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Inverse Problems in Science and Engineering

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Impact force reconstruction using the regularizedWiener filter method

Fergyanto E. Gunawan

To cite this article: Fergyanto E. Gunawan (2015): Impact force reconstruction using theregularized Wiener filter method, Inverse Problems in Science and Engineering, DOI:10.1080/17415977.2015.1101760

To link to this article: http://dx.doi.org/10.1080/17415977.2015.1101760

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING, 2015http://dx.doi.org/10.1080/17415977.2015.1101760

Impact force reconstruction using the regularizedWiener filtermethod

Fergyanto E. Gunawan

Binus Graduate Programs, Bina Nusantara University, Jakarta, Indonesia

ABSTRACT

There are cases in engineering applicationswhere the involved impactforces are impractical or difficult to be directly measured. In suchcases, the impact forces may inversely be reconstructed on the basisof the associated-induced elastic responses. However, the inverseimpact force reconstruction is an ill-posed problem. So far, variousmethods have been proposed. Many require complex mathematicaltreatments; thus, they are impractical for experimental engineers.In this article, we adopt the Wiener filter method and introduce aregularization technique so that themethod can provide an optimumsolution which balances the aspects of the minimum noise and theminimum residual. The proposed method is evaluated using threeexamples of the impact problems. The evaluation reveals that theregularized Wiener filter method is less sensitive to the regularizationparameter unlike many of the existing methods. Therefore, themethod is more suitable for practical applications.

ARTICLE HISTORYReceived 7 December 2014;accepted 26 September2015

KEYWORDSWiener filter; impact forceidentification; frequencyresponse function; ill-posedproblems; conditionnumber; singular valuedecomposition; conjugategradient method; truncatedsingular valuedecomposition; numericalregularization

AMS SUBJECTCLASSIFICATIONS01A23; 45B67

1. Introduction

The inverse reconstruction of a dynamic force is important for various engineeringapplications where the data of the dynamic force are required, but direct measurementsare impractical or difficult. In the inverse reconstruction, the dynamic force is inferredfrom other measurable quantities. Those quantities, for example, are the induced elasticresponses such as the structural displacements, velocities or strains.

Some example scenarios where the inverse reconstruction is preferable are of thefollowing. The first scenario is in the case of a lateral impact of a small particle with aplate.[1] The second is in the case of a bird striking an aircraft fuselage.[2] The third is inthe case of the main-bearing loads of a diesel engine.[3] As the final scenario is the case ofthe distributed forces acting on a cutting tool.[4]

Despite of its importance, the field of inverse reconstruction of the dynamic force is arelatively young field.[5]. The number of publications in the field is rather limited.

The inverse reconstruction problem is challenging from the mathematical perspectivebecause of the following reasons.

CONTACT Fergyanto E. Gunawan fgunawan@binus.edu© 2015 Taylor & Francis

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0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9(a) (b)1

Forc

e (N

)

Time (µs)0 20 40 60 80 100

−150

−100

−50

0

50

100

150

Forc

e (N

)

Time (µs)

Figure 1. An example result of the noise amplification by the direct deconvolution without a numericalregularization. (a) The actual impact force and (b) the noise dominated impact force.

Consider a structural system with an operator L that maps the input dynamic-forcevector �f (t), the unknown variable, to the output structural response vector �e(t):

L[�f (t)] = �e(t). (1)

Via a perturbation analysis, we can show that

‖��f (t)‖‖�f (t)‖ ≤ cond(L)

‖��e(t)‖‖�e(t)‖ , (2)

where cond(L) = ‖L‖‖L−1‖ and the norm of the linear operator is defined by ‖L‖ =max ‖Lx‖/‖x‖ given any vector �x �= �0.

Expression (2) implies that small perturbation in the data, ��e(t), will be amplified asmuch as the condition number of the linear operator L. The inverse reconstruction ofthe dynamic force in the form of Equation (1) is ill-posed because the problem has largecondition number, cond.L/. The condition number is in order of 1 × 106 for the casesconsidered in this work.

The current work is limited to the case where L is a convolution operator; thus, theproblem can be written as ∫ t

0K(t, τ)f (τ ) dτ = e(t), (3)

where the right-hand side e(t) and the kernel K(t, τ) are given, and f (t) is the unknownto be sought. On the basis of the Riemann-Lebesque lemma, Refs. [6] and [7] have shownthat the deconvolution process, i.e. that of computing f (t) from e(t) will amplify the highfrequency components. To better appreciate this problem, in Figure 1, we compare anactual impact force to that directly estimated by a numerical deconvolution without anyregularization. The result is a severe erroneous solution shown by Figure 1(b). The exactsolution is in Figure 1(a). The noise dominant solution shows no indication of the actualsolution and completely indicates the characteristics of the noise.

Recently, Ref. [5] provides an overview regarding the force reconstruction techniques.The reference divides the existing reconstruction methods into three categories: direct,

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 3

regularization and probabilistic/statistical. The regularization category involves Tikhonovregularization, frequency range truncation methods, optimization methods and weightedbasis functions. The probabilistic/statistical category involves recurrence plots, Bayesianmethods and adaptive estimation.

