This paper presents a study of multi-objective optimization of elastic beams with minimum weight and radiated sound power. The goal of this research is to discover the potentials to design multi-objective optimal elastic structures for better acoustic performance. We discuss various structural-acoustic properties of the Pareto solutions of the multi-objective optimization problem (MOP). We have found that geometrical and dynamic constraints can substantially reduce the volume fraction of feasible solutions in the design space, which can make it difficult to search for the optimal solutions. Several case studies with different boundary conditions are studied to demonstrate the multi-objective optimal designs of the structure.

References

1.
Koopmann
,
G. H.
, and
Fahnline
,
J. B.
,
1997
,
Designing Quiet Structures: A Sound Power Minimization Approach
,
Academic Press
, London.
2.
Marburg
,
S.
,
2002
, “
Developments in Structural-Acoustic Optimization for Passive Noise Control
,”
Arch. Comput. Methods Eng.
,
9
(
4
), pp.
291
370
.
3.
Adali
,
S.
,
1983
, “
Pareto Optimal Design of Beams Subjected to Support Motions
,”
Comput. Struct.
,
16
(
1
), pp.
297
303
.
4.
Eschenauer
,
H.
,
Kneppe
,
G.
, and
Stenvers
,
K. H.
,
1986
, “
Deterministic and Stochastic Multiobjective Optimization of Beam and Shell Structures
,”
J. Mech. Transm. Autom. Des.
,
108
(
1
), pp.
31
37
.
5.
Denli
,
H.
, and
Sun
,
J. Q.
,
2007
, “
Optimization of Boundary Supports for Sound Radiation Reduction of Vibrating Structures
,”
ASME J. Vib. Acoust.
,
130
(
1
), p.
011007
.
6.
Purekar
,
A. S.
, and
Pines
,
D. J.
,
2000
, “
Detecting Damage in Non-Uniform Beams Using the Dereverberated Transfer Function Response
,”
Smart Mater. Struct.
,
9
(
4
), p.
429
.
7.
Joshi
,
P.
,
Mulani
,
S. B.
, and
Kapania
,
R. K.
,
2015
, “
Multi-Objective Vibro-Acoustic Optimization of Stiffened Panels
,”
Struct. Multidiscip. Optim.
,
51
(
4
), pp.
835
848
.
8.
Kodiyalam
,
S.
,
Adali
,
S.
, and
Sadek
,
I. S.
,
1992
, “
Multiobjective Design Optimization of Continuous Beams by Numerical Methods
,”
Eng. Comput.
,
9
(
5
), pp.
539
546
.
9.
Au
,
F. T. K.
,
Zheng
,
D. Y.
, and
Cheung
,
Y. K.
,
1999
, “
Vibration and Stability of Non-Uniform Beams With Abrupt Changes of Cross-Section by Using C1 Modified Beam Vibration Functions
,”
Appl. Math. Modell.
,
23
(
1
), pp.
19
34
.
10.
Lee
,
S. Y.
, and
Hsiao
,
J. Y.
,
2002
, “
Free In-Plane Vibrations of Curved Nonuniform Beams
,”
Acta Mech.
,
155
(
3
), pp.
173
189
.
11.
Marburg
,
S.
,
Dienerowitz
,
F.
,
Fritze
,
D.
, and
Hardtke
,
H.-J.
,
2006
, “
Case Studies on Structural-Acoustic Optimization of a Finite Beam
,”
Acta Acust. Acust.
,
92
(
3
), pp.
427
439
.https://www.researchgate.net/publication/233524811_Case_Studies_on_Structural-Acoustic_Optimization_of_a_Finite_Beam
12.
Chen
,
L.-Y.
, and
Wang
,
D.-Y.
,
2008
, “
Structural-Acoustic Optimization of Stiffened Panels Based on a Genetic Algorithm
,”
J. Mar. Sci. Appl.
,
6
(
4
), pp.
55
61
.
13.
Sun
,
J. Q.
,
1995
, “
Vibration and Sound Radiation of Non-Uniform Beams
,”
J. Sound Vib.
,
185
(
5
), pp.
827
843
.
14.
Ho
,
S. H.
, and
Chen
,
C. K.
,
1998
, “
Analysis of General Elastically End Restrained Non-Uniform Beams Using Differential Transform
,”
Appl. Math. Modell.
,
22
(
4–5
), pp.
219
234
.
15.
Fliege
,
J.
, and
Svaiter
,
F. B.
,
2000
, “
Steepest Descent Methods for Multicriteria Optimization
,”
Math. Methods Oper. Res.
,
51
(
3
), pp.
479
494
.
16.
Custódio
,
A. L.
,
Madeira
,
J. F. A.
,
Vaz
,
A. I. F.
, and
Vicente
,
L. N.
,
2011
, “
Direct Multisearch for Multiobjective Optimization
,”
SIAM J. Optim.
,
21
(
3
), pp.
1109
1140
.
17.
Ringkamp
,
M.
,
Ober-Blöbaum
,
S.
,
Dellnitz
,
M.
, and
Schütze
,
O.
