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RESEARCH PAPERS: Conduction

# Multiquadric Collocation Method for Time-Dependent Heat Conduction Problems With Temperature-Dependent Thermal Properties

[+] Author and Article Information
Somchart Chantasiriwan

Faculty of Engineering, Thammasat University, Pathum Thani 12121, Thailandsomchart@engr.tu.ac.th

J. Heat Transfer 129(2), 109-113 (Apr 30, 2006) (5 pages) doi:10.1115/1.2401617 History: Received June 13, 2005; Revised April 30, 2006

## Abstract

The multiquadric collocation method is a meshless method that uses multiquadrics as its basis function. Problems of nonlinear time-dependent heat conduction in materials having temperature-dependent thermal properties are solved by using this method and the Kirchhoff transformation. Variable transformation is simplified by assuming that thermal properties are piecewise linear functions of temperature. The resulting nonlinear equation is solved by an iterative scheme. The multiquadric collocation method is tested by a heat conduction problem for which the exact solution is known. Results indicate satisfactory performance of the method.

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## Figures

Figure 1

Variations of the reciprocal of thermal diffusivities of materials A, B, and C with Kirchhoff transformation variable

Figure 2

Domain of test problem is a 0.12m×0.12m square. The left side is the Neumann boundary and the other three sides are the Dirichlet boundary. Black circles indicate locations of 24 boundary test nodes.

Figure 3

Variations with time of average error for 36 interior test nodes corresponding to different shape parameters of multiquadrics

Figure 4

Variations with time of average error for 36 interior test nodes corresponding to different grid spacings and time steps

Figure 5

Variations with time of average error for 36 interior test nodes corresponding to different node arrangements. The solid line represents the uniform arrangement, whereas five other lines represent five random arrangements.

Figure 6

Comparison of average error in computed temperatures for 36 interior test nodes, average error in computed temperatures for 6 boundary test nodes, and average error in computed heat flux for 18 boundary test nodes

Figure 7

Comparison of average errors for 121 interior test nodes by the multiquadric collocation method (MCM) and the finite difference method (FDM)

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