Ultrasonics 64 (2016) 25–42

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Ultrasonics

journal homepage: www.elsevier .com/locate /ul t ras

A quantitative method for evaluating numerical simulation accuracy oftime-transient Lamb wave propagation with its applications to selectingappropriate element size and time step

http://dx.doi.org/10.1016/j.ultras.2015.07.0070041-624X/� 2015 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: State Key Laboratory for Manufacturing SystemEngineering, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an710049, China.

E-mail addresses: wx72420@163.com (X. Wan), xugh@mail.xjtu.edu.cn (G. Xu),zhangq@mail.xjtu.edu.cn (Q. Zhang), meptse@cityu.edu.hk (P.W. Tse), xy_thh@163.com (H. Tan).

Xiang Wan a,c, Guanghua Xu a,b,⇑, Qing Zhang a, Peter W. Tse c, Haihui Tan a

a School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, Chinab State Key Laboratory for Manufacturing System Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, Chinac Department of Systems Engineering and Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China

a r t i c l e i n f o

Article history:Received 29 January 2015Received in revised form 29 May 2015Accepted 13 July 2015Available online 26 July 2015

Keywords:Cross correlation analysisGVE (group velocity error)MACCC (maximum absolute value of crosscorrelation coefficient)Simulation accuracy evaluationQuantitative method

a b s t r a c t

Lamb wave technique has been widely used in non-destructive evaluation (NDE) and structural healthmonitoring (SHM). However, due to the multi-mode characteristics and dispersive nature, Lamb wavepropagation behavior is much more complex than that of bulk waves. Numerous numerical simula-tions on Lamb wave propagation have been conducted to study its physical principles. However,few quantitative studies on evaluating the accuracy of these numerical simulations were reported.In this paper, a method based on cross correlation analysis for quantitatively evaluating the simula-tion accuracy of time-transient Lamb waves propagation is proposed. Two kinds of error, affecting theposition and shape accuracies are firstly identified. Consequently, two quantitative indices, i.e., theGVE (group velocity error) and MACCC (maximum absolute value of cross correlation coefficient)derived from cross correlation analysis between a simulated signal and a reference waveform, are pro-posed to assess the position and shape errors of the simulated signal. In this way, the simulationaccuracy on the position and shape is quantitatively evaluated. In order to apply this proposedmethod to select appropriate element size and time step, a specialized 2D-FEM program combinedwith the proposed method is developed. Then, the proper element size considering different elementtypes and time step considering different time integration schemes are selected. These results provedthat the proposed method is feasible and effective, and can be used as an efficient tool for quantita-tively evaluating and verifying the simulation accuracy of time-transient Lamb wave propagation.

� 2015 Elsevier B.V. All rights reserved.

1. Introduction

Compared with point-to-point inspection using traditionalultrasonic bulk waves, Lamb wave technique provides acost-effective inspection method in non-destructive evaluation(NDE) and structural health monitoring (SHM), as Lamb wave isable to travel a very long distance with little energy loss and canbe used to interrogate physically inaccessible areas of structuresand components. However, due to the multi-mode characteristicsand dispersive nature, Lamb wave propagation behavior is much

more complex than that of bulk waves. Understanding physicalprinciples of Lamb wave propagation is indispensable for fullyexploiting the advantages of Lamb wave technique and thus bene-ficial to its applications in NDE and SHM.

Analytical approaches [1–4] have been used to resolve Lambwaves propagation problems. In these references, explicit expres-sions, known as the Rayleigh–Lamb frequency equations, werederived and dispersion curves, a fundamental way of describingLamb wave propagation in a specified structure, could be easilyplotted based on these equations. Giurgiutiu [5,6] published aclosed equation of time-transient Lamb wave displacementresponse under the excitation of piezoelectric wafer active sensor(PWAS). Although analytical approaches are precise, they are onlyapplicable to simple and regular structures.

Besides analytical approaches, a great number of numericalsimulation methods [7–10] have been employed to study theLamb wave propagation. Willberg et al. [10] reviewed the

26 X. Wan et al. / Ultrasonics 64 (2016) 25–42

state-of-the-art numerical simulation methods used in guidedwave-based SHM. These approaches include finite element analy-sis [11–24], finite difference equations [25–27], finite strip element[28,29], boundary element method [30,31,56], semi-analyticalfinite element analysis [32–40], global matrix methods [41],spectral element approaches [42–48], mass-spring latticemodels [49,50], the local interaction simulation approach (LISA)[51–53], finite cell method [54], and the spectral cell method[55]. The use of these numerical methods providing diverse solu-tions to time-transient Lamb wave response under certain condi-tions, is beneficial to understanding the fundamental physicalprinciples of Lamb wave propagation, and paves the way forexpanding the applications of Lamb wave technique in variousindustries.

A successful numerical solution to a practical problem shouldbe composed of not only numerical method itself but also mea-surement strategy for the evaluation of simulation accuracy.However, according to our literature review, considerable atten-tion has been focused on the simulation using different numericalmethods, whereas accuracy evaluation has received little atten-tion. Some authors [17] just pointed out that their simulationresults were comparable with previous published studies; someauthors [19,40,43] only evaluated the simulation accuracy bychecking the time of flight of the simulated wave packets visually.And in most of the numerical simulation research work, the sim-ulation accuracy of time-transient Lamb wave propagation isevaluated as follows. First, simulation waveform is superimposedwith the corresponding analytical waveform [16,18,20,33,34] orexperimental signals [8,24,27] in one figure. Then the simulationaccuracy is qualitatively verified by visually checking and com-paring the difference between them. At present, few quantitativeanalysis on evaluating simulation accuracy has been reported.Willberg et al. [44] and Duczek et al. [48] developed and com-pared higher order finite element schemes for simulating Lambwave propagation and presented a convergence indicator toquantify numerical accuracy and performance. In their studies,the convergence indicator is computed from Hilbert transformand is actually used for quantifying the group velocity accuracy(namely the position accuracy in this paper). However, the shapeaccuracy was not considered in their investigations. In this paper,a quantitative method based on cross correlation analysis is pro-posed. Cross correlation analysis is used to measure the similaritybetween two signals at different times. Although cross correlationanalysis has been widely used in many fields [57–59], it is thefirst time to be applied to quantitatively study the simulationaccuracy evaluation of time-transient Lamb waves propagation.In the previous studies [60,61], wavelet analysis has been usedto compute the group velocity. However, it is quite complicatedto evaluate the similarity between two signals using waveletanalysis. Therefore, compared with wavelet analysis, our pro-posed method using cross correlation analysis is quite efficientand easy to implement, as two indicators to evaluate numericalsimulation accuracies including the position accuracy and shapeaccuracy can be calculated simultaneously.

Two kinds of error, influencing the position and shape accura-cies of simulated waveform are firstly identified. Consequently,two quantitative indices, i.e., the GVE (group velocity error) andMACCC (maximum absolute value of cross correlation coefficient)derived from cross correlation analysis between simulated signaland reference waveform are thus proposed to assess the positionand shape errors of the simulated signal. As the proposed methodis based on cross correlation analysis, a reference waveform or sig-nal derived from the corresponding analytical or experimentalstudy is a prerequisite.

