Application of Finite-Difference Residual Correction Method for Nonlinear Heat Transfer Problems

In recent years, there has been a great deal of interest in developing meshless methods for computational fluid dynamics (CFD) applications. In this paper, a meshless finite difference method is developed for solving conjugate heat transfer problems in complex geometries. Traditional finite difference methods (FDMs) have been restricted to an orthogonal or a body-fitted distribution of points. However, the Taylor series upon which the FDM is based is valid at any location in the neighborhood of the point about which the expansion is carried out. Exploiting this fact, and starting with an unstructured distribution of mesh points, derivatives are evaluated using a weighted least squares procedure. The system of equations that results from this discretization can be represented by a sparse matrix. This system is solved with an algebraic multigrid (AMG) solver. The implementation of Neumann, Dirichlet and mixed boundary conditions within this framework is described, as well as the handling of conjugate heat transfer. The method is verified through application to classical heat conduction problems with known analytical solutions. It is then applied to the solution of conjugate heat transfer problems in complex geometries, and the solutions so obtained are compared with more conventional unstructured finite volume methods. Metrics for accuracy are provided and future extensions are discussed.

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A Space-Time Meshless Method for Heat Transfer Problems With High Discontinuities

Volume 4: Heat and Mass Transfer Under Extreme Conditions; Environmental Heat Transfer; Computational Heat Transfer; Visualization of Heat Transfer; Heat Transfer Education and Future Directions in Heat Transfer; Nuclear Energy ◽

10.1115/ht2013-17018 ◽

2013 ◽

Author(s):

Arthur Da Silva ◽

Tonino Sophy ◽

Ali Kribèche

Keyword(s):

Heat Transfer ◽

Finite Difference ◽

Heat Source ◽

Weight Function ◽

Meshless Method ◽

Space Time ◽

Transient Heat Transfer ◽

Source Power ◽

Spurious Oscillations ◽

Transfer Problems

The aim of this research is the development of a space-time driscretization method based on Diffuse Approximation Meshless method. This method, devoted to transient heat transfer problems presenting high temporal discontinuities, avoids any Finite-Difference time stepping procedure. The space-time discretization proposed here seems to be convenient for continuous transient heat transfer. Nevertheless, for problems including temporal discontinuities, some spurious oscillations, whose amplitudes depend on source power, appear. A new weight function respecting the principle of causality, based on a modification of the involved node’s selection and a normalisation of the distances, is developed. The use of this new weight function both improves the accuracy and vanishes the oscillations. The method is validated by a source free transient heat transfer problem presenting convective exchanges. Then problems including a constant and a discontinuous heat source are solved. Temperatures fields obtained when using the classical weight function are closer to those obtained with a backward Finite Difference scheme when the heat source is continuous. In case of discontinuous sources, when using the classical weight function, temperature fields present some spurious oscillations which disappear when choosing the new one. The proposed method associated to a grid refinement procedure will lead to adaptive grids in space and/or time, independently.

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