A spacetime discontinuous Galerkin method for hyperbolic heat conduction

Computer Methods in Applied Mechanics and Engineering

Volumn 198, Issue 2, 2008, Pages 194-209

  Miller, S T ^a   Haber, R B ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Adaptive analysis; Discontinuous Galerkin; Finite element; Hyperbolic heat conduction; Shock capturing

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; GALERKIN METHODS; HEAT CONDUCTION; HEATING; MATHEMATICAL OPERATORS; MODELS; NUMERICAL ANALYSIS;

ADAPTIVE ANALYSIS; DISCONTINUOUS GALERKIN; FINITE ELEMENT; HYPERBOLIC HEAT CONDUCTION; SHOCK CAPTURING;

FINITE ELEMENT METHOD;

EID: 54949132501     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2008.07.016     Document Type: Article

References (57)

1. Bubnov V.A. Wave concepts in the theory of heat. Int. J. Heat Mass Transf. 19 (1976) 175-184
2. Joseph D.D., and Preziosi L. Heat waves. Rev. Mod. Phys. 61 1 (1989) 41-73
3. Joseph D.D., and Preziosi L. Addendum to the paper "Heat waves". Rev. Mod. Phys. 62 2 (1990) 375-391
4. Ackerman C.C., Berman B., Fairbank H.A., and Guyer R.A. Second sound in solid helium. Phys. Rev. Lett. 16 18 (1966) 789-791
5. Jackson H.E., Walker C.T., and McNelly T.F. Second sound in NaF. Phys. Rev. Lett. 25 1 (1970) 26-28
6. Narayanamurti V., and Dynes R.C. Observation of second sound in bismuth. Phys. Rev. Lett. 28 22 (1972) 1461-1465
7. Maxwell J.C. On the dynamical theory of gases. Philos. Trans. Roy. Soc. Lond. 157 (1867) 49-88
8. 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 (1958) 431-433
10. Vernotte P. Les paradoxes de la théorie continue de l'équation de la chaleur. C.R. Acad. Sci. 246 22 (1958) 3154-3155
11. Dreyer W., and Struchtrup H. Heat pulse experiments revisited. Continuum Mech. Thermodyn. 5 (1993) 3-50
12. Özisik M.N., and Tzou D.Y. On the wave theory in heat conduction. ASME J. Heat Transf. 116 (1994) 535-536
13. Chandrasekharaiah D.S. Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51 12 (1998) 705-729
14. Chandrasekharaiah D.S. Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39 3 (1986) 355-376
15. Coleman B.D., Hrusa W.J., and 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 (1986) 267-289
16. Glass D.E., Özisik M.N., McRae D.S., and Vick B. On the numerical solution of hyperbolic heat conduction. Numer. Heat Transf. 8 (1985) 497-504
17. Glass D.E., Özisik M.N., and McRae D.S. Hyperbolic heat conduction with radiation in an absorbing and emitting medium. Numer. Heat Transf. 12 (1987) 321-333
18. Liu L.H., Tan H.P., and Tong T.W. Non-fourier effects on transient temperature response in semitransparent medium caused by laser pulse. Int. J. Heat Mass Transf. 44 (2001) 3335-3344
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.
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.
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.
22. Yang H.Q. Characteristics-based, high-order accurate and nonoscillatory numerical method for hyperbolic heat conduction. Numer. Heat Transf.: Part B - Fund. 18 (1990) 221-241
23. Yang H.Q. Solution of two-dimensional hyperbolic heat conduction by high-resolution numerical methods. Numer. Heat Transf.: Part A - Appl. 21 (1992) 333-349
24. Shen W., and Han S. Numerical solution of two-dimensional axisymmetric hyperbolic heat conduction. Comput. Mech. 29 (2002) 122-128
25. Shen W., and Han S. A numerical solution of two-dimensional hyperbolic heat conduction with non-linear boundary conditions. Heat Mass Transf. 39 (2003) 499-507
26. Goodman J.B., and LeVeque R.J. On the accuracy of stable schemes for 2D scalar conservation laws. Math. Comput. 45 171 (1985) 15-21
27. LeVeque R.J. Finite Volume Methods for Hyperbolic Problems, Cambridge texts in Applied Mathematics (2002), Cambridge University Press
28. Chen H.T., and Lin J.Y. Numerical analysis for hyperbolic heat conduction. Int. J. Heat Mass Transf. 36 11 (1992) 2891-2898