Among the above numerical approaches, FEM is the most pop-ular and widely used technique, as it is very straightforward,

easy-to-learn and convenient to select general simulation platform(e.g. commercial FEM software). However, in order to achieve accu-rate FEM simulation, selecting appropriate parameters of elementsize and time step is of great importance. In this paper, the pro-posed quantitative simulation accuracy evaluation method isapplied to selecting proper element size regarding different ele-ment types and time step regarding different time integrationschemes.

Up to now, there is no FEM software offering simulation accu-racy evaluation function, let alone quantitative accuracy evaluationfunction. In this paper, an innovative FEM program integratingsimulation accuracy quantitative evaluation function for 2Dtime-transient Lamb wave propagation is developed. This softwareis designed to provide a platform for applying the proposed quan-titative simulation accuracy evaluation method to selecting properparameters of element size and time step.

Regarding the selection of the parameter of element size, var-ious researchers have their own choices. Alleyne and Cawley [62]reported that using quadrilateral elements, substantially morethan the threshold of 8 elements per wavelength is a good limitfor accurate modeling of wave propagation problem; Xu [23] usedan element size corresponding to 10 elements per wavelength;Moser [20], Gresil [18], Shen and Giurgiutiu [63] and Wan [64]employed an element size equal to 1/20 of the shortest wave-length of interest. In these studies, the authors just used one kindof element type, and did not quantitatively verify the simulationaccuracy. In general, accurate numerical simulation can beachieved by reducing element size. However, small element sizeresults in much computation time. Therefore, selecting an appro-priate element size is still an open problem. Recently, Willberget al. [44] and Duczek et al. [48] studied the optimal element sizefor high-order finite element schemes. In this paper, the depen-dence of GVE and MACCC on element size considering differentelement types is studied and only low-order element types areconsidered. An appropriate element size regarding element typesis selected when both the position and shape accuracies reach ahigh level.

Just like the parameter of element size, the selection of anappropriate time step is also still worth discussing. Bathe [65] pro-posed that the maximum time step satisfying stability for an expli-cit time integration method is given by DI/c, where DI is theelement size and c refers to the wave speed of the fastest wavemode. Moser [20], Gresil [18], Shen and Giurgiutiu [63] and Wan[64] used the expression Dt = 1/(20 ⁄ fmax) to select time step,where Dt is the time step and fmax refers to the highest frequencyof Lamb wave mode. In these studies, the authors just used onekind of time integration scheme, and did not quantitatively vali-date the simulation accuracy. In this paper, the dependence ofGVE and MACCC on time step regarding different time integrationschemes is studied. An appropriate time step regarding time inte-gration schemes is selected when both the position and shapeaccuracies approach a high level.

The rest of this paper is organized as follows. Section 2introduces the basic theories including cross correlationanalysis, analytical model of dispersion curves of Lamb waves,theoretical analysis of time-transient Lamb wave propagationunder the excitation of PWAS, and finite element method fortime-transient Lamb propagation simulation. In Section 3, aquantitative method for evaluating the simulation accuracy isproposed and an innovative software providing both finite ele-ment simulation function and accuracy quantitative evaluationfunction for 2D time-transient Lamb wave propagation is devel-oped. In Section 4, the applications to selecting proper elementsize regarding different element types and time step regardingdifferent time integration schemes are presented. Conclusionsare drawn in section 5.

-a a

y=d

y=-d

2d x

y

τ

Fig. 1. Distributed shear stress introduced to the upper surface of the plate.

X. Wan et al. / Ultrasonics 64 (2016) 25–42 27

2. Methodology

2.1. Cross correlation analysis

Cross correlation analysis is used to measure the similarity oftwo waveforms as a function of a time-lag applied to one of them.Suppose two temporal waveforms are denoted by x(t) and y(t), thecross correlation function is expressed in Eq. (1) [57,58]

RxyðsÞ ¼1T

Z T

0xðtÞyðt þ sÞdt; ð1Þ

where s represents a time-lag applied to y(t); T is the observationtime of x(t) and y(t); the correlation function Rxy(s) is a functionof s.

In practical applications, it is better to normalize cross correla-tion function for two time series using Eq. (2) [57,58]

RxyðsÞ ¼RxyðsÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Rxxð0ÞRyyð0Þp : ð2Þ

The normalized quantity RxyðsÞ will vary between �1 and 1. A valueof RxyðsÞ ¼ 1 indicates that at the alignment s, the two temporalwaveforms have the identical shape; while a value RxyðsÞ ¼ �1 rep-resents that they have the same shape except that they have oppo-site signs; and a value RxyðsÞ ¼ 0 illustrates that they are completelyuncorrelated. In practice, the normalized quantity RxyðsÞ is usuallyreferred to ‘‘cross correlation coefficient’’ and an absolute value ofthe cross correlation coefficient greater than 0.9 indicates a prettygood similarity.

After calculating the normalized cross correlation function, themaximum absolute value of cross correlation coefficient (MACCC)indicating best similarity of two signals can be derived and thealignment s0, at which the MACCC is achieved, is the time delaybetween these two waveforms. And the time delay s0 is used tocalculate the GVE.

2.2. Analytical analysis of basic theory of Lamb wave

In this section, we will simply present the analytical analysis ofdispersion curves [3,4] of Lamb wave and the theoretical analysisof time-transient Lamb wave propagation [5,6] under PWASexcitation.

2.2.1. Analytical analysis of dispersion curves of Lamb wavesIn a traction-free, homogeneous and isotropic plate, the

homogenous solution can be derived by applying the stress-freeboundary conditions at the upper and lower surfaces of thestructure. There are two groups of waves, symmetric andanti-symmetric, in which the normal displacement of the particlesis symmetric or anti-symmetric with respect to the median planeof the plate, satisfying the Navier wave governing equation andthe boundary conditions. Equations known as Rayleigh–Lamb fre-quency relations for both symmetric and anti-symmetric wavesare expressed as [3,4]

tan qdtan pd

¼ � 4pqk2

ðq2 � k2Þ2 ; ð3Þ

tan qdtan pd

¼ �ðq2 � k2Þ

2

4pqk2 ; ð4Þ

where 2d is the thickness of a plate, k the wave number, x theangular frequency, and cL and cT the longitudinal and transversewave velocity, respectively; p and q are described by the followingequations [3,4]

p2 ¼ x2

c2L

� k2; q2 ¼ x2

c2T

� k2: ð5Þ

Dispersion curves including phase velocity and group velocity dis-persion curves, can be plotted from the calculated results of Eqs.(3) and (4). Dispersion curve is used to determine the velocity (orvelocities) at which a wave of a particular frequency will propagatewithin the plate. Theoretical value of phase velocity or group veloc-ity at any particular frequency of any specific mode can be obtainedfrom dispersion curves.

