In this study, a new coupled surface shape design (SSD) methodology named direct design method is presented for the solution of problems containing different types of convection heat transfer in which a specific distribution of either heat flux or temperature is given instead of the shape of a boundary. In the proposed method, the governing equation, without using any mathematical transformation for the physical domains, is manipulated so that the grid generation, solving fluid flow, and heat transfer as well as shape updating can all be carried out simultaneously. Five different inverse shape design problems containing different types of convection heat transfer are solved by the proposed method. All the problems are also solved using the ball-spine algorithm (BSA), which is a recently developed de-coupled algorithm, for the sake of comparison. In all problems, the effects of using different under-relaxation parameters are investigated and the capability of both approaches is compared with each other. The results show that the proposed coupled method can solve the problems better than the BSA in the sense that the direct design method converges sooner than the BSA when the same under-relaxation parameter is used for both methods. Also, it is shown that the computational cost of solving a SSD problem using the direct design method is slightly greater than solving an analysis problem.

References

References
1.
Gosselin
,
L.
,
Tye-Gingras
,
M.
, and
Mathieu-Potvin
,
F.
,
2009
, “
Review of Utilization of Genetic Algorithms in Heat Transfer Problems
,”
Int. J. Heat Mass Transfer
,
52
(
9–10
), pp.
2169
2188
.
2.
Huang
,
C. H.
, and
Wang
,
S. P.
,
1999
, “
A Three-Dimensional Inverse Heat Conduction Problem in Estimating Surface Heat Flux by Conjugate Gradient Method
,”
Int. J. Heat Mass Transfer
,
42
(
18
), pp.
3387
3403
.
3.
Cheng
,
C. H.
, and
Wu
,
C. Y.
,
2000
, “
An Approach Combining Body Fitted Grid Generation and Conjugate Gradient Methods for Shape Design in Heat Conduction Problems
,”
Numer. Heat Transfer, Part B
,
37
(
1
), pp.
69
83
.
4.
Lan
,
C. H.
,
Cheng
,
C. H.
, and
Wu
,
C. Y.
,
2001
, “
Shape Design for Heat Conduction Problems Using Curvilinear Grid Generation Conjugate Method and Redistribution Method
,”
Numer. Heat Transfer, Part A
,
39
(
5
), pp.
487
510
.
5.
Zhou
,
J.
,
Zhang
,
Y.
,
Chen
,
J. K.
, and
Feng
,
Z. C.
,
2010
, “
Inverse Estimation of Surface Heating Condition in a Three-Dimensional Object Using Conjugate Gradient Method
,”
Int. J. Heat Mass Transfer
,
53
(
13–14
), pp.
2643
2654
.
6.
Huang
,
C.
, and
Lee
,
C.
,
2015
, “
An Inverse Problem to Estimate Simultaneously Six Internal Heat Fluxes for a Square Combustion Chamber
,”
Int. J. Therm. Sci.
,
88
, pp.
59
76
.
7.
Huang
,
C. H.
, and
Chen
,
W. C.
,
2000
, “
A Three-Dimensional Inverse Forced Convection Problem in Estimating Surface Heat Flux by Conjugate Gradient Method
,”
Int. J. Heat Mass Transfer
,
43
(
17
), pp.
3171
3181
.
8.
Cheng
,
C. H.
, and
Chang
,
M. H.
,
2003
, “
Shape Design for a Cylinder With Uniform Temperature Distribution on the Outer Surface by Inverse Heat Transfer Method
,”
Int. J. Heat Mass Transfer
,
46
(
1
), pp.
101
111
.
9.
Katamine
,
E.
,
Kawase
,
Y.
, and
Azegami
,
H.
,
2008
, “
Shape Optimization of Thermal Forced Convection Fields
,”
Heat Transfer-Asian Res.
,
37
(
5
), pp.
313
328
.
10.
Park
,
H. M.
, and
Chung
,
O. Y.
,
1999
, “
An Inverse Natural Convection Problem of Estimating the Strength of a Heat Source
,”
Int. J. Heat Mass Transfer
,
42
(
23
), pp.
4259
4273
.
11.
Prud'homme
,
M.
, and
Hung Nguyen
,
T.
,
2001
, “
Solution of Inverse Free Convection Problems by Conjugate Gradient Method: Effects of Rayleigh Number
,”
Int. J. Heat Mass Transfer
,
44
(
11
), pp.
2011
2027
.
12.
