×

Multiple spatial and temporal scales method for numerical simulation of non-Fourier heat conduction problems: multidimensional case. (English) Zbl 1213.80023

The primary objective of this paper is to simulate the phenomenon of non-Fourier heat conduction in periodic heterogeneous materials by the multiple spatial and temporal scales method to study the multidimensional effects. Two different kinds of scales, the amplified spatial scale and the reduced temporal scale are introduced. Numerical examples demonstrate the validity of the high-order non-local model.

MSC:

80A20 Heat and mass transfer, heat flow (MSC2010)
80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
80M35 Asymptotic analysis for problems in thermodynamics and heat transfer
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cattaneo, C.: A form of heat conduction equation which eliminates the paradox of instantaneous propagation, C. R. Acad. sci. 247, 431-433 (1958) · Zbl 1339.35135
[2] Vernotte, P.: LES paradoxes de la théorie continue de l’équation de la chaleur, C. R. Acad. sci., 3153-3155 (1958) · Zbl 1341.35086
[3] Ozisik, M. N.; Tzou, D. Y.: On the wave theory in heat conduction, J. heat transfer 116, 526-535 (1994)
[4] Tzou, D. Y.: A unified field approach for heat conduction macro- to micro-scales, J. heat transfer 117, 8-16 (1995)
[5] Hien, T. D.; Kleiber, M.: Stochastic finite element modelling in linear transient heat transfer, Comput. meth. Appl. mech. Eng. 144, 111-124 (1997) · Zbl 0890.73066
[6] Kamiński, M.; Hien, T. D.: Stochastic finite element analysis of transient heat transfer in composite material with interface defects, Arch. mech. 51, No. 3-4, 399-418 (1999) · Zbl 0991.74070
[7] Jiaung, W. S.; Ho, J. R.; Kuo, C. P.: Study of heat transfer in multilayered structure within the framework of dual-phase-lag heat conduction model using lattice Boltzmann method, Int. J. Heat mass transfer 46, 55-69 (2003) · Zbl 1027.80515
[8] Bargmann, S.; Steinmann, P.: Finite element approaches to non-classical heat conduction in solids, Comput. model. Eng. sci. 9, No. 2, 133-150 (2005) · Zbl 1232.80004
[9] Bakhvalov, N. S.; Panasenko, G. P.: Homogenization: averaging processes in periodic media, (1989) · Zbl 0692.73012
[10] Benssousan, A.; Lions, J. L.; Papanicolaou, G.: Asymptotic analysis for periodic structures, (1978) · Zbl 0404.35001
[11] Sanchez-Palencia, E.: Non-homogeneous media and vibration theory, (1980) · Zbl 0432.70002
[12] Zohdi, T. I.: Homogenization methods and multiscale modeling, Solids & structure 2, 407-430 (2004)
[13] Zhang, H. W.; Galvanetto, U.; Schrefler, B. A.: Local analysis and global nonlinear behaviour of periodic assemblies of bodies in elastic contact, Comput. mech. 24, No. 4, 217-229 (1999) · Zbl 0967.74047
[14] Zhang, H. W.; Schrefler, B. A.: Global constitutive behaviour of periodic assemblies of inelastic bodies in contact, Mech. compos. Mater. struct. 7, No. 4, 355-382 (2000)
[15] Gambin, B.; Kroner, E.: High order terms in the homogenized stress-strain relation of periodic elastic media, Phys. stat. Sol. 51, 513-519 (1989)
[16] Boutin, C.; Auriault, J. L.: Rayleigh scattering in elastic composite materials, Int. J. Eng. sci. 31, No. 12, 1669-1689 (1993) · Zbl 0780.73021
[17] Fish, J.; Chen, W.: Uniformly valid multiple spatial-temporal scale modeling for wave propagation in heterogeneous media, Mech. compos. Mater. struct. 8, No. 2, 1-19 (2001)
[18] Fish, J.; Chen, W.: Higher-order homogenization of initial/boundary-value problem, J. eng. Mech, ASCE 127, No. 12, 1223-1230 (2001)
[19] Ladevèze, P.; Nouy, A.: A multiscale computational method with time and space homogenization, C. R. Mecanique 330, 683-689 (2002) · Zbl 1177.74316
[20] Fish, J.; Chen, W.; Nagai, G.: Non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case, Int. J. Numer. meth. Eng. 54, No. 3, 347-363 (2002) · Zbl 1034.74030
[21] Kamiński, M.: Multiscale homogenization of n-component composites with semi-elliptical random interface defects, Int. J. Solids struct. 42, No. 11 – 12, 3571-3590 (2005) · Zbl 1127.74346
[22] Boutin, C.: Microstructural influence on heat conduction, Int. J. Heat mass transfer 38, No. 17, 3181-3195 (1995) · Zbl 0924.73019
[23] Yu, Q.; Fish, J.: Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem, Int. J. Solids struct. 39, 6429-6452 (2002) · Zbl 1032.74627
[24] Kamiński, M.: Homogenization of transient heat transfer problems for some composite materials, Int. J. Eng. sci. 41, 1-29 (2003) · Zbl 1211.80043
[25] Zhang, H. W.; Zhang, S.; Guo, X.; Bi, J. Y.: Multiple spatial and temporal scales method for numerical simulation of nonclassical heat conduction problems: one dimensional case, Int. J. Solids struct. 42, 877-899 (2005) · Zbl 1125.80321
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.