In addition, the reference also identifies some aspects in the field that may of interestfor further development. One of the aspects is that some existing methods do not clearlydiscuss the method to determine the proper regularization parameter. The other aspect isthat some proposed methods are rather impractical for application.

In our opinion, it is also necessary to provide an insight on how the trend of the forcereconstruction methods evolves across the time. We observe that, as also observed byRef. [8], up to year 2005, majority of the solution methods were made on the frequencydomain. These can be seen in Refs. [9–19]. The frequency domain approach is rarelydeployed these days, except Ref. [20] where the force acting on a plate is reconstructed bysolving the frequency-domain algebraic equation using theMoore-Penrose pseudoinversemethod.

A brief description regarding those existing methods is of the following. In Ref. [9], thestructural equation ofmotionwas expressed in the frequency domain as amultiplication ofthematrix of accelerationmobilities and the distributed-dynamic force. The dynamic forcewas reconstructed using the pseudo-inverse of the acceleration matrix, which consisted ofthe structural vibrationmodes. Themodeswere established by themeans ofmodal analysis.Ref. [11] proposed the use of non-causal Wiener filter to minimize the effect of noise onthe reconstructed impact force. No explicit numerical regularization was used; thus, theapproach was limited to the case where the level of ill-posedness is rather low. Ref. [12]used a small random number to overcome singular spectra; the caveats of this methodwill be discussed extensively in Section 2.1. Furthermore, Refs. [13] and [17] deployed theprevious method to a frame structure case and a multiple impact-force case. In Ref. [14],weighting factors were added to the structural frequency domain response function, andlow-magnitude in-operation loads were constructed. In Ref. [16], the structural equationof motion in the integral convolution form was solved by applying the Laplace transform.If we define the Laplace transform as F(s) = ∫ +∞

−∞ e−st f (t) dt where s = α + |γ , Ref.[16] controlled the regularization level via the scalar variable α. Finally, Ref. [19] proposedthe least-squared method and was applied to the frequency-domain structural responsematrix.

The recent force reconstruction methods often involve the finite element discretizationfor the structural governing dynamics.[21–23] To deploy thesemethods, engineers requirenot only knowledge of the experimental method, but also of the computational mechanics.

Somehow the mathematical formula of the force reconstruction can be expressednaturally in the form of the convolution integral equation where the structural dynamicresponse is simply encapsulated into the structural impulse response function avoidingmanymechanical details such as the structuralmechanical properties and vibrationmodes.Within this conception, we only focus on how to reconstruct the force for given a set ofstructural response data.

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By deploying the convolution theorem,[24] the integral convolution equation can beexpressed as a multiplication in the frequency domain:

∫ t

0h(t − τ)f (τ ) dτ = e(t) ≡ H(ω) · F(ω) = E(ω). (4)

The frequency domain variable is related to that in time with:

H(ω) =∫ +∞

−∞h(t) exp ( − jωt) dt. (5)

The convolution theorem could be the reason why earlier publications preferring thefrequency domain approach.

The integral convolution equation can also be discretized in time. By applying theRiemann’s approximation, the equation can be written as

N−1∑k=0

h(i − k)f (k) = e(i), for i = 0, . . . ,N − 1, (6)

where f (i) is the force at time ti = i ·�t, and�t is the sampling time. Equation 6 is writtenmore compact in the matrix-vector form as:

Hf = e, (7)

whereH is the non-causal convolution matrix of

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

h(t0) 0 0 0 0h(t1) h(t0) 0 0 0

... h(t1) h(t0) 0 0

h(tN−2)... h(t1) h(t0) 0

h(tN−1) h(tN−2)... h(t1) h(t0)

⎤⎥⎥⎥⎥⎥⎥⎥⎦

. (8)

There are many benefits expressing the force reconstruction problem in form of Equa-tion (7).Manynumerical regularization techniques have beendeveloped for the expression;see, for example, Ref. [25]. Thus, the existing techniques are readily to be deployed.

The time domain expression is also beneficial because the physical behaviour of thestructure is better understood in the domain. These knowledge can be used as prioriinformation to regulate the solution numerically. In general, a better solution can beobtained by using more priori information. We took advantages of this fact and proposeda number of time domain regularization techniques.[1,26–28] In addition, a number ofrecent publication is also within this category, for examples, Refs. [29–32]. Ref. [29]proposed a regularization by limiting the number of vibration modes. Refs. [30] and [32]proposed generalized forms the Tikhonov filter factor. In addition, [31] proposed an itera-tive solution method that inherently regulates the solution, so called Landweber iteration,and evaluated the solution optimality condition by Morozov’s discrepancy principle.

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 5

(a) (b)

Figure 2. An example result of the noise amplification by the direct deconvolution without a numericalregularization. (a) Impact force F(ω) and (b) elastic E(ω) and impulse H(ω) responses.

