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Meshless Local Petrov-Galerkin Method for Three-Dimensional Heat Transfer Analysis

[+] Author and Article Information
Jun Tian

Department of Mechanical Engineering and Aerospace Engineering,  University of Miami, Coral Gables, FL 33146jtian0417@gmail.com

Singiresu S. Rao

Department of Mechanical Engineering and Aerospace Engineering,  University of Miami, Coral Gables, FL 33146srao@miami.edu

J. Heat Transfer 134(11), 112701 (Sep 24, 2012) (9 pages) doi:10.1115/1.4006845 History: Received June 22, 2011; Accepted April 25, 2012; Published September 24, 2012; Online September 24, 2012

Abstract

A meshless local Petrov-Galerkin (MLPG) method is proposed to obtain the numerical solution of nonlinear heat transfer problems. The moving least squares scheme is generalized to construct the field variable and its derivatives continuously over the entire domain. The essential boundary conditions are enforced by the direct scheme. By defining a radiation heat transfer coefficient, the nonlinear boundary value problem is solved as a sequence of linear problems each time updating the radiation heat transfer coefficient. The matrix formulation is used to drive the equations for a three dimensional nonlinear coupled radiation heat transfer problem. By using the MPLG method, along with the linearization of the nonlinear radiation problem, a new numerical approach is proposed to find the solution of the coupled heat transfer problem. A numerical study of the dimensionless size parameters for the quadrature and support domains is conducted to find the most appropriate values to ensure convergence of the nodal temperatures to the correct values quickly. Numerical examples are presented to illustrate the applicability and effectiveness of the proposed methodology for the solution of one-, two-, and three-dimensional heat transfer problems involving radiation with different types of boundary conditions. In each case, the results obtained using the MLPG method are compared with those given by the finite element method (FEM) method for validating the results.

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Figures

Figure 7

Comparison of MLPG results with FEM results along the fin

Figure 8

Two-dimensional model and meshing (numbers indicate dimensions in cm)

Figure 9

Specific lines used for comparison of results

Figure 10

Comparison of MLPG results with FEM results

Figure 11

Rectangular and circular support domains (N—50th node; Q—quadrature point)

Figure 12

Nodal temperature along lines AD and CD

Figure 13

Three-dimensional example with six bounding surfaces

Figure 1

Support domain Ωs of a quadrature point

Figure 2

One-dimensional fin

Figure 3

Nodal temperatures in the first three iterations (quadratic basis)

Figure 4

Results from different αq(dmax)

Figure 5

Results from different αs

Figure 6

Nodal temperatures of three iterations

Figure 14

Effect of αq(dmax) on nodal results

Figure 15

Effect of αs on nodal results

Figure 16

Comparison of MLPG results with FEM results

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