Accepted Manuscript

Comparison of different higher order finite element schemes for the simulation

of Lamb waves

C. Willberg, S. Duczek, J.M. Vivar Perez, D. Schmicker, U. Gabbert

PII: S0045-7825(12)00199-5

DOI: http://dx.doi.org/10.1016/j.cma.2012.06.011

Reference: CMA 9748

To appear in: Comput. Methods Appl. Mech. Engrg.

Received Date: 21 November 2011

Revised Date: 8 June 2012

Accepted Date: 13 June 2012

Please cite this article as: C. Willberg, S. Duczek, J.M. Vivar Perez, D. Schmicker, U. Gabbert, Comparison of

different higher order finite element schemes for the simulation of Lamb waves, Comput. Methods Appl. Mech.

Engrg. (2012), doi: http://dx.doi.org/10.1016/j.cma.2012.06.011

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Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25

Computer methods inapplied mechanics and

engineering

Comparison of different higher order finite element schemes for thesimulation of Lamb waves

C. Willberg∗, S. Duczek∗, J. M. Vivar Perez, D. Schmicker, U. Gabbert

Institut for Mechanics, Otto-von-Guericke-University of Magdeburg, Germany-39106 Magdeburg.

Abstract

Structural Health Monitoring (SHM) applications call for both efficient and powerful numerical tools to predict the behavior ofultrasonic guided waves. When considering waves in thin-walled structures, so called Lamb waves, conventional linear or quadraticpure displacement finite elements soon reach their limits. The mesh density required to obtain good quality solutions has to berather fine both spatial and temporal. This results in enormous computational costs (computational time and memory storagerequirements) when ultrasonic wave propagation problems are solved in the time domain. To resolve this issue several higherorder finite element methods with polynomial degreesp > 2 are proposed. The objective of the current article is to developsuch higher order schemes and to verify their capabilities with respect to accuracy and numerical performance. To the best of theauthors’ knowledge such comparison has not been reported in literature, yet. Specifically, spectral elements based on Lagarangepolynomials (SEM), p-elements using the normalized integrals of the Legendre polynomials (p-FEM) and isogeometric elementsutilizing non-uniform rational B-splines (NURBS, N-FEM) are discussed in this paper. By solving a two-dimensional benchmarkproblem, their advantages and drawbacks with respect to Lamb wave propagation are highlighted. The results of the convergencestudies are then used to derive guidelines for estimating the optimal element size for a given finite element type and polynomialdegree template. These findings serve the purpose to determine the optimal mesh configuration a priori and thus, save a considerableamount of computational effort. The proposed guideline is then tested on a three-dimensional structure with a conical hole showingan excellent agreement with the predicted behaviour.

c© 2011 Published by Elsevier Ltd.

Keywords: Lamb wave analysis, higher order finite elements, isogeometric finite elements, p-FEM, spectral finite elements,convergence studies

1. Introduction

The application of elastic guided waves to inspect structures has a long history, and is nowadays widely employedfor online monitoring purposes [1, 2]. In 1917 Horace Lamb mathematically predicted a special type of these wavesoccurring in thin-walled designs [3]. Named after its discoverer, Lamb waves refer to elastic perturbations propagat-ing in elastic solid plates (or layers) with free boundaries. The direction of the displacements is both parallel to themidplane of the plate and perpendicular to it [4]. Two basic types of Lamb wave modes can be distinguished in anhomogenous, isotropic plate, namely symmetric and anti-symmetric ones. For each excitation frequency a number ofpropagating modes exists. They correspond to the solution of the mathematical model description of Lambs problem

∗Corresponding authors.Email addresses:christian.willberg@ovgu.de (C. Willberg),sascha.duczek@st.ovgu.de (S. Duczek)

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 2

[2]. Both modes of these waves are highly dispersive [5] and can furthermore convert into each other [6]. Despite theircomplex propagation characteristics there are certain properties which make them interesting for SHM applicationsand account for their wide spread use. Firstly, their small wavelengths, in a higher frequency range, and secondly, onlya slight loss of amplitude magnitude make them very popular for online monitoring applications. Small wavelengthsare required to ensure the interaction of Lamb waves with structural damages, like cracks or flaws. The geometricalattenuation is only proportional to 1/

