×

A spacetime discontinuous Galerkin method for hyperbolic heat conduction. (English) Zbl 1194.80118

Summary: Non-Fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation. For applications involving very short length or time scales, hyperbolic conduction models better represent the physical thermal transport processes. This paper reviews the Maxwell-Cattaneo-Vernotte modification of the Fourier conduction law and describes its implementation within a spacetime discontinuous Galerkin (SDG) finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time. A causal, advancing-front meshing procedure enables a patch-wise solution procedure with linear complexity in the number of spacetime elements. An \(h\)-adaptive scheme and a special SDG shock-capturing operator accurately resolve sharp solution features in both space and time. Numerical results for one spatial dimension demonstrate the convergence properties of the SDG method as well as the effectiveness of the shock-capturing method. Simulations in two spatial dimensions demonstrate the proposed method’s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place.

MSC:

80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
80A20 Heat and mass transfer, heat flow (MSC2010)

Software:

Triangle
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bubnov, V. A., Wave concepts in the theory of heat, Int. J. Heat Mass Transf., 19, 175-184 (1976) · Zbl 0316.35045
[2] Joseph, D. D.; Preziosi, L., Heat waves, Rev. Mod. Phys., 61, 1, 41-73 (1989) · Zbl 1129.80300
[3] Joseph, D. D.; Preziosi, L., Addendum to the paper “Heat waves”, Rev. Mod. Phys., 62, 2, 375-391 (1990)
[4] Ackerman, C. C.; Berman, B.; Fairbank, H. A.; Guyer, R. A., Second sound in solid helium, Phys. Rev. Lett., 16, 18, 789-791 (1966)
[5] Jackson, H. E.; Walker, C. T.; McNelly, T. F., Second sound in NaF, Phys. Rev. Lett., 25, 1, 26-28 (1970)
[6] Narayanamurti, V.; Dynes, R. C., Observation of second sound in bismuth, Phys. Rev. Lett., 28, 22, 1461-1465 (1972)
[7] Maxwell, J. C., On the dynamical theory of gases, Philos. Trans. Roy. Soc. Lond., 157, 49-88 (1867)
[8] C. Cattaneo, On the conduction of heat, Atti del Seminario Matematico e Fisico dell’Universitá di Modena 3 (3).; C. Cattaneo, On the conduction of heat, Atti del Seminario Matematico e Fisico dell’Universitá di Modena 3 (3).
[9] Cattaneo, C., Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, C.R. Acad. Sci., 247, 4, 431-433 (1958) · Zbl 1339.35135
[10] Vernotte, P., Les paradoxes de la théorie continue de l’équation de la chaleur, C.R. Acad. Sci., 246, 22, 3154-3155 (1958) · Zbl 1341.35086
[11] Dreyer, W.; Struchtrup, H., Heat pulse experiments revisited, Continuum Mech. Thermodyn., 5, 3-50 (1993)
[12] Özisik, M. N.; Tzou, D. Y., On the wave theory in heat conduction, ASME J. Heat Transf., 116, 535-536 (1994)
[13] Chandrasekharaiah, D. S., Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev., 51, 12, 705-729 (1998)
[14] Chandrasekharaiah, D. S., Thermoelasticity with second sound: a review, Appl. Mech. Rev., 39, 3, 355-376 (1986) · Zbl 0588.73006
[15] Coleman, B. D.; Hrusa, W. J.; Owen, D. R., Stability of equilibrium for a nonlinear hyperbolic system describing heat propagation by second sound in solids, Arch. Ration. Mech. Anal., 94, 3, 267-289 (1986) · Zbl 0621.73132
[16] Glass, D. E.; Özisik, M. N.; McRae, D. S.; Vick, B., On the numerical solution of hyperbolic heat conduction, Numer. Heat Transf., 8, 497-504 (1985)
[17] Glass, D. E.; Özisik, M. N.; McRae, D. S., Hyperbolic heat conduction with radiation in an absorbing and emitting medium, Numer. Heat Transf., 12, 321-333 (1987)
[18] Liu, L. H.; Tan, H. P.; Tong, T. W., Non-fourier effects on transient temperature response in semitransparent medium caused by laser pulse, Int. J. Heat Mass Transf., 44, 3335-3344 (2001) · Zbl 1013.80011
[19] J. Zhang, J.J. Zhao, High accuracy stable numerical solution of 1D microscale heat transport equation, Technical Report 297-00, Department of Computer Science, University of Kentucky, Lexington, KY, 2000.; J. Zhang, J.J. Zhao, High accuracy stable numerical solution of 1D microscale heat transport equation, Technical Report 297-00, Department of Computer Science, University of Kentucky, Lexington, KY, 2000.
