Wavelet-based iterative data enhancement for implementation in purification of modal frequency for extremely noisy ambient vibration tests in Shiraz-Iran
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
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.
S W Doebling, C R Farrar, M B Prime. A summary review of vibration-based damage identification methods. Journal of Shock and Vibration, 1998, 30(2): 91–105 https://doi.org/10.1177/058310249803000201
2
S W Doebling, C R Farrar, M B Prime, D W Shevitz. Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in Their Vibration Characteristics: A Literature Review. OSTI.GOV Technical Report LA-13070-MS ON: DE96012168; TRN: 96:003834. 1996
3
T H Le, Y Tamura. Modal identification of ambient vibration structure using frequency domain decomposition and wavelet transform. In: Proceedings of the 7th Asia-Pacific Conference on Wind Engineering. Taipei, China: APCWE, 2009
4
K K Wijesundara, C Negulescu, E Foerster, D Monfort Climent. Estimation of modal properties of structures through ambient excitation measurements using continuous wavelet transform. In: Proceedings of 15WCEE. Lisbon: SPES, 2012, 26: 15–18
I Harik, D Allen, R Street, M Guo, R Graves, J Harison, M Gawry. Free and ambient vibration of Brent-Spence Bridge. Journal of Structural Engineering, 1997, 123(9): 1262–1268 https://doi.org/10.1061/(ASCE)0733-9445(1997)123:9(1262)
7
C Farrar, G James. System identification from ambient vibration measurements on a bridge. Journal of Sound and Vibration, 1997, 205(1): 1–18 https://doi.org/10.1006/jsvi.1997.0977
8
D M Siringoringo, Y Fujino. System identification of suspension bridge from ambient vibration response. Engineering Structures, 2008, 30(2): 462–477 https://doi.org/10.1016/j.engstruct.2007.03.004
9
H Sohn. A Review of Structural Health Monitoring Literature: 1996–2001. Los Alamos National Laboratory Report. 2004
10
T Kijewski, A Kareem. Wavelet transforms for system identification in civil engineering. Computer-Aided Civil and Infrastructure Engineering, 2003, 18(5): 339–355 https://doi.org/10.1111/1467-8667.t01-1-00312
11
J Lin, L Qu. Feature extraction based on Morlet wavelet and its application for mechanical fault diagnosis. Journal of Sound and Vibration, 2000, 234(1): 135–148 https://doi.org/10.1006/jsvi.2000.2864
12
K F Al-Raheem, A Roy, K P Ramachandran, D K Harrison, S Grainger. Rolling element bearing faults diagnosis based on autocorrelation of optimized: Wavelet de-noising technique. International Journal of Advanced Manufacturing Technology, 2009, 40(3–4): 393–402 https://doi.org/10.1007/s00170-007-1330-3
13
B Yan, A Miyamoto, E Brühwiler. Wavelet transform-based modal parameter identification considering uncertainty. Journal of Sound and Vibration, 2006, 291(1–2): 285–301 https://doi.org/10.1016/j.jsv.2005.06.005
14
X Jiang, H Adeli. Pseudospectra, MUSIC, and dynamic wavelet neural network for damage detection of highrise buildings. International Journal for Numerical Methods in Engineering, 2007, 71(5): 606–629 https://doi.org/10.1002/nme.1964
15
R A Osornio-Rios, J P Amezquita-Sanchez, R J Romero-Troncoso, A Garcia-Perez. MUSIC-ANN analysis for locating structural damages in a truss-type structure by means of vibrations. Computer-Aided Civil and Infrastructure Engineering, 2012, 27(9): 687–698 https://doi.org/10.1111/j.1467-8667.2012.00777.x
16
L Carassale, F Percivale. POD-based modal identification of wind-excited structures. In: Proceedings of the 12th International Conference on Wind Engineering. Cairns, 2007, 1239–1246
17
Y Tamura. Advanced Structural Wind Engineering. Tokyo: Springer, 2013, 347–376
