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.
Issue Section:
Research Papers
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.2012.03.0202.
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
.10.1115/1.40050193.
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.2012.08.0084.
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.2013.02.0095.
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.2012.09.0096.
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.2010.08.0097.
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.33508.
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.2012.08.0029.
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.63611.
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
.10.1115/1.291298812.
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.2009.06.01313.
Bojanov
, B.
, and Petrov
, P.
, 2005
, “Gaussian Interval Quadrature Formulae for Tchebycheff Systems
,” SIAM J. Numer. Anal.
, 43
(2
), pp. 787
–795
.10.1137/04060652114.
Garloff
, J.
, 2009
, “Interval Gaussian Elimination With Pivot Tightening
,” SIAM J. Matrix Anal. Appl.
, 30
(4
), pp. 1761
–1772
.10.1137/08072962115.
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.2007.06.02216.
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.229817.
McWilliam
, S.
, 2001
, “Anti-Optimisation of Uncertain Structures Using Interval Analysis
,” Comput. Struct.
, 79
(4
), pp. 421
–430
.10.1016/S0045-7949(00)00143-718.
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.28119.
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
.10.1115/1.402620920.
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-X21.
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.2012.12.00322.
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.2009.02.00923.
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.2011.08.02924.
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.2012.08.01025.
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.2012.03.02426.
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.2006.07.00127.
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
.10.1115/1.299876328.
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.2009.01.00130.
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.2012.10.02031.
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
.10.1115/1.402485332.
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.2012.12.02833.
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
.10.1115/1.402305434.
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.2003.12.08935.
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.246336.
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-VCopyright © 2014 by ASME
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