numerical time-domain modeling of lamb wave propagation

Research ArticleNumerical Time-Domain Modeling of Lamb Wave PropagationUsing Elastodynamic Finite Integration Technique

Hussein Rappel,1 Aghil Yousefi-Koma,1 Jalil Jamali,2 and Ako Bahari3

1 Center of Advanced Systems and Technologies (CAST), School of Mechanical Engineering, College of Engineering,University of Tehran, Tehran 14399 57131, Iran

2Department of Mechanical Engineering, Islamic Azad University, Shushtar Branch, Shushtar, Iran3 School of Railway Engineering, Iran University of Science and Technology, Tehran 16846 13114, Iran

Correspondence should be addressed to Aghil Yousefi-Koma; [email protected]

Received 19 October 2012; Accepted 19 November 2012; Published 10 July 2014

Academic Editor: Hamid Mehdigholi

Copyright © 2014 Hussein Rappel et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a numericalmodel of lambwave propagation in a homogenous steel plate using elastodynamic finite integrationtechnique (EFIT) as well as its validation with analytical results. Lamb wave method is a long range inspection technique whichis considered to have unique future in the field of structural health monitoring. One of the main problems facing the lamb wavemethod is how to choose the most appropriate frequency to generate the waves for adequate transmission capable of properlypropagating in the material, interfering with defects/damages, and being received in good conditions. Modern simulation toolsbased on numerical methods such as finite integration technique (FIT), finite element method (FEM), and boundary elementmethod (BEM) may be used for modeling. In this paper, two sets of simulation are performed. In the first set, group velocities oflamb wave in a steel plate are obtained numerically. Results are then compared with analytical results to validate the simulation. Inthe second set, EFIT is employed to study fundamental symmetric mode interaction with a surface braking defect.

1. Introduction

Lamb wave testing technique is increasingly used for assess-ing defects in thin-wall structures like plate and pipes [1–3]. Lamb waves are elastic waves whose wavelength is inthe same order as thickness of the structure [4]. One of themain advantages of lamb wave technique is that it allowslong-range inspection in contrast to traditional ultrasonictesting, where the coverage is limited to a small area invicinity of each transducer. Lamb waves were first describedtheoretically by Horace Lamb in 1917 [5]. These wavesarise from coupling between shear and longitudinal wavesreflected at the top and bottom edges of a thin wall structure[6]. Lamb wave theory can be found in a number of textbooks [7]. Defects such as corrosion and fatigue cracks causechanges in effective thickness and local material propertiesand therefore measurement of variations in lamb wavepropagation can be used to assess the integrity of plate [1].Successful usage of lambwaves in an inspection system needsto understand its schemes of propagation in a waveguide and

its scattering at defects. Thus, there is an increasing demandfor powerful, flexible, and accurate simulation techniques.First works on numerical simulation of ultrasonic waves weredone by Harumi (1986) and Yamawaki and Saito (1992) whocalculated and visualized bulk wave propagation [8]. Now,numerical simulation of lamb waves is possible. Commontechniqueswhich are used to simulate lambwave propagationare finite difference time domain (FDTD) [9], finite elementmethod (FEM) [5], boundary element method (BEM) [10],elastodynamic finite integration technique (EFIT) [11, 12],and specialized methods for guided wave calculations suchas hybrid methods [13] and semianalytical finite elementmethod (SAFEM) [8].

In this work, calculations are based on elastodynamicfinite integration technique; historically, finite integrationtechnique was introduced by Weiland in electrodynamics.Fellinger and Langenberg used Weiland’s idea for governingequations of ultrasonic waves in solid, calling it EFIT [14].EFIT is a grid based numerical time-domain method, usingvelocity-stress formalism, and easily treats with different

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2 Shock and Vibration

boundary conditions which are essential to model ultrasonicwave propagation [12]. Because of its relative simplicity andflexibility, Schubert et al. used EFIT equations to cylindricalcoordinates (CEFIT) to simulate axisymmetric wave propa-gation in pipes with a 2D grid [15]. Schubert also used finiteintegration technique to simulate elastic wave propagation inporous concrete and showed efficiency of EFIT to model adiverse range of applications [16].

Two sets of simulation results are presented in this workusing a program developed in MATLAB environment. Inthe first one, lamb wave propagation in a 2D steel plate isdiscussed. Results are then compared with analytical resultsto validate the accuracy of modeling and, in the secondexample, interaction lamb wave with a surface breakingdefect is investigated.

