Wavelet-based iterative data enhancement for implementation in purification of modal frequency for extremely noisy ambient vibration tests in Shiraz-Iran

doi:10.1007/s11709-019-0605-8

Front. Struct. Civ. Eng.

2020, Vol. 14

Issue (2) : 446-472 https://doi.org/10.1007/s11709-019-0605-8

RESEARCH ARTICLE

Wavelet-based iterative data enhancement for implementation in purification of modal frequency for extremely noisy ambient vibration tests in Shiraz-Iran

Hassan YOUSEFI¹(

), Alireza TAGHAVI KANI², Iradj MAHMOUDZADEH KANI², Soheil MOHAMMADI¹

¹. High Performance Computing Lab, School of Civil Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran
². School of Civil Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

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Abstract

The main purpose of the present study is to enhance high-level noisy data by a wavelet-based iterative filtering algorithm for identification of natural frequencies during ambient wind vibrational tests on a petrochemical process tower. Most of denoising methods fail to filter such noise properly. Both the signal-to-noise ratio and the peak signal-to-noise ratio are small. Multiresolution-based one-step and variational-based filtering methods fail to denoise properly with thresholds obtained by theoretical or empirical method. Due to the fact that it is impossible to completely denoise such high-level noisy data, the enhancing approach is used to improve the data quality, which is the main novelty from the application point of view here. For this iterative method, a simple computational approach is proposed to estimate the dynamic threshold values. Hence, different thresholds can be obtained for different recorded signals in one ambient test. This is in contrast to commonly used approaches recommending one global threshold estimated mainly by an empirical method. After the enhancements, modal frequencies are directly detected by the cross wavelet transform (XWT), the spectral power density and autocorrelation of wavelet coefficients. Estimated frequencies are then compared with those of an undamaged-model, simulated by the finite element method.

Keywords ambient vibration test high level noise iterative signal enhancement wavelet cross and autocorrelation of wavelets

Corresponding Author(s): Hassan YOUSEFI

Just Accepted Date: 23 February 2020 Online First Date: 08 April 2020 Issue Date: 08 May 2020

Cite this article:

Hassan YOUSEFI,Alireza TAGHAVI KANI,Iradj MAHMOUDZADEH KANI, et al. Wavelet-based iterative data enhancement for implementation in purification of modal frequency for extremely noisy ambient vibration tests in Shiraz-Iran[J]. Front. Struct. Civ. Eng., 2020, 14(2): 446-472.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-019-0605-8
https://academic.hep.com.cn/fsce/EN/Y2020/V14/I2/446

Fig.1 Ammonium Nitrate periling tower.

Fig.2 Recording locations and directions: (a) locations of data recording; (b) recording directions “1” and “2”. (unit: mm)

Fig.3 Some examples of concrete destruction.

Fig.4 Recorded accelerations at different stations: (a) signal P21; (b) signal P31; (c) signal P51; (d) signal PP12; (e) signal P72.

Fig.5 Empirical evaluation of p values based on SNR-p curves for data P12. (a) SNR-p curve by the TV regularization; (b) SNR-p curve by the L² regularization; (c) SNR-p curve by the Sobolev regularization.

Fig.6 Residual noise of the regularization-based denoising and corresponding energy in the wavelet space; plotted ranges are

0.02 M a x [| W Ψ | 2] ≤ | W Ψ | 2 ≤ M a x [| W Ψ | 2]

. (a) The residual noise obtained by the L² regularization (by using

Ω (f) L 2

); (b) the residual noise obtained by the Sobolev regularization (by using

Ω (f) S o b o l e v

); (c) the time-period representation in the wavelet space of the residual noise obtained by

Ω (f) L 2

; (d) the time-period representation in the wavelet space of the residual noise obtained by

Ω (f) S o b o l e v

Tab.1 Different denoising approaches with corresponding thresholding method and estimating noise level

Fig.7 Denoising of the P51 signal with different one-step wavelet-based denoising methods using Symlet[12] wavelet where N_d =13. (a) denoising with GCV method; (b) denoising with GCVLevel method; (c) denoising with SURE method; (d) denoising with SURELevel method; (e) denoising with SUREhrink method; (f) denoising with Universal method; (g) denoising with UniversalLevel method; (h) denoising with VisuShrink method; (i) denoising with VisuShrinkLevel method.