√r [7], wherer is the travelled distance from the source. As a consequence

Lamb wave based damage detection devices are a very attractive and a common choice for SHM systems [8].The simulation of ultrasonic Lamb wave propagation is a highly demanding task from a computational point of view.It requires both a fine temporal [9] and spatial [10, 11] discretization to capture the different wave modes. In thisregard analytical methods [12], semi-analytical and wave finite element methods (SAFE, WFE) [5, 13, 14, 15] offerfast and accurate results. Therefore, they are frequently used to calculate dispersion diagrams. But these methods arenot able to analyse complex three-dimensional geometries and arbitrary perturbations of the waveguide. The wavepropagation in structures containing failures, such as delaminations, cracks or other defects, are hard to be describedanalytically. Also the WFE-approach, which is more flexible as the SAFE-method, requires a periodic structure to beemployable [5]. Furthermore, these methods are numerically more expensive if the global behaviour of the structureis to be analysed, as the computational effort per degree-of-freedom is higher compared to conventional FEM [16].An extension of semi-analytical finite element methods has been published by Gopalakrishnan et al. [17, 18]. Thisapproach can be thought of as finite element method formulated in the frequency domain. While linear wave analysisof simple geometries is shown to be solved very efficiently even for the higher order modes, non-linear effects likethe contact between debonded surfaces and delaminations cannot be treated, since the problem is solved in frequencydomain - Fast Fourier transform (FFT) is only viable for linear systems. Moreover, if transient time-domain solutionsare wanted the calculation time increases significantly in order to avoid wrap around errors [16].Considering problems dealing with three-dimensional, complex geometries it is in general inevitable to implementdiscretization techniques in all three spatial directions and also in time. When dealing with thin-walled structures anatural approach is to deploy finite shell elements to discretize the model. This type of finite elements is founded ona dimensionally reduced theory and has numerical advantages when thin-walled component parts are to be examined.Approaches of this kind have been proposed by Ostachowicz et al. [19, 20, 21] and Fritzen et al. [22, 23]. Since thevariation of the displacement field over the thickness of the structure is neglected, the symmetric Lamb wave modescannot be resolved. Additionally, they have the drawback that multi-layered materials and complex three-dimensionalstress states arising at welded joints or rivets, for example cannot be captured easily. Hence, approaches based onutilizing higher order shell finite elements [19, 24, 25, 26] are also not suitable if all observed phenomena are to becaptured.While certain representatives of three-dimensional approaches like the local interaction simulation approach (LISA)by Lee et al. [27] or the mass-spring lattice model (MSLM) by Yim et al. [28] are also confined to non-complex ge-ometries, finite element methods (FEM) in general offer a broader variety of applications and are not limited to specialassumptions, such as e.g. material parameters or geometrical regularities. Since the convergence rate of conventionallower order FEM formulations (h-FEM) is rather low, also when dealing with Lamb wave propagation problems, inrecent years the focus of research has shifted to the implementation of high-order shape functions. In the literaturea large variety of higher order shape-functions has been proposed, such as the Lagrange-polynomials on the Gauss-Lobatto-Legendre grid also referred to asSpectral Element Method(SEM) in the time-domain [29, 21, 30] (the termSEM is used according to Ostachowicz et al. [26]), the normalized integrals of the Legendre-polynomials resultingin the hierarchical p-FEM [31, 32, 33, 34, 35], and the application of Non-Uniform Rational B-Splines (NURBS)termedN-FEM [36, 37, 38]. Heretofore, the SEM has been used almost exclusively for high frequency wave propa-gation problems, and the other two mentioned approaches have been principally utilized for static problems includingnon-linear analyses, plasticity etc.Since our goal is to describe the Lamb wave behavior in arbitrary geometries while confining the computational coststo a realizable extent, in our opinion only the higher order finite element methods are a recommendable choice. Themutual benefits and drawbacks of the SEM, p-FEM and N-FEM regarding the application to Lamb wave propagationproblems have to the best of our knowledge not been analysed so far. Consequently the objective of this paper is togive a quantitative comparison of these methods in the time domain. The benchmark problem is a two-dimensionalLamb wave propagation in an isotropic plate. This example has been chosen as an analytical solution is given in theliterature [2]. The intention of this benchmark is the characterization of the convergence properties and the numerical

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 3

effort of the three proposed higher order finite element approaches. This enables the user to quantify the performanceand the accuracy of these methods in analyzing Lamb wave propagation problems for SHM applications. While dif-ferent spatial discretization techniques are tested, the same time integration scheme is applied to all analyzed cases,ensuring a good comparability of the results. Hence, the computational times are not directly evaluated, since theapplied time-integration scheme is not necessarily best suited for each of the analysed finite element approaches. Onthis account, we compare the degrees-of-freedom required to achieve a certain level of accuracy, by the different ap-proaches. In addition, the number of non-zero elements in the system matrices are examined, measuring the memorystorage requirements of each method.The paper is divided into three main parts. In the first part the main principles of the finite element method and theanalysed shape function types, namely the Lagrange polynomials, the normalized integrals of the Legendre polyno-mials and the NURBS are introduced. Then, the model setup is described and the results of the convergence studyare interpreted and discussed. To this end, optimal discretization schemes for the three higher order FE-approachesare proposed. These schemes are applied to a three-dimensional problem for verification reasons. Finally, the paperis concluded and an outlook to future research activities is given.

2. Finite element equations

The basis of the finite element developments is the variational formulation corresponding to Naviers equations,namely Hamiltons principle. It states that the motion of the system within the time interval [t1, t2] is such, that underinfinitesimal variation of the displacements the Hamiltonian actionS vanishes, meaning that the motion of the systemtakes the path of the stationary action [39]

δS = δ

t2∫

t1

(L + W) dt = 0 . (1)

HereL represents the Lagrangian function, andW the work done by the external forces. The Lagrangian is the sumof the kinetic energy and the potential strain energy. After some calculus and the substitution of Hookes law into Eq.(1), we obtain

0 = −∫

V

[ρδuT u + δεTCε

]dV

︸ ︷︷ ︸δL

+

∫

V

δuTFV dV +

∫

S1

δuTFS1 dS1 +

n∑

i=1

δuTi Fi

︸ ︷︷ ︸δW

, (2)

whereρ is the mass density,ε andσ are the vectors of mechanical strains and stresses in Voigt-notation, respectively[40]. C denotes the elasticity matrix andu stands for the acceleration vector. The displacement fieldu(x, t) is approx-imated by the product of the space-dependent shape function matrixN(x) and a time-dependent vector of unknownsU(t),

u(x, t) = N(x)U(t). (3)

The mechanical strain is defined as

ε = DNU = BU, (4)

introducing the strain-displacement matrixB = DN, whereD is a linear differential operator relating strains anddisplacements. With the aid of Eq. (2), (4) and (3) and the reasoning that Hamiltons principle has to be satisfied forall variationsδu = N δU we obtain the well known system of equations, describing the motion of a body

ρ

∫

V

NTNdV

︸ ︷︷ ︸M

U +

∫

V

BTCBdV

︸ ︷︷ ︸K

U =

∫

V

NTFVdV +

∫

S1

NTFS1dS1 + NTFP

︸ ︷︷ ︸f

. (5)

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 4

Introducing the mass matrixM , the stiffness matrixK and the load vector of the external forcesf , Eq. (5) reads

MU + KU = f . (6)

For further explanations, e.g. including the influence of damping in the finite element method, the reader is referredto standard text books on this subject by Zienkiewicz [40], Bathe [41] and Hughes [42], for instance.