18
M Meo, G Zumpano, X Meng, E Cosser, G Roberts, A Dodson. Measurements of dynamic properties of a medium span suspension bridge by using the wavelet transforms. Mechanical Systems and Signal Processing, 2006, 20(5): 1112–1133 https://doi.org/10.1016/j.ymssp.2004.09.008
19
J Lardies, S Gouttebroze. Identification of modal parameters using the wavelet transform. International Journal of Mechanical Sciences, 2002, 44(11): 2263–2283 https://doi.org/10.1016/S0020-7403(02)00175-3
20
X He, B Moaveni, J P Conte, A Elgamal, S F Masri. Modal identification study of Vincent Thomas bridge using simulated wind-induced ambient vibration data. Computer-Aided Civil and Infrastructure Engineering, 2008, 23(5): 373–388 https://doi.org/10.1111/j.1467-8667.2008.00544.x
21
Y C Ni, X L Lu, W S Lu. Field dynamic test and Bayesian modal identification of a special structure—The Palms Together Dagoba. Structural Control and Health Monitoring, 2016, 23(5): 838–856 https://doi.org/10.1002/stc.1816
22
F L Zhang, C E Ventura, H B Xiong, W S Lu, Y X Pan, J X Cao. Evaluation of the dynamic characteristics of a super tall building using data from ambient vibration and shake table tests by a Bayesian approach. Structural Control and Health Monitoring, 2017, 25(2): 1– 18
23
N Kang, H Kim, Sunyoung Choi & Seongwoo Jo, J S Hwang, E Yu. Performance evaluation of TMD under typhoon using system identification and inverse wind load estimation. Computer-Aided Civil and Infrastructure Engineering, 2012, 27(6): 455–473 https://doi.org/10.1111/j.1467-8667.2011.00755.x
24
H Wenzel, D Pichler. Ambient Vibration Monitoring. Vienna: John Wiley & Sons, 2005
25
J M W Brownjohn. Structural health monitoring of civil infrastructure. Philosophical Transactions of the Royal Society A, 2007, 365(1851): 589–622 https://doi.org/10.1098/rsta.2006.1925
26
X H He, X G Hua, Z Q Chen, F L Huang. EMD-based random decrement technique for modal parameter identification of an existing railway bridge. Engineering Structures, 2011, 33(4): 1348–1356 https://doi.org/10.1016/j.engstruct.2011.01.012
27
C S Huang, S L Hung, C I Lin, W C Su. A wavelet-based approach to identifying structural modal parameters from seismic response and free vibration data. Computer-Aided Civil and Infrastructure Engineering, 2005, 20(6): 408–423 https://doi.org/10.1111/j.1467-8667.2005.00406.x
28
S Ivanovic, M D Trifunac, E I Novikova, A A Gladkov, M I Todorovska. Instrumented 7-Storey Reinforced Concrete Building in Van Nuys, California: Ambient Vibration Survey Following the Damage from the 1994 Northridge Earthquake. Report No. CE 9903. 1999
29
S S Ivanovic, M D Trifunac, M I Todorovska. Ambient vibration tests of structures—A review. ISET Journal of Earthquake Technology, 2000, 37: 165–197
30
J M W Brownjohn, A De Stefano, Y L Xu, H Wenzel, A E Aktan. Vibration-based monitoring of civil infrastructure: Challenges and successes. Journal of Civil Structural Health Monitoring, 2011, 1(3–4): 79–95 https://doi.org/10.1007/s13349-011-0009-5
31
G D Roeck. The state-of-the-art of damage detection by vibration monitoring: The SIMCES experience. Structural Control and Health Monitoring, 2003, 10(2): 127–134 https://doi.org/10.1002/stc.20
32
D He, X Wang, M I Friswell, J Lin. Identification of modal parameters from noisy transient response signals. Structural Control and Health Monitoring, 2017, 24(11): 1–10 https://doi.org/10.1002/stc.2019
33
J N Juang, R S Pappa. Effects of noise on modal parameters identified by the eigensystem realization algorithm. Journal of Guidance, Control, and Dynamics, 1986, 9(3): 294–303 https://doi.org/10.2514/3.20106
