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Non-convex sparse optimization-based impact force identification with limited vibration measurements

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Abstract

Impact force identification is important for structure health monitoring especially in applications involving composite structures. Different from the traditional direct measurement method, the impact force identification technique is more cost effective and feasible because it only requires a few sensors to capture the system response and infer the information about the applied forces. This technique enables the acquisition of impact locations and time histories of forces, aiding in the rapid assessment of potentially damaged areas and the extent of the damage. As a typical inverse problem, impact force reconstruction and localization is a challenging task, which has led to the development of numerous methods aimed at obtaining stable solutions. The classical ℓ2 regularization method often struggles to generate sparse solutions. When solving the under-determined problem, ℓ2 regularization often identifies false forces in non-loaded regions, interfering with the accurate identification of the true impact locations. The popular ℓ1 sparse regularization, while promoting sparsity, underestimates the amplitude of impact forces, resulting in biased estimations. To alleviate such limitations, a novel non-convex sparse regularization method that uses the non-convex ℓ1–2 penalty, which is the difference of the ℓ1 and ℓ2 norms, as a regularizer, is proposed in this paper. The principle of alternating direction method of multipliers (ADMM) is introduced to tackle the non-convex model by facilitating the decomposition of the complex original problem into easily solvable subproblems. The proposed method named ℓ1–2-ADMM is applied to solve the impact force identification problem with unknown force locations, which can realize simultaneous impact localization and time history reconstruction with an under-determined, sparse sensor configuration. Simulations and experiments are performed on a composite plate to verify the identification accuracy and robustness with respect to the noise of the ℓ1−2-ADMM method. Results indicate that compared with other existing regularization methods, the ℓ1–2-ADMM method can simultaneously reconstruct and localize impact forces more accurately, facilitating sparser solutions, and yielding more accurate results.

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Abbreviations

ADMM:

Alternating direction method of multipliers

BVID:

Barely visible impact damage

GRE:

Global relative error

IRF:

Impulse response function

LE:

Localization error

LRE:

Local relative error

MC-GSURE:

Monte Carlo generalized stein unbiased risk estimate

PRE:

Peak relative error

SISO:

Single-input single-output

SNR:

Signal-noise ratio

a(t):

Impulse response function

a ij(t):

Impulse response function between the output position i and the input location j

a ij(ω):

Frequency response function between the output position and the input location j

A :

Transfer matrix of the multiple-input multiple-output dynamic system

A s :

Transfer matrix of the single-input single-output dynamic system

B :

Amplitude of the Gaussian-shaped impact force

c :

All-ones vector

C :

Damping matrix

e :

Random noise in measurements

E :

Elastic modulus

f i :

Elements in the vector f

f(t):

Impact force excitation function

f :

Force vector of the multiple-input multiple-output dynamic system

f̂ :

Estimated vector of f

f p :

Actual force vector at the impact position p

f̂ p :

Estimated force vector at the impact position p

f̂ ML :

Maximum likelihood estimation of the vector f

f s :

Force vector of the single-input single-output dynamic system

g(f):

General representation function of penalty terms

G :

Shear modulus

h :

An additional vector for variable splitting

ĥ :

Estimated vector of h

i :

Looping variable within the summation operation

I :

Identity matrix

k :

Number of iterations

K :

Stiffness matrix

m :

Number of measurement responses

M :

Mass matrix

n :

Number of impact force excitations

n f :

Total number of potential impact force locations

N :

Data length of the discretized impulse response function

N max :

Maximum number of iterations

O(nN):

Computational complexity of n × N

p :

Serial number of the location subjected to impact force

q :

Norm parameter defined in ℝ+*

Q :

Projection matrix

r :

Gaussian white noise vector

\(\cal{R}(f)\) :

General expression for calculating the norm of vector f

s(t):

System response

s :

Response vector of the multiple-input multiple-output dynamic system

s̃ :

Noisy response vector

s i :

Response vector at a certain position i

s s :

Response vector of the single-input single-output dynamic system

t :

Time

t 0 :

Occurrence time instant of the impact

Δt :

Sampling interval

T :

Impact duration

u :

Sufficient statistic of the model Eq. (8)

w max :