3. Shape functions

3.1. Spectral finite element method and Lagrange polynomials

Spectral finite elements are based on a set of Lagrange polynomials utilizing a specific nodal distribution on theinterval [−1,+1]. The (p+ 1) one-dimensional basis functions are formally defined by

NLagrange, pn (ξ) =

p+1∏

j=1, j,n

ξ − ξ j

ξn − ξ j, n = 1, 2, . . . , p+ 1. (7)

Therein the nodal distribution considered in this articleξi with i = 1, . . . , (p+ 1) is

ξi =

−1 if i = 1ξ

Lo,p−10,i−1 if 2 ≤ i < p+ 1+1 if i = p+ 1

. (8)

Determined by the set of rootsξLo,p−10,a with a = 1, . . . , (p− 1) of the (p− 1)-order Lobatto polynomial

Lop−1(ξ) =1

2pp!

dp+1

dξp+1

[(ξ2 − 1

)p]. (9)

This nodal distribution, referred to as Gauss-Lobatto-Legendre (GLL) grid, offers in conjunction with the GLL-quadrature rule the possibility to diagonalize the mass matrix in an elegant way [29, 43, 30]. This lumping procedureprovides significant savings in computational time if an explicit time integration scheme is used, while much lessmemory is required to save the inverted mass matrix. In addition, it is shown in [30] that in case of a mesh with undis-torted elements the diagonalization also minimizes the round-off errors. Spectral elements employing the GLL-gridhave been proposed by Komatitsch and Tromp [29], Zak et al. [43], Kudela et al. [19], Peng et al. [44] and Ha andChang [45] in order to mention just a few.

3.2. p-Version of the finite element method and normalized integrals of the Legendre polynomials

To construct higher order ansatz functions according to the p-version of FEM Szabo and Babuska have favoreda hierarchic basis, meaning that all shape functions of lower order are included in the set of higher order shapefunctions. The presentation of those shape functions follows [32] and [33] closely. Generally speaking, hierarchicshape functions are based on the normalized integrals of the Legendre polynomials.Following the approach first established by Szabo and Babuska [33] and seized by many others [46, 47, 32, 48, 49]it is shown how hierarchic basis functions can be generated up to any desired polynomial degree. The p-version ofthe finite element method presented here is based on Legendre polynomialsLn(x). They satisfy Bonnets recursionformula

Ln+1(x) =1

(n+ 1)[(2n+ 1)xLn(x) − nLn−1(x)] , n = 1, 2, 3, . . . − 1 ≤ x ≤ 1 (10)

which is very well suited for a computer implementation. Heren denotes the polynomial degree. If Equation (10) isused to generate the Legendre polynomials, the recursion must start with the first two polynomials

L0(x) = 1, (11)

L1(x) = x. (12)

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 5

In order to construct shape functions forp-elements the following normalized integrals of the Legendre polynomialsΦn(ξ) (modified Legendre polynomials) are used

Φn(ξ) =

√2n− 1

2

ξ∫

x=−1

Ln−1(x)dx=1

√4n− 2

[Ln(ξ) − Ln−2(ξ)

], n = 2, 3, . . . . (13)

Thus, the value of the modified Legendre polynomials is equal to zero at the boundariesξ = ∓1. The linear Lagrangepolynomials constitute the first two shape functionsNLegendre

1 (ξ) andNLegendre2 (ξ) as

NLegendre1 (ξ) = NLagrange,1

1 (ξ) =1− ξ

2, (14a)

NLegendre2 (ξ) = NLagrange,1

2 (ξ) =1+ ξ

2. (14b)

They are commonly called nodal modes or nodal shape functions. They can be thought of as being external, sincethey are responsible for ensuringC0-continuity between adjacent elements. Higher order ansatz functionsNn(ξ) ofpolynomial degreep are generated from Equation (13) in the following way

NLegendren (ξ) = Φn−1(ξ), n = 3, 4, . . . , p+ 1. (15)

They are also referred to as internal modes or bubble modes, because the corresponding degrees-of-freedom are notdirectly related to the internal degrees-of-freedom of adjacent finite elements

NLegendren (∓1) = 0, n = 3, 4, . . . , p+ 1. (16)

An important property of the hierarchic set of shape functions, which it derives from the orthogonality of the Legendrepolynomials, is the condition

+1∫

−1

dNLegendren

dξdNLegendre

m

dξdξ =

1 if n = m,

0 if n , m.(17)

This property results in a sparse structure of the stiffness matrix, which is the reason why the normalized integralsand not the polynomials themselves are used to construct shape functions for higher order approaches. The numericalintegration is done using a Gaussian quadrature rule withp+ 1 points in every direction. More efficient schemes (e.g.vector integration proposed by Hinnant [50]) are out of the scope of this article.

3.3. Isogeometric finite element method and non-uniform rational B-splines

In the following section, the shape functions that define an isogeometric element are introduced. Fig. 1 shows anexample of a two-dimensional isogeometric finite elements. The geometry is described using non-uniform rationalB-splines (NURBS) in a cartesian coordinate system [x1, x2]. NURBS are originally defined in parameter coordinates[β, ξ]. To perform a numerical integration (Gauss quadrature) a third set of coordinates, the so called local coordinates,[β, ξ], is introduced. Consequently, two Jacobian transformation matricesJ1 andJ2 have to be taken into account (seeFig. 1). Even though in the literature more efficient quadrature schemes are proposed for NURBS [51], here thestandard Gauss quadrature is applied, since the numerical integration is not in the scope of our current research. Theover-integration does not worsen the quality of the solution nor its convergence behaviour.

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 6

-1, -1

1, 1

1, -1

-1, 1

x1

x2

β

ξ

ηΩe

Ωe

Ωe

J−11

J−12

γ

Figure 1: Finite element using NURBS shape functions.

A B-spline basis is comprised of piece-wise polynomials joined with prescribed continuity conditions. To definea one-dimensional B-spline of polynomial orderp, one needs to understand the notion of a knot vectorV. A knotvector is a set of coordinates in a parametric space, written as

V = [β0, β1, β2, .., βncont+p, βncont+p+1] with βi ≤ βncont+p+1 , (18)

wherei is the knot index,i = 0, 1, ..., ncont + p + 1, βi is the ith knot andncont is the total number of control points[52]. There are various ways to define B-spline basis functions, but for computer implementation the application ofa recurrence formula is an appropriate way [53, 54]. The first order basis functionsR(β)i,0 of the polynomial degreep = 0 are

R(β)i,0 =

1, if β ∈ [βi , βi+1)0, otherwise.

