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.

References

References
1.
Belytschko
,
T.
,
Lu
,
Y. Y.
, and
Gu
,
L.
,
1994
, “
Element Free Galerkin Methods
,”
Int. J. Numer. Methods Eng.
,
37
(
2
),
pp.
229
256
.10.1002/nme.1620370205
2.
Liu
,
G. R.
,
2003
,
Mesh Free Methods: Moving beyond the Finite Element Method
, 2nd ed.,
CRC Press
,
Boca Raton, FL
.
3.
He
,
Z.
,
Li
,
P.
,
Zhao
,
J. Y.
, and
Chen
,
H.
,
2011
, “
A Meshless Galerkin Least-Square Method for the Helmholtz Equation
,”
Eng. Anal. Boundary Elem.
,
35
(
6
),
pp.
868
878
.10.1016/j.enganabound.2011.01.010
4.
Wang
,
C. A.
,
Sadat
,
H.
,
Ledez
,
V.
, and
Lemonnier
,
D.
,
2010
, “
Meshless Method for Solving Radiative Transfer Problems in Complex Two-Dimensional and Three-Dimensional Geometries
,”
Int. J. Therm. Sci.
,
49
(
12
),
pp.
2282
2288
.10.1016/j.ijthermalsci.2010.06.024
5.
Atluri
,
S. N.
, and
Zhu
,
T. L.
,
2000
, “
The Meshless Local Petrov-Galerkin (MLPG) Approach for Solving Problems in Elasto-Statics
,”
Comput. Mech.
,
25
(
2/3
),
pp.
169
179
.10.1007/s004660050467
6.
Wu
,
S. C.
,
Zhang
,
H. O.
, and
Tang
,
Q.
,
2009
, “
Meshless Analysis of the Substrate Temperature in Plasma Spraying Process
,”
Int. J. Therm. Sci.
,
48
(
4
),
pp.
674
681
.10.1016/j.ijthermalsci.2008.06.011
7.
Wu
,
X. H.
, and
Tao
,
W. Q.
,
2010
, “
A Stabilized Method for Steady State Incompressible Fluid Flow Simulation
,”
J. Comput. Phys.
,
229
(
22
),
pp.
8564
8577
.10.1016/j.jcp.2010.08.001
8.
Sentruk
,
U.
,
2011
, “
Modeling Nonlinear Waves in a Numerical Wave Tank With Localized Meshless RBF Method
,”
Comput. Fluids
,
44
(
1
),
pp.
221
228
.10.1016/j.compfluid.2011.01.004
9.
Kuo
,
S. H.
,
2006
, “
A Meshless, High-Order Integral Equation Method for Smooth Surfaces With Application to Biomolecular Electrostatics
,”
Ph.D thesis
,
Massachusetts Institute of Technology
,
Cambridge, MA
.
10.
Singh
,
I. V.
,
2003
, “
Heat Transfer Analysis of Two-Dimensional Fins Using a Meshless Element Free Galerkin Method
,”
Numer. Heat Transfer, Part A
,
44
(
1
),
pp.
73
84
.10.1080/713838174
11.
Singh
,
A.
,
Singh
,
I. V.
, and
Prakash
,
R.
,
2006
, “
Numerical Solution of Temperature–Dependent Thermal Conductivity Problems Using a Meshless Method
,”
Numer. Heat Transfer, Part A
,
50
(
2
),
pp.
125
145
.10.1080/10407780500507111
12.
Liu
,
Y.
,
Zhang
,
X.
, and
Lu
,
M. W.
,
2005
, “
A Meshless Method Based on Least-Squares Approach for Steady- and Unsteady-State Heat Conduction Problems
,”
Numer. Heat Transfer, Part A
,
47
(
3
),
pp.
257
275
.10.1080/10407790590901648
13.
Lin
,
H.
, and
Atluri
,
S. N.
,
2000
, “
Meshless Local Petrov-Galerkin (MLPG) Method for Convection-Diffusion problem
,”
Comput. Model Eng. Sci.
,
1
(
2
),
pp.
45
60
.
14.
Sadat
,
H.
,
Dubus
,
N.
,
Gbahoue
,
L.
, and
Sophy
,
T.
,
2006
, “
On the Solution of Heterogeneous Heat Conduction Problems by a Diffuse Approximation Meshless Method
,”
Numer. Heat Transfer, Part B
,
50
(
6
),
pp.
491
498
.10.1080/10407790600710184
15.
Qian
,
L. F.
, and
Batra
,
R. C.
,
2005
, “
Three Dimensional Transient Heat Conduction in a Functionally Graded Thick Plated With a High Order Plate Theory and a Meshless Local Petrov-Galerkin Method
,”
Comput. Mech.
,
35
(
3
),
pp.
214
226
.10.1007/s00466-004-0617-6
16.
Zhang
,
X. H.
,
Ouyang
,
J.
, and
Zhang
,
L.
,
2009
, “
Matrix Free Meshless Method for Transient Heat Conduction Problems
,”
Int. J. Heat Mass Tranfer
,
52
(
7-8
),
pp.
2161
2165
.10.1016/j.ijheatmasstransfer.2008.11.010
17.
Lancaster
,
P.
, and
Salkauskas
,
K.
,
1981
, “
Surfaces Generated by Moving Least Squares Methods
,”
Am. Math. Soc.
,
37
(
7
),
pp.
141
158
.10.1090/S0025-5718-1981-0616367-1
18.
Singh
,
I. V.
,
2004
, “
A Numerical Solution of Composite Heat Transfer Problems Using Meshless Method
,”
Int. J. Heat Mass Tranfer
,
47
(
10-11
),
pp.
2133
2138
.10.1016/j.ijheatmasstransfer.2003.12.013
19.
Atluri
,
S. N.
, and
Shen
,
S. P.
,
2002
,
The Meshless Local Petrov-Galerkin (MLPG) Methods
,
Tech Science Press
,
Encino, CA
.
20.
Rao
,
S. S.
,
2011
,
The Finite Element Method in Engineering
, 5th ed.,
Elsevier Butterworth-Heinemann
,
Boston, MA
.
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