Element with the largest absolute value in the vector w

w :

Intermediate vector defined as w = f(k+1) + z(k)

x λ(u):

Solution result of Eq. (8) when f = u

y(f):

Proximal operator

δ :

Small positive parameter

δ :

Lagrange multiplier vector

ε :

Iteration termination threshold

λ :

Regularization parameter

ρ :

A positive penalty parameter

σ :

Standard deviation of the vector s

σn :

Standard deviation of noise in the measurements

τ :

Time delayed operator

ν :

Possion’s ratio

Γ:

Threshold value defined as Γ = λ/ρ

References

  1. Wu J, Xu X B, Liu C, Deng C, Shao X Y. Lamb wave-based damage detection of composite structures using deep convolutional neural network and continuous wavelet transform. Composite Structures, 2021, 276: 114590

    Article  Google Scholar 

  2. Gu X J, Su X Z, Wang J, Xu Y J, Zhu J H, Zhang W H. Improvement of impact resistance of plain-woven composite by embedding superelastic shape memory alloy wires. Frontiers of Mechanical Engineering, 2020, 15(4): 547–557

    Article  Google Scholar 

  3. Han D, Jia X, Zhang H J, Gao X G, Han X, Sun L, Zheng Z K, Zhang L, Wang F, Song Y D. Foreign object damage and postimpact tensile behavior of plain-woven SiC/SiC composites. Composite Structures, 2022, 295: 115767

    Article  Google Scholar 

  4. Singh T, Sehgal S. Structural health monitoring of composite materials. Archives of Computational Methods in Engineering, 2022, 29(4): 1997–2017

    Article  Google Scholar 

  5. Chen X F, Wang S B, Qiao B J, Chen Q. Basic research on machinery fault diagnostics: past, present, and future trends. Frontiers of Mechanical Engineering, 2018, 13(2): 264–291

    Article  Google Scholar 

  6. Liu R X, Dobriban E, Hou Z C, Qian K. Dynamic load identification for mechanical systems: a review. Archives of Computational Methods in Engineering, 2022, 29(2): 831–863

    Article  MathSciNet  Google Scholar 

  7. Wang L, Liu Y R, Xu H Y. Review: recent developments in dynamic load identification for aerospace vehicles considering multi-source uncertainties. Transactions of Nanjing University of Aeronautics & Astronautics, 2021, 38(2): 271–287 (in Chinese)

    Google Scholar 

  8. Jacquelin E, Bennani A, Hamelin P. Force reconstruction: analysis and regularization of a deconvolution problem. Journal of Sound and Vibration, 2003, 265(1): 81–107

    Article  Google Scholar 

  9. Li Q F, Lu Q H. Impact localization and identification under a constrained optimization scheme. Journal of Sound and Vibration, 2016, 366: 133–148

    Article  Google Scholar 

  10. Yan G, Sun H, Büyüköztürk O. Impact load identification for composite structures using Bayesian regularization and unscented Kalman filter. Structural Control and Health Monitoring, 2017, 24(5): e1910

    Article  Google Scholar 

  11. Aucejo M, De Smet O, Deü J F. On a space-time regularization for force reconstruction problems. Mechanical Systems and Signal Processing, 2019, 118: 549–567

    Article  Google Scholar 

  12. Rezayat A, Nassiri V, De Pauw B, Ertveldt J, Vanlanduit S, Guillaume P. Identification of dynamic forces using group-sparsity in frequency domain. Mechanical Systems and Signal Processing, 2016, 70–71: 756–768

    Article  Google Scholar 

  13. Candelieri A, Archetti F. Sparsifying to optimize over multiple information sources: an augmented gaussian process based algorithm. Structural and Multidisciplinary Optimization, 2021, 64(1): 239–255

    Article  MathSciNet  Google Scholar 

  14. Figueiredo M A T, Nowak R D, Wright S J. Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE Journal of Selected Topics in Signal Processing, 2007, 1(4): 586–597

    Article  Google Scholar 

  15. Kim S J, Koh K, Lustig M, Boyd S, Gorinevsky D. An interior-point method for large-scale ℓ1-regularized least squares IEEE Journal of Selected Topics in Signal Processing, 2007, 1(4): 606–617