Tab.2 Effects of denoising with different thresholding methods by the Symlet[12] and N_d =13

Fig.8 Variations of SNR against for P12 signal with different one-step wavelet-based denoising methods, where N_d=13. (a) soft and hard denoising with Symlet [12]; (b) soft and hard denoising with Db [8]; (c) soft and hard denoising with BattleLemarie[8].

Tab.3 Effects of denoising with the GCV method for “Db” wavelets, where Nd=13

Tab.4 Effects of denoising with the GCV method for “Symlet” wavelets, where Nd=13

Tab.5 Effects of denoising with the GCV method for “BattleLemarie” wavelets, where Nd=13

Tab.6 Effect of decomposition levels (Nd ) with Symlet [12] wavelet

Fig.9 The iterative thresholding for P12 where

C n j

=1.8, Symlet [12] and Nd=13. The last row contains final denoised signal and estimated noise.

Fig.10 The iterative thresholding for P12 where

C n j

=2, Symlet [12] and Nd=13. The last row contains final denoised signal and estimated noise.

Fig.11 Determination of thresholds for

C n j

for different enhanced data; SNRs and PSNRs are obtained by the iterative denoising method by the wavelet Symlet [12], Nd=13 and ten iterations. (a) SNR-

C n j

for P12; (b) PSNR-

C n j

for P12; (c) SNR-

C n j

for P51; (d)PSNR-

C n j

for P51; (e)SNR-

C n j

for P31; (f)PSNR-

C n j

for P31; (g)SNR-

C n j

for P72; (h)PSNR-

C n j

for P72; (i)SNR-

C n j

for P21; (j)PSNR-

C n j

for P21.

Tab.7 Iterative denoising of data P12 with parameters:

C n j = 1.75

and N_d=13

Fig.12 Iterative denoising of the signal P12 with ten iterations; both denoised signal and estimated noise are provided at each iteration (rows 1–5). The last row contains final denoised signal and estimated noise; evaluations are obtained with: Symlet[12],

C n j = 1.75

and N_d=13.

Fig.13 Energy Density for remaining noise after ten iterations of the peeling algorithm where Nd=13; the energies are evaluated by the complex Morlet wavelet with parameters

υ

_b=2 and

υ

_c=1.75. Energies are presented for the range

0.02 M a x [| W Ψ | 2] ≤ | W Ψ | 2 ≤ M a x [| W Ψ | 2]

. (a) noise of P12 in the wavelet space where

C n j = 1.75

; (b) noise of P51 in the wavelet space where

C n j = 1.90

; (c) noise of P31 in the wavelet space where

C n j = 1.90

; (d) noise of P72 in the wavelet space where

C n j = 1.85

; (e) noise of P21 in the wavelet space where

C n j = 1.90

Fig.14 Powers and energies of WTs of enhanced signals P21 and P31 and corresponding XWT and spectral power of XWT; plotted ranges are

0.02 M a x [| W Ψ | 2] ≤ | W Ψ | 2 ≤ M a x [| W Ψ | 2]

and

0.02 M a x [| W Ψ |] ≤ | W Ψ | ≤ M a x [| W Ψ |]

. (a) The spectral power of the enhanced data P21, evaluated by

| W Ψ (P 21) | 2

; (b) the density of energy for the enhanced data P21,

| W Ψ (P 21) | 2

; (c) the spectral power of the enhanced data P31, evaluated by

| W Ψ (P 31) | 2

; (d) the density of energy of the enhanced data P31,

| W Ψ (P 31) | 2

; (e) the spectral power of XWT for the enhanced data P21 and P31; (f) the density of

| X W Ψ |

for the enhanced data P21 and P31.