2.2.2. Analytical analysis of time-transient Lamb waves propagationunder PWAS excitation in plates

In recent years, a lot of researchers [5,6,66–73] have appliedPWAS transducers to the generation and detection of Lamb waveas a result of its numerous advantages over other transducers.PWAS is an inexpensive, nonintrusive, un-obtrusive, and minimallyinvasive device that can be surface-mounted on existing structuresor inserted between the layers of lap joints [68]. In Refs. [5,6],Giurgiutiu published a closed form of time-transient Lamb wavedisplacement response under PWAS excitation. In this section,we just give a brief introduction to the analytical results. Fordetailed descriptions, please refer to Refs. [5,6,74].

Suppose a PWAS with a width of 2a is applied to the uppersurface of a plate with a thickness of 2d, and a distributed shearforce is assumed to be applied to the area covered by the PWASin the upper surface of the plate. The PWAS is under a harmonicloading, and the shear stress introduced to the upper surface ofthe plate can be expressed as s = s0(x)eixt, where s0(x) is the dis-tribution function of the shear stress. The harmonic distributedshear force on the upper surface of the plate is illustrated inFig. 1 [74].

The distributed shear stress can be split into the symmetric andanti-symmetric components as shown in Fig. 2 [5]. A pair ofself-equilibrating shear stress, having the same amplitude s/2,with one shear stress pointing to the positive x axis direction,and the other pointing to the negative x axis direction, is intro-duced to the lower surface of the plate. The shear stress on theupper surface is divided into two equal parts, with the same ampli-tudes s/2 and the same directions of the positive x axis.

The integral form of closed time-transient response to a PWASexcitation on the upper surface of a plate is derived by combiningthe symmetric and anti-symmetric response together, as expressedin Eq. (6) [5]

uxðx; tÞjy¼d ¼1

2p�12l

Z 1

�1

1k

~saðkÞNsðkÞDsðkÞ

þ ~saðkÞNAðkÞDAðkÞ

� �eiðkx�xtÞdk;

ð6Þ

where l, k and x refers to the Lame constant, wavenumber andangular frequency, respectively; ~saðkÞ, Ns(k), Ds(k), NA(k) and DA(k)are described in the following equations [5]

-a a

y=d

y=-d

2d x

y

τS =τ/2

τS =τ/2-a a

y=d

y=-d

2d x

y

τA =τ/2

τA =τ/2

(a) (b)

Fig. 2. Distributed shear stress on the upper surface of the plate split into: (a) symmetric loading and (b) anti-symmetric loading.

28 X. Wan et al. / Ultrasonics 64 (2016) 25–42

~saðkÞ ¼Z 1

�1s0ðxÞ½Hðxþ aÞ � Hðx� aÞ�e�ikxdx; ð7Þ

NsðkÞ ¼ kqðk2 þ q2Þ cos pd cos qd; ð8Þ

DsðkÞ ¼ ðk2 � q2Þ2

cos pd sin qdþ 4k2pq sin pd cos qd; ð9Þ

NAðkÞ ¼ kqðk2 þ q2Þ sin pd sin qd; ð10Þ

DAðkÞ ¼ ðk2 � q2Þ2

sin pd cos qdþ 4k2 cos pd sin qd; ð11Þ

where H(x) is the Heaviside step function, and p and q are expressedin Eq. (5).

The functions Ns(k), Ds(k), NA(k) and DA(k) are also dependent onx though not explicitly illustrated. The integral of Eq. (6) issingular at the roots of Ds(k) and DA, which are the symmetricand anti-symmetric eigenvalues of the Rayleigh–Lamb equation,

i.e., kS0; k

S1; k

S2; . . ., and k A

0 ; kA1 ; k

A2 ; . . .. The number of eigenvalues that

exist for a given x will vary according to different frequencies. Theintegral of Eq. (6) is calculated by the Cauchy’s residue theorem,using a contour consisting of a semicircle in the upper half of thecomplex k plane and the real axis as shown in Fig. 3 [5]. Onlypositive wavenumbers are included.

For ideal bonding between PWAS and the structure, Eq. (7) canbe written as Eq. (12) [5]. Thus, after using the Cauchy’s residuetheorem, Eq. (6) becomes Eq. (13) [5]

~saðkÞ ¼ as0ð�2i sin kaÞ; ð12Þ

Fig. 3. Contour for calculating the integral form of complete response to a PWASexcitation on the top surface of a plate.

uxðx; tÞjy¼d ¼ �as0

lX

kS

sin kSa

kS

NsðkSÞD0sðk

SÞeiðkSx�xtÞ

� as0

lX

k A

sin k Aa

k A

NAðk AÞD0Aðk

AÞeiðk Ax�xtÞ: ð13Þ

Using Eq. (13), theoretical waveform of time-transient responseunder a PWAS excitation can be obtained.

2.3. Finite element method for time-transient Lamb wave propagation

2.3.1. Time-transient Lamb wave propagation simulation using finiteelement method

Time-transient Lamb wave propagation is a dynamic problem.Basic process for dynamic finite element method is elaborated inthe following steps.

(a) Preprocessing: inputting necessary data, e.g., geometry,material properties and boundary conditions.

(b) Discretization: different element types can be used to dividethe simulating domain into a mesh of finite elements.

(c) Calculating element matrices and force vector: Eqs. (14–17)[75] are used to calculate element mass matrix Me, elementstiffness matrix Ke, element damping matrix Ce and elementvector of external force Fe respectively for each element.

Z

Me ¼

Ve

qNT NdV ; ð14Þ

Z

Ke ¼

Ve

BT DBdV ; ð15Þ

Z

Ce ¼

Ve

lNT NdV ; ð16Þ

Z Z

Fe ¼

Se

NT TdSþVe

NT fdV ; ð17Þ

where Ve is element domain, Se element boundary for naturalconditions, q the density and l the Lame constant; N, B, D, Tand f refers to shape function matrix, strain displacementmatrix, constitutive matrix, surface traction and externalbody force, respectively; and the superscript T denotes matrixtranspose.

(d) Assembling: global matrices and force vector are obtained

by assembling the corresponding element matrices and ele-ment force vector together using Eq. (18) whereby equationof dynamic equilibrium in matrix form is given in Eq. (19).

X. Wan et al. / Ultrasonics 64 (2016) 25–42 29

X X X X

Table 1Shape functions of 4 types of element.

M ¼e

Me;K ¼e

Ke;C ¼e

Ce; F ¼e

Fe; ð18Þ

Shape function at node i of the element Element type

M€uþ C _uþ Ku ¼ F; ð19Þ

N1 ¼ nN2 ¼ gN3 ¼ 1� n� g

where u, _u and €u are the displacement, velocity and accelera-

tion vectors. In this study, damping is not considered.(e) Solving the dynamic equilibrium equation: various time

N1 ¼ 0:25ð1� nÞð1� gÞN2 ¼ 0:25ð1þ nÞð1� gÞN3 ¼ 0:25ð1þ nÞð1þ gÞN4 ¼ 0:25ð1� nÞð1þ gÞ

N1 ¼ ð2n� 1Þn N2 ¼ ð2g� 1ÞgN3 ¼ ð2ð1� n� gÞ � 1Þð1� n� gÞN4 ¼ 4ng N5 ¼ 4gð1� n� gÞN6 ¼ 4nð1� n� gÞ

N1 ¼ �ð1� nÞð1� gÞð1þ nþ gÞ=4N2 ¼ ð1þ nÞð1� gÞðn� g� 1Þ=4N3 ¼ ð1þ nÞð1þ gÞðnþ g� 1Þ=4N4 ¼ ð1� nÞð1þ gÞðg� n� 1Þ=4N5 ¼ ð1� n2Þð1� gÞ=2 N6 ¼ ð1þ nÞð1� g2Þ=2N7 ¼ ð1� n2Þð1þ gÞ=2 N8 ¼ ð1� nÞð1� g2Þ=2

integration schemes can be used.(f) Postprocessing: listing or graphically displaying the

expected solutions.