2. The Elastodynamic Finite IntegrationTechnique for Linear Elastics

2.1. Governing Equations. The governing equations of elasticwaves in a general media are the Cauchy equation of motionand equation of deformation rate. These equations are givenin integral form for a finite volume𝑉and surface 𝑆 as follows:

𝜕

𝜕𝑡∫𝑉

𝜌V𝑖𝑑𝑉 = ∫

𝑆

𝑇𝑖𝑗𝑛𝑖𝑑𝑆 + ∫

𝑉

𝑓𝑖𝑑𝑉, (1)

𝜕

𝜕𝑡∫𝑉

𝑠𝑖𝑗𝑘𝑙𝑇𝑘𝑙𝑑𝑉 =1

2∫𝑆

(V𝑖𝑛𝑗+ V𝑗𝑛𝑖) 𝑑𝑆, (2)

where V is the particle velocity vector, 𝑇 is stress tensor, 𝜌is density, 𝑛 is the outward normal vector on surface 𝑆, 𝑓 isthe body force vector, and 𝑠 is the compliance tensor. Theinverse of 𝑠 is the stiffness tensor 𝑐. Thus, using stiffnesstensor, deformation rate equation can be expressed in anotherform. Consider

𝜕

𝜕𝑡∫𝑉

𝑇𝑘𝑙𝑑𝑉 =1

2∫𝑆

𝑐𝑘𝑙𝑖𝑗(V𝑖𝑛𝑗+ V𝑗𝑛𝑖) 𝑑𝑆 = ∫

𝑆

𝑐𝑘𝑙𝑖𝑗

V𝑖𝑛𝑗𝑑𝑆.

(3)

In the case of isotropic material 𝑐 can be written as [17]

𝑐𝑖𝑗𝑘𝑙= 𝜆𝛿𝑖𝑗𝛿𝑘𝑙+ 𝜇 (𝛿

𝑖𝑘𝛿𝑗𝑙+ 𝛿𝑖𝑙𝛿𝑗𝑘) , (4)

where 𝜆 and 𝜇 are lame constants.

2.2. Spatial Discretized Form of Two Dimensional EFIT.Consider the Cartesian coordinate {𝑥, 𝑦, 𝑧} and ultrasonicwave which propagates in two dimensional 𝑥𝑧-plane. Toapply FIT to (1) and (2), squares shown in Figure 1 are usedas integral volume 𝑉, assuming constant V and 𝑇 for eachvolume.

orPseudomaterial cellsTrue material cells(coincide with Txx and Tzzintegration cells)

Txz integration cell�x integration cell�z integration cell

x

z(x1, z1)

(x0, z0) (x2, z2)

(x3, z3)

Figure 1: Definition of integration cells for stress and velocity com-ponents. The geometry consists of four true material cells and fourpseudomaterial cells [12].

The final results for discretized form are

𝜕V𝑥(𝑥0, 𝑧0)

𝜕𝑡

=1

𝜌[𝑇𝑥𝑥(𝑥0+ (Δ𝑥/2) , 𝑧0) − 𝑇𝑥𝑥 (𝑥0 − (Δ𝑥/2) , 𝑧0)

Δ𝑥

+𝑇𝑥𝑧(𝑥0, 𝑧0+ (Δ𝑧/2)) − 𝑇𝑥𝑧 (𝑥0, 𝑧0 − (Δ𝑧/2))

Δ𝑧

+𝑓𝑥

Δ𝑥Δ𝑧] .

(5)

A same manner of integration equation (1) about a V𝑧

integration cell centered at (𝑥1, 𝑧1) results in

𝜕V𝑧(𝑥1, 𝑧1)

𝜕𝑡

=1

𝜌[𝑇𝑥𝑧(𝑥1+ (Δ𝑥/2) , 𝑧1) − 𝑇𝑥𝑧 (𝑥1 − (Δ𝑥/2) , 𝑧1)

Δ𝑥

+𝑇𝑧𝑧(𝑥1, 𝑧1+ (Δ𝑧/2)) − 𝑇𝑧𝑧 (𝑥1, 𝑧1 − (Δ𝑧/2))

Δ𝑧

+𝑓𝑧

Δ𝑥Δ𝑧] .

(6)

Shock and Vibration 3

Excitation force Sensor

50mm

50mm

l = 300mm

(a)

Excitation force Sensor 50mm50mm

l = 300mm

(b)

Figure 2: Steel sheet with length 𝑙 = 300mm and thickness 𝑑 = 2mm. (a) Excitation pattern for symmetric mode. (b) Excitation pattern foraxisymmetric mode.