(19)

The basis functionsRi,p(β) of higher ordersp > 0 are defined as

Ri,p(β) =β − βi

βi+p − βiRi,p−1(β) +

βi+p+1 − β

βi+p+1 − βi+1Ri+1,p−1(β) . (20)

The indicesi andp denote thei th basis function of polynomial orderp. Utilizing the B-spline basis functionsRi,p(β),the NURBS basis function can be expressed in the following form

NNURBS,pi (β) =

Ri,p(β)wincont∑

j=1Rj,p(β)wj

, i = 1, 2, . . . , p+ 1, (21)

wherewi are weights corresponding to each functionRi,p. An arbitrary NURBS curve is expressed as [55]

X(β) =ncont∑

i=1

NNURBS,pi (β)Pi . (22)

X is the position vector of the described curve andPi are control points in global cartesian coordinates [x1, x2]. ANURBS curve can be interpreted as projection of a B-spline curve fromRn+1 on a defined surface inRn [56]. Thisprojection is controlled by weight parameterswi . If these weights are non-negative the curve lies in the convex hullof the control polygon [52]. Another feature of NURBS is that they are projection invariant. To develop a two-dimensional finite element we need a two dimensional NURBS formulation. For the derivation of these functions werefer to Cottrell, Piegl, et al. [56, 55].

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 7

3.4. Comparison of ansatz functions

In this subsection basic properties of the chosen higher order shape functions are identified and compared to oneanother. The curves in Figure 2 illustrate the presented shape functions of an one-dimensional element correspondingto the polynomial degrees 2, 3 and 4. While the Lagrange and modified Legendre graphs only cover the domain of oneelement, the NURBS plots involve two elements in order to show their transition behavior at the elements boundary.One easily notes the higher order of continuity (Cp−1) at the boundary.On the left column, it can be seen that the standard Lagrange polynomials using a GLL nodal distribution do notexceed the value of+1, which is the reason why for interpolation problems utilizing this no nodal distribution noRunge-phenomenon is encountered. The approach based on these polynomials is commonly referred to as SEM. Itshould be mentioned that when using this formulation the solution converges exponentially. Furthermore, these poly-nomials minimize the interpolation error in the Lebesgue norm. However, if the discretization is changed both themass and stiffness matrix have to be re-computed.The modified Legendre polynomials, constituting the basis for the p-FEM, are depicted in the middle column. Againthe convergence rate is spectral when applying p-version finite elements. Another feature is that they constitute ahierarchical basis. That is to say, shape functions of lower polynomial degrees are also present in higher order ap-proaches. This is an attractive feature of this shape functions set, since the hierarchy of the basis functions has also animmediate consequence on the the structure of the resulting system matrices. All system matrices corresponding tothe polynomial orders 1 top− 1 are sub-matrices of the system matrix corresponding to polynomial degreep. Sincethe degrees-of-freedom do not retain a physical meaning a more advanced post-processing is needed in comparison tothe SEM-approach. From this modal formulation a nodal one can be obtained using a conversion matrix as suggestedby Pozrikidis [57].The right column displays the NURBS basis functions, which have higher order continuity between element bound-aries as can be seen atβ = 0.5. This is achieved by relating several internal degrees-of-freedom of adjacent elementsexplicitly. The higher continuity further guarantees a faster convergence of the N-FEM solution compared to theother two approaches. Thus, less degrees-of-freedom are required. Regarding the interpolation error NURBS are ableto represent any geometry exactly and hence interpolation errors are negligible. In addition, all function values arepositive in the element domain and each graph has at most one local maximum while the sum of all polynomials atan arbitrary locationβ is equal to one. An unfavourable property is a higher condition number of the stiffness matrixcompared to SEM and p-FEM, which is disadvantageous if iterative equation solver are [32].

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 8

-0.4

0

0.4

0.8

1.2

-0.8 -0.4 0 0.4 0.8

ξ

(a) Lagrangep = 2

-0.4

0

0.4

0.8

1.2

-0.8 -0.4 0 0.4 0.8

ξ

(b) modified Legendrep = 2

00.20.40.60.8

1

0 0.2 0.4 0.6 0.8 1

β

(c) NURBSp = 2

-0.4

0

0.4

0.8

1.2

-0.8 -0.4 0 0.4 0.8

ξ

(d) Lagrangep = 3

-0.4

0

0.4

0.8

1.2

-0.8 -0.4 0 0.4 0.8

ξ

(e) modified Legendrep = 3

00.20.40.60.8

1

0 0.2 0.4 0.6 0.8 1

β

(f) NURBS p = 3

-0.4

0

0.4

0.8

1.2

-0.8 -0.4 0 0.4 0.8

ξ

(g) Lagrangep = 4

-0.4

0

0.4

0.8

1.2

-0.8 -0.4 0 0.4 0.8

ξ

(h) modified Legendrep = 4

00.20.40.60.8

1

0 0.2 0.4 0.6 0.8 1

β

(i) NURBS p = 4

Figure 2: Comparison of different order ansatz function sets. The dashed central line in the figures of the third columnconcerning the NURBS shape functions marks the boundary of two adjacent elements. Observe the higher continuity(Cp−1) between adjacent isogeometric finite elements, while spectral- and p-element shape functions exhibit onlyC0-continuity at their element boundaries.

In contrast to the SEM-approach, the degrees-of-freedom of the p-version of FEM and N-FEM do not describenodal displacements. Hence, for these two methods a more involved post-processing is needed to evaluate the me-chanical displacement field, for instance. In Table 1 a comparison regarding the properties of the presented one-dimensional shape function sets is presented. The selection of which method to deploy is application dependent andis one of the objectives of this work.

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 9

Table 1: Comparison of the properties of different types of ansatz functions for one-dimensional case.

