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Annals of the Assembly for International Heat Transfer Conference 13

ISBN 1-56700-225-0 / CD 1-56700-226-9

Volumes per year:

various

Year 2006

DOI: 10.1615/IHTC13.p9

A FAST MATRIX-FREE IMPLICIT UNSTRUCTURED-HYBRID ALGORITHM FOR MODELLING NON-LINEAR HEAT CONDUCTION

Arnaud G. Malan
Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria 0002, Rep. of South Africa

Josua Meyer
University of Pretoria

ABSTRACT

A matrix-free implicit algorithm is developed for the fast and efficient solution of the non-linear diffusion equation on unstructured-hybrid meshes. The aforementioned captures the physics prevalent in plasticity (structural mechanics), moisture transport in porous materials and non-linear heat conduction. This paper will be restricted to the latter. The boundary condition types supported are prescribed temperature, convection and radiation type Neumann conditions. A compact edge-based vertex-centered finite volume technique is employed for spatial discretization purposes, while implicit temporal discretization is effected. The resulting system of equations is Newton-linearized, where approximate analytical expressions for the Jacobian terms are developed. The discrete system is solved via a preconditioned Generalised Minimal Residual (GMRES) algorithm, where Lower-Upper Symmetric Gauss-Seidel (LU-SGS) is used as preconditioner. The computational performance of the developed algorithm is demonstrated by application to the modelling of steady 2D non-linear heat conduction problems. The calculated Jacobian terms are stored in the interest of computational speed, with an associated additional overall memory cost of less than 10%. Even for small problems, the developed solver outperforms Jacobi with local time-stepping in terms of computational time by a factor ranging between 30 and 1000, while having a total memory requirement of 1.9 times the aforementioned. Further, as compared to the LU-SGS and GMRES methods, the proposed solver methodology offers a speed-up in solution time of between one and two orders of magnitude.