The acoustic field with convex parameters widely exists in the engineering practice. The vertex method and the anti-optimization method are not considered as appropriated approaches for the response analysis of acoustic field with convex parameters. The shortcoming of the vertex method is that the local optima out of vertexes cannot be identified. The disadvantage of the anti-optimization method is that the analytical formulation of response may be not obtained. To analyze the acoustic field with convex parameters efficiently and effectively, a first-order convex perturbation method (FCPM) and a second-order convex perturbation method (SCPM) are presented. In FCPM, the response of the acoustic field with convex parameters is expanded with the first-order Taylor series. In SCPM, the response of the acoustic field with convex parameters is expanded with the second-order Taylor series neglecting the nondiagonal elements of Hessian matrix. The variational bounds of the expanded responses in FCPM and SCPM are yielded by the Lagrange multiplier method. The accuracy and efficiency of FCPM and SCPM are investigated by numerical examples.

References

References
1.
Hurtado
,
J. E.
, and
Alvarez
,
D. A.
,
2012
, “
The Encounter of Interval and Probabilistic Approaches to Structural Reliability at the Design Point
,”
Comput. Methods Appl. Mech. Eng.
,
225–228
, pp.
74
94
.10.1016/j.cma.2012.03.020
2.
Seçgin
,
A. J.
,
Dunne
,
F.
, and
Zoghaib
,
L.
,
2012
, “
Extreme-Value-Based Statistic Bounding of Low, Mid, and High Frequency Responses of a Forced Plate With Random Boundary Conditions
,”
ASME J. Vib. Acoust.
,
134
(
2
), p.
021003
.10.1115/1.4005019
3.
Pascual
,
B.
, and
Adhikari
,
S.
,
2012
, “
Combined Parametric-Nonparametric Uncertainty Quantification Using Random Matrix Theory and Polynomial Chaos Expansion
,”
Comput. Struct.
,
112–113
, pp.
364
379
.10.1016/j.compstruc.2012.08.008
4.
Pettersson
,
P.
,
Doostan
,
A.
, and
Nordström
,
J.
,
2013
, “
On Stability and Monotonicity Requirements of Finite Difference Approximations of a Class of Stochastic Conservation Laws With Random Viscosity
,”
Comput. Methods Appl. Mech. Eng.
,
258
, pp.
134
151
.10.1016/j.cma.2013.02.009
5.
Sarrouy
,
E.
,
Dessombz
,
O.
, and
Sinou
,
J. J.
,
2013
, “
Piecewise Polynomial Chaos Expansion With an Application to Brake Squeal of a Linear Brake System
,”
J. Sound Vib.
,
332
(
3
), pp.
577
594
.10.1016/j.jsv.2012.09.009
6.
Kamiński
,
M. M.
, and
Świta
,
P.
,
2011
, “
Generalized Stochastic Finite Element Method in Elastic Stability Problems
,”
Comput. Struct.
,
89
(
11–12
), pp.
1241
1252
.10.1016/j.compstruc.2010.08.009
7.
Kamiński
,
M. M.
,
2012
, “
Probabilistic Entropy in Homogenization of the Periodic Fiber-Reinforced Composites With Random Elastic Parameters
,”
Int. J. Numer. Methods Eng.
,
90
(
8
), pp.
939
954
.10.1002/nme.3350
8.
Kamiński
,
M.
, and
Solecka
,
M.
,
2013
, “
Optimization of the Truss-Type Structures Using the Generalized Perturbation-Based Stochastic Finite Element Method
,”
Finite Elem. Anal. Des.
,
63
, pp.
69
79
.10.1016/j.finel.2012.08.002
9.
Ben-Haim
,
Y.
, and
Elishakoff
,
I.
,
1990
,
Convex Models of Uncertainties in Applied Mechanics
,
Elsevier Science
,
Amsterdam, Netherlands
.
10.
Qiu
,
Z.
,
2003
, “
Comparison of Static Response of Structures Using Convex Models and Interval Analysis Method
,”
Int. J. Numer. Methods Eng.
,
56
(
12
), pp.
1735
1753
.10.1002/nme.636
11.
Wang
,
X.
,
Elishakoff
,
I.
, and
Qiu
,
Z.