2.3.2. Element typesDuring the application of the proposed method to element size

selection, 4 types of element, i.e., three-node triangular element(T3), four-node rectangular element (Q4), six-node triangular ele-ment (T6) and eight-node rectangular element (Q8) illustrated inFig. 4, are considered. In finite element method, the purpose is tofind the field variables at nodal points through rigorous analysis,assuming that at any point within the element basic variable is afunction of values at nodal points of the element. The functionrelating the field variable at any point inside the element to thefield variables of nodal points is referred to shape function. Takingdisplacement as the field variable, the relationship is described inEqs. (20) and (21) [75]

u ¼Xm

i¼1

Niui; ð20Þ

v ¼Xm

i¼1

Niv i; ð21Þ

where u and v denote horizontal and vertical displacement vectors;ui and vi are horizontal and vector displacements at node i; Ni is theshape function at node i and m the number of the nodes of theelement.

The shape function responds to each type of element is illus-trated in Table 1 [76].

2.3.3. Time integration schemesThe dynamic equilibrium Eq. (19) is usually solved using time

integration schemes, which can be categorized into explicit andimplicit methods. Explicit methods use the differential equationat current time t to predict a solution at time t + Dt and the solu-tion of a set of linear equations is not involved at each time step;while implicit methods attempt to satisfy the differential equationat time t after the solution at time t � Dt is found and the solutionof a set of linear equations at each time step is required. The centraldifference method is one of the most frequently used explicitmethods and it is conditionally stable. The Newmark method, theHoubolt method and the Wilson-h method belong toimplicit approaches. When the parameters a and b satisfying

b P 0:5 and a P 0:25ð0:5þ bÞ2, the Newmark method is uncondi-tionally stable; the Houbolt method is unconditionally stable;when the parameter h satisfying h P 1:37, the Wilson-h methodis unconditionally stable [77]. In this paper, a = b = 0.5 is selected

T3 Q4

Fig. 4. 4 types o

in the Newmark method and h = 1.4 is used in the Wilson-h method.When applying the proposed method to time step selection, these 4time integration schemes are taken into consideration.

2.3.3.1. The central difference method. The central differencemethod is summarized in the following steps [75]

Step 1: Initialize the initial displacement u0, and velocity _u0.Step 2: Solve for the initial acceleration €u0 from Eq. (22)

T

f element.

€u0 ¼ M�1ðF0 � C _u0 � Ku0Þ: ð22Þ

Step 3: Select the time step Dt.Step 4: Compute the fictitious displacement at time Dt using

Eq. (23)

u�Dt ¼ u0 � ðDtÞ _u0 þDt2

2€u0: ð23Þ

Step 5: Calculate the effective mass matrix by equation (24)

~M ¼ 1Dt2 M þ 1

2DtC: ð24Þ

6 Q8

30 X. Wan et al. / Ultrasonics 64 (2016) 25–42

Step 6: Repeat from step 7 to step 9 for each time step.Step 7: Compute the effective force vector through Eq. (25)

� � � �

~Ft ¼ Ft � K � 2

Dt2 M ut �1

Dt2 M � 12Dt

C ut�Dt:

ð25Þ

Step 8: Find the displacement at time t + Dt from Eq. (26)

utþDt ¼ ~M�1~Ft: ð26Þ

Step 9: If required, derive the acceleration and the velocity attime t from Eqs. (27) and (28)

€ut ¼1

Dt2 ðutþDt � 2ut þ ut�DtÞ: ð27Þ

_ut ¼1

2DtðutþDt � ut�DtÞ: ð28Þ

2.3.3.2. The Newmark method. The Newmark method is expressedin the following steps [65]

Step 1: Initialize the initial displacement u0, and velocity _u0.Step 2: Solve for the initial acceleration €u0 employing Eq. (22).Step 3: Select the time step Dt, parameters a and b.Step 4: Compute the effective stiffness matrix ~K using Eq. (29)

~K ¼ K þ 1aDt2 M þ b

aDtC: ð29Þ

Step 5: Repeat from step 6 to step 8 for each time step.Step 6: Calculate the effective force vector at time t + Dt through

Eq. (30)

� �� �

~FtþDt ¼ FtþDt þ

1aDt2 ut þ

1aDt

_ut þ1

2a� 1 €ut M

þ baDt

ut þba� 1

� �_ut þ

Dt2

ba� 2

� �€ut

� �C: ð30Þ

Step 7: Find the displacement at time t + Dt from Eq. (31)

utþDt ¼ eK�1eF tþDt: ð31Þ

Step 8: Derive the acceleration and the velocity at time t + Dtfrom equations (32) and (33)

€utþDt ¼1

aDt2 ðutþDt � utÞ �1

aDt_ut �

12a� 1

� �€ut : ð32Þ

_utþDt ¼ _ut þ ð1� bÞðDtÞ€ut þ bðDtÞ€utþDt: ð33Þ

2.3.3.3. The Houbolt method. The Houbolt approach is elaboratedin the following steps [65]

Step 1: Initialize the initial displacement u0, and velocity _u0.Step 2: Solve for the initial acceleration €u0 employing Eq. (22).Step 3: Select the time step Dt.Step 4: Use a special starting procedure to compute uDt and u2Dt.Step 5: Compute the effective stiffness matrix ~K using Eq. (34)

~K ¼ K þ 2Dt2 M þ 11

6DtC: ð34Þ

Step 6: Repeat from step 7 to step 9 for each time step.

Step 7: Calculate the effective force vector at time t + Dt throughEq. (35)

� �

eF tþDt ¼ FtþDt þ5

Dt2 ut �4

Dt2 ut�Dt þ1

Dt2 ut�2Dt M

þ 12Dt

ut �3

2Dtut�Dt þ

13Dt

ut�2Dt

� �C: ð35Þ

Step 8: Find the displacement at time t + Dt from Eq. (31).Step 9: If required, derive the acceleration and the velocity at

time t + Dt from Eqs. (36) and (37)

€utþDt ¼1

Dt2 ð2utþDt � 5ut þ 4ut�Dt � ut�2DtÞ: ð36Þ

_utþDt ¼1

6Dtð11utþDt � 18ut þ 9ut�Dt � 2ut�2DtÞ: ð37Þ

2.3.3.4. The Wilson-h method. The Wilson-h method is illustrated inthe following steps [65]

Step 1: Initialize the initial displacement u0, and velocity _u0.Step 2: Solve for the initial acceleration €u0 employing Eq. (22).Step 3: Select the time step Dt and set h = 1.4.

