impact load identification for composite structures using...

Impact load identification for composite structures using Bayesianregularization and unscented Kalman filter

Gang Yan1, Hao Sun2,*,† and Oral Büyüköztürk2

1State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics,Nanjing 210016, China

2Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

SUMMARY

In structural health monitoring of composite structures, one important task is to detect and identify the low-velocityimpact events, which may cause invisible internal damages. This paper presents a novel approach for simulta-neously identifying the impact location and reconstructing the impact force time history acting on a compositestructure using dynamic measurements recorded by a sensor network. The proposed approach consists of twoparts: (1) an inner loop to reconstruct the impact force time history and (2) an outer loop to search for the impactlocation. In the inner loop, a newly developed inverse analysis method with Bayesian inference regularization isemployed to solve the ill-posed impact force reconstruction problem using a state-space model. In the outer loop,a nonlinear unscented Kalman filter (UKF) method is used to recursively estimate the impact location by minimiz-ing the error between the measurements and the predicted responses. The newly proposed impact load identifica-tion approach is illustrated by numerical examples performed on a composite plate. Results have demonstratedthe effectiveness and applicability of the proposed approach to impact load identification. Copyright © 2016 JohnWiley & Sons, Ltd.

Received 15 January 2016; Revised 13 May 2016; Accepted 13 June 2016

KEY WORDS: Bayesian regularization; composite structures; impact force reconstruction; impact location identifi-cation; unscented Kalman filter

1. INTRODUCTION

In aerospace industry, composite materials have been widely used as primary components for both com-mercial and military vehicles. One of the major concerns for design of composite structures is the hiddendamages caused by low-velocity impacts, which are difficult to detect and might lead to significantreduction of structural integrity (e.g., progressive growth of impact damage might result in catastrophicstructural failure). Recent advances in sensing technologies along with the developments in computa-tion have resulted in a significant interest in investigating and developing structural health monitoring(SHM) technologies that can be integrated into composite structures as a built-in diagnosis system[1,2]. For composite structures, to accurately evaluate the damage extent and the residual strength,the first task of an efficient and reliable SHM system is to detect and identify the impact load wheneveran impact event happens.

In general, identifying an impact load consists of two main aspects: (1) identifying the impact loca-tion and (2) reconstructing the impact force time history. A number of studies have been conducted andreported in literature on impact load identification for composite structures during the past two decades[3–21]. To identify the impact location alone, a straightforward approach is triangulation using the time

*Correspondence to: Hao Sun, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology,Cambridge, MA 02139, USA.†E-mail: [email protected]

STRUCTURAL CONTROL AND HEALTH MONITORINGStruct. Control Health Monit. (2016)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/stc.1910

Copyright © 2016 John Wiley & Sons, Ltd.

information of impact-induced stress waves [3–8]. For example, the impact location can be identifiedby solving a set of nonlinear equations describing the relationship among the impact location, wavevelocities and time-of-flights (TOFs). These equations can be solved by traditional optimization algo-rithms [3,4] or intelligent heuristic algorithms such as the genetic algorithm (GA) [5,6]. However, onedifficult problem involved in triangulation for composite structures is to obtain the direction-dependentwave velocities introduced by anisotropic properties and structural complexity. Recently, algorithmscombined with sensor cluster arrangement have been proposed to simultaneously triangulate the im-pact location and estimate the direction-dependent wave velocities [7,8]. Machine learning methodswere also employed for impact localization of composite structures [9–11]. For example, LeClerc et al.[9] used two neural network approaches, i.e., regression and classification, to localize impacts on a full-scale aircraft component made of composites. Sharif-Khodaei et al. [10] used numerical simulations toestablish a neural network meta-model of a stiffened composite panel for localizing impacts with var-ious energies. The advantage of machine learning methods is that they can handle impact localizationproblems for complex structures, such as a full scale UAV composite wing box structure [11], whilethe drawback is obvious that they require a large collection of impact data sets for training tests or nu-merical simulations.

To reconstruct the impact force time history, besides the machine learning methods such as theneural network [12], most of the methods reported in literature can be generally categorized asmodel-based methods. The models characterize the dynamic properties of composite structures underimpact, which can be obtained through experiments [13,14], analytical solutions [15,16] or numericalmethods [17–21]. By comparing the measured outputs with the estimated responses from a model, im-pact force time history can be reconstructed by different kinds of algorithms. For example, Choi andChang [15] developed a state space dynamic model based on Bernolli-Euler beam theory and applieda smoother/filter algorithm to reconstruct the impact force time history acting on a beam structure. Later,Tracy and Chang [16] and Seydel and Chang [17] successfully modified and improved the model andthe algorithm to identify impact forces acting on a laminated composite plate and a stiffened compositepanel, respectively. By employing the model proposed by Seydel and Chang [17], Yan and Zhou [18]proposed a GA-based approach for impact force identification with simplified force profile described byseveral parameters. Hu et al. [19] employed the finite element method to model the relationship betweenthe unknown force and the structural response, in which the impact force was represented by the com-bination of a series of Chebyshev basis functions. A quadratic programming optimization method wasthen employed to obtain the coefficients of the Chebyshev functions. Recently, with the concern ofuncertainties involved in impact force reconstruction, uncertainty analysis methods have been used toextend the aforementioned methods. For instance, Sun et al. [20] performed an uncertainty analysis be-fore identifying the impact for a stiffened composite panel. By reducing the material uncertainty througha model updating procedure, this method can accurately reconstruct the impact force time history. Yan[21] proposed a Bayesian approach for impact identification considering measurement and modelinguncertainties, where a Markov chain Monte Carlo (MCMC) algorithm was used to obtain samples ofthe impact parameters. The results have demonstrated that rather than pinpointing a single solutionusing deterministic approaches, the Bayesian approach provides the probability distribution of theimpact parameters, giving a possible uncertainty analysis to the estimates [22–24].