Fig.15 Powers and energies of WTs of enhanced signals P21 and P51 and corresponding XWT and spectral power of XWT; plotted ranges are

0.02 M a x [| W Ψ | 2] ≤ | W Ψ | 2 ≤ M a x [| W Ψ | 2]

and

0.02 M a x [| W Ψ |] ≤ | W Ψ | ≤ M a x [| W Ψ |]

. (a) The spectral power of the enhanced data P51, evaluated by

| W Ψ (P 51) | 2

; (b) the density of energy for the enhanced data P51,

| W Ψ (P 51) | 2

; (c) the spectral power of XWT for the enhanced data P21 and P51; (d) the density of

| X W Ψ |

for the enhanced data P21 and P51.

Fig.16 Powers and energies of WTs of enhanced signals P12 & P72 and corresponding XWT and spectral power of XWT; plotted ranges are

0.02 M a x [| W Ψ | 2] ≤ | W Ψ | 2 ≤ M a x [| W Ψ | 2]

and

0.02 M a x [| W Ψ |] ≤ | W Ψ | ≤ M a x [| W Ψ |]

. (a) The spectral power of the enhanced data P12, evaluated by

| W Ψ (12) | 2;

(b) the density of energy for the enhanced data P12,

| W Ψ (12) | 2;

; (c) The spectral power of the enhanced data P72, evaluated by

| W Ψ (72) | 2;

; (d) the density of energy for the enhanced data P72,

| W Ψ (72) | 2;

; (e) the spectral power of XWT for the enhanced data P12 and P72; (f) the density of

| X W Ψ |

for the enhanced data P12 and P72.

Fig.17 Autocorrelations of wavelet coefficients of the enhanced data. (a) results for the enhanced P21; (b) results for the enhanced P31; (c) results for the enhanced P51; (d) results for the enhanced P12; (e) results for the enhanced P72.

Tab.8 Comparison of modal periods from the ambient vibration test and the FE model [¹¹⁷]