Step 4: Compute the effective stiffness matrix eK using Eq. (38)

eK ¼ K þ 6h2Dt2

M þ 3hDt

C: ð38Þ

Step 5: Repeat from step 6 to step 8 for each time step.Step 6: Calculate the effective force vector at time t + hDt

through Eq. (39)

� �

~FtþhDt ¼ FtþhDt þ

6h2Dt2

ut þ6

hDt_ut þ 2€ut M

þ 3hDt

ut þ 2 _ut þhDt2

€ut

� �C: ð39Þ

Step 7: Find the displacement at time t + hDt from Eq. (40)

utþhDt ¼ eK�1eF tþhDt : ð40Þ

Step 8: Derive the acceleration, the velocity and the displace-ment at time t + Dt from Eqs. (41) and (43)

€utþDt ¼6

h3Dt2ðutþhDt � utÞ �

6h2Dt

_ut þ 1� 3h

� �€ut : ð41Þ

_utþDt ¼ _ut þDt2ð€utþDt þ €utÞ: ð42Þ

utþDt ¼ ut þ _utDt þ Dt2

6ð€utþDt þ 2€utÞ: ð43Þ

3. The proposed quantitative simulation accuracy evaluationmethod

3.1. The proposed quantitative method for evaluating simulationaccuracy of time-transient Lamb wave propagation

The diagram of the proposed quantitative simulation accuracyevaluation method is illustrated in Fig. 5.

Fig. 5. The flow chart of the proposed quantitative numerical simulation accuracy evaluation method.

X. Wan et al. / Ultrasonics 64 (2016) 25–42 31

3.1.1. Identifying two kinds of errorTwo kinds of major error influencing the simulation accuracy

are identified. The first discrepancy is the wave packet position,namely the position error, which is explicitly shown by the timegap between the simulated wave packet and the correspondingreference wave packet. An example illustrating time gap betweena simulated waveform and its corresponding reference signal areobviously shown in Fig. 6. The other major difference is the shape,namely the shape error, which can be clearly observed by compar-ing the shape between the simulated waveform and the corre-sponding reference signal (analytical signal in this paper). A

simulated waveform and its corresponding analytical temporalwaveform are illustrated in Fig. 7(a) and (b). It is obvious thatthe difference of the shape between these two waveforms cannotbe neglected.

3.1.2. Proposing two quantitative indicesTwo identified kinds of error, i.e., the shape and the position

error affecting the position and shape accuracies contribute tothe simulation accuracy. In order to evaluate these two kinds oferror, two quantitative indices, i.e., the GVE and MACCC areproposed.

Fig. 6. A tone burst consisting of 8 cycles with the center frequency of 100 kHz isapplied simultaneously to 2 PWASes which are bounded to the upper and lowersurface of a 4 mm thick aluminum plate respectively to excite only S0 mode Lambwaves, and the displacement signals are monitored at top surface of the plate200 mm away from the actuators. The black curve denotes the temporal waveformobtained from finite element analysis using 3 nodes triangular elements with theelement size of 8 mm and the central difference integration method with the timestep of 0.1 us and the red curve represents the theoretical signal. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

32 X. Wan et al. / Ultrasonics 64 (2016) 25–42

3.1.3. Using cross correlation analysis to calculate two quantitativeindices

In this section, cross correlation analysis is applied to simulta-neously calculate the two quantitative indices. The reference signalis obtained from analytical study described in Section 2.2. The cal-culation of two quantitative indices, i.e., the GVE and MACCC isillustrated in Fig. 8 and elaborated in the following two steps:

Step 1: Cross correlation analysis is conducted between thenumerical simulation signal and the reference waveform.After performing the analysis, the MACCC and the time gap tgap

can be derived directly and simultaneously. The time gap tgap isused to calculate the GVE.Step 2: Calculate the GVE using the formulation given inEq. (44).

GVE ¼ jCgn � CgrjCgr

� 100%; ð44Þ

where Cgn and Cgr refers to the group velocity of the simulatedwaveform and the reference signal, respectively. Cgr is derived fromthe group velocity dispersion curves. Suppose s and t denote thewave packet travelling distance and the arriving time of the refer-ence wave-packet respectively, and s is known variable. Therefore,using Eqs. (45) and (46), Cgn can be obtained

Fig. 7. A tone burst consisting of 8 cycles with the center frequency of 100 kHz is applied4 mm thick aluminum plate respectively to excite only S0 mode Lamb waves, and the disactuators: (a) finite element analysis using 3 nodes triangular elements with the element(b) the analytical signal.

s ¼ Cgr � t; ð45Þ

s ¼ Cgn � ðt � tgapÞ; ð46Þ

where the positive sign ‘+’ is selected when the simulated wavepacket lags behind the reference wave packet, otherwise the nega-tive sign ‘�’ is selected.

3.2. A specialized FEM program providing simulation accuracyquantitative evaluation function

Up to now, there is no FEM software offering numerical simula-tion accuracy evaluation function, let alone quantitative accuracyevaluation function. In this section, a specialized FEM programintegrating simulation accuracy quantitative evaluation functionfor 2D time-transient Lamb wave propagation is developed. Thissoftware is designed to provide a platform for applying the pro-posed method to select appropriate element size and time step.The diagram for this specialized FEM program is illustrated inFig. 9.

4. applications to select proper element size and time step

4.1. Finite element simulation setup

4.1.1. Dispersion curvesThe phase and group velocity dispersion curves, according to

the theoretical study described in Section 2.2.1, for a 4-mm thickaluminum plate are illustrated in Fig. 10(a) and (b), respectively.Due to the multi-mode nature of Lamb wave, exciting a singlemode Lamb wave at low frequency range is preferable and hasbeen widely used in NDE and SHM. Therefore, our study is focusedonly on fundamental symmetric mode (S0) and antisymmetricmode (A0) at low frequencies.

4.1.2. Finite element simulation modelIn this article, a two-dimensional plane strain model is consid-

ered with loads and boundary conditions illustrated in Fig. 11. Asshown in the figure, a single S0 mode can be excited by applyingtwo distributed shear forces with the same direction simultane-ously to the upper and lower surfaces of the plate; while a singleA0 mode is excited by applying two distributed shear forces withthe opposite direction at the same time, where f1(t) and f2(t)denote the distributed shear forces on the upper and lower sur-faces, and 2a, 2d and l refers to the width of a PWAS, the platethickness and Lamb wave propagation distance, respectively. Atthe right end, fixed boundary condition is applied. The setup of this

simultaneously to 2 PWASes which are bounded to the upper and lower surface of aplacement signals are monitored at top surface of the plate 200 mm away from thesize of 4 mm and the Wilson-h time integration scheme with the time step of 1.6 us,

Fig. 8. The flow chart of calculating two quantitative indices.

X. Wan et al. / Ultrasonics 64 (2016) 25–42 33

finite element model fulfills the assumptions of the analyticalstudy described in Section 2.2.2.