In addition to the aforementioned techniques, many model-based impact force reconstructionmethods have been developed by using inverse analysis based on deconvolution [25,26]. Usually,the deconvolution methods are considered as “straightforward”, which employ impulse responses (orcalled Green functions) to reconstruct the impact force time history. However, the deconvolution-basedimpact force reconstruction is typically a well-known ill-posed inverse problem which is vulnerable tonoise and whose solution may be unstable if there exists. In this case, regularization is usually requiredto solve the inverse problem so as to obtain a stable bounded solution. The regularization can be aphysical constraint in which the impact force must be non-negative or just be a mathematical compro-mise [27]. The most widely used mathematical regularization in inverse impact force analyses is theTikhonov regularization which relaxes the ill-conditioning by balancing the original objective functionand a smoothness condition on the solution. The major difficulty of applying the Tikhonov regulariza-tion lies in how to efficiently find the optimal regularization parameter. A number of choices areavailable in literature, e.g., the L-curve method [28], the S-curve method [29], and the generalized

G. YAN, H. SUN AND O. BÜYÜKÖZTÜRK

Copyright © 2016 John Wiley & Sons, Ltd. Struct. Control Health Monit. (2016)DOI: 10.1002/stc

cross-validation (GCV) method [30]. These methods operate on the basis of the singular value decom-position (SVD) of the coefficient matrix and are computationally effective when the scale/size of theformulated linear equations is not very large. Nevertheless, when the matrix size increases, the compu-tational efficiency decreases dramatically [31]. Large values of the matrix size bring severe computa-tion burden when using the L-curve and GCV methods, especially in an iterative optimization processwhere repeated forward analyses are required. To overcome the deficiencies of the traditionalTikhonov regularization, Jin and Zou [32,33] developed a regularization approach based on Bayesianinference which can adaptively determine the regularization parameter and detect the noise level in adata-driven manner. Yuen et al. [34,35] proposed a Bayesian probabilistic method for online estima-tion of the process noise and measurement noise parameters for Kalman filter, in which the noiseparameters are comparable to the regularization parameter mentioned above. By employing theBayesian inference regularization, Sun and his colleagues [31,36,37] successfully identified thetraffic-induced excitations of a truss bridge and the moving vehicle axle loads on a beam-type bridge,demonstrating the potential of Bayesian regularization in solving ill-posed inverse problems in SHM.

The aim of this paper is to present a novel approach to simultaneously identify the impact locationand reconstruct the impact force time history for composite structures. The proposed approach com-bines the Bayesian inference regularization method with a nonlinear unscented Kalman filter (UKF)to inversely reconstruct the impact force time history while recursively estimating the impact locationfor composite structures. The rest of this paper is structured as follows. Section 2 describes a state-space model for modeling the dynamics of a composite plate under impact, which is reformulatedfor impact force reconstruction. In Section 3, an inverse analysis for impact force reconstruction basedon Bayesian inference regularization is presented. A procedure for identification of the impact locationby using a nonlinear UKF method is described in Section 4. The flow chart of the proposed impact loadidentification algorithm is given in Section 5. Numerical studies using noisy synthetic data areperformed in Section 6 to verify the effectiveness of the proposed impact load identification method.Finally, conclusions are summarized in Section 7.

2. STATE-SPACE MODEL FOR COMPOSITE PLATE UNDER IMPACT

As illustrated in Figure 1, a rectangular composite plate with a symmetric layup is considered in thisstudy. We assume that only one point impact perpendicular to the plate exists and the overall structuralresponse is linear though small local damage may occur in the impact location. A nonlinear forwardmodel considering the damage process at the impact location is not employed because it is computa-tionally cumbersome for a composite impact load identification problem. Although the simplificationof using linear responses inevitably introduces modelling error, the effect of possible damage confinedto a small local region on the far field responses may be considered negligible.

In this paper, we employ the assumed modes method, which is a Rayleigh-Ritz solution in terms ofenergy with assumed shape functions that satisfy the geometric boundary conditions, to model the dy-namic responses of the composite plate under the point impact [17]. Based on the classical laminatetheory [38], the governing equation can be formulated as

Figure 1. Illustration of a rectangular composite plate under impact.