,
2012
, “
Handling High-Dimensional Problems With Multi-Objective Continuation Methods Via Successive Approximation of the Tangent Space
,”
Eng. Optim.
,
44
(
9
), pp.
1117
1146
.
18.
Schütze
,
O.
,
Mostaghim
,
S.
,
Dellnitz
,
M.
, and
Teich
,
J.
,
2003
, “
Covering Pareto Sets by Multilevel Evolutionary Subdivision Techniques
,”
Second International Conference on Evolutionary Multi-Criterion Optimization
(
EMO
), Faro, Portugal, Apr. 8–11, pp.
118
132
.
19.
Hernández
,
C.
,
Naranjani
,
Y.
,
Sardahi
,
Y.
,
Liang
,
W.
,
Schütze
,
O.
, and
Sun
,
J.-Q.
,
2013
, “
Simple Cell Mapping Method for Multi-Objective Optimal Feedback Control Design
,”
Int. J. Dyn. Control
,
1
(
3
), pp.
231
238
.
20.
Naranjani
,
Y.
,
Hernández
,
C.
,
Xiong
,
F.-R.
,
Schütze
,
O.
, and
Sun
,
J.-Q.
,
2013
, “
A Hybrid Algorithm for the Simple Cell Mapping Method in Multi-Objective Optimization
,”
EVOLVE—A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation IV
,
M.
Emmerich
, A. Deutz, O. Schuetze, Th. Bäck, E. Tantar, A. A. Tantar, P. Del Moral, P. Legrand, P. Bouvry, and C. A. Coello Coello, eds.,
Springer International Publishing
,
Berlin
, pp.
207
223
.
21.
Deb
,
K.
,
Pratap
,
A.
,
Agarwal
,
S.
, and
Meyarivan
,
T.
,
2002
, “
A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II
,”
IEEE Trans. Evol. Comput.
,
6
(
2
), pp.
182
197
.
22.
Coello
,
C. A. C.
,
Pulido
,
G. T.
, and
Lechuga
,
M. S.
,
2004
, “
Handling Multiple Objectives With Particle Swarm Optimization
,”
IEEE Trans. Evol. Comput.
,
8
(
3
), pp.
256
279
.
23.
Amuso
,
V. J.
, and
Enslin
,
J.
,
2007
, “
The Strength Pareto Evolutionary Algorithm 2 (SPEA2) Applied to Simultaneous Multi-Mission Waveform Design
,” International Waveform Diversity and Design Conference (
WDDC
), Pisa, Italy, June 4–8, pp.
407
417
.
24.
Naranjani
,
Y.
,
Hernández
,
C.
,
Xiong
,
F.-R.
,
Schütze
,
O.
, and
Sun
,
J.-Q.
,
2016
, “
A Hybrid Method of Evolutionary Algorithm and Simple Cell Mapping for Multi-Objective Optimization Problems
,”
Int. J. Dyn. Control
, epub.
25.
Coello
,
C. A. C.
, and
Lechuga
,
M. S.
,
2002
, “
MOPSO: A Proposal for Multiple Objective Particle Swarm Optimization
,” Congress on Evolutionary Computation (
CEC
), Honolulu, HI, May 12–17, Vol.
2
, pp.
1051
1056
.
26.
Poli
,
R.
,
Kennedy
,
J.
, and
Blackwell
,
T.
,
2007
, “
Particle Swarm Optimization
,”
Swarm Intell.
,
1
(
1
), pp.
33
57
.
27.
Coello
,
C. A. C.
,
Lamont
,
G. B.
, and
van Veldhuizen
,
D. A.
,
2007
,
Evolutionary Algorithms for Solving Multi-Objective Problems
,
Springer
,
New York
.
28.
Hsu
,
C. S.
,
1985
, “
A Discrete Method of Optimal Control Based Upon the Cell State Space Concept
,”
J. Optim. Theory Appl.
,
46
(
4
), pp.
547
569
.
29.
Han
,
S. M.
,
Benaroya
,
H.
, and
Wei
,
T.
,
1999
, “
Dynamics of Transversely Vibrating Beams Using Four Engineering Theories
,”
J. Sound Vib.
,
225
(
5
), pp.
935
988
.
30.
Majkut
,
L.
,
2009
, “
Free and Forced Vibrations of Timoshenko Beams Described by Single Difference Equation
,”
J. Theor. Appl. Mech.
,
47
(
1
), pp.
193
201
.http://www.warminski.pollub.plwww.ptmts.org.pl/Majkut-1-09.pdf
31.
Lamancusa
,
J.
,
1993
, “
Numerical Optimization Techniques for Structural-Acoustic Design of Rectangular Panels
,”
Comput. Struct.
,
48
(
4
), pp.
661
675
.
32.
Klaerner
,
M.
,
Wuehrl
,
M.
,
Kroll
,
L.
, and
Marburg
,
S.
,
2017
, “
FEA-Based Methods for Optimising Structure-Borne Sound Radiation
,”
Mech. Syst. Signal Process.
,
89
, pp.
37
47
.
33.
Bathe
,
K. J.
,
1995
,
Finite Element Procedures
,
Prentice Hall
, Upper Saddle River, NJ.
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