4.1.3. Excitation signalA tone burst consisting of N cycles with a specified center fre-

quency f0 is used as the excitation signal and it is formulated inEq. (47). An example of the excitation temporal waveform and itsfrequency spectrum are illustrated in Fig. 12(a) and (b),respectively.

f ðtÞ ¼ F0 sinð2pf 0tÞ � ðsinðpf 0t=NÞÞ2; ð47Þ

where F0 refers to the amplitude of the excited signal.

4.1.4. Thresholds of the two quantitative indicesIn order to apply the proposed quantitative method to selecting

proper element size and time step, thresholds of these two quanti-tative indices are set. From an engineering point of view, a thresh-old value of 1% and 0.95 is set for the GVE and MACCC respectively.The GVE is used to evaluate the position accuracy. Low values ofGVE indicate that the simulation position accuracy is high, whereashigh values of GVE represent that the simulation position accuracyis low. The MACCC is used to assess the shape accuracy. TheMACCC lies between 0 and 1. The value of MACCC approaching 1indicates high simulation shape accuracy, whereas the value ofthe MACCC approaching 0 denotes low simulation shape accuracy.

4.2. Application to selecting appropriate element size regardingelement types

Element size is expressed by Eq. (48)

DI ¼ kmin

M; ð48Þ

where DI refers to element size; M is a positive integer; and kmin isconstant determined by Eq. (49)

kmin ¼Cph

f max; ð49Þ

where Cph represents the phase velocity and fmax is estimated by theupper limit of the major lobe of the frequency spectrum of a excita-tion waveform of interest as illustrated in Fig. 12(b). In the case ofnonlinear effects [78–82], in which high-order harmonics will begenerated in the received waveform, fmax should be the upper limitof the major lobe of high-order harmonics.

The purpose of this section is to select an appropriate positiveinteger of M regarding element types.

4.2.1. The influence of element distortion on the simulation accuracyBefore investigating the influence of element size on the simu-

lation accuracy, the influence of element distortion referring tohigh values of the ratio of the longest to the shortest element edge(aspect ratio) is firstly studied. We define the normal Q4 elementsas ‘‘undistorted elements’’ and Q4 elements with an aspect ratio of4 as ‘‘distorted elements’’. Parameters for studying the influence ofelement distortion on the GVE and MACCC are shown in Table 2. In

Fig. 9. The diagram for the developed specialized FEM program integrating accuracy quantitative evaluation function.

(a) (b)

0 500 1000 1500 20000

5000

10000

15000

f (kHz)

Phas

e ve

loci

ty (m

/s)

anti-symmetricsymmetric

0 500 1000 1500 20000

1000

2000

3000

4000

5000

6000

f (kHz)

Gro

up V

eloc

ity (m

/s)

anti-symmetricsymmetric

Fig. 10. Dispersion curves for a 4 mm thick aluminum plate: (a) phase velocity and (b) group velocity.

34 X. Wan et al. / Ultrasonics 64 (2016) 25–42

Fig. 11. Two dimensional model with loads and boundary conditions. Two distributed shear force f1(t) and f2(t), with f1(t) = f2(t) for exciting a single fundamental symmetricmode (S0) and f1(t) = �f2(t) for the excitation of a single fundamental anti-symmetirc mode (A0).

Fig. 12. Excitation signal with N = 8 and f0 = 200 kHz: (a) temporal waveform and (b) frequency spectrum and the red line denotes the upper limit of the frequencybandwidth. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2Parameters used in finite element simulations for studying the influence of element distortion.

Mode Q4 elements f0 (kHz) fmax (kHz) Cph (m/s) kmin (mm) DI ¼ kminM (mm) Dt (s)

M = 10 M = 20 M = 40

S0 Undistorted 100 125 5387 43.09 DIx = 4 DIx = 2 DIx = 1 5e�8DIy = 4 DIy = 2 DIy = 1

S0 Distorted 100 125 5387 43.09 DIx = 16 DIx = 8 DIx = 4 5e�8DIy = 4 DIy = 2 DIy = 1

A0 Undistorted 100 125 1881 15.05 DIx = 1 DIx = 0.5 DIx = 0.25 5e�8DIy = 1 DIy = 0.5 DIy = 0.25

A0 Distorted 100 125 1881 15.05 DIx = 4 DIx = 2 DIx = 1 5e�8DIy = 1 DIy = 0.5 DIy = 0.25

Fig. 13. The variations of two indices with M (representing different element size) regarding element distortion: (a) the GVE and (b) the MACCC.

X. Wan et al. / Ultrasonics 64 (2016) 25–42 35

Table 2, DIx and DIy refer to the length of elements along the x andy directions.

The variations of GVE and MACCC with the element size consid-ering element distortion are illustrated in Fig. 13. In Fig. 13(a), it isobviously that the position accuracy is very sensitive to element

distortion, which is in accordance with the result reported in theprevious studies [10,44]. In these articles, the authors proposedhigh-order element types due to the high sensitivity to elementdistortion of low-order elements. In Fig. 13(b), it is illustrated thatthe shape accuracy is not so sensitive to element distortion. From

36 X. Wan et al. / Ultrasonics 64 (2016) 25–42

Fig. 13(a) and (b), it is also shown that higher accuracy can bederived for S0-mode Lamb wave than for A0-mode Lamb wave.

4.2.2. Appropriate element size selection regarding element types foraccurate finite element simulations of time-transient S0-mode Lambwave at the center frequency of 100 kHz

Finite element simulations using T3, Q4, T6 and Q8 elementtypes with an element size of kmin=5, kmin=10, kmin=20 and kmin=40of time-transient Lamb wave propagation are first conducted. Inthese simulations, only S0-mode Lamb wave at the center fre-quency of 100 kHz is excited. In order to make sure that these sim-ulations are converged and reliable, the Newmark time integrationscheme is used and the time step Dt is set to 1/(100fmax). Theparameters used in the simulations for studying dependence ofthe GVE and MACCC on element size are shown in Table 3.Time-transient displacement responses are acquired for cross cor-relation analysis. Analytical waveform of time-transient Lambwave response at the center frequency of 100 kHz is also obtainedas a reference signal.