We believe the force reconstruction is a part of the experimental approach. Thus, themethod should be able be deployed with the existing experimental hardware withoutcomplex computational algorithms. In this work, we adopt the Wiener filter method andpropose a regularization technique so that the method can provide an optimum solutionwhich balances the aspects of theminimumnoise and theminimumresidual. The proposedapproach requires minimum computation, fits with the common existing experimentalhardwares and is more suitable for practical applications

2. Method

2.1. Previous developments of impact force reconstruction in the frequencydomain, their limitations and potential improvement

For the purpose of impact force reconstruction, Equation (4) should be recast into

F(ω) = E(ω)

H(ω). (9)

For all physically-realizable systems, the magnitude of the spectra |F(ω)|, |E(ω)|, and|H(ω)| should approach zero as ω increases. These characteristics are shown in Figure 2.The typical force spectra |F(ω)| are shown in the left panel, labelled with ‘Exact Solution’.The typical impulse response function |H(ω)| and the elastic response spectra |E(ω)|are shown in the right panel, labelled with ‘Impulse Response’ and ‘Exact Response’,respectively.

The problem associated with Equation (9) arises when the elastic response is contam-inated with noise, which are often unavoidable in the actual measurement. The typicalnoise affected elastic response is shown in the right panel of Figure 2, labelled with ‘ExactResponse + White Noise’.

For this illustration, we note that the noise is assumed to be random, but followedan independent and identical statistical distribution (i.i.d) function. The distribution isnormal N(0, σ 2) where the standard deviation σ is set as a small fraction of the maximumelastic response.

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(a) (b)

Figure 3. The results of the Monte Carlo simulation for reconstructing impact force using formulaF(ω) = E(ω) · H(ω)/[H(ω) · H(ω)+ Rn]with random Rn. (a) An acceptable reconstructed impact force,(b) an unacceptable reconstructed impact force.

The direct inversion of Equation (9) provides a solution F(ω) that is heavily affected bythe noise. The solution is shownon the left panel of Figure 2with the label ‘NoiseDominantSolution’. In the time domain, this solution is shown in Figure 1. Figure 2 shows that thereconstructed impact force spectra are accurate only in the narrow low frequency region.Outside the region, the reconstructed impact force spectra resemble those of the noisespectra.

Figure 2 clearly demonstrates the ill-posedness of the impact force reconstructionproblem in the frequency domain. The estimated impact force spectra on low frequencyare accurate, but on high frequency, they are inaccurate.

Two aspects contribute to the inaccurate impact force spectra. The first is that on highfrequency, the magnitude of the noise spectra is higher than the structural elastic responsespectra. The second is that the spectra of the impulse response function on high frequencyare small; the spectra decrease continuously by increasing the frequency. Thus, dividing theelastic response spectra by the impulse response function spectra increases the contributionof the noise spectra in the solution. In fact, the noise spectra are significantly higher thanthe exact solution spectra on the high frequency region as can be seen on the left panel.

Ref. [12] addresses the above problem with

F(ω) = E(ω) · H(ω)

H(ω) · H(ω) + Rn, (10)

where H(ω) is the complex conjugate ofH(ω) andRn is a small amount of randomnoise. Inaddition, they note that ‘the addition of noise to the denominator does not, of itself, replacethe lost information, but rather it removes the potential singularity at those frequencieswhere H(ω) becomes quite small’.

One can think ofRn as a fudge factor, which is introduced to patch the singular spectra ofH(ω) · H(ω). Rationally, the value of Rn should be within the range of max

[H(ω) · H(ω)

]and min

[H(ω) · H(ω)

]. A high value of Rn, but lower than max

[H(ω) · H(ω)

], means

more regularization. In general, more regularization produces smoother solution.

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 7

(a)

(c)

(b)

(d)

Figure 4. The results of the impact force reconstruction using formula F(ω) = E(ω) · H(ω)/[H(ω) ·H(ω) + Rn] with a fixed Rn. (a) The optimum reconstructed impact force with Rn = 3.3358 × 10−11,(b) the estimation error as a function of Rn, (c) trade-off curve and (d) the optimum Rn in H(ω) · H(ω)

spectra.

Figure 5. The linear convolution model with additive noise.

The idea that Rn is random is dubious as it may take a large value as well as a small one.And, if Rn value is small when theH(ω) · H(ω) spectra is singular, then Rn will not be ableto patch the singularity, and the division of Equation (10) will amplify the noise. Thus, therandom Rn has a chance of failure to patch the singularity.

To demonstrate the above assessment, we use the Monte Carlo simulation in conjunc-tion with Equation (10). The simulation generates a set of random numbers and each

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number is assigned to Rn. The random value Rn is generated from the normal distributionN(0, σ 2), where σ is fixed to a small value.

The Monte Carlo simulation provides impact forces with various degree of accuracy.Some are accurate, and some are inaccurate. Figure 3 shows one of the accurate solutionsand one of the inaccurate solution. This empirical finding demonstrates that the randomRn may fail.