Park
,
H. M.
, and
Shin
,
H. J.
,
2003
, “
Shape Identification for Natural Convection Problems Using the Adjoint Variable Method
,”
J. Comput. Phys.
,
186
(
1
), pp.
198
211
.
13.
Park
,
H. M.
, and
Yoon
,
T. Y.
,
2000
, “
Solution of the Inverse Radiation Problem Using a Conjugate Gradient Method
,”
Int. J. Heat Mass Transfer
,
43
(
10
), pp.
1767
1776
.
14.
Hosseini Sarvari
,
S. M.
, and
Mansouri
,
S. H.
,
2004
, “
Inverse Design for Radiative Heat Source in Two-Dimensional Participating Media
,”
Numer. Heat Transfer, Part B
,
46
(
3
), pp.
283
300
.
15.
Salinas
,
C. T.
,
2010
, “
Inverse Radiation Analysis in Two-Dimensional Gray Media Using the Discrete Ordinates Method With a Multidimensional Scheme
,”
Int. J. Therm. Sci.
,
49
(
2
), pp.
302
310
.
16.
Demeulenaere
,
A.
, and
Van Den Braembussche
,
R. A.
,
1998
, “
Three-Dimensional Inverse Method for Turbomachinery Blading Design
,”
ASME J. Turbomach.
,
120
(
2
), pp.
247
255
.
17.
De Vito
,
L.
,
Van den Braembussche
,
R. A.
, and
Deconinck
,
H.
,
2003
, “
A Novel Two-Dimensional Viscous Inverse Design Method for Turbomachinery Blading
,”
ASME J. Turbomach.
,
125
(
2
), pp.
310
316
.
18.
Barger
,
R. L.
, and
Brooks
,
C. W.
,
1974
, “
A Streamline Curvature Method for Design of Supercritical and Subcritical Airfoils
,” NASA Langley Research Center, Hampton, VA, Technical Report No.
NASA TN D-7770
.https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19740024303.pdf
19.
Garabedian
,
P.
, and
McFadden
,
G.
,
1982
, “
Design of Supercritical Swept Wings
,”
AIAA J.
,
20
(
2
), pp.
289
291
.
20.
Malone
,
J.
,
Vadyak
,
J.
, and
Sankar
,
L. N.
,
1987
, “
Inverse Aerodynamic Design Method for Aircraft Components
,”
J. Aircr.
,
24
(
1
), pp.
8
9
.
21.
Safari
,
M.
,
Nili-Ahmadabadi
,
M.
,
Ghaei
,
A.
, and
Shirani
,
E.
,
2014
, “
Inverse Design in Subsonic and Transonic External Flow Regimes Using Elastic Surface Algorithm
,”
Comput. Fluids
,
102
, pp.
41
51
.
22.
Nikfar
,
M.
,
Ashrafizadeh
,
A.
, and
Mayeli
,
P.
,
2015
, “
Inverse Shape Design Via a New Physical-Based Iterative Solution Strategy
,”
Inverse Probl. Sci. Eng.
,
23
(
7
), pp.
1138
1162
.
23.
Nili-Ahmadabadi
,
M.
,
Durali
,
M.
,
Hajilouy
,
A.
, and
Ghadak
,
F.
,
2009
, “
Inverse Design of 2D Subsonic Ducts Using Flexible String Algorithm
,”
Inverse Probl. Sci. Eng.
,
17
(
8
), pp.
1037
1057
.
24.
Nili-Ahmadabadi
,
M.
,
Hajilouy
,
A.
,
Ghadak
,
F.
, and
Durali
,
M.
,
2010
, “
A Novel 2-D Incompressible Viscous Inverse Design Method for Internal Flows Using Flexible String Algorithm
,”
ASME J. Fluids Eng.
,
132
(
3
), p.
031401
.
25.
Nili-Ahmadabadi
,
M.
,
Ghadak
,
F.
, and
Mohammadi
,
M.
,
2013
, “
Subsonic and Transonic Airfoil Inverse Design Via Ball-Spine Algorithm
,”
Comput. Fluids
,
84
, pp.
87
96
.
26.
Vaghefi
,
N. S.
,
Nili-Ahmadabadi
,
M.
, and
Roshani
,
M. R.
,
2012
, “
Optimal Design of Axe-Symmetric Diffuser Via Genetic Algorithm and Ball-Spine Inverse Design Method
,”
ASME
Paper No. IMECE2012-87924.
27.
Madadi
,
A.
,
Kermani
,
M. J.
, and
Nili-Ahmadabadi
,
M.