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82	P Areias, T Rabczuk, D Dias-da-Costa. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137 https://doi.org/10.1016/j.engfracmech.2013.06.006
83	P Areias, T Rabczuk. Finite strain fracture of plates and shells with configurational forces and edge rotations. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122 https://doi.org/10.1002/nme.4477
84	P Areias, M Msekh, T Rabczuk. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143 https://doi.org/10.1016/j.engfracmech.2015.10.042
85	P Areias, T Rabczuk. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41 https://doi.org/10.1016/j.finel.2017.05.001
86	P Areias, J Reinoso, P P Camanho, J César de Sá, T Rabczuk. Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Engineering Fracture Mechanics, 2018, 189: 339–360 https://doi.org/10.1016/j.engfracmech.2017.11.017
87	C Anitescu, M N Hossain, T Rabczuk. Recovery-based error estimation and adaptivity using high-order splines over hierarchical T-meshes. Computer Methods in Applied Mechanics and Engineering, 2018, 328: 638–662 https://doi.org/10.1016/j.cma.2017.08.032
88	T Chau-Dinh, G Zi, P S Lee, T Rabczuk, J H Song. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92–93: 242–256 https://doi.org/10.1016/j.compstruc.2011.10.021
89	P R Budarapu, R Gracie, S P Bordas, T Rabczuk. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148 https://doi.org/10.1007/s00466-013-0952-6
90	H Talebi, M Silani, S P Bordas, P Kerfriden, T Rabczuk. A computational library for multiscale modeling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071 https://doi.org/10.1007/s00466-013-0948-2
91	P R Budarapu, R Gracie, S W Yang, X Zhuang, T Rabczuk. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143 https://doi.org/10.1016/j.tafmec.2013.12.004
92	H Talebi, M Silani, T Rabczuk. Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92 https://doi.org/10.1016/j.advengsoft.2014.09.016
93	F Amiri, D Millán, Y Shen, T Rabczuk, M Arroyo. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109 https://doi.org/10.1016/j.tafmec.2013.12.002
94	P Areias, T Rabczuk, M Msekh. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350 https://doi.org/10.1016/j.cma.2016.01.020
95	H Ren, X Zhuang, Y Cai, T Rabczuk. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476 https://doi.org/10.1002/nme.5257
96	H Ren, X Zhuang, T Rabczuk. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782 https://doi.org/10.1016/j.cma.2016.12.031
97	T Rabczuk, P Areias, T Belytschko. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548 https://doi.org/10.1002/nme.2013
98	T Rabczuk, T Belytschko. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799 https://doi.org/10.1016/j.cma.2006.06.020
99	T Rabczuk, R Gracie, J H Song, T Belytschko. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81: 48–71
100	T Rabczuk, S Bordas, G Zi. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23–24): 1391–1411 https://doi.org/10.1016/j.compstruc.2008.08.010
101	T Rabczuk, G Zi, S Bordas, H Nguyen-Xuan. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455 https://doi.org/10.1016/j.cma.2010.03.031
102	T Rabczuk, T Belytschko. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343 https://doi.org/10.1002/nme.1151
103	T Rabczuk, T Belytschko, S Xiao. Stable particle methods based on Lagrangian kernels. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12–14): 1035–1063 https://doi.org/10.1016/j.cma.2003.12.005
104	F Amiri, C Anitescu, M Arroyo, S P A Bordas, T Rabczuk. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 2014, 53(1): 45–57 https://doi.org/10.1007/s00466-013-0891-2
105	T J Hughes, J A Cottrell, Y Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4135–4195 https://doi.org/10.1016/j.cma.2004.10.008
106	N Nguyen-Thanh, H Nguyen-Xuan, S P A Bordas, T Rabczuk. Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids. Computer Methods in Applied Mechanics and Engineering, 2011, 200(21–22): 1892–1908 https://doi.org/10.1016/j.cma.2011.01.018
107	V P Nguyen, C Anitescu, S P Bordas, T Rabczuk. Isogeometric analysis: An overview and computer implementation aspects. Mathematics and Computers in Simulation, 2015, 117: 89–116 https://doi.org/10.1016/j.matcom.2015.05.008
108	H Ghasemi, H S Park, T Rabczuk. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258 https://doi.org/10.1016/j.cma.2016.09.029
109	N Nguyen-Thanh, J Kiendl, H Nguyen-Xuan, R Wüchner, K U Bletzinger, Y Bazilevs, T Rabczuk. Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47–48): 3410–3424 https://doi.org/10.1016/j.cma.2011.08.014
110	N Nguyen-Thanh, N Valizadeh, M Nguyen, H Nguyen-Xuan, X Zhuang, P Areias, G Zi, Y Bazilevs, L De Lorenzis, T Rabczuk. An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291 https://doi.org/10.1016/j.cma.2014.08.025
111	N Nguyen-Thanh, K Zhou, X Zhuang, P Areias, H Nguyen-Xuan, Y Bazilevs, T Rabczuk. Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 1157–1178 https://doi.org/10.1016/j.cma.2016.12.002
112	N Vu-Bac, T Duong, T Lahmer, X Zhuang, R Sauer, H Park, T Rabczuk. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Computer Methods in Applied Mechanics and Engineering, 2018, 331: 427–455 https://doi.org/10.1016/j.cma.2017.09.034
113	H Ghasemi, H S Park, T Rabczuk. A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 47–62 https://doi.org/10.1016/j.cma.2017.12.005
114	S S Ghorashi, N Valizadeh, S Mohammadi, T Rabczuk. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146 https://doi.org/10.1016/j.compstruc.2014.09.017
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