Analysis and Modelling of Non-Fourier Heat Behavior Using the Wavelet Finite Element Method

Abstract

Non-Fourier heat behavior is an important issue for film material. The phenomenon is usually observed in some laser induced thermal responses. In this paper, the non-Fourier heat conduction problems with temperature and thermal flux relaxations are investigated based on the wavelet finite element method and solved by the central difference scheme for one- and two-dimensional media. The Cattaneo–Vernotte model and the Dual-Phase-Lagging model are used for finite element formulation, and a new wavelet finite element solving formulation is proposed to address the memory requirement problem. Compared with the current methodologies for the Cattaneo–Vernotte model and the Dual-Phase-Lagging model, the present model is a direct one which describe the thermal behavior by one equation about temperature. Compared with the wavelet method proposed by Xiang et al., the developed method can be used for arbitrary shapes. In order to address the efficient computation problems for the Dual-Phase-Lagging model, a novel iteration updating methodology is also proposed. The proposed iteration algorithms on time avoids the use the global stiffness matrix, which allows the efficient calculation for title issue. Numerical calculations have been conducted in the manner of comparisons with the classical finite element method and spectral finite element method. The comparisons from accuracy, efficiency, flexibility, and applicability validate the developed method to be an effective and alternative tool for material thermal analysis.

Non-Fourier Heat Conduction of Nano-Films under Ultra-Fast Laser

“…The observation that there is a significant difference between the Fourier and non-Fourier results emphasizes the significance of a non-Fourier model for similar problems. But the CV model only considers the heat flux density vector phase retardation, so the problem of insufficient physical basis still exists [26][27][28]. Later, Tzou [29,30] proposed the DPL model to reveal the microscopic heat conduction inside the medium from a macroscopic perspective, and to explain the hysteresis time of the heat flux vector and temperature gradient.…”

Fractional Order Dual-Phase-Lag Model of Heat Conduction in a Composite Spherical Medium

“…The dual-phase-lag heat shown in Equation ( 5 ) has been applied in mathematical modeling of the heat transfer in functionally graded materials [ 13 , 14 , 15 ], ultrafast pulse-laser heating problems [ 16 , 17 ], porous media [ 18 , 19 , 20 ], nanocomposites [ 21 , 22 ], and living tissue [ 23 , 24 , 25 ]. If in Equation ( 5 ), then the classical parabolic heat conduction equation is obtained.…”

Two-field mixed hp-finite elements for time-dependent problems in the refined theories of thermodynamics

“…The philosophy of using low-order polynomials over successively finer meshes is called h-type approximation technique [10,11,12,13]. In most cases [14,15,16,17], that h-type approach is utilised for the Maxwell-Cattaneo-Vernotte-(MCV, with κ 2 = 0 in Eq. ( 2)), or for the dual-phase-lag (DPL) equations, which models have much less practical interest.…”