Efficient computation of wave propagation along axisymmetric pipesunder non-axisymmetric loading

Zahra Heidary, Didem Ozevin n

Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, IL 60607, United States

a r t i c l e i n f o

Article history:Received 6 November 2013Received in revised form19 March 2014Accepted 1 April 2014Available online 4 May 2014

Keywords:Non-axisymmetric loadingSpectral elementAxisymmetric geometry

a b s t r a c t

This paper presents an efficient formulation of the problem of wave propagation along the length ofaxisymmetric pipes under non-axisymmetric loading such as leaks or new cracks so that wavecharacteristics in a pipe can be identified without the excessive computational time associated withmost current 3D modeling techniques. The axisymmetric geometry of the pipe is simplified by reducingthe problem to 2D while the non-axisymmetric loading is represented by the summation of Fourierseries. Since the pipe stiffness matrix as conventionally formulated represents the greatest singlecomputational load, the strain–displacement matrix is partitioned in such a way that numericalintegration components are decoupled from θ (the angular parameter) and n (the number of Fourierterms). A single numerical integration of the strain–displacement matrix is performed and utilized for allthe iterations of Fourier terms to represent the non-axisymmetric load. The numerical formulation isconducted using spectral elements, which also reduce computational time since these elements yield adiagonal mass matrix. The computational efficiency of the developed method is compared withconventional finite element tools.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

The main wave propagation based Structural Health Monitoring(SHM) methods for pipelines are guided wave ultrasonics andacoustic emission. Guided wave ultrasonics relies on capturing thereflected wave energy from a defect after introducing a perturba-tion signal using ultrasonic transducers [1,2]. If less-dispersiveguided modes are selected to transmit and receive the signal,long-range pipes can be monitored using a limited set of transdu-cers. The acoustic emission (AE) method relies on propagatingelastic waves emitted from newly formed damage surfaces suchas active cracks and leaks. Crack growth causes sudden stress–strainchange in its vicinity, which generates a wideband step function. Aleak causes turbulence at its location, which generates continuousemissions. The AE method may be based on elastic waves propagat-ing through the pipe material [3,4] or acoustic waves propagatingthrough the material inside the pipe [5]. For an effective andaccurate monitoring approach, wave characteristics such as thedispersion curves under buried or fluid filled conditions and theattenuation profile should be known prior to the implementation ofan SHM method. However, experimental simulations of differentpipe geometries and conditions are generally not possible. Wave

propagation in pipes is a complex phenomenon due to the excita-tion of multi-mode waves, which must be superimposed to providean overall solution. Analytical solutions of governing differentialequations are not applicable when the pipe geometry becomescomplex with the presence of defects, coatings and internalmaterials [6,7], buried conditions [8], and pipe bends [9]. Themodeling of wave propagation is important for quantitative under-standing of damage mechanics and the identification of theSHM system characteristics (e.g. frequency selection, sensor posi-tion [10]).

Wave propagation in pipes can be numerically modeled as 2Dor 3D [11]. The 3D wave propagation problems of hollow circularcylinders including non-axisymmetric wave modes are formulatedby Gazis [12]. If the problem requires modeling high frequencywaves in a large-scale structure, the 3D model becomes compu-tationally expensive. Therefore, it is imperative to reduce themathematical problem to 2D or implement semi-analytical finiteelement formulation [13,14] for reducing the computational load.When the structure and loading are axisymmetric, the structuralmodel can be reduced to a 2D problem as displacements andstresses are independent of θ (the angular parameter). There areseveral other methods for reducing the computational time of highfrequency wave propagation in hollow structures. Benmeddouret al. [15] developed a three dimensional hybrid method whichcombined a classical FE method and normal mode expansiontechnique in order to study the interaction of guided waves with

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/finel

Finite Elements in Analysis and Design

http://dx.doi.org/10.1016/j.finel.2014.04.0010168-874X/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ1 312 413 3051.E-mail address: dozevin@uic.edu (D. Ozevin).

Finite Elements in Analysis and Design 86 (2014) 81–88

non-axisymmetric cracks in cylinders. Zhou et al. [16] utilized thenumerical eigenmode extraction method applicable to wave pro-pagation in periodic structures. Mazzotti et al. [17] applied a SemiAnalytical Finite Element (SAFE) method to study the influence ofprestressing load on the dispersion of viscoelastic pressurizedpipe. Bai et al. [14] constructed an elastodynamic steady-stateGreen's function based on modal data determined from thespectral decomposition of a circular laminated piezoelectric cylin-der using semi-analytical finite element formulation. Zhuang et al.[13] proposed integral transform and forced vibration as twodifferent methods both based on the same set of eigenvalue datato construct a steady-state Green's function for laminated circularcylinder. Gsell et al. [18] discretized the displacement equationsdirectly using the finite difference method, which reduced thecomputational time by 25%.