29. Chen H.T., and Lin J.Y. Analysis of two-dimensional hyperbolic heat conduction problems. Int. J. Heat Mass Transf. 37 1 (1993) 153-164
30. Carey G.F., and Tsai M. Hyperbolic heat transfer with reflection. Numer. Heat Transf. 5 (1982) 309-327
31. Manzari M.T., and Manzari M.T. A mixed approach to finite element analysis of hyperbolic heat conduction problems. Int. J. Numer. Methods Heat Fluid Flow 8 1 (1998) 83-96
32. Manzari M.T., and Manzari M.T. On numerical solution of hyperbolic heat conduction. Commun. Numer. Methods Engrg. 15 (1999) 853-866
33. Xu B.Q., Shen Z.H., Lu J., Ni X.W., and 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 (2003) 4963-4968
34. Ai X., and Li B.Q. A discontinuous finite element method for hyperbolic thermal wave problems. J. Engrg. Comput. 21 5 (2004) 577-597
35. Ai X., and Li B.Q. Numerical simulation of thermal wave propagation during laser processing of thin films. J. Electron. Mater. 34 5 (2005) 583-591
36. Wu W., and Li X. Application of the time discontinuous Galerkin finite element method to heat wave simulation. Int. J. Heat Mass Transf. 49 (2006) 1679-1684
37. Li X., Yao D.M., and 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 (2003) 1775-1800
38. Jamet P. Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal. 15 5 (1978) 912-928
39. Eriksson K., Johnson C., and Larsson S. Adaptive finite element methods for parabolic problems, VI: analytic semigroups. SIAM J. Numer. Anal. 35 4 (1998) 1315-1325
40. Zhang M.P., and Shu C.W. An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci. 13 3 (2003) 395-413
41. Zienkiewicz O.C., Taylor R.L., Sherwin S.J., and Peiro J. On discontinuous Galerkin methods. Int. J. Numer. Methods Engrg. 58 8 (2003) 1119-1148
42. Kulkarni D.V., Rovas D.V., and Tortorelli D.A. Discontinuous Galerkin framework for adaptive solution of parabolic problems. Int. J. Numer. Methods Engrg. 70 1 (2007) 1-24
43. Chrysafinos K., and Walkington N.J. Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 1 (2006) 349-366
44. Palaniappan J., Miller S.T., and Haber R.B. Sub-cell shock capturing and spacetime interface tracking for nonlinear conservation laws. Int. J. Numer. Methods Fluids 57 (2008) 1115-1135 10.1002/fld.1850
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.
46. Abedi R., Petracovici B., and Haber R.B. A spacetime discontinuous Galerkin method for elastodynamics with element-wise momentum balance. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3247-3273
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.
48. Abedi R., Chung S.-H., Hawker M., Palaniappan J., Thite S., and Haber R. Modeling evolving discontinuities with spacetime discontinuous Galerkin methods. In: Combescure A., de Borst R., and Belytschko T. (Eds). IUTAM Symposium on Discretization Methods for Evolving Discontinuities vol. 5 (2007), Springer 59-87
49. Spivak M. Calculus on Manifolds (1965), W.A. Benjamin, Inc., New York
50. Fleming W.H. Functions of Several Variables (1964), Addison-Wesley
51. Palaniappan J., Haber R.B., and Jerrard R.L. A spacetime discontinuous Galerkin method for scalar conservation laws. Comput. Methods Appl. Mech. Engrg. 193 (2004) 3607-3631
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.
53. Üngör A., and Sheffer A. Pitching tents in spacetime: mesh generation for discontinuous Galerkin method. Int. J. Found. Comput. Sci. 13 2 (2002) 201-221
54. Shewchuk J.R. Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. In: Lin M.C., and Manocha D. (Eds). Applied Computational Geometry: Towards Geometric Engineering. Lecture Notes in Computer Science vol. 1148 (1996), Springer-Verlag, Berlin 203-222
55. Baumeister K.J., and Hamill T.D. Hyperbolic heat conduction equation - a solution for the semi-infinite body problem. ASME J. Heat Transf. 117 (1969) 256-263
56. Tzou D.Y. Macro- to Microscale Heat Transfer (1997), Taylor & Francis
57. Zhou Y., Garland M., and Haber R.B. Pixel-exact rendering of spacetime finite element solutions. Proc. IEEE Visual. 2004 (2004) 425-432