Rational to think that a fixed value of Rn, instead of random, should definitely close allsingular spectra, and this value should lie within the interval

{min

[H(ω) · H(ω)

], max[

H(ω) · H(ω)] }

. However, the fixed value may interfere non-singular spectra if the valueis much bigger than H(ω) · H(ω).

To evaluate the above idea, we reconstruct the impact force using Equation (10) andRn is varied within the previously-stated interval. For each value of Rn, we estimate thesolution f (t) as well as the residual term

Residual = h(t) ∗ f (t) − e(t), (11)

where ∗ denotes a convolution.The results of the study where Rn is fixed and used as the regularization parameter are

shown in Figure 4. In the figure, Panel (a) shows the most optimum solution that obtainedwhen the regularization parameter Rn is set to 3.3358× 10−11. This optimum solution hasthe estimation error of 23.5%. The error is defined as

Estimation Error(%) = ‖f (t) − f (t)‖2‖f (t)‖2 × 100, (12)

where f (t) is the exact solution.The value of the regularization parameter Rn associated with the optimum solution is

obtained from a trade-off between the residual term, ‖h(t)∗ f (t)− e(t)‖2, and the solutionsmoothness, ‖f (t)‖2. For the detail of the technique, consult Ref. [25]. This trade-off isgraphically shown on Panel (c). The most optimum solution, marked by a bullet, residesat the point nearest to the graph origin. This solution satisfies the Tikhonov optimumcriterion:

min ‖h(t) ∗ f (t) − e(t)‖2 subject to ‖f (t)‖2 ≤ α, (13)

where α is a scalar regularization parameter.Figure 4(d) shows the plot ofH(ω) · H(ω) as a function of frequency. The optimum Rn

is also shown in the figure. The optimum Rn is a few order higher than the spectra on thefrequencies higher than 150 kHz. Thus, Rn patches all singular spectra in the frequencyrange.

Besides the above approach, Ref. [11] demonstrated that when the level of the sin-gularity is not severe, the following formula is reasonably accurate. The impact force isreconstructed with

F(ω) = G(ω) · E(ω) (14)

where G(ω) = Seh(ω)/See(ω), Seh(ω) = E(ω) · E(ω), and See(ω) = E(ω) · H(ω).

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 9

2.2. Previous developments of impact force reconstruction in the time domain

Some previous proposals of the impact force inverse reconstruction in the time domainwill be discussed in this section. The solutions of those proposals will be compared thatobtained by the current method.

2.2.1. Truncated singular value decompositionThe truncated singular value decomposition (TSVD) is relatively simple and has been usedto solve various ill-posed inverse problems.[25,33] To apply the method, we should firstdecompose the square matrixH into its singular value and vector components:

H =n∑

j=1

ujσjvTj , (15)

where uj and vj is the left and right singular vectors, and σj is the singular value. ThematrixH has a size of n × n.

The TSVD solution for the impact force is written as

f =k∑

j=1

uTj e

σivj, (16)

where k is the TSVD regularization parameter, and it is selected to satisfy:

min ‖f‖2 subject to min ‖Hk f − e‖2. (17)

The matrixHk = ∑kj=1 ujσjv

Tj .

2.2.2. Tikhonov regularizationRef. [34] has deployed Tikhonov regularization for the impact force reconstruction. TheTikhonov regularization solution is written

f =n∑

j=1

fiuTj e

σivj, (18)

where fi is the Tikhonov filter factor and is computed by fi = σ 2i /(σ 2

i + λ2). Theregularization parameter λ is varied between the smallest singular value σn and the largestσ1. The optimum λ is one that minimizes

‖H f − e‖22 + λ‖f‖22. (19)

2.2.3. Iterative regularizationIterative regularization of the conjugate-gradient (CG)method for the impact force recon-struction is discussed by Refs. [35,36]. The search for the optimal solution goes accordingto Algorithm 1. If the data are well-scaled, the iteration should be terminated when thefollowing condition meets. The effect of the scaling in general to the optimum solution is

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Data: Initializationf0r0 = Hf0 − e ;d0 = HT r0 ;

for i = 0, 1, … dok = 1 ;

αk = HT r(k−1)

Hd(k−1) ;

f (k) = f (k−1) + αkd(k−1) ;r(k) = r(k−1) − αkHd(k−1) ;

fik = ‖HT r(k)‖22‖HT r(k−1)‖22

;

d(k) = HT r(k) + βkd(k−1) ;k = k + 1 ;

end

Algorithm 1: Conjugate-Gradient Least-Squares (CGLS) algorithm applied to thenormal equation .HTH/ f = HT e and produce an optimal solution in the least-squaresense

discussed in Ref. [37].