IMPACT LOAD IDENTIFICATION FOR COMPOSITE STRUCTURES


f tð Þδ x� ximpact; y� yimpact� �

¼ ρh∂2w∂t2

þ ∂2

∂x2D11

∂2w∂x2

þ D12∂2w∂y2

þ 2D16∂2w∂x∂y

� �

þ ∂2

∂y2D12

∂2w∂x2

þ D22∂2w∂y2

þ 2D26∂2w∂x∂y

� �

þ2∂2

∂x∂yD16

∂2w∂x2

þ D26∂2w∂y2

þ 2D66∂2w∂x∂y

� �(1)

where f(t) is the impact force, (ximpact, yimpact) is the impact location, Dij are elements of the bendingstiffness matrix D, w is the out-of-plane displacement, ρ is the density, and h is the thickness of thelaminate.

To eliminate the spatial dependence, each term of Equation (1) is multiplied by a vector of shapefunctions, ϕ(x, y), and integrated over the dimensions X by Y in x- and y-directions, respectively. Bychoosing ϕ(x, y) to satisfy different boundary conditions, Equation (1) can be simplified as

f tð Þϕ ximpact; yimpact� �

¼ ∫X0 ∫Y0

ρh∂2w∂t2

ϕ

þ D11∂2w∂x2

þ D12∂2w∂y2

þ 2D16∂2w∂x∂y

� �∂2ϕ∂x2

þ D12∂2w∂x2

þ D22∂2w∂y2

þ 2D26∂2w∂x∂y

� �∂2ϕ∂y2

þ2 D16∂2w∂x2

þ D26∂2w∂y2

þ 2D66∂2w∂x∂y

� �∂2ϕ∂x∂y

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

dydx (2)

The selection of shape functions depends on the boundary conditions, and the details can be foundin references [39,40].

The approximation of the displacement w by Nm terms of shape functions yields

w x; y; tð Þ≅∑Nm

i¼1wi tð Þϕi x; yð Þ (3)

Therefore, the equation of motion can be written as follows

M∂2w∂t2

þKw ¼ Lf tð Þ (4)

where w is the generalized displacement vector and w ¼ w1 w2 ⋯ wNm½ �T ; L, M, K are thegeneralized force location, mass and stiffness matrices, respectively, i.e.,

Li ¼ ϕi ximpact; yimpact� �

(5)

Mij ¼ ρh∫X0 ∫Y0ϕiϕjdydx (6)

Kij ¼ ∫X0 ∫Y0

D11∂2ϕi

∂x2∂2ϕj

∂x2þ D12

∂2ϕi

∂y2∂2ϕj

∂x2þ ∂2ϕi

∂x2∂2ϕj

∂y2

!

þD16∂2ϕi

∂x∂y∂2ϕj

∂x2þ 2

∂2ϕi

∂x2∂2ϕj

∂x∂y

!þ D22

∂2ϕi

∂y2∂2ϕj

∂y2

D26 2∂2ϕi

∂x∂y∂2ϕj

∂y2þ 2

∂2ϕi

∂x∂y∂2ϕj

∂y2

!þ 4D66

∂2ϕi

∂x∂y∂2ϕj

∂x∂y

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;dydx (7)

Defining the state vector z ¼ w _w� �T

, Equation (4) can then be formulated in the state space form

_z ¼ Azþ Bf tð Þ (8)

where



A ¼ 0 I

�M�1K 0

� and B ¼ 0

M�1L

� (9)

The impact response can be written in the output form as follows

y ¼ Cz (10)

where C is the output matrix and y is the output response. Here, we assume the measurement at eachsensor location is the strain combination εxx+ εyy on the surface of the composite plate, which is an in-variant variable that can be easily collected by biaxial strain gauges or circular piezoelectric sensors[16]. In this case, the output matrix C can be written as

C ¼C1 0

C2 0

⋮ ⋮CNo 0

26664

37775 (11)

in which No is the total number of sensors, and for each sensor Cs (s=1, 2,…,No) is a row vector withthe ith element as (∂2ϕi/∂x2 + ∂2ϕi/∂y2)h/2.

Assuming zero order hold for the input, the system is converted from the continuous form to thediscrete from, namely,

z nþ 1ð Þ ¼ ϕz nð Þ þ Γf nð Þ (12)

y nð Þ ¼ Cz nð Þ (13)

where ϕ=exp(ATs), Γ ¼ ∫Ts

0 exp Atð ÞdtB, Ts is the discrete sampling time and n represents the sam-pling points at time tn= nTs (n=0, 1,…,Ns) in which Ns is the total number of sampling points. Pro-vided that the system has zero initial conditions, the substitution of Equation (12) into (13) to solvefor y(n) yields

Y ¼ HF (14)

with

Y ¼ y 1ð Þ; y 2ð Þ;…; y Ns � 1ð Þ; y Nsð Þf g (15)

