10]

P. G. Ciarlet, Basic error estimates for elliptic problems, In: Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions eds., vol II 1991, 17-351.

[11]

M. DreherR. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Lett., 22 (2009), 1374-1379.  doi: 10.1016/j.aml.2009.03.010.

[12]

M. A. EzzatA. S. El-Karamany and S. M. Ezzat, Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer, Nuc. Eng. Des., 252 (2012), 267-277.  doi: 10.1016/j.nucengdes.2012.06.012.

[13]

M. Fabrizio and F. Franchi, Delayed thermal models: Stability and thermodynamics, J. Thermal Stresses, 37 (2014), 160-173.  doi: 10.1080/01495739.2013.839619.

[14]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses 15 (1992), 253-264. doi: 10.1080/01495739208946136.

[15]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.

[16]

M. A. HaderM. A. Al-Nimr and B. A. Abu Nabah, The dual-phase-lag heat conduction model in thin slabs under a fluctuating volumetric thermal disturbance, Int. J. Thermophys., 23 (2002), 1669-1680.  doi: 10.1023/A:1020754304107.

[17]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[18]

A. MagañaA. Miranville and R. Quintanilla, On the stability in phase-lag heat conduction with two temperatures, J. Evol. Equations, 18 (2018), 1697-1712.  doi: 10.1007/s00028-018-0457-z.

[19]

A. MagañaA. Miranville and R. Quintanilla, On the time decay in phase-lag thermoelasticity with two temperatures, Electronic Res. Arch., 27 (2019), 7-19.  doi: 10.3934/era.2019007.

[20]

A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150.  doi: 10.1007/s00245-010-9114-9.

[21]

S. MukhopadhyayR Prasad and R. Kumar, On the theory of two-temperature thermoelasticity with two phase-lags, J. Thermal Stresses, 34 (2011), 352-365.  doi: 10.1080/01495739.2010.550815.

[22]

M. A. OthmanW. M. Hasona and E. M. Abd-Elaziz, Effect of rotation on micropolar generalized thermoelasticity with two temperatures using a dual-phase-lag model, Canadian J. Physics, 92 (2014), 149-158.  doi: 10.1139/cjp-2013-0398.

[23]

R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, J. Non-Equilib. Thermodyn., 27 (2002), 217-227.  doi: 10.1515/JNETDY.2002.012.

[24]

R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, J. Thermal Stresses, 31 (2008), 260-269.  doi: 10.1080/01495730701738272.

[25]

R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278.  doi: 10.1080/01495730903310599.

[26]

R. Quintanilla and P. M. Jordan, A note on the two-temperature theory with dual-phase-lag decay: Some exact solutions, Mech. Research Comm., 36 (2009), 796-803.  doi: 10.1016/j.mechrescom.2009.05.002.

[27]

R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM J. Appl. Math., 66 (2006), 977-1001.  doi: 10.1137/05062860X.

[28]

R. Quintanilla and R. Racke, A note on stability of dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213.  doi: 10.1016/j.ijheatmasstransfer.2005.10.016.

[29]

R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag heat conduction, Proc. Royal Society London A Math. Phys. Eng. Sci., 463 (2007), 659-674.  doi: 10.1098/rspa.2006.1784.

[30]

R. Quintanilla and R. Racke, A note on stability in three-phase-lag heat conduction, Int. J. Heat Mass Transfer, 51 (2008), 24-29.  doi: 10.1016/j.ijheatmasstransfer.2007.04.045.

[31]

R. Quintanilla and R. Racke, Spatial behavior in phase-lag heat conduction, Differ. Integral Equ., 28 (2015), 291-308. 

[32]

S. A. Rukolaine, Unphysical effects of the dual-phase-lag model of heat conduction, Int. J. Heat Mass Transfer, 78 (2014), 58-63.  doi: 10.1016/j.ijheatmasstransfer.2014.06.066.

[33]

D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16.  doi: 10.1115/1.2822329.