,
2011
, “
Application of an Inverse Design Method to Meet a Target Pressure in Axial-Flow Compressors
,”
ASME
Paper No. GT2011-46091.
28.
Madadi
,
A.
,
Kermani
,
M. J.
, and
Nili-Ahmadabadi
,
M.
,
2014
, “
Application of the Ball-Spine Algorithm to Design Axial-Flow Compressor Blade
,”
Sci. Iran.
,
21
(
6
), pp.
1981
1992
.http://scientiairanica.sharif.edu/article_3590.html
29.
Madadi
,
A.
,
Kermani
,
M. J.
, and
Nili-Ahmadabadi
,
M.
,
2014
, “
Applying the Ball-Spine Algorithm to the Design of Blunt Leading Edge Airfoils for Axial Flow Compressors
,”
J. Mech. Sci. Technol.
,
28
(
11
), pp.
4517
4526
.
30.
Madadi
,
A.
,
Kermani
,
M. J.
, and
Nili-Ahmadabadi
,
M.
,
2014
, “
Aerodynamic Design of S-Shaped Diffusers Using Ball–Spine Inverse Design Method
,”
ASME J. Eng. Gas Turbines Power
,
136
(
12
), p.
122606
.
31.
Shumal
,
M.
,
Nili-Ahmadabadi
,
M.
, and
Shirani
,
E.
,
2016
, “
Development of the Ball-Spine Inverse Design Algorithm to Swirling Viscous Flow for Performance Improvement of an Axisymmetric Bend Duct
,”
Aerosp. Sci. Technol.
,
52
, pp.
181
188
.
32.
Hoghooghi
,
H.
,
Nili-Ahmadabadi
,
M.
, and
Manshadi
,
M. D.
,
2016
, “
Optimization of a Subsonic Wind Tunnel Nozzle With Low Contraction Ratio Via Ball-Spine Inverse Design Method
,”
J. Mech. Sci. Technol.
,
30
(
5
), pp.
2059
2067
.
33.
Mayeli
,
P.
,
Nili-Ahmadabadi
,
M.
, and
Besharati-Foumani
,
H.
,
2016
, “
Inverse Shape Design for Heat Conduction Problems Via Ball Spine Algorithm
,”
Numer. Heat Transfer, Part B
,
69
(
3
), pp.
249
269
.
34.
Mayeli
,
P.
,
Nili-Ahmadabadi
,
M.
,
Pirzadeh
,
M. R.
, and
Rahmani
,
P.
,
2017
, “
Determination of Desired Geometry by a Novel Extension of Ball Spine Algorithm Inverse Method to Conjugate Heat Transfer Problems
,”
Comput. Fluids
,
154
, pp.
390
406
.https://doi.org/10.1016/j.compfluid.2016.05.022
35.
Hesami
,
H.
, and
Mayeli
,
P.
,
2016
, “
Development of the Ball-Spine Algorithm for the Shape Optimization of Ducts Containing Nanofluid
,”
Numer. Heat Transfer, Part A
,
70
(
12
), pp.
1371
1389
.http://dx.doi.org/10.1080/10407782.2016.1243976
36.
Stanitz
,
J. D.
,
1988
, “
A Review of Certain Inverse Methods for the Design of Ducts With 2 or 3-Dimensional Potential Flow
,”
ASME Appl. Mech. Rev.
,
41
(
6
), pp.
217
238
.
37.
Nelson
,
C. D.
, and
Yang
,
T.
,
1977
, “
Design of Branched and Unbranched Axially Symmetrical Ducts With Specified Pressure Distribution
,”
AIAA J.
,
15
(
9
), pp.
1272
1277
.
38.
Chaviaropoulos
,
P.
,
Dedoussis
,
V.
, and
Papailion
,
K. D.
,
1995
, “
On the 3-D Inverse Potential Target Pressure Problem—Part I: Theoretical Aspects and Method Formulation
,”
J. Fluid Mech.
,
282
, pp.
131
146
.
39.
Chaviaropoulos
,
P.
,
Dedoussis
,
V.
, and
Papailion
,
K. D.
,
1995
, “
On the 3-D Inverse Potential Target Pressure Problem—Part II: Numerical Aspects and Application to Duct Design
,”
J. Fluid Mech.
,
282
, pp.
147
162
.https://doi.org/10.1017/S0022112095000073
40.
Zannetti
,
L.
,
1986
, “
A Natural Formulation for the Solution of Two–Dimensional or Axis–Symmetric Inverse Problems
,”
Int. J. Numer. Methods Eng.
