Keywords: non-linear heat diffusion, multigrid methods, hysteresis.

In this paper a transient non-linear heat diffusion problem is studied. Utilizing the advantages of the multigrid technique a coarse level iterating algorithm is introduced. Applying a hierarchical algorithm the solution time can be reduced. The proposed method has been verified by comparing to other numerical schemes. Non-linear diffusion with periodic boundary conditions and inhomogeneous initial conditions are also analysed.

The analyses of transient thermal responses in composite materials with variable thermal properties could be of great importance in various engineering and scientific fields. The thermal diffusivity is fundamentally related to the microstructure and mineralogical composition. Several experiments prove that with temperature changing the thermal conductivity or diffusivity of composite materials shows hysteresis like behaviour. The hysteresis of diffusivity can be thought as a consequence of the temperature changing induced micro-structural transformation. For example hysteresis of thermal diffusivity of silica based materials is reported in reference [1].

Modelling of transient thermal responses of materials with non-linear properties means the numerical solution of a nonlinear parabolic differential equation. In the two dimensional case, considering a rectangular domain with length , by spatial coordinates , the diffusion equation has the form of

where is the dimensionless temperature, is the thermal diffusivity, is a thermal source. The hysteresis of thermal diffusivity has been approximated by the scalar Preisach hysteresis model described in reference [2].

Differentiation by the Crank-Nicolson scheme is second-order accurate in time and space. The solution domain has been discretised by a Cartesian mesh with uniform mesh size . The discretised form of Equation (15) with space and time-averaged diffusivity yields an algebraic equation system

where the first superscript refers to the time step, the second superscript to the iteration cycle and the subscript to the level of spatial resolution. The kernels of and should be changed point-by-point, depending on the diffusivity.

In this work a coarse level iterating multigrid technique (CLI-MG) is suggested by the authors. In the iteration process induced by the nonlinearity, the diffusivity has to be newly calculated for all fine grid points. Hysteresis like temperature dependence of the diffusivity can dramatically increase the solution time because of handling at each grid point the local memory. To reduce the solution time, before the iteration, the discretised problem (16) should be reduced to the next coarser grid, with mesh size . The non-linear iteration has been evaluated on this coarse resolution. After the non-linear iteration the diffusivity and the temperature can be interpolated to the fine grid. The prolonged diffusivity needs smoothing, to remove the high frequency component originated from the prolongation. The interpolated temperature could be seen as a good initial solution to the grid level . To evaluate the fine grid solution , some multigrid V-cycles have to be applied, until the convergence criterion is reached. During the fine grid V-cycles the diffusivity remains unchanged.

The coarse level iterating hierarchical algorithm consists of two types of multigrid methods. In the non-linear part a full multigrid (FMG) algorithm works. On the fine level multigrid V-cycles (MG-V) are applied. The multigrid techniques have been described for example in reference [3]. Numerical tests prove that the method shows a good agreement with the fine grid iteration solutions when the problem is smooth enough.

Thermal processes in materials with temperature dependent thermal properties can significantly differ from processes that considered constant thermal parameters. The proposed algorithm allows determining how the two-dimensional temperature fields could vary by time in non-linear and also non-symmetric cases of heat diffusion.

1

W.N. Santos, J.B. Baldo, R. Taylor, "Effect of SiC on the thermal diffusivity of silica-based materials", Materials Research Bulletin, 35, 2091-2100, 2000. doi:10.1016/S0025-5408(00)00428-1

2

A.Iványi, "Hysteresis models in electromagnetic computation", Akadémiai Kiadó, Budapest, 1997.

3

P. Wesseling, "An Introduction to Multigrid Methods", Wiley, Chichester, 1992.

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