SEM p-FEM N-FEM

Naming convention NLagrange, pn NLegendre, p

n NNURBS, pn

Inter element continuity C0 C0 Cp−1 or less

Standard element domain W = [−1,1] W = [−1,1] W = [0,0.5] orW = [0.5,1]

Interpretation of the degrees-of-freedom

mechanical dis-placements

unknowns of shapefunction ansatz

control pointdisplacements

Common degrees-of-freedomof adjacent elements

1 1 p or less

Function values can be negative Yes Yes No

For a detailed discussion of two- and three-dimensional shape functions the reader is referred to the monographby Ostachowicz et al. [26] for the time-domain spectral element approach, to the works of Szabo and Babuska [33]for the p-version of the finite element method and to the textbook by Cottrell et al. [56] for isogeometric analysis,respectively.

4. Numerical model and methodology to interpret the quality of the FEM solution

4.1. Model definition

In order to compute ultrasonic wave propagation efficiently, it is essential to have a guideline on how to choose thepolynomial degree and/or the mesh density for a given problem. In the subsequent convergence studies the influenceof the discretization on the quality of the numerical solution is shown. For this purpose, a two-dimensional planestrain model is considered. Physical damping is not included as it does not influence the convergence behaviour. Thegeometry and the boundary conditions are depicted in Fig. 3.

x2

x1

d

la

lp

lb

A BF1(t)

F2(t)

Figure 3: Two-dimensional model with loads and boundary conditions for the convergence study. Two point forcesF1(t) = F sinωt sin2

(ωt2n

)andF2(t) = aF1(t) are applied, witha = 1 for the excitation of a purely symmetric Lamb

wave mode (S0) anda = −1 if the anti-symmetric mode (A0) is considered. The dimensions of the aluminium (seeTable 2) plate are:la = 100mm, lb = 200mm, l p = 500mm, d = 2mm.

The length of the aluminum made plate is given asl p = 0.5m, the thickness asd = 2mm. The material propertiesare summarized in Table 2. The length of the plate guarantees that no reflection from the right boundary are affectingthe signals at the points of measurement during the simulation time.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 10

Table 2: Material data for aluminum.

Youngs modu-lus (E)

Poissonsratio (ν)

Massdensity(ρ)

Longitudinalspeed (c1)

Transversalspeed (c2)

7× 1010N/m2 0.33 2700kg/m3 6197m/s 3121m/s

Lamb waves are excited using a pair of concentrated line loads on both, the top and the bottom surfaces of the plate.In the two-dimensional model these loads are modeled as point forces acting in thex2-direction. Their time-dependentamplitudes follow a sine burst signal given by

F(t) = F sinωt sin2(ωt2n

), (23)

whereω = 2π f denotes the central circular frequency. This kind of pulse has the advantage that the frequency contentis narrow-banded. The number of cycles within the signaln determines the width of the excited frequency bandaround the central frequencyf . For a narrow frequency bandwidth, n has to be chosen high. Applying concentratedloads at the top and bottom surfaces allows us to exploit the advantages of a mono-modal excitation, which meansthat only a single mode is present at a time iff ∙ d < 1.5 MHzmm(see Fig. 4). In order to generate a signal containingonly theA0-mode both forces have to act in the same direction, meaning they have to be in-phase. If the two loads areout-of-phase a pureS0-mode is generated. An excitation frequency off = 477.5kHz is chosen because around thisfrequency value the dispersion-effect is comparatively low for both modes regarding a plate thickness ofd = 2mm(see Fig. 4). Furthermore, to keep the frequency range rather narrow-banded the excitation signal contains 32 cycles(n = 32).

A1

A0

2.40.955

S0

f d [MHzmm]

c p[k

m/s]

2.521.510.50

10

8

6

4

2

0

(a) phase velocity

A1

A0

2.40.955

S0

f d [MHzmm]

c g[k

m/s]

2.521.510.50

5

4

3

2

1

0

(b) group velocity

Figure 4: Phase and group velocity dispersion curves for an aluminum plate (see Table 2); dashed curves denote theanti-symmetric modes (Ai) and solid lines the symmetric modes (Si). The vertical dashed line corresponds to thecentral frequency-thickness product utilized to excite the Lamb wave propagation.

At the left boundary (x1=0) of the model symmetric boundary condition are applied in order to exploit the sym-metry of the problem. This set up fulfills the assumptions made in the analytical solution published by Giurgiutiu[2]. The convergence behavior of the numerical results is evaluated with respect to this reference solution. The timehistory of the displacement field is saved at the location of the two points A (uA) and B (uB) at the top surface (Fig.3). The first one is located atx1 = la and the second atx1 = lb.

4.2. Methodology to evaluate the quality of the numerical results

To determine the quality of the finite element solution the time-of-flighttc of the propagating Lamb wave packetbetween the points A and B is utilized (Fig. 5). The time-of-flight computed using the finite element method (tcnum,type)

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 11

is compared to the value given by the analytical solution (tcana). In order to extract this value from the finite elementor analytical solution, the time signal of the displacement at the regarded point (A or B) is subjected to an Hilberttransform

HA,B(u(t)) =1π

∞∫

−∞

uA,B(τ) ∙1

t − τdτ. (24)

Using the Hilbert transformHA,B of the time-dependent signaluA,B, the envelopeA,B of the time displacement historycan be computed. The time-of-flight is then evaluated by comparing the position of the centroid of the envelop of thetime signal at these two points. The envelop can be computed deploying the following relation

eA,B(t) =√

HA,B(uA,B(t))2 + uA,B(t)2 (25)

and the coordinate of its centroid is obtained computing the statical moment of the envelop

tA,B =

tend∫

0

eA,B(t) ∙ t dt

tend∫

0

eA,B(t) dt

. (26)

The time-of-flighttc between points A and B serves as a measure of the quality of the numerical results. It can becomputed as the difference

tc = tB − tA . (27)

Basically using this methodology the comparison of time-of-flights equals the evaluation of resulting group speeds,since dispersive effects are almost excluded due to the narrow bandwidth of the excitation signal and due to the con-stant distance between the evaluation points. However, the remaining small dispersive effects enter the analytically(tcana) and numerically (tcnum,type) computed time-of-flight parameters to an equal extent, hence the presented conver-gence indicator is able to quantify the Lamb wave behaviour even for very low orders of the errors magnitude.

