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

doi:10.1007/s11465-023-0762-2

Front. Mech. Eng.

2023, Vol. 18

Issue (3) : 46 https://doi.org/10.1007/s11465-023-0762-2

RESEARCH ARTICLE

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

Lin CHEN^1,², Yanan WANG^1,²(

), Baijie QIAO^1,², Junjiang LIU^1,², Wei CHENG^1,², Xuefeng CHEN^1,²

¹. National Key Laboratory of Aerospace Power System and Plasma Technology, Xi’an Jiaotong University, Xi’an 710049, China
². School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China

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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 $ℓ 2$ regularization method often struggles to generate sparse solutions. When solving the under-determined problem, $ℓ 2$ regularization often identifies false forces in non-loaded regions, interfering with the accurate identification of the true impact locations. The popular $ℓ 1$ 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 $ℓ 1 − 2$ penalty, which is the difference of the $ℓ 1$ and $ℓ 2$ 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 $ℓ 1 − 2 -ADMM$ 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 $ℓ 1 − 2 -ADMM$ method. Results indicate that compared with other existing regularization methods, the $ℓ 1 − 2 -ADMM$ method can simultaneously reconstruct and localize impact forces more accurately, facilitating sparser solutions, and yielding more accurate results.

Keywords impact force identification inverse problem sparse regularization under-determined condition alternating direction method of multipliers

Corresponding Author(s): Yanan WANG

Just Accepted Date: 04 July 2023 Issue Date: 13 October 2023

Cite this article:

Lin CHEN,Yanan WANG,Baijie QIAO, et al. Non-convex sparse optimization-based impact force identification with limited vibration measurements[J]. Front. Mech. Eng., 2023, 18(3): 46.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-023-0762-2
https://academic.hep.com.cn/fme/EN/Y2023/V18/I3/46

Fig.A1

Fig.A2

Fig.1 3D plots of different penalties and their corresponding contour maps: (a) 3D plots of

? 2

norm,

? 1

norm,

? 1 ? 2

norm, and

? 0

norm and (b) corresponding contour maps.

Fig.A3

Tab.1 Material properties of the composite plate

Fig.2 Composite plate with two opposite edges clamped: (a) schematic diagram of an applied impact force on the plate and (b) distribution of strain sensors and potential impact locations. Fifteen potential impact locations are considered, and the selected measurement positions are

S 8

S 10

S 14

, and

S 18

Fig.3 Reconstructed impact time histories at all monitored locations at 20 dB noise level when (a) real impact acts on

P 3

and (b) real impact acts on

P 10

. ADMM: alternating direction method of multipliers.

Fig.4 Impact force identification results denoted by accuracy indicators at eight randomly chosen locations at 20 dB noise level by

? 1 ? 2

-alternating direction method of multipliers (ADMM),

? 1

-ADMM, and

? 2

-ADMM: (a) global relative error, (b) local relative error, (c) peak relative error, and (d) localization error.

Position	Method	GRE/%	LRE/%	PRE/%			LE/%	Computing time/s
Position	Method	GRE/%	LRE/%	Sine	Triangle	Gaussian	LE/%	Computing time/s
P₉	$? 1 ? 2$ -ADMM	10.53	8.64	3.89	4.58	3.85	15.45	280.07
	$? 1$ -ADMM	15.03	11.94	9.33	7.37	6.69	21.92	292.65
	$? 2$ -ADMM	85.90	71.88	70.29	73.42	72.85	84.48	96.88
P₁₁	$? 1 ? 2$ -ADMM	11.03	9.59	5.35	2.80	1.49	16.11	272.45
	$? 1$ -ADMM	15.14	13.21	10.30	5.98	4.21	20.78	273.10
	$? 2$ -ADMM	94.21	88.95	89.29	86.83	87.50	90.85	40.14

Tab.2 Identification accuracy indicators and computing time of

? 1 ? 2

-ADMM,

? 1

-ADMM, and

? 2

-ADMM for the multimorphology continuous impact force at different locations at 20 dB noise level

Fig.5 Identification results of the multimorphology continuous impact force at

P 9

under 20 dB noise level by

? 1 ? 2

-alternating direction method of multipliers (ADMM),

? 1

-ADMM, and

? 2

-ADMM: (a) time history reconstruction and (b) localization results.

Fig.6 Identification results of the multimorphology continuous impact force at

P 11

under 20 dB noise level by

? 1 ? 2

-alternating direction method of multipliers (ADMM),

? 1

-ADMM, and

? 2

-ADMM: (a) time–history reconstruction and (b) localization results.

Fig.7 Four accuracy indicators of identification results for the same single impact force via

? 1 ? 2

-alternating direction method of multipliers (ADMM),

? 1

-ADMM, and

? 2

-ADMM at different noise levels: (a) impact at

P 1

and (b) impact at

P 9

. SNR: signal-noise ratio.

