Abstract
In this paper, a model validation framework is proposed and applied to a large vibro-acoustic finite element (FE) model of a passenger car. The framework introduces a p-box approach with an efficient quantification scheme of uncertainty sources and a new area metric which is relevant to the responses in the frequency domain. To prioritize the input uncertainties out of the enormous FE model, the experts’ knowledge is utilized to select candidate input parameters which have large potential influences on the response of interests (ROI) among several thousands of input parameters. Next, a variance-based sensitivity analysis with an orthogonal array is introduced in effort to quantify the influence of the selected input parameters on the ROIs. The employment of the eigenvector dimension reduction method and orthogonal combinations of interval-valued input parameters provides the p-box of the ROI even if the size of the FE model is very large. A color map and the u-pooling of the p-boxes over the frequency band as well as the p-box at different frequencies are introduced to assess the model error and quantitative contributions of the aleatory and the epistemic input uncertainties to the overall variability of the ROIs in the frequency domain. After assessing the model error, the FE model is updated. It was found that the sensitivity results and the experts’ knowledge about the associated components effectively determine the modifications of the component models and the input parameter values during the updating process.
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Babuska I, Oden JT (2004) Verification and validation in computational engineering and science: basic concepts. Comput Methods Appl Mech Eng 193:4057–4066. doi:10.1016/j.cma.2004.03.002
Barthelmann V, Novak E, Ritter K (2000) High dimensional polynomial interpolation on sparse grids. Adv Comput Math 12:273–288
Campbell K (2006) Statistical calibration of computer simulations. Reliab Eng Syst Saf 91:1358–1363. doi:10.1016/j.ress.2005.11.032
Choi KK, Kim N-H (2006) Structural sensitivity analysis and optimization 1: linear systems. Springer Science & Business Media
Choi KK, Shim I, Wang S (1997) Design sensitivity analysis of structure-induced noise and vibration. J Vib Acoust 119:173–179
Choi S-K, Grandhi RV, Canfield RA, Pettit CL (2004) Polynomial chaos expansion with latin hypercube sampling for estimating response variability. AIAA J 42:1191–1198
Choi S-K, Grandhi RV, Canfield RA (2007) Reliability-based structural design. Springer
Durand J-F, Soize C, Gagliardini L (2008) Structural-acoustic modeling of automotive vehicles in presence of uncertainties and experimental identification and validation. J Acoust Soc Am 124:1513–1525. doi:10.1121/1.2953316
Ferson S, Ginzburg LR (1996) Different methods are needed to propagate ignorance and variability. Reliab Eng Syst Saf 54:133–144. doi:10.1016/S0951-8320(96)00071-3
Ferson S, Oberkampf WL, Ginzburg L (2008) Model validation and predictive capability for the thermal challenge problem. Comput Methods Appl Mech Eng 197:2408–2430. doi:10.1016/j.cma.2007.07.030
Gerstner T, Griebel M (1998) Numerical integration using sparse grids. Numer Algorithm 18:209–232
Hu C, Youn BD (2011a) Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct Multidiscip Optim 43:419–442
Hu C, Youn BD (2011b) An asymmetric dimension-adaptive tensor-product method for reliability analysis. Struct Saf 33:218–231
Jung BC, Lee D, Youn BD, Lee S (2011) A statistical characterization method for damping material properties and its application to structural-acoustic system design. J Mech Sci Technol 25:1893–1904
Kennedy MC, O’Hagan A (2001) Bayesian calibration of computer models. J R Stat Soc Ser B (Stat Methodol) 63:425–464
Kim NH, Dong J, Choi KK, Vlahopoulos N, Ma Z-D, Castanier M, Pierre C (2003) Design sensitivity analysis for sequential structural–acoustic problems. J Sound Vib 263:569–591
Kroese DP, Taimre T, Botev ZI (2011) Handbook of Monte Carlo methods vol 706. Wiley
Kwon J-H, Lee D (2015) Variability analysis of vibrational responses in a passenger car considering the uncertainties of elastomers. Proc Inst Mech Eng C J Mech Eng Sci. doi:10.1177/0954406215612816
Lardeur P, Scigliano R, Scionti M (2013) Verification and validation for the vibration study of automotive structures modelled by finite elements. J Strain Anal Eng Des 48:59–72. doi:10.1177/0309324712466508
Lee D, Ahn T-S (2014) A boundary element model for acoustic responses in the ear canal and its statistical validation and updating. J Mech Sci Technol 28:1203–1217
Lee D, Ahn T-S (2015) Statistical calibration of a finite element model for human middle ear. J Mech Sci Technol 29:2803–2815. doi:10.1007/s12206-015-0609-9
Lee D, Hwang I-S (2011) Analysis on the dynamic characteristics of a rubber mount considering temperature and material uncertainties. Trans Comput Struct Eng Inst Korea 24:383–389
Oberkampf WL, Roy CJ (2010) Verification and validation in scientific computing. Cambridge University Press, Cambridge
Oberkampf WL, Trucano TG, Hirsch C (2004) Verification, validation, and predictive capability in computational engineering and physics. Appl Mech Rev 57:345–384
Phadke MS (1989) Quality engineering using robust design. Prentice Hall, London
Rao CR (1947) Factorial experiments derivable from combinatorial arrangements of arrays. Suppl J R Stat Soc 9:128–139. doi:10.2307/2983576
Roy R (1990) A primer on the Taguchi method. Van Norstrand Reinhold, New York
Roy CJ, Oberkampf WL (2011) A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing. Comput Methods Appl Mech Eng 200:2131–2144. doi:10.1016/j.cma.2011.03.016
Saltelli A et al. (2008) Global sensitivity analysis: the primer. John Wiley & Sons
Scigliano R, Scionti M, Lardeur P (2011) Verification, validation and variability for the vibration study of a car windscreen modeled by finite elements. Finite Elem Anal Des 47:17–29
Soize C (2013) Stochastic modeling of uncertainties in computational structural dynamics—recent theoretical advances. J Sound Vib 332:2379–2395. doi:10.1016/j.jsv.2011.10.010
Trucano TG, Swiler LP, Igusa T, Oberkampf WL, Pilch M (2006) Calibration, validation, and sensitivity analysis: what’s what. Reliab Eng Syst Saf 91:1331–1357. doi:10.1016/j.ress.2005.11.031
Xiong Y, Chen W, Tsui K-L, Apley DW (2009) A better understanding of model updating strategies in validating engineering models. Comput Methods Appl Mech Eng 198:1327–1337. doi:10.1016/j.cma.2008.11.023
Xu H, Rahman S (2004) A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics. Int J Numer Methods Eng 61:1992–2019
Youn B, Xi Z, Wang P (2008) Eigenvector dimension reduction (EDR) method for sensitivity-free probability analysis. Struct Multidiscip Optim 37:13–28. doi:10.1007/s00158-007-0210-7
Youn BD, Jung BC, Xi Z, Kim SB, Lee WR (2011) A hierarchical framework for statistical model calibration in engineering product development. Comput Methods Appl Mech Eng 200:1421–1431. doi:10.1016/j.cma.2010.12.012
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This work was supported by the Industrial Strategic Technology Development Program (Grant No. 10048305), funded by the Ministry of Trade, Industry & Energy (MI, Korea).
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Lee, D., Kim, N.H. & Kim, HS. Validation and updating in a large automotive vibro-acoustic model using a P-box in the frequency domain. Struct Multidisc Optim 54, 1485–1508 (2016). https://doi.org/10.1007/s00158-016-1427-0
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DOI: https://doi.org/10.1007/s00158-016-1427-0