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0 6.0 8.0 10 12 14

u[ 10−

4m]

t[10−4s

]

tc

u1 at point A u1 at point B

Figure 5: Time-of-flight as difference of the centroid of two Hilbert envelopes at point A and B for theS0-mode.

4.3. Discretization set up

In order to isolate the influence of the discretization inx1- andx2-direction with respect to the quality of the results,two different convergence studies are executed.Firstly, the discretization in the direction of the travelling wave is investigated (see section 5.1) using a variablenumber of degrees-of-freedom inx1-direction and a fixed amount inx2-direction. It has been found that it is sufficient

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 12

to discretize the thickness of the plate utilizing one finite element with a fixed polynomial degree ofpx2 = 4 [30].Thus, the discretization over the thickness of the plate will not noticeable pollute the computed time signal, as thedisplacement field through the thickness of the plate is well resolved by utilizing a quartic polynomial. To determinethe influence of the discretization inx1-direction the number of finite elements in this direction is varied using ah-refinement for the spectral and p-elements and ak-refinement [58] for the isogeometric elements. This refinementis repeated for various polynomial degreespx1 = 2, 3 . . . , 6. To evaluate the quality of the numerical solution ofthe different finite element approaches the relative error depending on the number of “nodes” per wavelength (χ) iscomputed. For the case of SEM these “nodes” correspond to actual nodes at known locations. When dealing withN-FEM or p-FEM they do not represent geometrical nodes. These “nodes” do not retain any physical meaning (seeTable 1) and are only a distinct measure of the mesh density. For a given model discretized with a regular mesh thevalue ofχ is determined using

χS0,A0 =degrees-of-freedom

2(px2 + 1)l p∙ λS0,A0. (28)

px2 denotes the polynomial degree inx2-direction,l p stands for the length of the plate andλS0 orλA0 are the wavelengthof the symmetric and anti-symmetric Lamb wave mode, respectively. The wavelengths of theA0- andS0-mode aretaken from the phase velocity curves (Fig. 4a) atf = 477.5kHz. The relative error is determined comparing thenumerical to the analytical solution. This can be expressed by

Erel =tcana − tcnum,type

tcana

∙ 100 [%]. (29)

Secondly, the impact of the discretization inx2-direction is investigated (see section 5.2). In order to quantify theinfluence of the discretization over the thickness of the plate, the numerical model is discretized utilizing a fixednumber of elements both inx1- andx2-direction. The discretization in the direction of the travelling wave is chosenaccording to the results of the first convergence study such that it ensures an unaffected evaluation of the computedresults. Thus, a discretization utilizing more than 20 “nodes” per wavelength and a polynomial degree ofpx1 = 6 ischosen. The convergence study is performed conducting ap-refinement withpx2 = 1, . . . , 8. To evaluate the qualityof the numerical solution of the different finite element approaches the relative error depending on the polynomialdegreepx2 is computed.After determining the optimal discretization parameters of the three presented finite element approaches these methodsare compared with standard linear elements, utilizing both fully integrated and selectively reduced integrated elements(see section 5.3).The numerical time integration scheme is only adjusted such that the effects of an inaccurate time integration arenegligible and thus barely contribute to the numerical errors. In this context, preliminary studies concerning differenttime integration schemes have shown that an explicit fourth order Runge-Kutta type time integration method offers thebest accuracy compared to a central difference and a standard Newmark method. Thus, for our convergence studiesthe ode45-solver provided in MATLAB is used. Hence, only the quality of the spatial discretization is evaluated.The results and conclusion are discussed in detail in the following section where also guidelines are derived to estimatethe quality of the numerical solution using different higher order approaches a priori.

5. Results and discussion

5.1. Discretization in the direction of the wave propagation

Fig. 6 and 7 display the results of the convergence study with respect to a varying polynomial degree in globalx1-direction. The first summarizes the findings obtained for theA0- and the latter for theS0-mode, respectively. Thecurves are steady except for small peaks experienced in the convergence curves of SEM and p-FEM. If in contrastto theCp−1-continuous isogeometric finite elements deployed in our convergence studiesC0-continuous ones areinvestigated the aforementioned behaviour is also observed. This behaviour can be attributed to local element eigen-frequencies, as conclusively explained in [30]. All curves converge faster for higher polynomial orders. It has to benoted that the convergence rate of N-FEM elements is significantly higher than for SEM and p-FEM ones. The lattertwo approaches show a very similar convergence behaviour. Additional studies indicate that the convergence rate of

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 13

C0-continuous isogeometric finite elements coincides with the results gained by using SEM and p-FEM. Thus, theconvergence rate merely depends on the highest complete polynomial ansatz, as known from static analysis [59]. Therecent findings are consistent with the statement given in [60] that the k-method has better approximation propertieson a per degrees-of-freedom basis than the p- or/and h-version of FEM.Considering the accuracy of the solution we observe that relative error reaches the same order of magnitude for allthree higher order finite element approaches. However, the p-elements offer the highest absolute accuracy concerningnumerical results of theA0-mode obtained for a polynomial degree distribution utilizingpx2 ≤ 4 andpx1 ≥ 4.Assuming that a relative error threshold of 1 % is acceptable from an engineering point of view all proposed higherorder finite element approaches need considerably less than 20 “nodes” per wavelength, typically mentioned in litera-ture [10]. Since for the considered excitation frequency (477.5kHz) the wavelength of theS0-mode is almost twice aslong as that of theA0-mode and the error threshold is met for similar values of “nodes” per wavelength (χA0 ≈ χS0).Thus the convergence curves of the anti-symmetric mode are generally of greater interest when judging the quality ofthe overall solution. In reality, both types of Lamb waves are excited even if collocated actuators are used. Then themesh parameters have to be chosen such that the anti-symmetric mode is resolved.In Fig. 8 the number of non-zero elements (nnz) of the system matricesK andM , for each polynomial degree whenreaching the mentioned threshold for the anti-symmetric mode case, is displayed. All values are normalized withrespect to the spectral element solution for the polynomial degreepx1 = 2. The number of non-zero elements servesas a measure of the memory storage requirements for each method. As can be inferred from the graphspx1 = 3 isthe optimal choice for all considered higher order approaches. Due to their fast convergence N-FEM elements requirethe least degrees-of-freedom which results in a reduced demand of memory to store the system matrices. The storagerequirements of the spectral finite elements and p-elements do not differ noticeably.