    Article  Google Scholar 

  16. Daubechies I, Defrise M, De Mol C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics, 2004, 57(11): 1413–1457

    Article  MathSciNet  MATH  Google Scholar 

  17. Ginsberg D, Fritzen C P. New approach for impact detection by finding sparse solution. In: Proceedings of ISMA. Leuven: Katholieke Universiteit Leuven, 2014, 2043–2056

    Google Scholar 

  18. Qiao B J, Zhang X W, Gao J W, Chen X F. Impact-force sparse reconstruction from highly incomplete and inaccurate measurements. Journal of Sound and Vibration, 2016, 376: 72–94

    Article  Google Scholar 

  19. Aucejo M, De Smet O. A space-frequency multiplicative regularization for force reconstruction problems. Mechanical Systems and Signal Processing, 2018, 104: 1–18

    Article  Google Scholar 

  20. Pan C D, Chen Z P. Elimination of accelerometer mass loading effects in sparse identification of impact forces. Mechanical Systems and Signal Processing, 2023, 191: 110178

    Article  Google Scholar 

  21. Xu Z B, Chang X Y, Xu F M, Zhang H. L1/2 regularization: a thresholding representation theory and a fast solver. IEEE Transactions on Neural Networks and Learning Systems, 2012, 23(7): 1013–1027

    Article  Google Scholar 

  22. Trehan D. Non-convex optimization: a review. In: Proceedings of 2020 the 4th International Conference on Intelligent Computing and Control Systems. Madurai: IEEE, 2020, 418–423

    Google Scholar 

  23. Zha Z Y, Wen B H, Yuan X, Ravishankar S, Zhou J T, Zhu C. Learning nonlocal sparse and low-rank models for image compressive sensing: nonlocal sparse and low-rank modeling. IEEE Signal Processing Magazine, 2023, 40(1): 32–44

    Article  Google Scholar 

  24. Zha Z Y, Wen B H, Yuan X, Zhou J T, Zhu C. Image restoration via reconciliation of group sparsity and low-rank models. IEEE Transactions on Image Processing, 2021, 30: 5223–5238

    Article  MathSciNet  Google Scholar 

  25. Chartrand R, Yin W T. Nonconvex sparse regularization and splitting algorithms. In: Glowinski R, Osher S J, Yin W T, eds. Splitting Methods in Communication, Imaging, Science, and Engineering. Cham: Springer, 2016, 237–249

    Chapter  MATH  Google Scholar 

  26. Qiao B J, Ao C Y, Mao Z, Chen X F. Non-convex sparse regularization for impact force identification. Journal of Sound and Vibration, 2020, 477: 115311

    Article  Google Scholar 

  27. Aucejo M, De Smet O. A generalized multiplicative regularization for input estimation. Mechanical Systems and Signal Processing, 2021, 157: 107637

    Article  Google Scholar 

  28. Liu J J, Qiao B J, Wang Y N, He W F, Chen X F. Non-convex sparse regularization via convex optimization for impact force identification. Mechanical Systems and Signal Processing, 2023, 191: 110191

    Article  Google Scholar 

  29. Esser E, Lou Y F, Xin J. A method for finding structured sparse solutions to nonnegative least squares problems with applications. SIAM Journal on Imaging Sciences, 2013, 6(4): 2010–2046

    Article  MathSciNet  MATH  Google Scholar 

  30. Yin P H, Lou Y F, He Q, Xin J. Minimization of ℓ1–2 for compressed sensing. SIAM Journal on Scientific Computing, 2015, 37(1): A536–A563

    Article  MathSciNet  MATH  Google Scholar 

  31. Sun H, Feng D M, Liu Y, Feng M Q. Statistical regularization for identification of structural parameters and external loadings using state space models. Computer-Aided Civil and Infrastructure Engineering, 2015, 30(11): 843–858

    Article  Google Scholar 

  32. Ginsberg D, Ruby M, Fritzen C P. Load identification approach based on basis pursuit denoising algorithm. Journal of Physics: Conference Series, 2015, 628(1): 012030

    Google Scholar 

  33. Boyd S, Parikh N, Chu E, Peleato B, Eckstein J. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 2011, 3(1): 1–122