F ¼ f 0ð Þ; f 1ð Þ; f 2ð Þ;…; f Ns � 2ð Þ; f Ns � 1ð Þf g (16)

H ¼

CΓ 0 0 ⋯ 0

CϕΓ CΓ 0 ⋯ 0

Cϕ2Γ CϕΓ CΓ ⋯ 0

⋮ ⋮ ⋮ ⋱ ⋮CϕNs�1Γ CϕNs�2Γ CϕNs�3Γ ⋯ CΓ

2666664

3777775 (17)

where Y is the assembled measured system response vector (e.g. strain), F is the assembled unknownimpact force vector, H is the lower block triangular Hankel matrix consisting of the system Markovparameters. From Equation (14), when the impact location (ximpact, yimpact) is provided, the unknownimpact force vector F could be determined form the measurement vector Y. Afterwards, we can re-construct the impact force time history f(t) from F.

3. BAYESIAN REGULARIZATION FOR IMPACT FORCE RECONSTRUCTION

In general, it is not straightforward to reconstruct the impact force time history by a direct matrixinversion of Equation (14), since the least-square problem may be ill-posed (e.g. H is singular) whenmeasurement noise and modeling error are present. In order to provide a bounded solution,



regularization should be applied while the ill-posed problem is solved by least square methods. Thewell-known Tikhonov regularization can be adopted to solve for F, namely,

F_¼ HTHþ ηI

��1HTY (18)

where η is the non-negative Tikhonov regularization parameter. It is noteworthy that by setting η=0,Equation (18) would reduce into the ordinary least squares solution.

Nonetheless, as aforementioned in the introduction, the difficulty of applying the Tikhonov regular-ization lies in how to efficiently find the optimal regularization parameter η. In this study, a statisticalanalysis within the context of Bayesian inference is employed to automatically determine an appropri-ate value of the regularization parameter η, and reconstruct the unknown impact force iterativelythrough a statistical Bayesian learning scheme. In detail, the unknown force vector F is modeled inthe posterior probability density function (PDF) p(F, σ2, λ2|Y) through hierarchical Bayesian modeling[36,41], given by

p F; σ2; λ2jY �∝p YjF; σ2 �

p Fjλ2 �p σ2 �

p λ2 �

(19)

where p(Y|F, σ2) is the likelihood function and p(F|λ2) is the prior PDF of F, expressed as

p YjF; σ2 �∝

1σNoNs

exp � 12σ2

HF� Yk k2� �

(20)

p Fjλ2 �∝

1

λNsexp � 1

2λ2Fk k2

� �(21)

where σ is the standard deviation of prediction error due to measurement noise and modeling error, λ isa scale parameter representing the force variance. For convenience, conjugate prior PDFs are adoptedfor the hyperparameters σ2 and λ2, which are modeled by the inverse Gamma distribution, written as

p σ2 �

∝βα00

Γ α0ð Þ σ�2 α0þ1ð Þe�β0σ

�2(22)

p λ2 �

∝βα11

Γ α1ð Þ λ�2 α1þ1ð Þe�β1λ

�2(23)

where {α0, β0,α1, β1} are non-negative hyperparameters of the conjugate prior PDFs.The substitution of Equations (21)-(23) into Equation (19) yields

p F; σ2; λ2jY �∝λ�2 α1þ1ð Þ�Ns

σ2 α0þ1ð Þ�NoNsexp � 1

2σ2HF� Yk k2 � 1

2λ2Fk k2 � β0σ

�2 � β1λ�2

� �(24)

The Bayesian inference approach maximizes the posterior PDF in Equation (24) so as to obtain an aposteriori estimate of the parameter set, namely

F_; σ⌢2

; λ⌢2

n o¼ arg max

F;σ2 ;;λ2f gp F; σ2; λ2jY ��

(25)

Finding the maximum of Equation (25) is equivalent to solve the three optimality equations,expressed as [31–33,36,37]

HTHþ σ_2

λ_2

I

!F_ �HTY ¼ 0

(26)

NoNs þ 2 α0 þ 1ð Þ½ �σ_2 � HT F_ �Y

�� 2 � 2β0 ¼ 0 (27)

F_�� 2 þ 2β1 � Ns þ 2 α1 þ 1ð Þ½ �λ⌢2 ¼ 0 (28)

The optimal solutions F_; σ_2

; λ_

2n o

can be obtained if Equations (26)-(28) are solved. A sequential



Bayesian learning algorithm is employed to interpret and solve the optimality equations. The regular-

ization parameter η can be determined using η ¼ σ_2

=λ_2. The details of the algorithm can be found in

reference [36]. Note that, to obtain a non-informative Bayesian learning process, the values of priorPDF hyperparameters are set to be as small as possible [36].