When the structure is axisymmetric and the load is non-axisymmetric, three displacement components in the radial, axialand circumferential directions exist [19,20]. To utilize the 2Daxisymmetric geometry of the pipe, the non-axisymmetric loadcan be expanded using Fourier series, and the structural responsecan be computed by superposing the solutions of each Fourierterm [21,22]. There are several examples in the literature relatedto applying Fourier series summation to model axisymmetricgeometries with non-axisymmetric loading such as Zhuang et al.[13], Bai et al. [14],Wunderlich et al. [23] and Bouzid et al. [24].However, in case where the Fourier expansion of the load functionrequires many harmonics to represent the non-axisymmetricloads such as concentrated loads, the 2D superposition methodcombined with conventional finite element formulation may notbe computationally efficient than 3D analyses [25,26]. Bathe [25]describes that the stiffness matrices corresponding to the differentharmonics can be decoupled due to the orthogonal properties oftrigonometric functions. However, to the best of our knowledge,there is no study that explicitly presents the mathematicalformulation. While numerical models are capable of simulatingvarious pipe geometries and conditions to deduce the waveformcharacteristics, existing numerical formulations for wave propaga-tion are computationally expensive, and not practical for modelinglong-range pipes.

In this paper, a detailed mathematical formulation is presented tomodel axisymmetric geometries with non-axisymmetric dynamicloads, specifically concentrated loads. The formulation is based onpartitioning the strain–displacement matrix in such a way that thenumerical integration terms are decoupled from the variables θ and n(the number of Fourier terms). Therefore, the strain–displacementmatrix is calculated only once and used for each Fourier termcalculation. Additionally, the numerical formulation is built usingspectral elements, which are special forms of finite elements that

yield a diagonal mass matrix and thereby provide highly efficientnumerical models for high frequency wave propagation [27,28,29].The nodal coordinates of the Lagrange shape functions are obtainedfrom the solutions of orthogonal polynomials. Moreover, one couldoptimize the computational expenditure by expanding the displace-ment field in the pipe over a series of normal modes.

The organization of this article is as follows: the discretization,spectral element formulation and the non-axisymmetric loadingformulation are described in Section 2. In Section 3, the 2Dnumerical results are compared with 3D finite element models.The computational efficiency of the 2D model is presented inSection 4. Finally, the major outcome of this study and future workare summarized in Section 5.

2. Method of analysis

2.1. Discretization

For the spectral element formulation, the discrete locations ofthe nodal coordinates are defined by the Gauss–Lobatto–Legendre(GLL) points using the selected Legendre polynomial degree,which defines the p refinement within the element. The GLLpoints are calculated by the roots of the following equation [30]:

ð1�ξ2ÞdPNðξÞdξ

¼ 0 ð1Þ

where PNðξÞ is the Legendre polynomial of degree N and ξA ½�1;1�. The basis function Nij ð ¼ 0;…;number of node Þ in 2D isexpressed using the Lagrange interpolation polynomials [31]

Nij ¼ ∏n

m¼ 0ma i

ξ�ξmξi�ξm

� �∏n

l¼ 0la j

η�ηlηj�ηl

!ð2Þ

where ξ and η are the natural coordinate systems in the elementand ξi and ηj are the coordinates of the ith and jth node in thedirections of ξ and η, respectively. Fig. 1 shows the Lagrangianshape function using GLL integration points for (ξ,η). The non-uniform distribution of nodal points reduces the presence ofRunge phenomenon [32].

2.2. Coordinate system and corresponding displacement fields

The cylindrical coordinates and the axis of revolution for pipegeometry are shown in Fig. 2(a). The axial, radial and circumfer-ential displacements are defined as w, u and v, respectively.The spectral element discretization of a cross sectional elementusing the 5th order Legendre polynomial is shown in Fig. 2b. Theaxis of revolution is z with the displacement component w.