34
S Dorvash, S N Pakzad. Effects of measurement noise on modal parameter identification. Smart Materials and Structures, 2012, 21(6): 065008 https://doi.org/10.1088/0964-1726/21/6/065008
35
P Li, S L J Hu, H J Li. Noise issues of modal identification using eigensystem realization algorithm. Procedia Engineering, 2011, 14: 1681–1689 https://doi.org/10.1016/j.proeng.2011.07.211
36
S Yoshitomi, I Takewaki. Noise-effect compensation method for physical-parameter system identification under stationary random input. Structural Control and Health Monitoring, 2009, 16(3): 350–373 https://doi.org/10.1002/stc.263
37
C S Huang, W C Su. Identification of modal parameters of a time invariant linear system by continuous wavelet transformation. Mechanical Systems and Signal Processing, 2007, 21(4): 1642–1664 https://doi.org/10.1016/j.ymssp.2006.07.011
38
B Yan, A Miyamoto. A comparative study of modal parameter identification based on wavelet and Hilbert-Huang transforms. Computer-Aided Civil and Infrastructure Engineering, 2006, 21(1): 9–23 https://doi.org/10.1111/j.1467-8667.2005.00413.x
39
W C Su, C S Huang, C H Chen, C Y Liu, H C Huang, Q T Le. Identifying the modal parameters of a structure from ambient vibration data via the stationary wavelet packet. Computer-Aided Civil and Infrastructure Engineering, 2014, 29(10): 738–757 https://doi.org/10.1111/mice.12115
40
W C Su, C Y Liu, C S Huang. Identification of instantaneous modal parameter of time-varying systems via a wavelet-based approach and its application. Computer-Aided Civil and Infrastructure Engineering, 2014, 29(4): 279–298 https://doi.org/10.1111/mice.12037
41
S L Chen, J J Liu, H C Lai. Wavelet analysis for identification of damping ratios and natural frequencies. Journal of Sound and Vibration, 2009, 323(1–2): 130–147 https://doi.org/10.1016/j.jsv.2009.01.029
42
T H Yi, H N Li, X Y Zhao. Noise smoothing for structural vibration test signals using an improved wavelet thresholding technique. Sensors (Basel), 2012, 12(8): 11205–11220 https://doi.org/10.3390/s120811205
43
N E Huang. Hilbert-Huang Transform and its Applications. Singapore: World Scientific, 2011, 1–26
44
A Teolis. Computational Signal Processing with Wavelets. Basel: Springer Science & Business Media, 2012
45
M Misiti, Y Misiti, G Oppenheim, J M Poggi. Wavelets and their Applications. Wiltshire: John Wiley & Sons, 2013
46
S Mallat. Wavelet Analysis & Its Applications. London: Academic Press, 1999
47
P Van Fleet. Discrete Wavelet Transformations: An Elementary Approach with Applications. New Jersey: John Wiley & Sons, 2011
48
M Jansen. Noise Reduction by Wavelet Thresholding. New York: Springer Science & Business Media, 2012
49
K P Soman. Insight into Wavelets: From Theory to Practice. New Delhi: PHI Learning Pvt. Ltd., 2010
50
X Jiang, S Mahadevan, H Adeli. Bayesian wavelet packet denoising for structural system identification. Structural Control and Health Monitoring, 2007, 14(2): 333–356 https://doi.org/10.1002/stc.161
51
R R Coifman, M V Wickerhauser. Adapted waveform “de-Noising” for medical signals and images. IEEE Engineering in Medicine and Biology Magazine, 1995, 14(5): 578–586 https://doi.org/10.1109/51.464774
52
R R Coifman, M V Wickerhauser. Experiments with adapted wavelet de-noising for medical signals and images. In: Time Frequency and Wavelets in Biomedical Signal Processing, IEEE press series in Biomedical Engineering. New York: Wiley-IEEE Press, 1998
53
L J Hadjileontiadis, S M Panas. Separation of discontinuous adventitious sounds from vesicular sounds using a wavelet-based filter. IEEE Transactions on Biomedical Engineering, 1997, 44(12): 1269–1281 https://doi.org/10.1109/10.649999