[20] J. Zhang, J.J. Zhao, Unconditionally stable finite difference scheme and iterative solution of 2D microscale heat transport equation, Technical Report 303-00, Department of Computer Science, University of Kentucky, Lexington, KY, 2000.; J. Zhang, J.J. Zhao, Unconditionally stable finite difference scheme and iterative solution of 2D microscale heat transport equation, Technical Report 303-00, Department of Computer Science, University of Kentucky, Lexington, KY, 2000.
[21] J. Zhang, J.J. Zhao, Iterative solution and finite difference approximations to 3D microscale heat transport equation, Technical Report 320-01, Department of Computer Science, University of Kentucky, Lexington, KY, 2000.; J. Zhang, J.J. Zhao, Iterative solution and finite difference approximations to 3D microscale heat transport equation, Technical Report 320-01, Department of Computer Science, University of Kentucky, Lexington, KY, 2000.
[22] Yang, H. Q., Characteristics-based, high-order accurate and nonoscillatory numerical method for hyperbolic heat conduction, Numer. Heat Transf.: Part B - Fund., 18, 221-241 (1990) · Zbl 0711.76093
[23] Yang, H. Q., Solution of two-dimensional hyperbolic heat conduction by high-resolution numerical methods, Numer. Heat Transf.: Part A - Appl., 21, 333-349 (1992)
[24] Shen, W.; Han, S., Numerical solution of two-dimensional axisymmetric hyperbolic heat conduction, Comput. Mech., 29, 122-128 (2002) · Zbl 1013.80008
[25] Shen, W.; Han, S., A numerical solution of two-dimensional hyperbolic heat conduction with non-linear boundary conditions, Heat Mass Transf., 39, 499-507 (2003)
[26] Goodman, J. B.; LeVeque, R. J., On the accuracy of stable schemes for 2D scalar conservation laws, Math. Comput., 45, 171, 15-21 (1985) · Zbl 0592.65058
[27] LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge texts in Applied Mathematics (2002), Cambridge University Press
[28] Chen, H. T.; Lin, J. Y., Numerical analysis for hyperbolic heat conduction, Int. J. Heat Mass Transf., 36, 11, 2891-2898 (1992) · Zbl 0775.73332
[29] Chen, H. T.; Lin, J. Y., Analysis of two-dimensional hyperbolic heat conduction problems, Int. J. Heat Mass Transf., 37, 1, 153-164 (1993)
[30] Carey, G. F.; Tsai, M., Hyperbolic heat transfer with reflection, Numer. Heat Transf., 5, 309-327 (1982)
[31] Manzari, M. T.; Manzari, M. T., A mixed approach to finite element analysis of hyperbolic heat conduction problems, Int. J. Numer. Methods Heat Fluid Flow, 8, 1, 83-96 (1998) · Zbl 0946.74064
[32] Manzari, M. T.; Manzari, M. T., On numerical solution of hyperbolic heat conduction, Commun. Numer. Methods Engrg., 15, 853-866 (1999) · Zbl 0952.65076
[33] Xu, B. Q.; Shen, Z. H.; Lu, J.; Ni, X. W.; Zhang, S. Y., Numerical simulation of laser-induced transient temperature field in film-substrate system by finite element method, Int. J. Heat Mass Transf., 46, 4963-4968 (2003) · Zbl 1048.80009
[34] Ai, X.; Li, B. Q., A discontinuous finite element method for hyperbolic thermal wave problems, J. Engrg. Comput., 21, 5, 577-597 (2004) · Zbl 1074.80004
[35] Ai, X.; Li, B. Q., Numerical simulation of thermal wave propagation during laser processing of thin films, J. Electron. Mater., 34, 5, 583-591 (2005)
[36] Wu, W.; Li, X., Application of the time discontinuous Galerkin finite element method to heat wave simulation, Int. J. Heat Mass Transf., 49, 1679-1684 (2006) · Zbl 1189.80047
[37] Li, X.; Yao, D. M.; Lewis, R. W., A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non-linear solids and saturated porous media, Int. J. Numer. Methods Engrg., 57, 1775-1800 (2003) · Zbl 1062.74621
[38] Jamet, P., Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain, SIAM J. Numer. Anal., 15, 5, 912-928 (1978) · Zbl 0434.65091
[39] Eriksson, K.; Johnson, C.; Larsson, S., Adaptive finite element methods for parabolic problems, VI: analytic semigroups, SIAM J. Numer. Anal., 35, 4, 1315-1325 (1998) · Zbl 0909.65063