4. UNSCENTED KALMAN FILTER FOR IMPACT LOCATION

In SHM, Bayesian filtering techniques, such as the well-known Kalman filter (KF) and its variants havebeen widely used for structural parameter identification, input estimation and damage detection withdynamic responses [42–44]. To deal with nonlinear system identification problems, the UKF hasdrawn increasing attention due to its capability to handle complex functional nonlinearities [45–47].It is founded on the intuition that “it is easier to approximate a probability distribution than it is to ap-proximate an arbitrary nonlinear function” [48,49]. Therefore, instead of direct linearization of the non-linear models using the first-order Taylor series expansion like the extended Kalman filter (EKF), theUKF employs a deterministic sampling approach, called the unscented transformation (UT), to capturethe mean and covariance of a Gaussian random variable used to approximate the state of the system witha set of sigma points [48]. In this study, to localize the impact event, i.e. determining (ximpact, yimpact), anonlinear UKF method is employed to recursively estimate the impact location using the measurementvector Y defined in Section 2.

To achieve this end, the first step is to define a state vector X= [ximpact, yimpact]T that consists of theunknown horizontal and vertical coordinates of the impact location. Then we formulate the relevantsystem equation (that describes the evolution of the state vector) and the measurement equation (thatdescribes the relationship between the hidden state vector and the observable measurement). Sincethe state vector X determines the impact location, it can be considered as time invariant (i.e. the impactlocation is a static point). Herein, the system equation can be written as

Xk ¼ Xk�1 (29)

where k is considered as an iteration number rather than a discrete time step typically used in filteringtechniques. Note that, at each iteration, the measurement Yk is kept consistently equal to Y (since thesame set of measurement is used in the iterations). We formulate Yk to be related to the state variableXk through a nonlinear measurement function hk which depends on the estimated impact location andthe reconstructed impact force time history at the kth iteration. To wit, the measurement equation canbe generally written as

Yk ¼ hk Xkð Þ þ υk (30)

in which a noise vector υ is introduced for taking into account uncertainties from modeling error andmeasurement noise. In UKF, υ is assumed to be a Gaussian type random vector with the same dimen-sion as Y, which is independent and identically distributed for all iterations. It is subjected to a zero-mean normal distribution p(υ) ~N(0,R), where R is an appropriate prescribed covariance matrix.

Considering the nonlinear function in Equation (30), we assume Xk and PXk are the mean and thecovariance of Xk, respectively. Noteworthy, UT uses 2N+1 sigma points χik and their associatedweights Wi to determine the mean Yk and the covariance PYk of Yk. These sigma points and theirweights are defined as following

χik ¼ Xk ;Wi ¼ KN þK; i ¼ 0

χik ¼ Xk þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN þKð ÞPXk

p� �i;W

i ¼ 12 N þKð Þ; i ¼ 1; 2; ⋯;N

χik ¼ Xk �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN þKð ÞPXk

p� �i;W

i ¼ 12 N þKð Þ; i ¼ N þ 1;N þ 2; ⋯; 2N

(31)

whereK ¼ ζ 2 N þ μð Þ � N is a scaling parameter; μ is a secondary scaling parameter which is usually

set to 0; ζ determines the spread of sigma points around X;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN þKð ÞPXk

p� �i is the ith row or column



of the matrix square root of N þKð ÞPXk ; Wi is the weight associated with the ith sigma point; and N is

the number of parameters to be estimated. It is noted that K is any positive or negative number satis-fying N þKð Þ≠0.

Given the measurement vector Y, the UKF for estimating the impact location vector X, shown inEquations (29) and (30), can then be written as

Yik ¼ hk χik�1

�(32)

Y⌢

k� ¼ ∑

2N

0WiYi

k (33)

PYkYk ¼ ∑2N

i¼0Wi Yi

k � Y⌢�k

� �Yi

kj � Y⌢�k

� �Tþ R (34)

PXkYk ¼ ∑2N

i¼0Wi χik�1 � X

⌢k�1

� �Yi

k � Y⌢�

k

� �T(35)

Kk ¼ PXkYkP�1YkYk

(36)

X⌢k ¼ X

⌢k�1 þKk Y� Y

⌢�k

� �(37)

Pk ¼ Pk�1 �KkPYkYkKTk (38)

The recursive process begins with a given initial value of the state vector X0 and its covariancematrix P0.

It is noteworthy that, in the impact force reconstruction, we found that the accuracy of the recon-structed force is highly dependent on the distance between the assumed impact location and the trueimpact location. With the increase of this distance, the magnitude of the reconstructed force decreasesdramatically. Outside a certain distance, the reconstructed forces are all nearly zero, leading to a totallyflat objective function which is defined as the ratio of the sum of error squares between the measure-ment and the predicted responses to the sum of squares of the measurement (see Equation (40)). There-fore, we employ a “divide-and-conquer” scheme in determining an admissible initial guess of theimpact location: (i) a grid with suitable spaced discrete points is firstly defined on the composite plate;(ii) the objective function values are only calculated at these grid points; (iii) the grid point with a min-imum objective function value is set as the initial estimate of impact location X0. Afterwards, the UKFis activated to recursively approach the true impact location. The details of implementation of discretesearching based on the “divide-and-conquer” grid points are illustrated in the numerical examples.