54
L J Hadjileontiadis, C N Liatsos, C C Mavrogiannis, T A Rokkas, S M Panas. Enhancement of bowel sounds by wavelet-based filtering. IEEE Transactions on Biomedical Engineering, 2000, 47(7): 876–886 https://doi.org/10.1109/10.846681
55
R Ranta, C Heinrich, V Louis-Dorr, D Wolf. Interpretation and improvement of an iterative wavelet-based denoising method. IEEE Signal Processing Letters, 2003, 10(8): 239–241 https://doi.org/10.1109/LSP.2003.814801
56
R Ranta, V Louis-Dorr, C Heinrich, D Wolf. Iterative wavelet-based denoising methods and robust outlier detection. IEEE Signal Processing Letters, 2005, 12(8): 557–560 https://doi.org/10.1109/LSP.2005.851267
G Peyré. Advanced Signal, Image and Surface Processing-Numerical Tours. Université Paris-Dauphine, 2010
59
A Grinsted, J C Moore, S Jevrejeva. Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics, 2004, 11(5/6): 561–566 https://doi.org/10.5194/npg-11-561-2004
60
J Rafiee, P W Tse. Use of autocorrelation of wavelet coefficients for fault diagnosis. Mechanical Systems and Signal Processing, 2009, 23(5): 1554–1572 https://doi.org/10.1016/j.ymssp.2009.02.008
61
X Jiang, H Adeli. Wavelet packet-autocorrelation function method for traffic flow pattern analysis. Computer-Aided Civil and Infrastructure Engineering, 2004, 19(5): 324–337 https://doi.org/10.1111/j.1467-8667.2004.00360.x
62
A Bruns. Fourier-, Hilbert-and wavelet-based signal analysis: Are they really different approaches? Journal of Neuroscience Methods, 2004, 137(2): 321–332 https://doi.org/10.1016/j.jneumeth.2004.03.002
63
M Le Van Quyen, J Foucher, J P Lachaux, E Rodriguez, A Lutz, J Martinerie, F J Varela. Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony. Journal of Neuroscience Methods, 2001, 111(2): 83–98 https://doi.org/10.1016/S0165-0270(01)00372-7
64
C Rainieri, G Fabbrocino. Operational Modal Analysis of Civil Engineering Structures. New York: Springer, 2014
65
J M W Brownjohn. Ambient vibration studies for system identification of tall buildings. Earthquake Engineering & Structural Dynamics, 2003, 32(1): 71–95 https://doi.org/10.1002/eqe.215
66
S Mahato, M V Teja, A Chakraborty. Adaptive HHT (AHHT) based modal parameter estimation from limited measurements of an RC—framed building under multi—component earthquake excitations. Structural Control and Health Monitoring, 2015, 22(7): 984–1001 https://doi.org/10.1002/stc.1727
67
Z K Peng, P W Tse, F L Chu. An improved Hilbert–Huang transform and its application in vibration signal analysis. Journal of Sound and Vibration, 2005, 286(1–2): 187–205 https://doi.org/10.1016/j.jsv.2004.10.005
68
W X Yang. Interpretation of mechanical signals using an improved Hilbert-Huang transform. Mechanical Systems and Signal Processing, 2008, 22(5): 1061–1071 https://doi.org/10.1016/j.ymssp.2007.11.024
69
C Bao, H Hao, Z X Li, X Zhu. Time-varying system identification using a newly improved HHT algorithm. Computers & Structures, 2009, 87(23–24): 1611–1623 https://doi.org/10.1016/j.compstruc.2009.08.016
70
Z Wu, N E Huang. Ensemble empirical mode decomposition: A noise-assisted data analysis method. Advances in Data Science and Adaptive Analysis, 2009, 1(1): 1–41 https://doi.org/10.1142/S1793536909000047
71
I Daubechies, J Lu, H T Wu. Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool. Applied and Computational Harmonic Analysis, 2011, 30(2): 243–261 https://doi.org/10.1016/j.acha.2010.08.002
72
E Brevdo, H T Wu, G Thakur, N S Fuckar. Synchrosqueezing and its applications in the analysis of signals with time-varying spectrum. Proceedings of the National Academy of Sciences of the United States of America, 2011, 93: 1079–1094