‖H fi − e‖22 + ‖fi‖22 ≤ ‖H fi+1 − e‖22 + ‖fi+1‖22 (20)

In the conjugate gradient algorithm, the iteration (i) is the regularization parameter. Itbegins with a strongly regularized solution; and then, it reduces the level of the regular-ization as the number of iterations increasing. Along the iteration, the objective function‖H fi − e‖22 + ‖fi‖22 will initially decrease until it hits the lowest value where the optimumsolution resides. Afterward, the value of the objective function will start to increase.

2.3. Mathematical model

Weconsider themodel inFigure 5where f (t)denotes the applied impact force,h(t)denotesthe impulse response function structure, e(t)denotes the structure actual response andn(t)denotes the measurement error. Thus, the measurement simply is a linear combination ofe(t) and n(t). We denote the measurement y(t).

Mathematically, the model can be written as

e(t) = h(t) ∗ f (t) (21)y(t) = h(t) ∗ f (t) + n(t) (22)

And, their frequency domain equivalence are:

E(ω) = H(ω) · F(ω) (23)Y(ω) = H(ω) · F(ω) + N(ω). (24)

The simplest approach to reconstruct is direct inverse filtering:

F(ω) = Y(ω)

H(ω)(25)

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 11

Substituting Equation (24) into Equation (25) yields

F(ω) = F(ω) + N(ω)

H(ω)(26)

And, as demonstrated earlier the spectraN(ω)/H(ω) is much higher than the spectra F(ω)

on the high frequencies region. Thus, F(ω) will be noise dominant.To avoid the issue, we seek the impact force that minimizes the error measure:

Mean Error = E{[

F(ω) − F(ω)]2}

, (27)

where E{·} is the expectation operator of the argument. The minimum condition gives

F =[

H(ω) · Sf (ω)

Sf (ω) · |H(ω)|2 + Sn(ω)

]E(ω)

=[

H(ω)

|H(ω)|2 + Sn(ω)/Sf (ω)

]E(ω)

=[

1H(ω)

· |H(ω)|2|H(ω)|2 + Sn(ω)/Sf (ω)

]E(ω)

= |H(ω)|2|H(ω)|2 + Sn(ω)/Sf (ω)

· E(ω)

H(ω). (28)

The variables Sn(ω) and Sf (ω) denote the power spectrum of the noise and the exactsolution; Sf (ω) = E|F(ω)|2 and Sn(ω) = E|N(ω)|2, where E denotes expectation. Theresult of Equation (28) is known as the Wiener filter.[38, Appendix 1]

The two power spectra are often unavailable. We approximate Sn(ω)/Sf (ω) with aconstant K ; thus, Equation (28) becomes

F(ω) = |H(ω)|2|H(ω)|2 + K

· E(ω)

H(ω). (29)

In this work, we use K as the regularization parameter and adjusting its value within therange of min[H(ω) · H(ω)] and H(0)2. The optimum value of K can be determined bymeans of the L-curve method.

3. Data analysis

3.1. Single degree of freedom system

3.1.1. Problem statementWeuse the case of the single degree of freedom system to evaluate and verify the current

method. We understand that the SDOF system response does not completely describe theimpact response of the continuous structure, but in a narrow frequency range, the SDOFsystemprovides a good approximation.[39] From themathematical stand point, the systemis rather ill-posed with the condition number around 1 × 105 for the current case.

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(a)

(c)

(e) (f)

(b)

(d)

Figure 6. TSVDmethod. (a) Over-regularized solution, (b) optimal solution, (c) under-regularized solutionand (d) solution versus residual norms.

The SDOF system consists of a massm, a spring k, and a damper ζ . The system impulseresponse function can be expressed as

h(t) = 1mωd

e−ζωnt sin (ωdt), (30)

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 13

(a)

(c)

(e)(f)

(b)

(d)

Figure 7. Tikhonov method. (a) Over-regularized solution, (b) estimation error, (c) optimal solution, (d)solution and residual norms, (e) under-regularized solution and (f) solution versus residual norms

where ωn and ωd denote the circular natural and circular damped frequencies. The bothparameters can be computed by ω2

n = k/m and ω2d = ω2

n(1 − ζ 2).

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(a)

(c)

(e)

(b)

(d)

(f)

Figure 8. Iterative regularization method. (a) Over-regularized solution, (b) estimation error, (c) optimalsolution, (d) solution and residual norms, (e) under-regularized solution and (f) solution versus residualnorms.

In the present study, the following data are selected: m = 1 kg, k = 1 × 1011 N/m,ζ = 0.2 and the final time of analysis is 100µs.

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 15

(a)

(c)

(e)

(b)

(d)

(e)

Figure 9.RegularizedWiener filtermethod. (a) Over-regularized solution, (b) estimation error, (c) optimalsolution, (d) solution and residual norms, (e) under-regularized solution and (f) solution vs. residualnorms.

