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Research Papers: Conduction

# A Meshless Finite Difference Method for Conjugate Heat Conduction Problems

[+] Author and Article Information
Chandrashekhar Varanasi1

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47906vvc@purdue.edu

Jayathi Y. Murthy

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47906jmurthy@ecn.purdue.edu

Sanjay Mathur

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47906smathur@purdue.edu

1

Corresponding author.

J. Heat Transfer 132(8), 081303 (Jun 09, 2010) (13 pages) doi:10.1115/1.4001363 History: Received April 15, 2009; Revised February 25, 2010; Published June 09, 2010; Online June 09, 2010

## Abstract

A meshless finite difference method is developed for solving conjugate heat transfer problems. Starting with an arbitrary distribution of mesh points, derivatives are evaluated using a weighted least-squares procedure. The resulting system of algebraic equations is sparse and is solved using an algebraic multigrid method. The implementation of the Neumann, Dirichlet, and mixed boundary conditions within this framework is described. For conjugate heat transfer problems, continuity of the heat flux and temperature are imposed on mesh points at multimaterial interfaces. 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. The method improves on existing meshless methods for conjugate heat conduction by eliminating spurious oscillations previously observed. Metrics for accuracy are provided and future extensions are discussed.

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## Figures

Figure 6

Problem 2–convergence rate for a nonuniform distribution of points

Figure 7

Problem 2–convergence rate for linear variation in conductivity

Figure 8

Problem 2–horizontal center line temperature profile for k versus T characteristic of air (nondimensionalized)

Figure 9

Problem 3–domain for two-dimensional conjugate conduction and one-dimensional temperature profile in a domain with k2/k1=1000

Figure 10

Problem 4–temperature distribution computed on a uniform arrangement of 1680 points k2/k1=2 and 1000

Figure 13

Problem 4–temperature profile at y=0.7 for k2/k1=2, 100, and 1000 (nonuniform point distribution)

Figure 14

Problem 4–variation in percent error in the heat balance with increasing density of points for different conductivity ratios (uniform point distribution)

Figure 1

Problem 1—variation in the rms percent error with the number of neighbors (σ=3)

Figure 2

Problem 1—variation in the rms percent error with sharpness of the Gaussian weight function (Nmin=7)

Figure 3

Problem 2—solution for the heat conduction problem obtained on a nonuniform distribution of 906 points

Figure 4

Problem 2—convergence rate for a uniform distribution of points

Figure 5

Problem 2—nonuniform distribution of 226 points used for the computation

Figure 11

Problem 4–temperature distribution computed on a nonuniform arrangement of 1680 points k2/k1=2 and 1000

Figure 12

Problem 4–temperature profile at y=0.7 for k2/k1=2, 100, and 1000 (uniform point distribution)

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