73
C A Perez-Ramirez, J P Amezquita-Sanchez, H Adeli, M Valtierra-Rodriguez, D Camarena-Martinez, R J Romero-Troncoso. New methodology for modal parameters identification of smart civil structures using ambient vibrations and synchrosqueezed wavelet transform. Engineering Applications of Artificial Intelligence, 2016, 48: 1–12 https://doi.org/10.1016/j.engappai.2015.10.005
74
C Li, M Liang. Time-frequency signal analysis for gearbox fault diagnosis using a generalized synchrosqueezing transform. Mechanical Systems and Signal Processing, 2012, 26: 205–217 https://doi.org/10.1016/j.ymssp.2011.07.001
75
Z Feng, X Chen, M Liang. Iterative generalized synchrosqueezing transform for fault diagnosis of wind turbine planetary gearbox under nonstationary conditions. Mechanical Systems and Signal Processing, 2015, 52–53: 360–375 https://doi.org/10.1016/j.ymssp.2014.07.009
76
W J Staszewski. Identification of damping in MDOF systems using time-scale decomposition. Journal of Sound and Vibration, 1997, 203(2): 283–305 https://doi.org/10.1006/jsvi.1996.0864
77
P Areias, T Rabczuk, P Camanho. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63 https://doi.org/10.1016/j.tafmec.2014.06.006
78
S Nanthakumar, T Lahmer, X Zhuang, G Zi, T Rabczuk. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24(1): 153–176 https://doi.org/10.1080/17415977.2015.1017485
79
K M Hamdia, M Silani, X Zhuang, P He, T Rabczuk. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227 https://doi.org/10.1007/s10704-017-0210-6
80
K M Hamdia, H Ghasemi, X Zhuang, N Alajlan, T Rabczuk. Sensitivity and uncertainty analysis for flexoelectric nanostructures. Computer Methods in Applied Mechanics and Engineering, 2018, 337: 95–109 https://doi.org/10.1016/j.cma.2018.03.016
81
N Vu-Bac, T Lahmer, X Zhuang, T Nguyen-Thoi, T Rabczuk. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31 https://doi.org/10.1016/j.advengsoft.2016.06.005
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
115
S Kumari, R Vijay. Effect of symlet filter order on denoising of still images. Advances in Computers, 2012, 3(1): 137–143 https://doi.org/10.5121/acij.2012.3112
116
P J Rousseeuw, A M Leroy. Robust Regression & Outlier Detection. Hoboken: John Wiley & Sons, 1987
117
J Samadi. Seismic Behavior of Structure-equipment in a petrochemical complex to evaluate vulnerability assessment: A case study. Thesis for the Master’s Degree. Tehran: Civil Engineering, University of Tehran, 2010
118
A Jensen, A la Cour-Harbo. Ripples in Mathematics: The Discrete Wavelet Transform. Heidelberg: Springer Science & Business Media. 2001
119
K Soman. Insight into Wavelets: From Theory to Practice. New Delhi: PHI Learning Pvt. Ltd., 2010
120
M V Wickerhauser. Adapted Wavelet Analysis: From Theory to Software. Natick: AK Peters/CRC Press, 1996
C M Stein. Estimation of the mean of a multivariate normal distribution. Annals of Statistics, 1981, 9(6): 1135–1151 https://doi.org/10.1214/aos/1176345632
123
H Yousefi, S S Ghorashi, T Rabczuk. Directly simulation of second order hyperbolic systems in second order form via the regularization concept. Communications in Computational Physics, 2016, 20(1): 86–135 https://doi.org/10.4208/cicp.101214.011015a
124
H Yousefi, A Noorzad, J Farjoodi. Multiresolution based adaptive schemes for second order hyperbolic PDEs in elastodynamic problems. Applied Mathematical Modelling, 2013, 37(12–13): 7095–7127 https://doi.org/10.1016/j.apm.2012.09.004
125
I W Selesnick, I Bayram. Total Variation Filtering, White paper, Connexions Web site. 2010