0 1 2 3 4 5 6 7 8 9 1002468

101214161820

Excitationregion

Phas

e velo

city

(mm

/𝜇s)

Frequency ∗ thickness (MHz∗mm)

(a)

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Gro

up v

eloci

ty (m

m/𝜇s)


(b)

Figure 3: Dispersion diagram for a plate steel (a) phase velocity curve and (b) group velocity curve.

Now, using the normal stress equations, integration of (3)about 𝑇

𝑥𝑥and 𝑇

𝑧𝑧centered at (𝑥

2, 𝑧2) yields

𝜕𝑇𝑥𝑥(𝑥2, 𝑧2)

𝜕𝑡

= (𝜆 + 2𝜇) [V𝑥(𝑥2+ (Δ𝑥/2) , 𝑧2) − V𝑥 (𝑥2 − (Δ𝑥/2) .𝑧2)

Δ𝑥]

+ 𝜆[V𝑧(𝑥2, 𝑧2+ (Δ𝑧/2)) − V𝑧 (𝑥2, 𝑧2 − (Δ𝑧/2))

Δ𝑧] ,

𝜕𝑇𝑧𝑧(𝑥2, 𝑧2)

𝜕𝑡

= (𝜆 + 2𝜇) [V𝑧(𝑥2, 𝑧2+ (Δ𝑧/2)) − V𝑧 (𝑥2, 𝑧2 − (Δ𝑧/2))

Δ𝑧]

+ 𝜆[V𝑥(𝑥2+ (Δ𝑥/2) , 𝑧2) − V𝑧 (𝑥2 − (Δ𝑥/2) , 𝑧2)

Δ𝑥] .

(7)Finally, integration of (3) over 𝑇

𝑥𝑧integration cell cen-

tered at (𝑥3, 𝑧3) the intersection for material cells results in

𝜕𝑇𝑥𝑧(𝑥3, 𝑧3)

𝜕𝑡

= 𝜇 [V𝑥(𝑥3, 𝑧3+ (Δ𝑧/2)) − V𝑥 (𝑥3, 𝑧3 − (Δ𝑧/2))

Δ𝑧

+V𝑧(𝑥3+ (Δ𝑥/2) , 𝑧3) − V𝑧 (𝑥3 − (Δ𝑥/2) , 𝑧3)

Δ𝑥] .

(8)

As shown in Figure 1, to simplify indexing into stress andvelocity arrays of staggered grids when programming the

numerics and to keep the same array sizes for all quantities,pseudomaterial cells are used. These cells have the samematerial properties as the true material they are added to butare not part of physical simulations.

2.3. Time Discretization. Central differences are used todiscretize the equations in time domain which results in thevelocity and stress components being staggered in time byΔ𝑡/2 [15]. Consider

V𝑛 = V𝑛−1 + Δ𝑡V̇𝑛−(1/2),

𝑇𝑛+(1/2)= 𝑇𝑛−(1/2)+ Δ𝑡�̇�

𝑛,

(9)

where Δ𝑡 is time interval, superscript 𝑛 is integer number oftime step, and dot {⋅} denotes the time differentiation.

Equations (5)–(8) are solved at all points in simulationspace and, by use of (9), the simulation proceeds in timein a “leap frogging” manner. A specific stability conditionand adequate spatial resolutionmust be satisfied to guaranteeEFIT convergence and accurate answers [15].

3. Propagation of Lamb Wave in a Steel Plate

In this part, the propagation of lamb wave in a steel plate issimulated using 2D-EFIT. The steel plate has the length 𝑙 =300mm and the thickness 𝑑 = 2mm. Table 1 shows materialproperties used in this paper.

As excitation source, point sources at top and bottomborders of plate are used. Figure 2 shows location of appliedloads.

Using excitation patterns shown in Figure 2 and disper-sion diagram for steel plate (Figure 3), singlemode lambwaveis generated which makes signal interpretation easier.


21.81.61.41.21

0.80.60.40.200 0.05 0.1 0.15 0.2 0.25 0.3

z(m

)

x (m)

×10−3 ×10−6

1.5

1

0.5

0

−0.5

−1

−1.5

−2

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3

x (m)

21.81.61.41.21

0.80.60.40.20

z(m

)

×10−3 ×10−5

1

0.5

0

−0.5

−1

(b)

Figure 4: Lamb wave propagation in a steel plate at time 𝑇 = 80 𝜇s. The snapshot represents normal component of particle velocity (V𝑧): (a)

symmetric mode and (b) axisymmetric mode.

Table 1: Material properties used for simulation.