The variations of two indices with the element size of differentelement types are shown in Fig. 14. In Fig. 14(a), for linear elementtypes (T3 and Q4), the GVE decreases dramatically to a quite lowlevel less than 1% (the threshold value of the GVE) when the ele-ment size approaches to kmin=20; while for higher order elementtypes (T6 and Q8), the GVE keeps at small values less than 1% whenthe element size changes from kmin=5 to kmin=40. In Fig. 14(b), forall 4 element types, the MACCC slightly increases as the elementsize becomes smaller. And it remains at a very high level above0.95 (the threshold value of the MACCC) when the element sizechanges from kmin=5 to kmin=40. From these 2 figures, several find-ings can be described as follows. First, using the GVE and MACCC toquantitatively assess the position and shape accuracies is feasible.Second, the position accuracy improves significantly to a high levelwhen the element size reaches to kmin=20 for linear element types(T3 and Q4); the position accuracy obtained by using four-noderectangular element type Q4 is higher than that of linear triangularelement type T3. For higher order element types (T6 and Q8), theposition accuracy improves as the element becomes finer, how-ever, it is not sensitive to the variation of the element size, and ithas approached to a very high level even the element size is a littlelarge equal to kmin=5. Third, values of the MACCC all above 0.95represents that these simulated signals have great similarity with

Table 3Parameters used in finite element simulations for studying dependence of the GVE and M

Mode Plate thickness (mm) f0(kHz) fmax (kHz) Cph (m/s)

S0 4 100 125 5387

Fig. 14. The variations of two indices with M (representing different elem

the corresponding analytical waveform. Even at a relatively largeelement size of kmin=5, a very high shape accuracy can be achievedfor all 4 element types. The overall tendency of the MACCC indi-cates that the shape accuracy increases quite slowly as the elementsize becomes smaller, and more importantly, the element size haslittle influence on the shape accuracy. These results provide guide-lines for selecting proper element size. For linear element types (T3and Q4), an element size equal to kmin=20 is selected for an accuratefinite element simulation, and this selected element size is inaccordance with previous studies [18,20,63,64]. For higher orderelement types (T6 and Q8), an element size of kmin=5 is selectedfor accurate simulation.

4.2.3. Verification the effectiveness of selected element size regardingelement types

The selected element size regarding element types is based ontime-transient S0-mode Lamb wave propagation at the center fre-quency of 100 kHz, the effectiveness of the selected element size isverified using S0-mode Lamb wave at other centered frequencies of100 kHz, 200 kHz and 300 kHz and A0-mode Lamb wave at thecenter frequency of 100 kHz. The parameters used in finite elementsimulations for verification are shown in Table 4.

Two quantitative indices calculated from the results of finiteelement analysis of S0-mode Lamb wave using the selected ele-ment size of each element type at centered frequencies of200 kHz, 300 kHz and 400 kHz are shown in Fig. 15. In Fig. 15(a),values of the GVE are all below 1% indicating that high positionaccuracy is obtained with the selected element size for each ele-ment type, and in Fig. 15(b), it is obviously illustrated that almostall MACCC values are above 0.95 representing that high shapeaccuracy is also derived using the selected element size.

The values of two indices calculated from results of finite ele-ment simulation of both S0 and A0 mode Lamb wave using theselected element size at the center frequency of 100 kHz regardingelement types are illustrated in Fig. 16. In Fig. 16(a) and (b), twofindings are obviously observed. First, high position and shapeaccuracies are derived for both S0 and A0 mode Lamb wave usingthe selected element size regarding different element types.Second, S0 mode is more accurately than A0 mode, which is accordwith the results obtained in the previous investigations [10,44].

Therefore, an element size of kmin=20 for linear element types(T3 and Q4) and kmin=5 for higher order element types (T6 and

ACCC on element size regarding element types.

kmin (mm) DI ¼ kminM (mm) Dt (s)

M = 5 M = 10 M = 20 M = 40

43.09 8 4 2 1 5e�8

ent size) regarding element types: (a) the GVE and (b) the MACCC.

Table 4Parameters used in finite element simulations for verifying recommended element size regarding element types.

Mode f0 (kHz) fmax (kHz) Cph (m/s) kmin (mm) DI ¼ kminM (mm) Dt (s)

T3 Q4 T6 Q8M = 20 M = 20 M = 5 M = 5

S0 200 250 5321 21.28 1 1 4 4 2.5e�8S0 300 375 5168 13.78 0.5 0.5 2 2 1.25e�8S0 400 500 4776 9.55 0.4 0.4 1.6 1.6 1e�8A0 100 125 1881 15.05 0.5 0.5 2 2 5e�8

Fig. 15. The values of two indices calculated from results of finite element simulation for verifying the recommended element size using S0 mode Lamb waves at othercentered frequencies of 200 kHz, 300 kHz and 400 kHz regarding element types: (a) the GVE and (b) the MACCC.

Fig. 16. The values of two indices calculated from results of finite element simulation using the recommended element size for both S0 and A0 mode Lamb waves at thecenter frequency of 100 kHz regarding element types: (a) the GVE and (b) the MACCC.

X. Wan et al. / Ultrasonics 64 (2016) 25–42 37

Q8) is adequate for accurate finite element simulation oftime-transient Lamb wave propagation at low frequencies.

4.3. Application to selecting appropriate time step regarding timeintegration schemes

In this section, the influence of time step on the simulationaccuracy is quantitatively investigated. Four different time integra-tion schemes including an explicit method namely the central dif-ference method, and three implicit methods which are theNewmark method, the Houbolt approach and the Wilson-h methodare considered.

Time step is expressed by Eq. (50)

Dt ¼ 1Nfmax

; ð50Þ

where Dt is time step, N a positive integer and fmax the upper limitof the frequency bandwidth of a waveform of interest as illustratedin Fig. 12(b).

Finite element simulations are conducted using the recom-mended element size of each element type, which means that an

element size of kmin=20 is used for linear element types (T3 andQ4) and an element size of kmin=5 for higher element types (T6and Q8).

4.3.1. Appropriate time step selection regarding time integrationschemes for accurate finite element simulations of time-transient S0-mode Lamb waves at the center frequency of 100 kHz

Finite element simulations using the central difference method,the Newmark method, the Houbolt approach and the Wilson-hmethod with a time step of 1/(5fmax), 1/(10fmax), 1/(20fmax),1/(50fmax), 1/(100fmax) and 1/(200fmax) on time-transient Lambwave propagation are first conducted. In these simulations, onlyS0-mode Lamb wave at the center frequency of 100 kHz is excited.The parameters used in these simulations are shown in Table 5.Analytical waveform of time-transient Lamb wave response atthe center frequency of 100 kHz is also obtained as a referencesignal.

The variations of GVE with time step of different time integra-tion schemes are shown in Fig. 17. Results calculated from simula-tions using T3, Q4, T6 and Q8 element types are illustrated inFig. 17(a –d), respectively. Several observations are elaborated as

Table 5Parameters used in finite element simulations for studying dependence of the GVE and MACCC on time step.

Mode f0 (kHz) fmax (kHz) DI ¼ kminM (mm) Dt ¼ 1

Nfmax(s)

T3, Q4 T6, Q8 N = 5 N = 10 N = 20 N = 50 N = 100 N = 200

S0 100 125 2 8 1.6e�6 8e�7 4e�7 1e�7 5e�8 2.5e�8

Fig. 17. The variations of the GVE with N (representing different time steps): (a) T3 element type, (b) Q4 element type, (c) T6 element type and (d) Q8 element type.

Fig. 18. The variations of the MACCC with N (representing different time steps): (a) T3 element type, (b) Q4 element type, (c) T6 element type and (d) Q8 element type.