In general, when loading has no symmetry and is defined byFourier series components, the displacement components at radial(u, axial (w and circumferential (v) directions can be defined in theform of Fourier expansion as [24]

u¼ ∑n ¼ 1

ðun cos nθþun sin nθÞ ð3Þ

v¼ ∑n ¼ 1

ðvn sin nθ�vn cos nθÞ ð4Þ

w¼ ∑n ¼ 1

ðwn cos nθþwn sin nθÞ ð5Þ

where n is the harmonic number. Single and double barred termsrepresent the symmetric and antisymmetric displacement ampli-tudes with respect to the plane θ¼0 [24].

If the applied load is symmetric about the plane θ¼0 andn¼1,2,3…, the terms of the single-barred series represent the

Fig. 1. Lagrangian 2D shape function using GLL integration points for the coordi-nate of (0.87, 0.87).

Z. Heidary, D. Ozevin / Finite Elements in Analysis and Design 86 (2014) 81–8882

displacement amplitudes, and double-barred components vanish.The double-barred terms are associated with additional antisym-metric loads. For instance, the n¼0 terms of the double-barredseries are used to represent a pure torque [24]. However, for anisotropic elastic material, symmetric and antisymmetric terms arecompletely uncoupled and can be handled separately. In the caseof antisymmetric harmonics, the stiffness matrix is identical to thesymmetric case, which explains the appearance of negative sign inEq. (4). Therefore, all the formulations are almost the same assymmetric case [18,24]. For the case of point loading at thecircumference of the pipe, shown in Fig. 3, loading becomessymmetric about the θ¼0 axis, and the displacement componentsare simplified as

u

v

w

8><>:

9>=>;¼∑

n

un cos nθ

vn sin nθ

wn cos nθ

8><>:

9>=>; ð6Þ

The concentrated load can be represented by various forms ofFourier series [e.g., 13, 33–35]. The Fourier series of the timedependent point load is defined as PNðθ; tÞ ¼ δNðθÞFðtÞ at r¼r0 and

z¼z0. The delta function δNðθÞ and the load function (3-cycle sinewave) F(t) are defined as

δNðθÞ ¼12π

sin ðNþ1=2Þθsin ð1=2θÞ ð7Þ

FðtÞ ¼ 10 sin ð2πf tÞ if tr3f

FðtÞ ¼ 0 if t43f

ð8Þ

2.3. The strain–displacement matrix

The approximation of the displacement fields for a 36-nodeelement shown in Fig. 2 can be written as

un

vnwn

8><>:

9>=>;¼

N1 0 00 N1 00 0 N1

�������N2 0 00 N2 00 0 N2

�������………

�������………

N36 0 00 N36 00 0 N36

�������264

375

� u1n v1n w1n u2n v2n w2n�� ��⋯ u36n v36n w36n

�� ��� �T ð9Þ

Fig. 2. (a) Coordinates and displacement components of a pipe model and (b) spectral element discretization of an element.

Fig. 3. The symmetry of the point load for θ¼0, (a) on a 3D view and (b) on a 2D view extending from –π to π.

Z. Heidary, D. Ozevin / Finite Elements in Analysis and Design 86 (2014) 81–88 83

Basis functions Ni are defined in Eq. (2). The element displace-ments are written in the form as

u

v

w

8><>:

9>=>;¼∑

n

un cos nθvn sin nθ

wn cos nθ

8><>:

9>=>;

¼∑n

N1 cos nθ 0 0 …0 N1 sin nθ 0 …0 0 N1 cos nθ …

264

375fdgn ð10Þ

where

fdgn ¼ u1n v1n w1n u2n v2n w2n j�� ⋯

� �T:

The three-dimensional expression of strains in cylindrical coordi-nates is given by [22]

ε¼

εr

εθ

εz

γrzγrθγzθ

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

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

¼

u;r

½uþv;θ�=rw;z

u;zþw;ru;θr þv;r�v=r

u;zþw;θ=r

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

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

ð11Þ

By substituting Eq. (10) into Eq. (11) and applying the operatormatrix ½∂�ε¼ ½∂�½Nn ¼ 1 Nn ¼ 2 Nn ¼ 3…�fdg ð12Þwhere

½∂� ¼

∂=∂r 0 01=r ∂=rð∂θÞ 00 0 ∂=∂z

∂=∂z 0 ∂=∂r1=ðr∂θÞ ∂=∂r�1=r 0

0 ∂=∂z ∂=rð∂θÞ

………………

��������������

26666666664

37777777775:

The contribution of nth harmonic to strain can be written as

fεgn ¼ ½B�nfdgn ð13Þwhere

εrn

εθn

εzn

γrznγrθnγzθn

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

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

¼

N1;r cos nθ 0 0ðN1=rÞ cos nθ ðnN1=rÞ cos nθ 0

0 0 N1;z cos nθN1;z cos nθ 0 N1;r cos nθ

�ðnN1=rÞ sin nθ ðN1;r�ðN1=rÞÞ sin nθ 00 N1;z sin nθ �ðnN1=rÞ sin nθ

………………

��������������

26666666664

37777777775

� u1n v1n w1n u2n v2n w2n�� ��⋯� �T ð14Þ

Eq. (14) indicates that the strain–displacement matrix B dependson r,z, θ and n.

2.4. Decoupling the stiffness matrix from θ and n

The elastic wave equation for isotropic, homogenous material isgoverned by the Navier equation [36]

μ∇2uþðμþλÞ∇∇Uu¼ ρ €u ð15Þwhere m and λ are Lamé constants and ρ is density. The variationalfinite element formulation of the elastic wave equation is obtainedthrough a dot product with a vector test function and integratedover the medium. The weak formulation of the elastic waveequation for isotropic medium using the divergence theoremresults in the following equation [37]:

ðλþμÞZΩð∇UuÞð∇UvÞdΩþμ

ZΩð∇u : ∇vÞdΩ¼ ρ∂tt

ZΩðuUvÞdΩ ð16Þ

where v is the test function, ∇u : ∇v¼∑ni ¼ 1∑

nj ¼ 1ð∇uÞijð∇vÞ0ij.

From the application of separation of variables and the Galerkinformulation, the following well-known differential equation forthe forced vibration F is obtained:

M €uþC _uþKu¼ F ð17Þ

The mass matrix M and the element stiffness matrix K areobtained using the shape functions in the local coordinate system(ξ,η) as [25]

K ¼ZAel

Z π

�πBTE Brdθ dA¼ ∬

1

�1

Z π

�πBTE B rdθ detJdξ dη ð18Þ

where E is the constitutive matrix contains the proper materialproperties and J is the Jacobian operator.

In order to calculate the structural response, one may computethe stiffness matrix for each Fourier harmonic of loading as B isfunctions of n and θ, and possibly different for each component ofthe total load. Therefore, superposition method becomes compu-tationally expensive particularly in the case of the concentratedload, which requires many harmonics.

To decouple the matrix from θ and n, B matrix is partitionedinto two blocks

B¼ A1

A2

� �ð19Þ

where

A1 ¼

N1;r cos nθ 0 0 …ðN1=rÞ cos nθ ðnN1=rÞ cos nθ 0 …

0 0 N1;z cos nθ …N1;z cos nθ 0 N1;r cos nθ …

266664

377775 ð20Þ

and

A2 ¼�ðnN1=rÞ sin nθ ðN1;r�ðN1=rÞÞ sin nθ 0 …

0 N1;z sin nθ �ðnN1=rÞ sin nθ …

" #

ð21Þ

A1 and A2 can be written as A1 ¼ BAnBC and A2 ¼ BBnBDwhere

BA ¼

N1;r 0 0 …ðN1=rÞ ðN1=rÞ 0 …

0 0 N1;z …N1;z 0 N1;r …

266664

377775 and BC ¼

1 0 0 ⋯0 n 0 ⋯0 0 1 ⋯⋮ ⋮ ⋮ ⋱

26664

37775 cos nθ

ð22Þ

BB ¼�ðN1=rÞ ðN1;r�ðN1=rÞÞ 0 …

0 N1;z �ðN1=rÞ …

" #and

BD ¼

n 0 0 ⋯0 1 0 ⋯0 0 n ⋯⋮ ⋮ ⋮ ⋱

26664

37775 sin nθ ð23Þ

where BA, BB are independent of n and θ.As a result of Eqs. (22) and (23), one can rewrite the strain–