,
22
(
2
), pp.
451
463
.
41.
Borges
,
J. E.
,
2007
, “
Computational Method for the Design of Ducts
,”
Comput. Fluids
,
36
(
2
), pp.
480
483
.
42.
Ashrafizadeh
,
A.
,
Raithby
,
G. D.
, and
Stubley
,
G. D.
,
2002
, “
Direct Design of Shape
,”
Numer. Heat Transfer, Part B
,
41
(
6
), pp.
501
520
.
43.
Ashrafizadeh
,
A.
,
Raithby
,
G. D.
, and
Stubley
,
G. D.
,
2003
, “
Direct Design of Ducts
,”
ASME J. Fluids Eng.
,
125
(
1
), pp.
158
165
.
44.
Ashrafizadeh
,
A.
,
Raithby
,
G. D.
, and
Stubley
,
G. D.
,
2004
, “
Direct Design of Airfoil Shape With a Prescribed Surface Pressure
,”
Numer. Heat Transfer, Part B
,
46
(
6
), pp.
505
527
.
45.
Ashrafizadeh
,
A.
, and
Raithby
,
G. D.
,
2006
, “
Direct Design Solution of the Elliptic Grid Generation Equations
,”
Numer. Heat Transfer, Part B
,
50
(
3
), pp.
217
230
.
46.
Taiebi-Rahni
,
M.
,
Ghadak
,
F.
, and
Ashrafizadeh
,
A.
,
2008
, “
A Direct Design Approach Using the Euler Equation
,”
Inverse Probl. Sci. Eng.
,
16
(
2
), pp.
217
231
.
47.
Ghadak
,
F.
,
Taiebi-Rahni
,
M.
, and
Ashrafizadeh
,
A.
,
2009
, “
Direct Design of Branched Ducts
,”
Sci. Iran., Trans. B
,
16
(
2
), pp.
111
120
.http://www.sid.ir/en/VEWSSID/J_pdf/95520092B09.pdf
48.
Ashrafizadeh
,
A.
,
Bavafa
,
B. A.
, and
Mayeli
,
P.
,
2015
, “
A New Co-Located Pressure-Based Discretization Method for the Numerical Solution of Incompressible Navier-Stokes Equations
,”
Numer. Heat Transfer, Part B
,
67
(
6
), pp.
563
589
.
49.
Nikfar
,
M.
, and
Ashrafizadeh
,
A.
,
2016
, “
A Coupled Element-Based Finite Volume Method for the Solution of Incompressible Navier-Stokes Equations
,”
Numer. Heat Transfer, Part B
,
69
(
5
), pp.
447
472
.
50.
Ashrafizadeh
,
A.
, and
Nikfar
,
M.
,
2016
, “
On the Numerical Solution of Generalized Convection Heat Transfer Problems Via the Method of Proper Closure Equations—Part I: Description of the Method
,”
Numer. Heat Transfer, Part A
,
70
(
5
), pp.
187
203
.
51.
Ashrafizadeh
,
A.
, and
Nikfar
,
M.
,
2016
, “
On the Numerical Solution of Generalized Convection Heat Transfer Problems Via the Method of Proper Closure Equations—Part II: Application to Test Problems
,”
Numer. Heat Transfer, Part A
,
70
(
5
), pp.
204
222
.
52.
Bejan
,
A.
,
2004
,
Convection Heat Transfer
,
3rd ed.
,
Wiley
,
New York
.
53.
Al-Amiri
,
A.
,
Khanafer
,
K.
,
Bull
,
J.
, and
Pop
,
I.
,
2007
, “
Effect of the Sinusoidal Wavy Bottom Surface on Mixed Convection Heat Transfer in a Lid-Driven Cavity
,”
Int. J. Heat Mass Transfer
,
50
(
10
), pp.
1771
1780
.
54.
Nikfar
,
M.
, and
Mahmoodi
,
M.
,
2012
, “
Meshless Local Petrov-Galerkin Analysis of Free Convection of Nanofluid in a Cavity With Wavy Side Walls
,”
Eng. Anal. Boundary Elem.
,
36
(
3
), pp.
433
445
.
55.
Arefmanesh
,
A.
, and
Nikfar
,
M.
,
2013
, “
Analysis of Natural Convection in a Nanofluid-Filled Triangular Enclosure Induced by Cold and Hot Sources on the Walls Using Stabilised MLPG Method
,”
Can. J. Chem. Eng.
,
91
(
10
), pp.
1711
1728
.
You do not currently have access to this content.