The system is subjected to an impact load having the half-sine profile with a loadingduration of 10µs, and the system elastic response e(t) is computed. Then, a small amount

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(a)

(c)

(e)

(b)

(d)

Figure 10. Wiener filter factor for various K ; K∗ = 1.482 is optimal. (a) K = 1.1318 × 10−5, (b)K = 0.058003, (c) K = 0.44461, (d) K∗ = 1.482 and (e) K = 12.8805.Note: The y-scale is changed to log-scale to accentuate the pass-band.

of the normally distributed pseudo-random noise having the signal-to-noise ratio of 10is superimposed to the elastic response data. To demonstrate the ill-posed nature of the

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 17

Figure 11. A quarter finite element model mesh of the impact of a sphere and a beam.

Figure 12. The strain-time history at the point below the impact site on the beam. The beam is impactedby a sphere.

present problem, we firstly reconstruct the impact force without any regularization. Theresult, as shown in Figure 1(b), is a noisy and unacceptable impact force.

3.1.2. The reconstructed impact force of the SDOF systemFirst, we should note the method to select the value for each regularization parameter.Table 1 provides a description of the regularization parameters for all methods and theirimplication. For the TSVD method, the regularization parameter k is discrete and variesfrom 1 to the number of the singular value components n. For the Tikhonov method, theregularization parameter λ is continuous and varies between the smallest singular value σn

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(a)

(c)

(e)

(b)

(d)

Figure 13. The results of the application of the regularized Wiener filter to reconstruct the impactforce acting on the beam. (a) Optimal solution, (b) over-regularization Solution, (c) estimation error, (d)solution versus residual norms and (e) K = 1.1719.

and the largest σ1. For the CG iterative method, the regularization parameter i is discreteand varies from 1 to the maximum number of iterations, which is usually 20–30. We

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 19

Table 1. Regularization parameters and their effects.

Regularization parameterMethod Symbol Description Smoother solution

TSVD k The number of SVD components Decreasing kTikhonov λ A fudge factor Increasing λ

CG i The number of CG iterations Decreasing iWiener K A fudge factor Increasing K

should note that the CG iterative methodmay converge rapidly with a quadratic rate; thus,only very few iterations are required to reach the optimum solution. For the regularizedWiener filter method, the regularization parameter K is continuous and varies betweenthe minimum of H(ω) · H(ω) and H(0) · H(0).

Figures 6–9 show the results of the TSVD, Tikhonov, iterative regularization andregularized Wiener’s filter methods. For each method, we observe the characteristics ofthe solution evolution, the residual norm (‖f ∗ h− e‖2), the solution norm (‖f ‖2), and theestimation error.

By observing these empirical evidences, we deduce the following conclusions.The estimation error, as a function of the regularization parameters, changes dramat-

ically for the TSVD, Tikhonov and iterative regularization methods. For example, theestimation error of theTSVDmethod, see Figure 6(b), drops from100% to about 20%as theregularization parameter k increases from 1 to 20, and when the regularization parameteris further increased to 40, the estimation increases to 100%. The same phenomenon is alsoobserved on the estimation errors of themethods of Tikhonov and iterative regularization.This phenomenonmeans that one has to carefully select the optimal regularization param-eter. The solution can quickly lose its accuracy when the regularization parameter missesits optimum value even slightly. However, for the method of the regularized Wiener filter,the change of the estimation error is relatively mild particularly when the regularizationparameter is higher than a certain value of K .

The common behaviour observed in many solutionmethods of the ill-posed problem isas follow. The residual norm ‖f ∗ h− e‖2 increases, and the solution norm ‖f ‖2 decreases,by increasing the regularization. This can be observed on Figures 6–8, panel (d), forthe methods of TSVD, Tikhonov, and the iterative regularization method. However, thiscommon behaviour is not strongly visible for the regularized Wiener filter method. Infact, the residual norm is almost constant as the regularization parameter increases. Thesolution norm changes visibly only for very small K and short interval.

On panel (f) of each figure, we can observe the L-curve of eachmethod. For themethodsofTSVD,Tikhonov and the iterative regularization, the solutiondramatically changes fromone extreme, where the solution is too far from the data, to another extreme, where thesolution is strongly dominated by the noise. As for the regularizedWiener filtermethod, seeFigure 9(f), the change is significantly less dramatic. It means that the regularized Wienerfilter method provides a bigger window of opportunity, with respect to the variation of theregularization parameter, to obtain the accurate solution. However, one has to avoid anextremely small value for the regularization parameter K .

The accuracy, in the sense of the definition (12), for the case of SDOF system of thecurrentmethod and the previousmethods is summarized in Table 2. In this particular case,

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20 F. E. GUNAWAN

Table 2. The comparison of the accuracy of the regularized Wiener filter, TSVD, Tikhonov and CGmethods in predicting the peak of the force and the entire force profile.

Estimation error (%)Method Entire solution Peak of the force

TSVD 15.61 2.64Tikhonov 19.61 8.44CGLS 17.35 5.76Wiener Filter 22.57 8.76

Figure 14. The schematic diagram of the air gun impact system.

Figure 15. The strain-time history at the point below the impact site on the beam. The experiment isreplicated for 10 times at a fixed impactor velocity.

the regularizedWiener filtermethod has slightly higher estimation error for the entire loadhistory; however, as for the peak of the force, its solution accuracy is rather comparableto that of the Tikhonov method. The best solution is that given by the TSVD method,and followed by the CGLS method. Important to note that these characteristics are notgenerally applicable.