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 14

10−3

10−2

10−1

100

101

102

103

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χA0 [−]

SEMp-FEMN-FEM

(a) px1=2

10−3

10−2

10−1

100

101

102

103

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χA0 [−]

SEMp-FEMN-FEM

(b) px1=3

10−3

10−2

10−1

100

101

102

103

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χA0 [−]

SEMp-FEMN-FEM

(c) px1=4

10−3

10−2

10−1

100

101

102

103

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χA0 [−]

SEMp-FEMN-FEM

(d) px1=5

10−3

10−2

10−1

100

101

102

103

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χA0 [−]

SEMp-FEMN-FEM

(e) px1=6

Figure 6: Convergence curve for theA0-mode.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 15

10−4

10−3

10−2

10−1

100

101

102

103

104

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χS0 [−]

SEMp-FEMN-FEM

(a) px1=2

10−4

10−3

10−2

10−1

100

101

102

103

104

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χS0 [−]

SEMp-FEMN-FEM

(b) px1=3

10−4

10−3

10−2

10−1

100

101

102

103

104

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χS0 [−]

SEMp-FEMN-FEM

(c) px1=4

10−4

10−3

10−2

10−1

100

101

102

103

104

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χS0 [−]

SEMp-FEMN-FEM

(d) px1=5

10−4

10−3

10−2

10−1

100

101

102

103

104

2 4 6 8 10 12 14 16 18 20

Ere

l[%

]

χS0 [−]

SEMp-FEMN-FEM

(e) px1=6

Figure 7: Convergence curve for theS0-mode.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 16

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

2 3 4 5 6

nnz(

K)

norm

aliz

ed

px1

SEMp-FEMN-FEM

(a) Normalized number of non zero elements (nnz) in K

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

2 3 4 5 6

nnz(

M)

norm

aliz

ed

px1

SEMp-FEMN-FEM

(b) Normalized number of non zero elements (nnz) in M

Figure 8: Normalized memory storage requirements for the models reaching the relative error threshold of 1 %.

5.2. p-Refinement over the thickness of the plate

After the optimal polynomial degree in the direction of the travelling wave has been determined a second conver-gence study is conducted in order to determine the optimal polynomial order in thickness direction of the plate. Thediscretization is set up as described in section 4.3. The results of the simulations are depicted in Fig. 9. The curvesare steady except for a small overshoot for spectral finite elements. It can be seen that the convergence rate of thep-version finite elements is the highest. Concerning the accuracy all proposed higher order methods reach a similarrelative error atpx2 = 6 regarding theA0-mode and atpx2 = 4 concerning theS0-mode. The difference in the minimalrelative error between SEM or NFEM and p-FEM is decreased for higher polynomial degrees inx2-direction untilboth approaches are virtually coincident. Taking a tolerance limit of 1% into account the anti-symmetric mode isresolved with a polynomial orderpx2 = 3 (Fig. 9a), while the symmetric mode demands a value ofpx2 = 2 (Fig.9b). Since, theA0-mode is primarily a flexural wave, higher polynomial degrees are needed to avoid stiffening effectscaused by the locking phenomenon, which is still present in the model for the lower order approximations.

10−4

10−3

10−2

10−1

100

101

1 2 3 4 5 6 7 8

Ere

l[%

]

px2 [−]

SEMp-FEMN-FEM

(a) A0-mode

10−4

10−3

10−2

10−1

100

101

1 2 3 4 5 6 7 8

Ere

l[%

]

px2 [−]

SEMp-FEMN-FEM

(b) S0-mode

Figure 9: Convergence study concerning the discretization in thickness direction of the plate

The combined results taken from Fig. 6 - 9, provide a good foundation to formulate a guideline about how anappropriate discretization is to be generated. In all cases a polynomial degree distribution ofpx1 = 3 andpx2 = 4 isa suitable choice to meet the chosen error criterion for both theA0- and theS0-mode and minimizing the amount of

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 17

non-zero elements in the system matrices. The solution depicted in Fig. 9a is dependent on both polynomial degreesused (px1 = 6, px2 = 1, . . . , 8). The results obtained utilizingpx2 = 3 barely meet the relative error threshold. Sincethe higher order mixed product terms of the shape functions - mixed product terms depend on both powersm, n ofthe one-dimensional ansatz functions - influence the accuracy of the computed results,px2 = 4 is chosen as optimaldegree instead ofpx2 = 3. Additionally, the effects of locking are kept to a minimum.

5.3. Comparison of the higher order approaches to conventional linear finite elementsIn this section the results obtained using the proposed higher order finite elements are compared to the results