[34]

W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelaticity, Acta Mech., 16 (1973), 83-117. 

[35]

Y. Zhang, Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues, Int. J. Heat Mass Transfer, 52 (2009), 4829-4834.  doi: 10.1016/j.ijheatmasstransfer.2009.06.007.

show all references

References:
[1]

I. A. Abdallah, Dual phase lag heat conduction and thermoelastic properties of a semi-infinite medium induced by ultrashort pulsed laser, Prog. Phys., 3 (2009), 60-63. 

[2]

S. Banik and M. Kanoria, Effects of three-phase-lag on two temperatures generalized thermoelasticity for an infinite medium with a spherical cavity, Appl. Math. Mech., 33 (2012), 483-498.  doi: 10.1007/s10483-012-1565-8.

[3]

K. BorgmeyerR. Quintanilla and R. Racke, Phase-lag heat conduction: Decay rates for limit problems and well-posedness, J. Evol. Equ., 14 (2014), 863-884.  doi: 10.1007/s00028-014-0242-6.

[4]

M. CampoJ. R. FernándezK. L. KuttlerM. Shillor and J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Engrg., 196 (2006), 476-488.  doi: 10.1016/j.cma.2006.05.006.

[5]

C. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée, C. R. Acad. Sci. Paris, 247 (1958), 431-433. 

[6]

P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, ZAMP-Z. Angew. Math. Phys., 19 (1968), 614-627.  doi: 10.1007/BF01594969.

[7]

P. J. Chen and W. O. Williams, A note on non-simple heat conduction, ZAMP-Z. Angew. Math. Phys., 19 (1968), 969-970.  doi: 10.1007/BF01602278.

[8]

P. J. ChenM. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, ZAMP-Z. Angew. Math. Phys., 20 (1969), 107-112.  doi: 10.1007/BF01591120.

[9]

S. K. R. Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238.  doi: 10.1080/01495730601130919.

[10]

P. G. Ciarlet, Basic error estimates for elliptic problems, In: Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions eds., vol II 1991, 17-351.

[11]

M. DreherR. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Lett., 22 (2009), 1374-1379.  doi: 10.1016/j.aml.2009.03.010.

[12]

M. A. EzzatA. S. El-Karamany and S. M. Ezzat, Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer, Nuc. Eng. Des., 252 (2012), 267-277.  doi: 10.1016/j.nucengdes.2012.06.012.

[13]

M. Fabrizio and F. Franchi, Delayed thermal models: Stability and thermodynamics, J. Thermal Stresses, 37 (2014), 160-173.  doi: 10.1080/01495739.2013.839619.

[14]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses 15 (1992), 253-264. doi: 10.1080/01495739208946136.

[15]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.

[16]

M. A. HaderM. A. Al-Nimr and B. A. Abu Nabah, The dual-phase-lag heat conduction model in thin slabs under a fluctuating volumetric thermal disturbance, Int. J. Thermophys., 23 (2002), 1669-1680.  doi: 10.1023/A:1020754304107.

[17]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[18]

A. MagañaA. Miranville and R. Quintanilla, On the stability in phase-lag heat conduction with two temperatures, J. Evol. Equations, 18 (2018), 1697-1712.  doi: 10.1007/s00028-018-0457-z.

[19]

A. MagañaA. Miranville and R. Quintanilla, On the time decay in phase-lag thermoelasticity with two temperatures, Electronic Res. Arch., 27 (2019), 7-19.  doi: 10.3934/era.2019007.

[20]

A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150.  doi: 10.1007/s00245-010-9114-9.

[21]

S. MukhopadhyayR Prasad and R. Kumar, On the theory of two-temperature thermoelasticity with two phase-lags, J. Thermal Stresses, 34 (2011), 352-365.  doi: 10.1080/01495739.2010.550815.

[22]

M. A. OthmanW. M. Hasona and E. M. Abd-Elaziz, Effect of rotation on micropolar generalized thermoelasticity with two temperatures using a dual-phase-lag model, Canadian J. Physics, 92 (2014), 149-158.  doi: 10.1139/cjp-2013-0398.