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 21

Figure 16. The optimal reconstructed impact force.

3.1.3. Characteristics of theWiener filter factorThe key factor of the use of the regularizedWiener filter to solve the ill-posed problem lieson the term |H(ω)|2

|H(ω)|2 + K. (31)

We call the termWiener filter factor, and the factor is plotted in Figure 10 for various valuesof the regularization parameter K . Specifically, we compare the Wiener filter factor withK = 1.482 and the finite impulse response (FIR) low-pass filter with Hamming windowhaving the cut-off frequency of 100 kHz and the order of 40 on Figure (10)(d).

We found that the Wiener filter factor is very much alike with the low-pass digitalfilter with a number of excellent characteristics from the digital filter perspective. TheWiener filter does not have ripple in the stop-band frequency. The Wiener filter does nothave ripple in the pass-band frequency. TheWiener filter has flat pass-band and almost flatstop-band. TheWiener filter does not shift the filtered signal; theHamming filter has linearphase shift. Thus, the Wiener filter is much more suitable for solving ill-posed problems.The Wiener filter has fast roll-off, but slightly slower than that of the Hamming filter.

3.2. Impact of a solid sphere onto a beam: finite element simulation

3.2.1. Problem statementIn this section, we evaluate the regularized Wiener filter method to reconstruct the forceoccurring on the lateral impact of a beam and a solid sphere. The impact event is shown inFigure 11. This case is clearly more realistic than the previous case. The present case takesinto account the non-linearity due to the contact between the two bodies. Figure 11 showsa quarter finite element model mesh of the case. The reader is recommended to consultRef. [40] regarding the finite element method.

The sphere and beam are assumed to be made of the same linear elastic material. Thematerial is steel with a Young modulus of 210GPa, a Poisson ratio of 0.3, and a density of7.8×10−6 kg=mm3. The beam is 350mm long, 25mmwide, and 15mm thick. The sphereis 18mm diametrical and hits the beam at the velocity of 10mm/ms.

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22 F. E. GUNAWAN

The impulse response function for the elastic strain at the bottom of the impact site inthe longitudinal direction is approximated by [1]:

h(t) =N∑i=1

ai sin(ωit

), (in 1/kN) (32)

where N = 4, ωi = 2π fi, fi = 1.0692, 5.3723, 7.4105, 11.962 kHz, and ai = 4.3467 ×10−6, 4.2514 × 10−6, 4.0394 × 10−7, 4.3723 × 10−6.

3.2.2. The reconstructed impact forceThe beam in the case described above has no damping and unconstrained. Thus, after theimpact, the beam oscillates harmonically; see Figure 12.

We should note that the regularizedWiener filter relies on the use of the Fourier integral.The integral requires that the signal y(t) to be integrated should satisfy the followingrequirement [41]: ∫ +∞

−∞|y(t)| dt < ∞. (33)

To enforce this condition, we multiply the both sides of Equation (4) with the exponentialwindow function of w(t) = exp ( − 10t). By applying the window function, the harmonicelastic response becomes a damped harmonic function, which is very similar to the casewhere the structure has a viscous damping.

The impact force reconstructed by the regularized Wiener filter is shown in Figure13(a) as a solid line. In the figure, we also plot the impact force computed by the finiteelement method as a broken line. In Figure 13(b), we plot the over-regularized solutionthat obtained at K = 12.9945, while the optimum regularization value is K = 1.1719.Figure 13(c) shows the change of the estimation error as a function of the regularizationparameter and panel (d) shows the plot of the solution norm versus the residual norm.

The reconstructed impact force is reasonably accurate but slightly underestimates theactual force by about 15.72% in the peak. The reconstructed force is followed by anoscillation that damped down within a short duration. As for the estimation error, weobserve that its behaviour is rather similar to that in the previous case. The change ofthe estimation error is not drastic. The trade-off curve, the curve of the solution normversus the residual norm, is also well established where the optimum solution can easilybe selected.

3.3. Impact of a solid sphere onto a beam: experimental approach

In this section, we evaluate the accuracy of the regularized Wiener filter method forreconstructing the impact force using data recorded from an impact experiment.

3.3.1. Experimental setupThe experimental set-up is schematically shown in Figure 14. The experimental apparatusconsists of a gas tank, a reduction valve, a reservoir, a solenoid valve, a barrel and aprojectile. The experiment uses helium gas to drive the projectile. Initially, the gas is storedin the gas tank, and then, is released to the reservoir. The use of the large reservoir is tostabilize the gas and to obtain a greater control on the pressure level of the gas. The solenoid

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 23

valve is used to regulate the flow of the helium gas from the reservoir into the barrel. Thebarrel has a length of 1m and an inner diameter of 20mm.