displacement matrix as

B¼ BAnBC

BBnBD

� �ð24Þ

The constitutive matrix E can be partitioned as

E¼E4n4 E2n4E2n4 E2n2

" #ð25Þ

Z. Heidary, D. Ozevin / Finite Elements in Analysis and Design 86 (2014) 81–8884

The following expression is then obtained

½B�T ½E�½B� ¼ BTc B

TA BT

DBTB

h in

E4n4 E2n4E2n4 E2n2

" #n

BABC

BBBD

� �ð26Þ

K ¼ ∬1

�1

Z π

�πBTEB r dθ detJdξ dη

¼Z Z Z 1

�1ðBT

c BTAE4x4BABCþBT

DBTBE2x4BABCþBT

c BTAE2x4BBBD

þBTDB

TBE2x2BBBDÞdetJdξ dη ð27Þ

The integrations of the trigonometric terms over the circumfer-ence result inZ π

�πcos 2nθdθ¼ π n a0 ð28Þ

Z π

�πsin 2nθdθ¼ π n a0 ð29Þ

Z π

�πsin nθ cos nθdθ¼ 0 for all n ð30Þ

Using the orthogonality properties of the trigonometric functionsand performing the circumferential integration, the stiffnessmatrix becomes

K ¼ πðk1þk2ÞdetJ r na0 ð31Þwhere

k1 ¼

1 0 0 ⋯0 n 0 ⋯0 0 1 ⋯⋮ ⋮ ⋮ ⋱

26664

37775nk11n

1 0 0 ⋯0 n 0 ⋯0 0 1 ⋯⋮ ⋮ ⋮ ⋱

26664

37775 ð32Þ

k2 ¼

n 0 0 ⋯0 1 0 ⋯0 0 n ⋯⋮ ⋮ ⋮ ⋱

26664

37775nk22n

n 0 0 ⋯0 1 0 ⋯0 0 n ⋯⋮ ⋮ ⋮ ⋱

26664

37775 ð33Þ

where

k11 ¼ ∬1

�1BTAE4x4BAdetJdξdη ð34Þ

k22 ¼ ∬1

�1BTBE2x2BBdetJdξdη ð35Þ

In these equations, k11 and k22 are independent of n and θ;therefore, they are computed only once for each Fourier iteration.

This property of the stiffness matrix significantly reduces thecomputational time of large structural problems when many termsof the Fourier series are summed to represent the applied load.

2.5. Numerical time integration

The explicit time integration method requires less computa-tional time as compared to the implicit methods [25]. The diagonalmass matrix obtained from the spectral element formulationallows the use of the explicit method. The stable time step istypically recommended as 1/(20f) where f is the maximumfrequency of interest. In this study, the time step is set as 1/(25f)to reduce the influence of time integration to the output wave-forms. As the spectral element formulation yields to diagonal massmatrix, reducing the time step does not influence the computa-tional time significantly. In order to find the computationalefficiency of the numerical formulation in comparison withcommercial finite element (FEM) software, Newmark's methodwith unconditionally stable condition (i.e., average acceleration) isimplemented to solve the numerical time integration.

3. Validation of numerical formulation

A 3D circular cylinder pipe shown in Fig. 4(a) is modeled usingconventional FEM tools in order to validate the numerical for-mulation. The pipe dimensions are 20 mm thickness, 50 mmradius and 500 mm length. The 3D geometry is meshed with theresolution of 1/20th of the wavelength using the wave velocity as3200 m s�1. The 2D geometry, shown in Fig. 4(b), is divided intosquare elements using 4th degree Legendre polynomial. The mate-rial properties are 200 GPa for Young's Modulus, 7850 kg m�3 fordensity and 0.33 for Poisson's ratio. The comparison is performedusing static point load and dynamic three-cycle sinusoidal pointload with 10 kHz frequency applied at the edge of the pipe. Theother end of the pipe is modeled as the fixed boundary condition.The boundaries along the internal and external walls of the pipe areconsidered as air/solid/air.

The displacement history at the edge of the pipe obtained fromthe 2D formulation is compared with the result obtained from 3Dmodel using COMSOL software in Fig. 5. The results of both modelsmatch well.

To validate the numerical formulation under dynamic pointload, the displacement histories of the 2D model using presentednumerical formulation and the 3D model are compared for threedifferent points along the pipe line as shown in Fig. 6.

Fig. 4. The validation model. (a) 3D FEM model and (b) 2D SEM model.

Z. Heidary, D. Ozevin / Finite Elements in Analysis and Design 86 (2014) 81–88 85

The displacement histories in radial and axial directions at theloading point for the 2D model and 3D model using COMSOLsoftware are shown in Fig. 7a. Both models agree in time domainand frequency domain Fig. 7b indicates the peak frequency close to10 kHz peak frequency, similar to the excitation frequency.