Fig.8 Experimental setup of the composite plate with two opposite edges clamped. Four sensors (

S 1

S 2

S 7

, and

S 9

) are selected to monitor the 15 potential impact locations

	Accuracy indicators and computing time
Method	Impact at P₁					Impact at P₁₅
	GRE/%	LRE/%	PRE/%	LE/%	Computing time/s	GRE/%	LRE/%	PRE/%	LE/%	Computing time/s
$? 1 ? 2$ -ADMM	5.99	5.90	0.86	2.26	93.83	5.28	4.63	0.83	6.16	267.35
$? 1 / 2$ -ADMM	6.44	5.44	1.15	7.22	253.91	6.15	4.96	1.32	4.56	376.05
$? 2 / 3$ -ADMM	8.10	5.89	2.23	12.70	261.04	5.92	4.98	1.21	4.04	298.46
$? 1$ -ADMM	7.96	7.27	3.71	8.86	101.11	7.59	6.06	3.15	11.37	217.71
$? 2$ -ADMM	75.89	61.63	66.46	77.97	46.24	86.91	77.31	75.77	83.09	71.51

Tab.3 Identification accuracy indicators and computing times of

? 1 ? 2

-ADMM,

? 1 / 2

-ADMM,

? 2 / 3

-ADMM,

? 1

-ADMM, and

? 2

-ADMM for the single-impact forces applied to

P 1

and

P 15

Fig.9 Monte Carlo generalized stein unbiased risk estimate (MC-GSURE) curves of

? 1 ? 2

-alternating direction method of multipliers in single-impact cases when exerted at: (a)

P 1

and (b)

P 15

Fig.10 Identification results of the impact force applied to

P 1

? 1 ? 2

-alternating direction method of multipliers (ADMM),

? 1

-ADMM,

? 2

-ADMM,

? 1 / 2

-ADMM, and

? 2 / 3

-ADMM: (a) time history reconstruction results, (b) localization results of

? 1 ? 2

-ADMM,

? 1

-ADMM, and

? 2

-ADMM, (c) localization results of

? 1 / 2

-ADMM and

? 2 / 3

-ADMM, and (d) convergence curves.

Fig.11 Identification results of the impact force applied to

P 15

? 1 ? 2

-alternating direction method of multipliers (ADMM),

? 1

-ADMM,

? 2

-ADMM,

? 1 / 2

-ADMM, and

? 2 / 3

-ADMM: (a) time history reconstruction results, (b) localization results of

? 1 ? 2

-ADMM,

? 1

-ADMM, and

? 2

-ADMM, (c) localization results of

? 1 / 2

-ADMM and

? 2 / 3

-ADMM, and (d) convergence curves.

	Accuracy indicators and computing time
Method	Impact at $P 5$						Impact at $P 6$
	GRE/%	LRE/%	PRE/%		LE/%	Computing time/s	GRE/%	LRE/%	PRE/%		LE/%	Computing time/s
	GRE/%	LRE/%	Peak1	Peak2	LE/%	Computing time/s	GRE/%	LRE/%	Peak1	Peak2	LE/%	Computing time/s
$? 1 ? 2$ -ADMM	11.76	9.88	0.48	0.93	18.17	219.71	5.94	4.95	0.40	4.29	7.76	229.85
$? 1 / 2$ -ADMM	19.80	17.74	6.84	15.35	15.60	287.43	11.49	8.94	4.88	5.26	11.44	282.67
$? 2 / 3$ -ADMM	16.46	14.35	7.89	11.67	21.25	188.96	10.17	8.12	2.81	5.17	10.43	193.90
$? 1$ -ADMM	13.57	11.29	4.56	4.71	21.25	189.58	8.53	7.22	4.62	8.83	8.56	213.31
$? 2$ -ADMM	85.39	74.96	77.90	77.02	82.19	45.69	82.20	68.81	65.50	69.85	81.10	101.03

Tab.4 Identification accuracy indicators and computing times of

? 1 ? 2

-ADMM,

? 1 / 2

-ADMM,

? 2 / 3

-ADMM,

? 1

-ADMM, and

? 2

-ADMM for the continuous-impact forces applied to

P 5

and

P 6

Fig.12 Identification results of the impact force applied to

P 5

? 1 ? 2

-alternating direction method of multipliers (ADMM),

? 1

-ADMM,

? 2

-ADMM,

? 1 / 2

-ADMM, and

? 2 / 3

-ADMM: (a) time history reconstruction results, (b) localization results of

? 1 ? 2

-ADMM,

? 1

-ADMM, and

? 2

-ADMM, (c) localization results of

? 1 / 2

-ADMM and

? 2 / 3

-ADMM, and (d) convergence curves.

Fig.13 Identification results of the impact force applied to

P 6

? 1 ? 2

-alternating direction method of multipliers (ADMM),

? 1

-ADMM,

? 2

-ADMM,

? 1 / 2

-ADMM, and

? 2 / 3

-ADMM: (a) time history reconstruction results, (b) localization results of

? 1 ? 2

-ADMM,

? 1

-ADMM, and

? 2

-ADMM, (c) localization results of

? 1 / 2

-ADMM and

? 2 / 3

-ADMM, and (d) convergence curves.

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
Variables
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 i 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
$f^$	Estimated vector of f
f_p	Actual force vector at the impact position p
$f^p$	Estimated force vector at the impact position p
$f^ML$	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
$h^$	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
$R (f)$	General expression for calculating the norm of vector f
s(t)	System response
s	Response vector of the multiple-input multiple-output dynamic system
$s ~$	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
t₀	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 $s$
σ_n	Standard deviation of noise in the measurements
τ	Time delayed operator
ν	Possion’s ratio
Γ	Threshold value defined as Γ = λ/ρ

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