The impactor is a steel sphere with a diameter of 18mm. The specimen size is 350 ×25× 15 in mm. Both are made of steel. The impactor velocity is measured and maintainedduring the test by controlling the gas pressure. A laser sensor is used to measure theimpactor velocity. The impact is repeated for 10 times at a constant gas pressure. Duringthe experiment, the sphere is directed to impact the centre of the specimen and thestrain response is measured at the other side; see Figure 11 for the impact site and themeasurement point. The flexural strain is measured using a strain gage.

The strain gage is connected to a Wheatstone bridge with 1/4-bridge configuration.Then, the bridge is connected to a signal conditioner and then, to a transient memory.The signal conditioner is that of CDV/CDA-700A manufactured by Kyowa. The transientmemory is TDS-700. The sampling time is 0.2µs and about 7000 data are recorded fromeach experiment. Thus, the data are recorded for about 1.4ms. The contact durationbetween the sphere and the specimen is less than 0.1ms according a finite elementsimulation. The finite element simulation is also used to verify the reconstructed impactforce as a direct measurement is difficult.

As the experiment results, Figure 15 shows the strain-time history for the repeatedimpact experiments. There is reasonable variation in the peak of the strain time historydespite the fact that the impact velocity is controlled. The first peaks in the strain-timehistory vary by about 80με between the highest and the lowest strains. For the inverseanalysis, we use the lowest, highest and averaged strains.

3.3.2. The reconstructed impact forceUsing the regularized Wiener filter method, we reconstruct the impact force on the basisof the strain data given in Figure 15. Figure 16 shows the three reconstructed impact forcesand the impact force calculated by the finite element method. The force labelled with‘MinimumStrain’ is that reconstructed using the lowest strain-time data. All reconstructedimpact forces underestimate the actual impact force. We assume that the finite-elementimpact force is the actual impact force. In their peaks, these reconstructed impact forcesare lower than the actual one by 47% for the minimum strain, 32% for the averaged strainand 22% for the maximum strain.

It is very difficult to identify the source of the discrepancy between the FEM impactforce and the reconstructed impact force. One potential source, we speculate, is that theimpact velocity is not exactly the same during the replication. In addition, the impact sites,according to the mark produced during the impact, are often slightly off the designedlocation. It is extremely difficult to maintain the impact site to be exactly at the centre ofthe specimen. Meanwhile, in the finite element simulation, the impact site is exactly at thespecimen centre. When the impact site is off the centre, we expect the induced strain willbe lower and consequently, the strain data produce a lower reconstructed impact force.

4. Conclusions

The inverse reconstruction of the impact force is important when the required forcedata are difficult to be directly measured. However, the reconstruction is an ill-posedproblem. In this work, we propose the regularized Wiener filter method, which is based

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on the non-causal Wiener filtering theory. The method utilizes the Wiener filter factor topatch the singular spectra across all frequencies in the domain. Using three examples, andcomparison with some popular methods, we find the regularized Wiener filter methodis better in the following aspects. The sensitivity of the solution accuracy is less in theregularized Wiener filter method and is more in the previous methods. In the Wienerfilter method, the estimation error increases slowly when the regularization parameter isbigger than its optimal value. The Wiener filter method is computationally efficient andhas potential for direct practical applications. Thus, this work has provided an opportunityto perform the impact force reconstruction in the field.

Disclosure statement

No potential conflict of interest was reported by the authors.

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Appendix 1. Derivation of Equation (28)Equation (24) gives us

Y(ω) = H(ω) · F(ω) + N(ω), (A1)

and we like to estimate F(ω) from Y(ω) such that the mean of the square error is minimum. Weexpress the estimate as

F(ω) = G(ω) · Y(ω), (A2)

and seek for G(ω) to provide the minimum mean-square error solution F(ω). The mean squareerror is written

Mean Error (ME) = E{[F(ω) − F(ω)]2

}. (A3)

Subsequently, we substitute Equation (A2) and then Equation (A1) into Equation (A3). We obtain:

ME = E {F(ω) − G(ω)[H(ω)F(ω) + N(ω)]} ,= [1 − G(ω)H(ω)][1 − G(ω)H(ω)]E {

F2(ω)}

− [1 − G(ω)H(ω)]G(ω)E{F(ω)N(ω)

}− G(ω)[1 − G(ω)H(ω)]E {

N(ω)F(ω)}

+ G(ω)G(ω)E{N2(ω)

},

= [1 − G(ω)H(ω)][1 − G(ω)H(ω)]Sf (ω)

+ G(ω)G(ω)Sn(ω).

We define E{F2(ω)

} = Sf (ω) and E{N2(ω)

} = Sn(ω), and assume E{F(ω)N(ω)

} = 0 andE

{N(ω)F(ω)

} = 0. Taking ∂ME/∂G(ω) = 0, we obtain

0 = G(ω)Sn(ω) − H(ω)[1 − G(ω)H(ω)]Sf (ω)

G(ω) = |H(ω)|2|H(ω)|2 + Sn(ω)/Sf (ω)

1H(ω)

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