    Article  MATH  Google Scholar 

  34. Meirovitch L. Elements of Vibration Analysis. New York: McGraw-Hill Book Co., 1986

    MATH  Google Scholar 

  35. Jankowski L. Off-line identification of dynamic loads. Structural and Multidisciplinary Optimization, 2009, 37(6): 609–623

    Article  Google Scholar 

  36. Zha Z Y, Liu X, Huang X H, Shi H L, Xu Y Y, Wang Q, Tang L, Zhang X G. Analyzing the group sparsity based on the rank minimization methods. In: Proceedings of 2017 IEEE International Conference on Multimedia and Expo. Hong Kong: IEEE, 2017, 883–888

    Chapter  Google Scholar 

  37. Zha Z Y, Zhang X G, Wu Y, Wang Q, Liu X, Tang L, Yuan X. Non-convex weighted ℓp nuclear norm based ADMM framework for image restoration Neurocomputing, 2018, 311: 209–224

    Article  Google Scholar 

  38. Li X W, Zhao H T, Chen Z, Wang Q B, Chen J A, Duan D P. Force identification based on a comprehensive approach combining Taylor formula and acceleration transmissibility. Inverse Problems in Science and Engineering, 2018, 26(11): 1612–1632

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang L, Liu J K, Lu Z R. Bandlimited force identification based on sinc-dictionaries and Tikhonov regularization. Journal of Sound and Vibration, 2020, 464: 114988

    Article  Google Scholar 

  40. Qiao B J, Liu J J, Liu J X, Yang Z B, Chen X F. An enhanced sparse regularization method for impact force identification. Mechanical Systems and Signal Processing, 2019, 126: 341–367

    Article  Google Scholar 

  41. Lou Y F, Yan M. Fast L1−L2 minimization via a proximal operator. Journal of Scientific Computing, 2018, 74(2): 767–785

    Article  MathSciNet  MATH  Google Scholar 

  42. Chang X T, Yan Y J, Wu Y F. Study on solving the ill-posed problem of force load reconstruction. Journal of Sound and Vibration, 2019, 440: 186–201

    Article  Google Scholar 

  43. Cao W F, Sun J, Xu Z B. Fast image deconvolution using closed-form thresholding formulas of Lq (q = 1/2, 2/3) regularization. Journal of Visual Communication and Image Representation, 2013, 24(1): 31–41

    Article  Google Scholar 

  44. Zha Z Y, Yuan X, Wen B H, Zhou J T, Zhang J C, Zhu C. A benchmark for sparse coding: when group sparsity meets rank minimization. IEEE Transactions on Image Processing, 2020, 29: 5094–5109

    Article  MathSciNet  MATH  Google Scholar 

  45. Eldar Y C. Generalized SURE for exponential families: applications to regularization. IEEE Transactions on Signal Processing, 2009, 57(2): 471–481

    Article  MathSciNet  MATH  Google Scholar 

  46. Donoho D L. De-noising by soft-thresholding. IEEE Transactions on Information Theory, 1995, 41(3): 613–627

    Article  MathSciNet  MATH  Google Scholar 

  47. Ramani S, Blu T, Unser M. Monte-carlo sure: a black-box optimization of regularization parameters for general denoising algorithms. IEEE Transactions on Image Processing, 2008, 17(9): 1540–1554

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52075414 and 52241502), and China Postdoctoral Science Foundation (Grant No. 2021M702595).

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Authors and Affiliations

  1. National Key Laboratory of Aerospace Power System and Plasma Technology, Xi’an Jiaotong University, Xi’an, 710049, China

    Lin Chen, Yanan Wang, Baijie Qiao, Junjiang Liu, Wei Cheng & Xuefeng Chen

  2. School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

    Lin Chen, Yanan Wang, Baijie Qiao, Junjiang Liu, Wei Cheng & Xuefeng Chen

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  1. Lin Chen
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  2. Yanan Wang
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Correspondence to Yanan Wang.

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Chen, L., Wang, Y., Qiao, B. et al. Non-convex sparse optimization-based impact force identification with limited vibration measurements. Front. Mech. Eng. 18, 46 (2023). https://doi.org/10.1007/s11465-023-0762-2

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