,
2008
, “
Experimental Data Have to Decide Which of the Non-Probabilistic Uncertainty Descriptions—Convex Modeling or Interval Analysis-to Utilize
,”
ASME J. Appl. Mech.
,
75
(
4
), p.
041018
.10.1115/1.2912988
12.
Hua
,
J.
, and
Qiu
,
Z.
,
2010
, “
Non-Probabilistic Convex Models and Interval Analysis Method for Dynamic Response of a Beam With Bounded Uncertainty
,”
Appl. Math. Model.
,
34
(
3
), pp.
725
734
.10.1016/j.apm.2009.06.013
13.
Bojanov
,
B.
, and
Petrov
,
P.
,
2005
, “
Gaussian Interval Quadrature Formulae for Tchebycheff Systems
,”
SIAM J. Numer. Anal.
,
43
(
2
), pp.
787
795
.10.1137/040606521
14.
Garloff
,
J.
,
2009
, “
Interval Gaussian Elimination With Pivot Tightening
,”
SIAM J. Matrix Anal. Appl.
,
30
(
4
), pp.
1761
1772
.10.1137/080729621
15.
Qiu
,
Z.
,
Xia
,
Y.
, and
Yang
,
J.
,
2007
, “
The Static Displacement and the Stress Analysis of Structures With Bounded Uncertainties Using the Vertex Solution Theorem
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
49–52
), pp.
4965
4984
.10.1016/j.cma.2007.06.022
16.
Guo
,
X.
,
Bai
,
W.
, and
Zhang
,
W.
,
2008
, “
Extreme Structural Response Analysis of Truss Structures Under Material Uncertainty Via Linear Mixed 0-1 Programming
,”
Int. J. Numer. Methods Eng.
,
76
(
3
), pp.
253
277
.10.1002/nme.2298
17.
McWilliam
,
S.
,
2001
, “
Anti-Optimisation of Uncertain Structures Using Interval Analysis
,”
Comput. Struct.
,
79
(
4
), pp.
421
430
.10.1016/S0045-7949(00)00143-7
18.
Chen
,
S.
,
Lian
,
H.
, and
Yang
,
X.
,
2002
, “
Interval Static Displacement Analysis for Structures With Interval Parameters
,”
Int. J. Numer. Methods Eng.
,
53
(
2
), pp.
393
407
.10.1002/nme.281
19.
Banerjee
,
S.
, and
Jacobi
,
A. M.
,
2014
, “
Determination of Transmission Loss in Slightly Distorted Circular Mufflers Using a Regular Perturbation Method
,”
ASME J. Vib. Acoust.
,
136
(
2
), p.
021013
.10.1115/1.4026209
20.
Qiu
,
Z.
, and
Elishakoff
,
I.
,
1998
, “
Antioptimization of Structures With Large Uncertain-But-Non-Random Parameters Via Interval Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
152
(
3–4
), pp.
361
372
.10.1016/S0045-7825(96)01211-X
21.
Xia
,
B.
,
Yu
,
D.
, and
Liu
,
J.
,
2013
, “
Interval and Subinterval Perturbation Methods for a Structural-Acoustic System With Interval Parameters
,”
J. Fluids Struct.
,
38
, pp.
146
163
.10.1016/j.jfluidstructs.2012.12.003
22.
Chen
,
S. H.
,
Ma
,
L.
,
Meng
,
G. W.
, and
Guo
,
R.
,
2009
, “
An Efficient Method for Evaluating the Natural Frequency of Structures With Uncertain-But-Bounded Parameters
,”
Comput. Struct.
,
87
(
9–10
), pp.
582
590
.10.1016/j.compstruc.2009.02.009
23.
Fujita
,
K.
, and
Takewaki
,
I.
,
2011
, “
An Efficient Methodology for Robustness Evaluation by Advanced Interval Analysis Using Updated Second-Order Taylor Series Expansion
,”
Eng. Struct.
,
33
(
12
), pp.
3299
3310
.10.1016/j.engstruct.2011.08.029
24.
Xia
,
B.
, and
Yu
,
D.
,
2012
, “
Interval Analysis of Acoustic Field With Uncertain-But-Bounded Parameters
,”
Comput. Struct.
,
112–113
, pp.
235
244
.10.1016/j.compstruc.2012.08.010
25.
Xia
,
B.
, and
Yu
,
D.