In addition to COMSOL software, the 3D model is solved usingANSYS program as an additional benchmark study. Both programshave different input variables, such as element type, solvermethod, which influence the output waveform. Element typesand time integration methods for COMSOL and ANSYS are sum-marized in Table 1. The element types of COMSOL and 2D modelsare tethrahedral, and spectral element, respectively. For ANSYS,there are two options: Solid285 (lower order 3D element with fournodes having four degrees of freedoms at each node) and Solid187(higher-order 3D element having ten nodes and three degrees offreedom at each node). The comparison of ANSYS, COMSOL and 2Dmodel shows that Solid187 results in better agreement with the2D model and the 3D COMSOL model. While the time integrationmethod of ANSYS and 2D model are the same as Newmark'smethod, COMSOL has options as generalized alpha and backwarddifferential formulation (BDF). The generalized alpha method is animplicit method, which intends to increase the numerical dampingwithout having effect on the order of accuracy. The BDF method isan implicit multi-step variable-order derivatives method, which isup to the fifth order due to the absolute stability region. Asundamped models are not simulated properly with BDF method,the generalized alpha method is selected in this study. Thegeneralized alpha has a variable as high frequency amplificationfactor in the range of 0–1. In this study, the high frequencyamplification factor is varied until the COMSOL result fits wellwith the ANSYS result.

The comparison of the axial displacement histories using the2D and 3D models at the point 1 is shown in Fig. 8. Although agood agreement is found between the 2D and 3D ANSYS andCOMSOL models, especially the arrival time, magnitude and shapeof the first wave cycle, there are slight differences after the secondcycle. The differences in element types and time integration

methods are considered as the main reasons of slight differencesin the solutions of three models. It is worth mentioning that the2D result relies on the expansion of Fourier series of delta functionin the θ direction which is known as non-convergent; thereforethe concentrated load was replaced by narrow rectangular pulsewith an intensity of (2r0θ0)�1 over the distance 2r0θ0 where the2θ0 is a length of small arc [13].

The comparison of the 2D and 3D models is extended to theradial displacement of further point than the excitation location(i.e., the mid-point), and the result is shown in Fig. 9. The 2Dmodel agrees well with the 3D models. The static and dynamicloading results validate the accuracy of the 2D axisymmetricmodel for non-axisymmetric loading using spectral elements andthe method of partitioning of strain–displacement matrix. In nextsection, efficiency of implementing 2D model formulated in thisstudy is discussed.

4. Efficiency of 2D spectral element model

There are two fundamental sources that influence the compu-tational time efficiency of the numerical model: extracting massand stiffness matrices using numerical techniques and performingnumerical time integration. The computational time efficiency ofthe 2D model is controlled by reduction in degrees of freedoms tocalculate stiffness and mass matrices, decoupling θ (the angularparameter) and n (the number of Fourier terms) from strain–displacement matrix and diagonal mass matrix that enhancesnumerical time integration.

The degrees of freedoms (dofs) needed for the 2D and 3D(COMSOL) models to reach 10 kHz frequency resolution are 600and 30885, respectively. There is significant reduction in the dofswhen the axisymmetric geometry is taken into consideration, andthe spectral element is selected to model the structural domain.The difference between the 2D and 3D models increases drama-tically when the required frequency resolution is increased. It isdifficult to separate the computational time of the 3D model forthe numerical calculations of the mass and stiffness matrices, andthe time integration. However, the comparison of overall compu-tational times of the 2D and 3D models shows the efficiency of the2D model developed in this paper. For instance, the difference inthe computational time in the 2D model is about 400 s when thefrequency is increased from 10 kHz to 20 kHz while it is 3000 s forFEM. It is recorded that while reducing the time step does notinfluence the computational time of the spectral element thecomputational time increases for the classical finite elementformulation.

5. Conclusions

This paper presents an efficient formulation of the problem ofwave propagation along the length of axisymmetric pipes under

Fig. 5. The static displacements along the pipe for the 2D SEM and 3D FEM (Comsol) models.

Fig. 6. Three measurement points of displacement histories.

Z. Heidary, D. Ozevin / Finite Elements in Analysis and Design 86 (2014) 81–8886

Fig. 7. (a) The displacement histories at the edge of the pipe for the 2D SEM and 3D FEM (COMSOL) models in radial and axial directions and (b) corresponding frequencyspectra of time domain solution.

Table 1Element types and time integration methods for COMSOL, ANSYS and 2D model.