5. FLOW CHART OF THE PROPOSED ALGORITHM

The flow chart of the proposed algorithm for composite structure impact load identification is given inFigure 2. It is seen that the proposed algorithm includes three major components: (A) the “divide-and-conquer” scheme with discrete grid points for determining the initial guess of impact location,(B) the Bayesian regularization process for impact load profile reconstruction, and (C) the UKF forimpact load localization. The convergence criterion for the UKF is expressed as follows

Xk � Xk�1k kXk�1k k < e (39)

where e is a small positive number (i.e. e=1×10� 3).



6. NUMERICAL STUDIES

6.1. Numerical example description

To demonstrate the effectiveness and performance of the proposed approach on impact load identifica-tion, numerical studies are conducted in this section. Synthetic responses are generated as the referencemeasurements using a finite element model by ABAQUS. Zero-mean white noises are added to thesynthetic data to simulate the measurement noise corruption. A comparison between the impactresponses using the finite element method and those by the assumed modes method in a state-space formpresented in Section 2 is performed herein. For convenience, in the following, the finite element impactmodel is called FE model, while the assumed modes method-based impact model in this study is simplycalled SS model (i.e. short for state space). It is noted that we purposely introduce uncertainties includ-ing both the measurement noise (white noise) and the modeling error (the discrepancy between FE andSS models) into the identification process to test the performance of the proposed algorithm.

The numerical example considered in this study is a square composite plate with dimensions of1000 mm × 1000 mm. The composite plate consists of T300 graphite fibers and QY8911 epoxy matrix,with the layups (laminate orientation code) being [45/90/-45/0/0/45/0/0/-45/0]s. The thickness of eachlamina is 0.125 mm, and all of the laminas are assumed to be perfectly bonded. The material propertiesof T300/QY8911 lamina are presented in Table I. The FE model is modeled by 1600 reduced

Figure 2. Flow chart of the proposed algorithm for impact load identification.

Table I. Material properties of T300/QY8911 lamina.

E1 (GPa) E2 (GPa) G12 (GPa) ν12 ρ (kg/m3)

135 8.8 4.47 0.3 1560



integration general purpose shell elements, S4R, with a uniform element size of 25 mm × 25 mm. Allof the four edges of the plate are clamped and the origin of the coordinate system is set at the left-bottom corner as shown in Figure 1. The first five natural frequencies of the plate are 24.6, 49.3,52.2, 74.6, 89.0 Hz, and the 20th natural frequency is 251 Hz. A set of four sensors, designated asS1, S2, S3 and S4, are employed to measure the strain responses under the impact. The coordinatesof S1, S2, S3 and S4 are (250, 250) mm, (750, 250) mm, (750, 750) mm and (250, 750) mm,respectively.

In the numerical studies, two impact force time histories, labeled as I1 and I2 as shown in Figure 3,are adopted, each of which are applied to three randomly selected impact locations forming 6 individ-ual impact events. These impact locations are labeled as L1 to L6, whose coordinates are given inTable II. Thus each impact event can be labeled as IpLq, where I stands for the impact force typeand L represents the impact location (p=1, 2; q=1, 2, …, 6), for convenience. In the dynamic analy-sis, the sampling frequency is set as 25 kHz corresponding to a sampling interval of 0.04 ms. This highsampling rate is required and sufficient for a smooth reconstruction of the impact load with a high dom-inant frequency. Figure 4(a) shows a comparison of the impact responses obtained by the FE model

Figure 3. Two impact force histories considered in the numerical studies.

Table II. Impact events considered in the numerical studies (Unit: mm).

Impact force I1 Location L1 Location L2 Location L3(350, 400) (125, 675) (700,500)

Impact force I2 Location L4 Location L5 Location L6(400, 600) (600, 800) (550, 275)

Figure 4. Comparison of impact responses by FE model and SS model.



and the SS model, when the impact force acts on location L1 (i.e., impact event I1L1), and Figure 4(b)shows the comparison when the impact force I2 acts on location L4 (i.e., impact event I2L4). It can beseen that the outputs of the SS model agree quite well with the FE model results, demonstrating themodeling approach using the Rayleigh-Ritz type assumed modes method is satisfactory. However,the discrepancy between the two sets of responses is obvious. Thus, when we use the white noisecontaminated FE responses as the synthetic measurement data for the numerical studies, an “inversecrime” (e.g., the model for synthetic data generation identical to the model for inverse analysis) isavoided. The purpose is to simulate a situation similar to the real operational application in which bothmeasurement noise and modeling error are taken into account in the identification process.