[40] Zhang, M. P.; Shu, C. W., An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations, Math. Models Methods Appl. Sci., 13, 3, 395-413 (2003) · Zbl 1050.65094
[41] Zienkiewicz, O. C.; Taylor, R. L.; Sherwin, S. J.; Peiro, J., On discontinuous Galerkin methods, Int. J. Numer. Methods Engrg., 58, 8, 1119-1148 (2003) · Zbl 1032.76607
[42] Kulkarni, D. V.; Rovas, D. V.; Tortorelli, D. A., Discontinuous Galerkin framework for adaptive solution of parabolic problems, Int. J. Numer. Methods Engrg., 70, 1, 1-24 (2007) · Zbl 1194.65115
[43] Chrysafinos, K.; Walkington, N. J., Error estimates for the discontinuous Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 44, 1, 349-366 (2006) · Zbl 1112.65086
[44] Palaniappan, J.; Miller, S. T.; Haber, R. B., Sub-cell shock capturing and spacetime interface tracking for nonlinear conservation laws, Int. J. Numer. Methods Fluids, 57, 1115-1135 (2008) · Zbl 1338.76046
[45] P.O. Persson, J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods, in: Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV, USA, 2006, pp. 5-18.; P.O. Persson, J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods, in: Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV, USA, 2006, pp. 5-18.
[46] Abedi, R.; Petracovici, B.; Haber, R. B., A spacetime discontinuous Galerkin method for elastodynamics with element-wise momentum balance, Comput. Methods Appl. Mech. Engrg., 195, 3247-3273 (2006) · Zbl 1130.74044
[47] R. Abedi, S.-H. Chung, J. Erickson, Y. Fan, M. Garland, D. Guoy, R. Haber, J. Sullivan, S. Thite, Y. Zhou, Space-time meshing with adaptive refinement and coarsening, in: Proceedings 20th Annual ACM Symposium on Computational Geometry, 2004, pp. 300-309.; R. Abedi, S.-H. Chung, J. Erickson, Y. Fan, M. Garland, D. Guoy, R. Haber, J. Sullivan, S. Thite, Y. Zhou, Space-time meshing with adaptive refinement and coarsening, in: Proceedings 20th Annual ACM Symposium on Computational Geometry, 2004, pp. 300-309. · Zbl 1422.65242
[48] Abedi, R.; Chung, S.-H.; Hawker, M.; Palaniappan, J.; Thite, S.; Haber, R., Modeling evolving discontinuities with spacetime discontinuous Galerkin methods, (Combescure, A.; de Borst, R.; Belytschko, T., IUTAM Symposium on Discretization Methods for Evolving Discontinuities, vol. 5 (2007), Springer), 59-87 · Zbl 1209.74040
[49] Spivak, M., Calculus on Manifolds (1965), W.A. Benjamin, Inc.: W.A. Benjamin, Inc. New York · Zbl 0141.05403
[50] Fleming, W. H., Functions of Several Variables (1964), Addison-Wesley
[51] Palaniappan, J.; Haber, R. B.; Jerrard, R. L., A spacetime discontinuous Galerkin method for scalar conservation laws, Comput. Methods Appl. Mech. Engrg., 193, 3607-3631 (2004) · Zbl 1077.65108
[52] J. Erickson, D. Guoy, J. Sullivan, A. Üngör, Building spacetime meshes over arbitrary spatial domains, in: Proceedings of the 11th International Meshing Roundtable (Sandia National Laboratories), 2002, pp. 391-402.; J. Erickson, D. Guoy, J. Sullivan, A. Üngör, Building spacetime meshes over arbitrary spatial domains, in: Proceedings of the 11th International Meshing Roundtable (Sandia National Laboratories), 2002, pp. 391-402.
[53] Üngör, A.; Sheffer, A., Pitching tents in spacetime: mesh generation for discontinuous Galerkin method, Int. J. Found. Comput. Sci., 13, 2, 201-221 (2002) · Zbl 1066.65138
[54] Shewchuk, J. R., Triangle: engineering a 2D quality mesh generator and Delaunay triangulator, (Lin, M. C.; Manocha, D., Applied Computational Geometry: Towards Geometric Engineering. Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science, vol. 1148 (1996), Springer-Verlag: Springer-Verlag Berlin), 203-222
[55] Baumeister, K. J.; Hamill, T. D., Hyperbolic heat conduction equation – a solution for the semi-infinite body problem, ASME J. Heat Transf., 117, 256-263 (1969)
[56] Tzou, D. Y., Macro- to Microscale Heat Transfer (1997), Taylor & Francis
[57] Zhou, Y.; Garland, M.; Haber, R. B., Pixel-exact rendering of spacetime finite element solutions, Proc. IEEE Visual., 2004, 425-432 (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.