Property ValueDensity 7700 (kg/m3)Elastic modulus 195GpaLame constant 𝜆 96.95GpaLame constant 𝜇 76.17Gpa

Gro

up v

eloci

ty (m

/s)

2D-EFITAnalytical

3200

3150

3100

3050

3000

2950

29000.6 0.7 0.8 0.9 1 1.1 1.2 1.3


Figure 5: Analytical group velocities comparison with 2D-EFITresults for fundamental axisymmetric mode.

Using 2D-EFIT code developed in MATLAB, propaga-tion of lamb wave in the steel plate is simulated. To guaranteestability and accuracy of results, Δ𝑥 and Δ𝑧 are chosen0.2mm and Δ𝑡 is 20 ns. The simulation results using EFIT-tool for symmetric and axisymmetric modes are presented inFigure 4, where the ultrasonic wave field in the plate at time𝑇 = 80 𝜇s is shown (excitation pulse is a raised cosine withfive cycles with center frequency of 500 kHz).

As shown in Figure 4, for the fundamental symmetricmode (𝑆

0), the lamb wave field is symmetric about half

plane line and, for the fundamental axisymmetricmode (𝐴0),

normal component of particle velocity V𝑧has the same value

for every particle with same longitudinal position. Fromdispersion curve, we find that 𝑆

0travels faster than𝐴

0which

is validated by simulation results (see Figure 4).

47004800490050005100520053005400

Gro

up v

eloci

ty (m

/s)

2D-EFITAnalytical

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3


Figure 6: Analytical group velocities comparison with 2D-EFITresults for fundamental symmetric mode.

Erro

r (%

)

SymmetricAxisymmetric

3.3

3.2

3.1

3

2.9

2.8

2.7

2.6

2.5

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3


Figure 7: Error comparison for symmetric and axisymmetricmodes.

In order to check EFIT accuracy, group velocitiesobtained from simulation are compared with analyticalresults at both symmetric and axisymmetric modes (Figures5 and 6).

Figures 5–7 show good agreement for simulation resultswith analytical ones; also Figure 7 shows error dependence

Shock and Vibration 5

149mm d = 2mm

d = 2mm l = 300mm

Figure 8: Schematic of model used for studying interaction lambwave with a defect in a steel plate.

Defect depth (mm)

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

RS 0/R

A0

Figure 9: Ratio of reflection coefficients as function of crack depth.

on frequency for axisymmetric mode is less than symmetricmode.

4. Reflection of the Fundamental SymmetricMode (𝑆

0) from a Defect

In this section, interaction of the 𝑆0mode with a defected

steel plate is analyzed. The results presented here were usedfor a sizing study of rectangular surface braking defect withdifferent depths and opening length 2mm on a steel plate(Figure 8).

The same method used in the proceeding section isused to generate single mode with center frequency of500 kHz. However, as the lamb wave interacts with a defect,the axisymmetric mode will be generated. To study lambwave interaction with a defect, the ratio of the maximumamplitudes for two modes 𝑅

𝑆0/𝑅𝐴0

is then calculated andcompared at different depths (Figure 9).

Figure 10 shows the ultrasonic wave field in the defectedplate at time 𝑇 = 60 𝜇s; the defect depth is 0.4mm. Asshown in Figure 10, because symmetric modes travel fasterthan axisymmetric ones,mode separation happens after lambwave interaction with defect.

5. Conclusion

EFIT was used for studying lamb wave propagation in a steelplate using a program developed in MATLAB environment.Two sets of simulation results were presented in this paper.In the first example, group velocities of lamb wave fordifferent frequencies were obtained using numerical signalsand then the results were compared with analytical results;the comparison shows, for both fundamental symmetric andaxisymmetric modes, the group velocity values are in goodagreement with theoretical ones. In the second example,reflection of 𝑆

0mode from a defect is studied and ratio of

×10−3

21.81.61.41.21

0.80.60.40.200 0.05 0.1 0.15 0.2 0.25 0.3

z(m

)

x (m)

z(m

)

×10−6

21.510.50−0.5−1−1.5−2−2.5

Figure 10: Lambwave propagation in a steel plate with defect at time𝑇 = 60 𝜇s. The snapshot represents normal component of particlevelocity (V

𝑧(m/s)).

reflection coefficients was obtained as a function of crackdepth which shows that as the crack depth increases the ratio𝑅𝑆0/𝑅𝐴0

increases. Each calculation presented in this paperwas done on ordinary PC (Core i5, 2.4GHz, 4GB RAM).

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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