6.2. Identification of impact locations

To identify the impact locations, a “divide-and-conquer” grid is first defined for preliminary discretesearch. The grid point with a minimum objective function value is set as the initial estimate X0 forthe UKF procedure. In the numerical studies, a uniformly spaced grid containing 36 (i.e. 6 × 6) gridpoints is defined as illustrated in Figure 5. The coordinates of the grid points in the four corners(i.e., points No. 1, 6, 31, 36) are (100, 100) mm, (900, 100) mm, (900, 900) mm and (100, 900)mm, respectively, and the spacing of the grid is 160 mm. The objective function used for discretesearch is defined as

obj ¼Y� Y

_xgrid; ygrid� �� 2Yk k2 (40)

in which Y is the assembled measurement vector as defined in Section 2, Y_

xgrid ; ygrid� �

is the pre-

dicted response vector obtained from the SS model with the reconstructed impact force using Bayesianregularization at grid point (xgrid, ygrid).

Figure 6 shows the localization results for impact event I1L1 when the measurement signal-to-noise-ratios (SNRs) are 20 dB and 10 dB, respectively. It can be seen from Figure 6(a) that only the grid pointsnear the true impact location have objective values less than one, meaning that the impact force recon-structed at the points outside a certain distance from the true impact location is zero or close to zero. Forthis impact event, the grid point with a minimum objective function values is No. 14 whose coordinatesare (260, 420) mm. We then assign its coordinates as the initial estimate for the UKF. Afterwards, theUKF recursively estimates the impact location as shown in Figure 6(b). It can be observed that theUKF converges quickly within 10 iterations. The estimated impact locations by UKF are (348, 401)mm for the case of 20 dB SNR and (348, 402) mm for the case of 10 dB SNR, respectively. It can be

Figure 5. Grid points for discrete search of impact location in the numerical studies.



seen that the localization results are quite accurate � the discrepancies are less than 3 mm between theestimated and the true locations.

Similarly, Figure 7 shows the localization results for impact event I2L4 when the measurement SNRsare 20 dB and 10 dB, respectively. In this case, as shown in Figure 7(a), the grid point which has aminimum objective function is numbered as 21 and its coordinates are (420, 580) mm. Then the UKFrecursively estimates the impact location given by Figure 7(b). The estimated impact locations byUKF are (398, 603) mm for the case of 20 dB SNR and (397, 604) mm for the case of 10 dB SNR,respectively. Again, we can observe that the convergence is quite fast yielding an accurate identificationof the locations, i.e., the errors are less than 5 mm between the estimated and the true locations.

In addition, similar results can be obtained for other impact events with different level of measure-ment noise. Figure 8 illustrates comparisons of the true impact locations and the identified ones withSNRs of 20 dB and 10 dB, respectively. The numeric results are also summarized in Table III. Fromboth Figure 8 and Table III, it can be observed that the proposed UKF combined with a discrete gridsearch is insensitive to measurement noise and can identify the impact locations quite accurately. Themaximum error of the identified locations is about 12 mm when the SNR is 10 dB, which demonstratesa quite accurate identification when considering uncertainties of both the modeling error and the mea-surement noise.

6.3. Reconstruction of impact force

As an inner loop, when the impact location is recursively updated, in each UKF iteration, the Bayesianregularization method is used to iteratively reconstruct the impact force time history at a temporarilyestimated impact location. The corresponding impact responses are also simultaneously predicted forUKF updating. As long as the final estimate of the impact location is obtained, the final reconstruction

Figure 7. Location results for impact event I2L4 (a) objective values at the discrete grid points (b) UKF estimates.

Figure 6. Location results for impact event I1L1 (a) objective values at the discrete grid points (b) UKF estimates.



of impact force time history is accomplished. In this study, the hyperparameters of the conjugate priorsused in the Bayesian regularization are given as follows: α0 =α1 = 2 and β0 =β1 = 1×10� 25. The initialvalues of σ2 and λ2 are assigned as 1×10� 12 and 100, respectively.

Figure 9(a) shows the reconstructed impact force for impact event I1L1 when the SNR is 20 dB. Thetrue impact force time history is also given in this figure for comparison purpose. It can be seen that thereconstructed impact force is in a good agreement with the true impact force despite the oscillationdiscrepancy exists. In general, the reconstructed shape, duration and peak value of the impact forcewell match those of the true impact load time history. It is noteworthy that the oscillations of the recon-structed load time history shown in Figure 9(a) are mainly caused by the measurement noise. Whennoise-free measurements are used to identify the impact load, a perfect agreement between the recon-structed load and the true load can be obtained. Figure 9(b) shows comparisons between the measuredimpact responses and the predicted responses. A good agreement can be observed between the mea-sured and the predicted responses. Moreover, Figure 10 show the reconstructed impact force timehistory as well as the predicted responses for impact event I1L1 when the SNR of measurement isincreased to be 10dB. Again, the reconstructed impact force matches the true one quite well.

Table III. Identification results for impact locations (Unit: mm).

Location L1 L2 L3 L4 L5 L6

True (350, 400) (125, 675) (700,500) (400, 600) (600, 800) (550, 275)SNR = 20 dB (348, 401) (121, 679) (704, 503) (398, 603) (601, 806) (551, 272)SNR = 10 dB (348, 402) (114, 680) (706, 504) (397, 604) (602, 807) (548, 272)

Figure 9. Force identified results for impact event I1L1 when SNR = 20dB (a) reconstructed force (b) predictedresponses.