38 X. Wan et al. / Ultrasonics 64 (2016) 25–42

X. Wan et al. / Ultrasonics 64 (2016) 25–42 39

follows. First, the overall trends of GVE with time step for these 4element types are quite similar showing that the position accuracyincreases as the time step becomes smaller. Second, the GVE valuescomputed from results of finite element simulations using the cen-tral difference method at the time step of 1/(5fmax), 1/(10fmax) and1/(20fmax) are not shown in these figures, because the central dif-ference method is not converged at these large time steps. Accord-ing to our simulation results, if N is larger than 24, the centraldifference method is converged. However, finite element simula-tions using implicit time integration methods are stable and con-verged at these large time steps. Third, for implicit timeintegration methods, the GVE drops dramatically to a very lowvalue less than 1% as the time step decreases from 1/(5fmax) to1/(50fmax), and the GVE reaches to a low saturation level whenthe time step is less than 1/(50fmax). For the explicit central differ-ence method, when the time step is less than 1/(50fmax), the GVEalso maintains at a low saturation level less than 1% as the implicitmethods.

The variations of MACCC with time step of different time inte-gration schemes are shown in Fig. 18. Results computed from sim-ulations using T3, Q4, T6 and Q8 element types are illustrated inFig. 18(a–d), respectively. Several findings are described as follows.First, the overall tendencies of the MACCC time step for these 4 ele-ment types are quite similar showing that the shape accuracyincreases as the time step becomes smaller. Second, as the central

Table 6Parameters used in finite element simulations for verifying recommended time step regar

Mode f0 (kHz) fmax (kHz) DI ¼ kminM (mm) Dt ¼ 1

Nfmax(s)

The central difference methT3, Q4 T6, Q8 N = 50

S0 200 250 1 4 5e�8S0 300 375 0.5 2 2.5e�8S0 400 500 0.4 1.6 1.25e�8

Fig. 19. The GVE calculated from results of finite element simulations at other centered felement type.

difference method is not converged at the large time step of1/(5fmax), 1/(10fmax) and 1/(20fmax), the MACCC values are notshown in these figures. Third, for implicit time integration meth-ods, the MACCC increases significantly to a very high value morethan 0.95 as the time step decreases from 1/(5fmax) to 1/(20fmax),and the MACCC approaches to a very high stable level when thetime step is less than 1/(50fmax). For the explict central differencemethod, when the time step is less than 1/(50fmax), the MACCC alsokeeps at a high stable level above 0.95 as the implicit methods.

According to our literature review as described in Section 1,most of the researchers [18,20,63]select time step using CFL crite-ria [21] after the element size DI is set. And the CFL criteria isdefined as Dt = DI/Cph, where Cph is the phase velocity. Accordingto the CFL criteria, if the element size DI is chosen as kmin=5, thenthe time step Dt is selected as 1/(5fmax), and time step 1/(20fmax)is selected for an element size of kmin=20, and so on. However,our observations show that when the time step approaches to1/(50fmax) regardless of element size and element type, accuratesimulations can be obtained. And our findings indicate that evenfor the implicit time integration methods, i.e., the Newmark, Hou-bolt and the Wilson-h methods, which are converged and stable atlarge time steps, in order to get accurate simulations, smaller timesteps are quite necessary which is consistent with conclusionsreported in Ref. [21]. Therefore, a time step of 1/(50fmax) is selectedfor accurate simulation of time-transient Lamb wave propagation.

ding time integration schemes.

od The Newmark method The Houbolt method The Wilson-hmethod.N = 50 N = 50 N = 50

5e�8 5e�8 5e�82.5e�8 2.5e�8 2.5e�81.25e�8 1.25e�8 1.25e�8

requencies: (a) T3 element type, (b) Q4 element type, (c) T6 element type and (d) Q8

Fig. 20. The MACCC calculated from results of finite element simulations at other centered frequencies: (a) T3 element type, (b) Q4 element type, (c) T6 element type and (d)Q8 element type.

40 X. Wan et al. / Ultrasonics 64 (2016) 25–42

4.3.2. Verification the effectiveness of selected time step regarding timeintegration schemes

The selected time step for these 4 time integration schemes isbased on S0-mode Lamb wave propagation at the center frequencyof 100 kHz, the effectiveness of the selected time step is verifiedusing S0-mode Lamb wave at other centered frequencies of200 kHz, 300 kHz and 400 kHz. The parameters used in the simula-tions for verification are shown in Table 6.

The GVE calculated from the results of finite element analysisusing the selected time step of different time integration schemesat centered frequencies of 200 kHz, 300 kHz and 400 kHz areshown in Fig. 19. Results computed from simulations using T3,Q4, T6 and Q8 element types are illustrated in Fig. 19(a–d),respectively. In Fig. 19, the values of GVE are all below 1% indicat-ing that high position accuracy is derived with the selected timestep. The MACCC results computed from simulations using theselected time step of different time integration schemes atcentered frequencies of 200 kHz, 300 kHz and 400 kHz fordifferent element types T3, Q4, T6 and Q8 are illustrated inFig. 20(a –d), respectively. The values of MACCC are almost above0.95 representing that high shape accuracy is also obtained byusing the selected time step. Therefore, a time step of 1/(50fmax)for accurate simulation of Lamb wave propagation is verified.When considering the computation cost, if a lumped mass matrixis used, the central difference method will be better, as onlymatrix–vector operations are needed for the time-integration.Otherwise, implicit methods are recommended because of theirunconditional stability property.

5. Conclusions

In this paper, a quantitative method based on cross correlationanalysis for assessing numerical simulation accuracy of

time-transient Lamb wave propagation is proposed. Cross correla-tion analysis is the first time to be applied to quantitatively studythe simulation accuracy. Two kinds of error, i.e., the position andshape errors affecting the position and shape accuracies are firstlyidentified. Consequently, two quantitative indices, i.e., the GVE andMACCC derived from cross correlation analysis between a simu-lated signal and its corresponding reference waveform are thusproposed to evaluate these two kinds of errors. In this paper, aninnovative matlab-based software integrating simulation accuracyquantitative evaluation function for 2D time-transient Lamb wavepropagation is developed. This specialized software is designed toprovide a platform to apply the proposed method to select appro-priate element size regarding different element types and timestep regarding different time integration schemes. According toour study results, an element size of kmin=20 for linear elementtypes (T3 and Q4) and kmin=5 for high-order element types (T6and Q8) is selected. A time step of 1/(50fmax) using these 4 timeintegration schemes, the explicit method, the central differenceapproach and the implicit methods, the Newmark method, theHoubolt method and Wilson-h method is selected. The applicationsto select proper element size and time step prove that theproposed quantitative simulation accuracy evaluation method isfeasible and effective. The proposed quantitative method is basedon the cross correlation analysis between the numerical simulationsignal and its corresponding reference waveform, and this refer-ence signal can be analytical or experimental results. Therefore,the major drawback of this proposed method is that a referenceresult is a prerequisite. Also in this paper, the selected elementsize regarding element types and time step regarding timeintegration schemes provide guidelines for accurate finite elementsimulation of time-transient Lamb wave propagating along normalstructures. And abnormality, i.e., cracks will be considered in thefuture study.

X. Wan et al. / Ultrasonics 64 (2016) 25–42 41

Acknowledgments

The work described in this paper was supported by grant fromthe Research Grants Council of the Hong Kong Special Administra-tive Region, China (Project No. CityU 122513) and the NationalScience and Technology Major Project (Project No.2014ZX04015041).

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