,
2012
, “
Modified Sub-Interval Perturbation Finite Element Method for 2D Acoustic Field Prediction With Large Uncertain-But-Bounded Parameters
,”
J. Sound Vib.
,
331
(
16
), pp.
3774
3790
.10.1016/j.jsv.2012.03.024
26.
Qiu
,
Z. P.
,
Ma
,
L. H.
, and
Wang
,
X. J.
,
2006
, “
Ellipsoidal-Bound Convex Model for the Non-Linear Buckling of a Column With Uncertain Initial Imperfection
,”
Int. J. Non-Linear Mech.
,
41
(
8
), pp.
919
925
.10.1016/j.ijnonlinmec.2006.07.001
27.
Wang
,
X.
,
Elishakoff
,
I.
,
Qiu
,
Z.
, and
Ma
,
L.
,
2009
, “
Comparisons of Probabilistic and Two Non-Probabilistic Methods for Uncertain Imperfection Sensitivity of a Column on a Nonlinear Mixed Quadratic-Cubic Foundation
,”
ASME J. Appl. Mech.
,
76
(
1
), p.
011007
.10.1115/1.2998763
28.
Pantelides
,
C. P.
, and
Ganzerli
,
S.
,
1998
, “
Design of Trusses Under Uncertain Loads Using Convex Models
,”
ASCE J. Struct. Eng.
,
124
(
3
), pp.
318
329
.10.1061/(ASCE)0733-9445(1998)124:3(318)
29.
Qiu
,
Z.
,
Ma
,
L.
, and
Wang
,
X.
,
2009
, “
Unified Form for Static Displacement, Dynamic Response and Natural Frequency Analysis Based on Convex Models
,”
Appl. Math. Model
,
33
(
10
), pp.
3836
3847
.10.1016/j.apm.2009.01.001
30.
Jiang
,
C.
,
Bi
,
R.
,
Lu
,
G.
, and
Han
,
X.
,
2013
, “
Structural Reliability Analysis Using Non-Probabilistic Convex Model
,”
Comput. Methods Appl. Mech. Eng.
,
254
, pp.
83
98
.10.1016/j.cma.2012.10.020
31.
Xia
,
B.
, and
Yu
,
D.
,
2013
, “
Response Probability Analysis of Random Acoustic Field Based on Perturbation Stochastic Method and Change-of-Variable Technique
,”
ASME J. Vib. Acoust.
,
135
(
5
), p.
051032
.10.1115/1.4024853
32.
Xia
,
B.
,
Yu
,
D.
, and
Liu
,
J.
,
2013
, “
Hybrid Uncertain Analysis for Structural-Acoustic Problem With Random and Interval Parameters
,”
J. Sound Vib.
,
332
(
11
), pp.
2701
2720
.10.1016/j.jsv.2012.12.028
33.
Xia
,
B.
,
Yu
,
D.
, and
Liu
,
J.
,
2013
, “
Probabilistic Interval Perturbation Methods for Hybrid Uncertain Acoustic Field Prediction
,”
ASME J. Vib. Acoust.
,
135
(
2
), p.
021009
.10.1115/1.4023054
34.
Shi
,
C.
,
Lu
,
J.
, and
Zhang
,
G.
,
2005
, “
An Extended Kuhn–Tucker Approach for Linear Bilevel Programming
,”
Appl. Math. Comput.
,
162
(
1
), pp.
51
63
.10.1016/j.amc.2003.12.089
35.
Wenterodt
,
C.
, and
von Estorff
,
O.
,
2009
, “
Dispersion Analysis of the Meshfree Radial Point Interpolation Method for the Helmholtz Equation
,”
Int. J. Numer. Methods Eng.
,
77
(
12
), pp.
1670
1689
.10.1002/nme.2463
36.
Babuka
,
I.
,
Ihlenburg
,
F.
,
Strouboulis
,
T.
, and
Gangaraj
,
S. K.
,
1997
, “
A Posteriori Error Estimation for Finite Element Solutions of Helmholtz’ Equation—Part II: Estimation of the Pollution Error
,”
Int. J. Numer. Methods Eng.
,
40
, pp.
3883
3900
.10.1002/(SICI)1097-0207(19971115)40:21<3883::AID-NME231>3.0.CO;2-V
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