Figure 8. Comparison of true impact locations and identified ones (a) SNR = 20 dB (b) SNR = 10 dB.



It is worthy to mention that the main advantage of Bayesian regularization over the traditionalTikhonov regularization is that the former method can automatically determine the optimal regulariza-tion parameter through adaptive Bayesian learning from the measured data while the Tikhonov regu-larization requires full SVD of a large size matrix occupying high computational cost and solutionstability issue. Figure 11 illustrates the determination process of the regularization parameter in thefinal impact force reconstruction for impact event I1L1 with SNRs of 20 dB and 10 dB, respectively.It can be seen that, starting from an initial guess, the Bayesian learning determines the optimal valuesof the regularization parameter quite efficiently.

Similar impact force reconstruction results can be obtained through the Bayesian inference regular-ization for impact event I2L4. Figure 12(a) and Figure 13(a) show the reconstructed impact forces withSNRs of 20 dB and 10 dB, respectively. Figure 12(b) and Figure 13(b) show the predicted responses incomparison with the measurements, corresponding to reconstructed impact forces in Figure 12(a) andFigure 13(a), respectively. Figure 14 illustrates the regularization parameter evolution in the finalimpact force reconstruction for impact event of I2L4. The overall identification results are as goodas those for impact event I1L1.

6.4. Comparison of the impact energy

For composite structures, the impact damage patterns and damage extents are tightly related to theimpact energy [50]. Thus, the impact energies of the identified and the measured impact force historiesare presented for comparison. According to the formula in reference [15], the impact energy can becalculated as follows

Figure 10. Force identified results for impact event I1L1 when SNR = 10 dB (a) reconstructed force (b) predictedresponses.

Figure 11. Determination of regularization parameter in force reconstruction for impact event I1L1.



Energy ¼ ∫f tð Þdw≈ ∑Ns�2

n¼0

f nþ 1ð Þ þ f nð Þ½ �2

w0 nþ 1ð Þ � w0 nð Þ½ � (41)

where w0 is the displacement at the impact point, Ns is the number of discrete sampling points in thetime domain as defined in Section 2. Figure 15 shows the predicted displacements at impact locations


Figure 14. Determination of regularization parameter in force reconstruction for impact event I2L4.




for both impact events I1L1 and I1L4. The reference measured responses are calculated at the exactimpact locations with the true impact force, while the predicted responses are obtained at the identifiedlocations shown in Table III with reconstructed impact forces depicted in Figures 9(a), 10(a), 12(a) and13(a), respectively. Then the predicted impact energies can be calculated using Equation (41). Thecomparison of impact energies between the measured and the identified impact forces are presented inFigure 16. The numeric results are also given in Table IV. From both Figure 16 and Table IV, it can beseen that the identified impact energies agree quite well with the corresponding measured ones underuncertainties due to both modeling error and measurement noise, further demonstrating the effective-ness of the proposed impact load identification method.

Figure 15. Comparison of measured and predicted displacement responses at impact locations for impact events(a) I1L1 (b) I2L4.

Figure 16. Comparison of measured and identified impact energies.

Table IV. Comparison of measured and identified impact energies (Unit: J).

Impact event I1L1 I1L2 I1L3 I2L4 I2L5 I2L6

True 5.52 2.57 5.52 4.45 3.91 3.70SNR = 20 dB 5.75 2.42 5.59 4.22 3.81 3.55SNR = 10 dB 5.29 2.37 5.34 3.96 3.64 3.37



7. CONCLUSIONS

This paper presents a novel approach for simultaneously identifying the impact location andreconstructing the impact force time history acting on a composite structure through dynamic strainmeasurements. The proposed algorithm consists of two parts: (1) an inner loop for the impact forcetime history reconstruction using a newly developed Bayesian inference regularization approach,and (2) an outer loop for the impact localization using a nonlinear UKF method with an initial guessfrom a “divide-and-conquer” discrete grid search. The performance of the proposed approach is vali-dated by numerical studies based on a composite plate. The results have shown that the proposed ap-proach can accurately identify the impact location and successfully reconstructs the impact force timehistories. The shape, duration and peak of the reconstructed forces as well as the impact energies agreewith those of the true force quite well. It is noteworthy that the identified impact energies are useful forpredicting impact damage in composite plates. The overall satisfactory performance of the proposedalgorithm illustrates its potential to be applied in impact load identification for composite structuresin real applications. Nevertheless, a possible direction of the future work could be the experimentalstudy and testing the applicability of the proposed algorithm to impact load identification for varioustypes of composite structures.

ACKNOWLEDGEMENTS

This research is supported by the Research Fund of State Key Laboratory of Mechanics and Control of MechanicalStructures (Nanjing University of Aeronautics and astronautics) (Grant No. 0515G01), and a project funded by thePriority Academic Program Development of Jiangsu Higher Education Institutions.

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