3D COMSOL model 3D ANSYS model 2D model

Element type Free tetrahederal Solid187 Spectral element (Polynomial order 4)Time integration method Generalized alpha Newmark's method Newmark's method

Fig. 8. Axial displacement histories of 2D SEM and 3D FEM models (COMSOL andANSYS) at point 1.

Fig. 9. Radial displacement histories of 2D SEM and 3D FEM models (COMSOL andANYSY) at mid-point.

Z. Heidary, D. Ozevin / Finite Elements in Analysis and Design 86 (2014) 81–88 87

non-axisymmetric loading. The axisymmetric geometry of thepipe is simplified by reducing the problem to 2D while the non-axisymmetric loading is represented by the summation of Fourierseries. The strain–displacement matrix is partitioned in such a waythat numerical integration components are decoupled from θ (theangular parameter) and n (the Fourier series terms). The numericalformulation is built using spectral elements. The orthogonalLegendre polynomials are selected to define the nodal points ofthe discretized finite element. The 2D numerical model is com-pared with the 3D model using conventional finite element soft-ware, and the accuracy of the 2D model is validated under staticand dynamic loading. Because of the numerical integration ele-ments of the stiffness matrix as independent from θ and n, and thediagonal mass matrix resulting from the use of spectral elements,long pipes could be modeled for high frequency wave propagation.As future work of this study, the numerical formulation will befurther optimized in terms of computational expenditure by usinga mixed finite element modal solution.

Acknowledgments

This research is based upon work supported by the NationalScience Foundation under Grant number ECCS 1125114. Anyopinions, findings and conclusions or recommendations expressedin this paper are those of the authors and do not necessarily reflectthe views of the National Science Foundation.

References

[1] J.L. Rose, D. Jiao, J. Spanner, Ultrasonic guided wave NDE for piping, Mater.Eval. 54 (1996) 1310–1313.

[2] R. Kirby, Z. Zletev, P. Mudge, On the scattering of torsional waves fromaxisymmetric defects in coated pipes, J. Sound Vib. 31 (2012) 3989–4004.

[3] T.M. Juliano, J.N. Meegoda, D.J. Watts, Acoustic emission leak detection on ametal pipeline buried in sandy soil, J. Pipeline Syst. Eng. Pract. 4 (2013)149–155.

[4] R.K. Miller, A.A. Pollock, D.J Watts, J.M. Carlyle, A.N. Tafure, J.J. Yezzi, Areference standard for the development of acoustic emission pipeline leakdetection technique, NDT & E Int. 32 (1999) 1–8.

[5] Y.A. Khulief, A. Khalifa, R.B. Mansour, M.A. Habib, Acoustic detection of leaks inwater pipelines using measurements inside pipe, J. Pipeline Syst. Eng. Pract. 3(2011) 47–54.

[6] K. Balk, J. Jiang, T.G. Leighton, Acoustic attenuation, phase and group velocitiesin liquid-filled pipes III: nonaxisymmetric propagation and circumferentialmodes in lossless conditions, JASA 133 (2013) 1225–1236.

[7] D. Qing-Tian, Y. Zhi-Chun, Wave propagation analysis in buried pipe conveyingfluid, Appl. Math. Model. 37 (2013) 6225–6233.

[8] M.F. Shehadeh, W. Abdou, J.A. Steel, R.L. Reuben, Aspects of acoustic emissionattenuation in steel pipes subject to different internal and external environ-ments, Part E: J. Process Mech. Eng. 222 (2007) 41–54.

[9] H. Nishino, K. Yoshida, H. Cho, M. Takemoto, Propagation phenomena ofwideband guided waves in a bended pipe, Ultrasonics 44 (2006) 1139–1143.

[10] W.B. Na, T. Kundu, Wave attenuation in pipes and its application in determin-ing axial spacing of monitoring sensors, Mater. Eval. (2002) 635–644.

[11] W. Luo, Ultrasonic Guided Waves in Wave Scattering in Viscoelastic CoatedHollow Cylinders (Ph.D. dissertation), The Pennsylvania State University, 2005.

[12] D.C. Gazis, Three-dimensional investigation of the propagation of waves inhollow circular cylinders I analytical foundation, JASA 31 (1959) 568–573.

[13] W. Zhuang, A.H. Shah, S.B. Dong, Elastodynamic Green's functions forlaminated anisotropic circular cylinders, J. Appl. Mech. (1999) 665–674.