[23]

R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, J. Non-Equilib. Thermodyn., 27 (2002), 217-227.  doi: 10.1515/JNETDY.2002.012.

[24]

R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, J. Thermal Stresses, 31 (2008), 260-269.  doi: 10.1080/01495730701738272.

[25]

R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278.  doi: 10.1080/01495730903310599.

[26]

R. Quintanilla and P. M. Jordan, A note on the two-temperature theory with dual-phase-lag decay: Some exact solutions, Mech. Research Comm., 36 (2009), 796-803.  doi: 10.1016/j.mechrescom.2009.05.002.

[27]

R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM J. Appl. Math., 66 (2006), 977-1001.  doi: 10.1137/05062860X.

[28]

R. Quintanilla and R. Racke, A note on stability of dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213.  doi: 10.1016/j.ijheatmasstransfer.2005.10.016.

[29]

R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag heat conduction, Proc. Royal Society London A Math. Phys. Eng. Sci., 463 (2007), 659-674.  doi: 10.1098/rspa.2006.1784.

[30]

R. Quintanilla and R. Racke, A note on stability in three-phase-lag heat conduction, Int. J. Heat Mass Transfer, 51 (2008), 24-29.  doi: 10.1016/j.ijheatmasstransfer.2007.04.045.

[31]

R. Quintanilla and R. Racke, Spatial behavior in phase-lag heat conduction, Differ. Integral Equ., 28 (2015), 291-308. 

[32]

S. A. Rukolaine, Unphysical effects of the dual-phase-lag model of heat conduction, Int. J. Heat Mass Transfer, 78 (2014), 58-63.  doi: 10.1016/j.ijheatmasstransfer.2014.06.066.

[33]

D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16.  doi: 10.1115/1.2822329.

[34]

W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelaticity, Acta Mech., 16 (1973), 83-117. 

[35]

Y. Zhang, Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues, Int. J. Heat Mass Transfer, 52 (2009), 4829-4834.  doi: 10.1016/j.ijheatmasstransfer.2009.06.007.

Figure 1.  Example 1: Asymptotic behavior of the numerical scheme
Figure 2.  Example 1: Energy evolution in absolute and semilogarithmic scales
Figure 3.  Example 2: Evolution in time of the temperature and inductive temperature at point $ {\boldsymbol{x}} = (4, 0.5) $ for different values of parameter $ m $
Figure 4.  Example 2: Evolution in time of the temperature and inductive temperature at point $ {\boldsymbol{x}} = (1, 0.5) $ for different values of parameter $ m $
Figure 5.  Example 2: Evolution in time of the horizontal and vertical displacements at point $ {\boldsymbol{x}} = (1, 0.5) $ for different values of parameter $ m $
Table 1.  Example 1: Numerical errors ($ \times 100 $) for some $ nd $ and $ k $
$ n_{el} \downarrow k \to $ 0.02 0.01 0.005 0.001 0.0001
8 0.0986766 0.0985441 0.0993091 0.1004371 0.1018615
16 0.0368700 0.0313313 0.0303459 0.0308803 0.0315329
32 0.0246722 0.0145489 0.0105503 0.0090745 0.0093450
64 0.0244400 0.0124476 0.0066725 0.0030114 0.0028566
128 0.0245822 0.0124424 0.0063666 0.0016409 0.0009504
$ n_{el} \downarrow k \to $ 0.02 0.01 0.005 0.001 0.0001
8 0.0986766 0.0985441 0.0993091 0.1004371 0.1018615
16 0.0368700 0.0313313 0.0303459 0.0308803 0.0315329
32 0.0246722 0.0145489 0.0105503 0.0090745 0.0093450
64 0.0244400 0.0124476 0.0066725 0.0030114 0.0028566
128 0.0245822 0.0124424 0.0063666 0.0016409 0.0009504
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