[14] H. Bai, E. Taciroglu, S.B. Dong, A.H. Shah, Elastodynamic Green's functions forlaminated piezoelectric cylinder, Int. J. Solids Struct. (2004) 6335–6350.

[15] F. Benmeddour, F. Treyssede, L. Laguerre, Numerical modeling of guided waveinteraction with non-axisymmetric cracks in elastic cylinders, Int. J. SolidsStruct. 48 (2011) 764–774.

[16] W.J. Zhou, M.N. Inchchou, J.M. Mencik, Analysis of wave propagation incylindrical pipes with local inhomogeneities, J. Sound Vib. 319 (2009)335–354.

[17] M. Mazzotti, A. Marzani, I. Bartoli, E. Viola, Guided waves dispersion analysisfor prestressed viscoelastic waveguides by means of the SAFE method, Int. J.Solids Struct. 49 (2012) 2359–2372.

[18] D. Gsell, T. Leutengger, J. Dual, Modeling three-dimensional elastic wavepropagation in circular cylindrical structures using a finite-differenceapproach, JASA 116 (2003) 3284–3293.

[19] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method for Solid andStructural Mechanics, Butterworth-Heinemann Inc., Oxford, United Kingdom,2000.

[20] S. Nagarajan, E. Popov, Elastic–plastic dynamic analysis solids of axisymmetricsolids, Comput. Struct. 4 (1974) 1117–1134.

[21] E.L. Wilson, Structural analysis of axisymmetric solids, AIAA J. 3 (1965)2269–2274.

[22] T. Ladefoged, Triangular ring element with analytic expressions for stiffnessand mass matrix, Comput. Methods Appl. Mech. Eng. 67 (1988) 171–187.

[23] W. Wunderlich, H. Cramer, H. Obrecht, Application of ring elements in thenonlinear analysis of shells of revolution under nonaxisymmetriv loading,Comput. Methods Appl. Mech. Eng. (1985) 259–275.

[24] D.A. Bouzid, B. Tiliouine, P.A. Vermeer, Exact formulation of interface stiffnessmatrix for axisymmetric bodies under non-axisymmetric loading, Comput.Geotech. 31 (2004) 75–87.

[25] R.D. Cook, D.S. Malkus, M.E. Plesha, Concepts and Applications of FiniteElement Analysis, Wiley & Sons Inc., New York, United States, 1989.

[26] K.J. Bathe, Finite Element Procedures, Pearson Education Inc., New Jersey,United States, 1995.

[27] Z. Heidary, D. Ozevin, The selection of spectral element polynomial orders forhigh frequency numerical wave propagation, in: T.Y. Yu, (Ed.), Proceedings ofthe SPIE NDE/Smart Structures International Conference, San Diego 8694,March 2013.

[28] O.Z. Mehdizadeh, M. Paraschivoiu, Investigation of a two-dimensional spectralelement method for Helmholtz's equation, J. Comput. Phys. 189 (2003)111–129.

[29] A. Zak, M. Radzienski, M. Krawczuk, W. Ostachowicz, Damage detectionstrategies based on propagation of guided elastic waves, Smart Mater. Struct.21 (2012) 1–18.

[30] P. Kudela, M. Krawczuk, W. Ostachowicz, Wave propagation modeling in 1Dstructures using spectral element method, J. Sound Vib. 300 (2007) 88–100.

[31] C. Pozrikidis, Introduction to Finite and Spectral Element Methods UsingMATLAB, Chapman & Hall/CRC, San Diego, USA, 2005.

[32] G. Dahlquist, A. Bjorck, Numerical Methods, Dover Publications, Inc., NY, 2003.[33] R.F. Hoskins, Delta Function: An Introduction to Generalised Functions, Hor-

wood Publishing Limited, West Sussex, England, 1999.[34] S.S. Rao, Vibration of Continuous Systems, John Wiley & Sons Inc., New Jersey,

USA, 2007.[35] A. Marzani, Time-transient response for ultrasonic guided waves propagating

in damped cylinders, Int. J. Solids Struct. 45 (2008) 6347–6368.[36] D. Achenbach, Wave Propagation in Elastic Solids, Elsevier Science Publishers,

Amsterdam, 1984.[37] J.D. De Basabe, M.K. Sen, Grid dispersion and stability criteria of some

common finite element methods for acoustic and elastic wave equations,Geophysics 72 (2007) T81–T95.

Z. Heidary, D. Ozevin / Finite Elements in Analysis and Design 86 (2014) 81–8888