computed deploying conventional linear 4-node quadrilateral finite elements. In literature it is widely accepted that 20nodes per wavelength are sufficient to obtain good accuracy even for conventional lower order finite elements [10, 11].In the course of this section we will examine and try to verify this statement. The relative error curves obtained uti-lizing the optimal polynomial degree template for the higher order approaches (SEM:px1 = 3, px2 = 4; p-FEM:px1 = 3, px2 = 4; N-FEM: px1 = 3, px2 = 4) are contrasted with the solutions computed deploying conventional fullyor selectively reduced integrated linear finite elements. The model depicted in Fig. 3 is discretized with different typesof linear finite elements and the obtained results are scrutinized. Type one has been labeled ”linear undistorted fullyintegrated”. This is to say that neither a hourglass control scheme nor linear finite elements having an aspect ratio ofhigher than 1.4 are utilized. If the aspect ratio exceeds a value of 1.4 the number of elements in thickness direction isincreased by one. Thus, up to 14 elements over the thickness are deployed to prevent element distortion (high aspectratios). The second numerical model using lower order finite elements is termed ”linear distorted fully integrated”.This model is similar to the first one, only that now aspect ratios of 5 or higher are allowed. In this case a mesh den-sity of 4 linear elements over the thickness is chosen. This corresponds to 5 nodes in thickness direction of the plate,which is identical to the higher order approaches utilizingpx2 = 4. The third model deploying conventional 4-nodequadrilateral elements is labeled ”linear undistorted reduced integrated”. In comparison to the first type the elementmatrices are numerically integrated utilizing a selective reduced integration scheme. That is to say, that only thoseterms of the total elastic potential referring to the shear energy are reduced integrated. This prevents the shear energyfrom being overestimated. Due to this numerical trick the stiffness matrix becomes rank-deficient and displacementmodes, needing no energy, can evolve. To avoid these zero energy modes an hourglass control algorithm based on theartificial stiffness method proposed in [61] is chosen. The solutions for this special type of integration are computedusing the commercial software package ABAQUS 6.7-2.The results of the verification process are depicted in Fig. 10a (A0-mode) and 10b (S0-mode). The convergence curvesof the three linear finite element models are steady, except for a peak for the selectively reduced integrated case. Anerror-reducing influence of low aspect ratios as well as linear finite element using a selectively reduced integrationin conjunction with a suitable hourglass control scheme is to be seen in the convergence characteristics of the anti-symmetric Lamb wave mode.Since theA0-mode resembles a flexural wave and linear finite elements are prone to shear locking effects a significantimprovement in the relative error can be observed when using linear finite elements that are either ”undistorted” and/orselectively reduced integrated. Utilizing those approaches the shear energy is not significantly overestimated. Regard-ing the convergence curves of the anti-symmetric mode, only undistorted linear finite elements are able to reach thedesignated error criterion of 1 % (see Fig. 10a).Although up to 14 finite elements are deployed over the thickness of the plate the solution still does suffer from thelocking effect and cannot reach an accuracy similar to the higher order schemes. The error in the case of 20 nodesper wavelength is equal to 3.63 % for the ”distorted” linear fully integrated finite elements and 2.24 % for the ”undis-torted” ones. Considering the anti-symmetric mode the reduced integration of the shear energy term pays off and aclear improvement of the results can be seen in Fig. 10a. The relative error compared to the fully integrated linearFE-model is drastically reduced to 0.51 %. However, the accuracy of all higher order finite element approaches is stillsuperior. By using a selectively reduced integrated linear finite element the 1 % relative error bound is reached forround about 14 nodes per wavelength.Scrutinizing theS0-mode neither a positive influence of low aspect ratios nor of an improvement due to selectivelyreduced integration can be observed. In the contrary the chosen hourglass control deteriorates the results with lessthan 17 nodes per wavelength.The convergence behaviour of the symmetric mode is almost equivalent for both fullyintegrated linear finite element approaches. For the reduced integrated elements the results for the symmetric Lambwave mode in the range of 5-17 nodes per wavelength are even worse than for fully integrated linear elements. The

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 18

explanation for this behaviour is based on the default hourglass control ABAQUS employs. If different schemes thanthe artificial stiffness method proposed in [61] are used the relative error can be reduced even further. Preliminarystudies have shown that an enhanced hourglass control algorithm is an appropriate choice. This approach is basedon the enhanced assumed strain and physical hourglass control methods proposed by Belytschko and Bindeman [62].Then all three curves corresponding to the finite elements being available in commercial software packages are vir-tually coincident. But the difference between a converged solution for 30 nodes per wavelength and the analyticalreference solution is not altered significantly. The relative error in the case of 20 nodes per wavelength of both dis-torted and undistorted elements is equal to 1.91 %. This behaviour was to be expected since theS0-mode is merelya longitudinal wave which is not significantly influenced by locking effects. The 1 % error boundary is reached at 26nodes per wavelength in all three cases.Regardless of the improved accuracy in comparison to the fully integrated finite elements the convergence rate ofthe A0- andS0-mode is still low. It is not enough to specify a certain number of nodes per wavelength, because thediscretization over the thickness of the plate has to be considered as well. Additionally, if conventional lower orderfinite elements are to be deployed a specification of the numerical integration method as well as the hourglass con-trol algorithm is required. In general, higher order finite elements are in any case significantly more accurate thanconventional lower order approaches.

10−2

10−1

100

101

102

5 10 15 20 25 30

Ere

l[%

]

χA0 [−]

(a) A0-mode

10−3

10−2

10−1

100

101

102

5 10 15 20 25 30

Ere

l[%

]

χS0 [−]

(b) S0-mode

SEM px1 = 3p-FEM px1 = 3N-FEM px1 = 3

linear undistorted fully integratedlinear distorted fully integrated

linear undistorted reduced integrated

Figure 10: Comparison of the convergence curves for the proposed higher order finite element method schemes(px2 = 4) and conventional linear finite elements.

6. Higher order Lamb wave modes

Although higher order Lamb waves modes (S1, A1) are at the moment not widely used to detect damages theycould be of interest for further applications. In the literature, the higher order modes are hardly captured by finiteelement models so far. To demonstrate that higher order finite element formulations are able to capture these modesthe excitation frequency is increased tof = 1.2 MHz. Here theA1-mode is excited as well (see the dispersion diagramFig. 4). The model setup is the same as described in section 4 Fig. 3. The collocated forces are driven in-phase toexploit a “mono-modal” excitation of the anti-symmetric modes. The results for the symmetric Lamb wave mode arenot displayed as only theS0-mode exists at the chosen center frequency. Hence, no new insight can be gained. The

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 19

group velocities (see Fig. 4b) and wavelengths (see Fig. 4a,cp = λ ∙ f ) are taken from the dispersion curves:

cgA0= 3100

ms

and λA0 = 2.283mm

cgA1= 3652

ms

and λA1 = 5.889mm.

In order to observe two distinct wave packets without the interference of reflections in the signal the dimensions ofthe plate have to be adjusted. The following values have been chosen:l p = 700mm, la = 100mmandlb = 400mm.According the results obtained in section 5 the following discretization is proposed for the Spectral Element Method:

• SEM: px1 = 3, px2 = 3, px3 = 6,χA0 = 8 (see Fig. 6b).

8 “nodes” per wavelength are chosen to ensure accurate results, since only the two basic modes have been the subjectof the convergence studies. The out-of-plane polynomial degree has been increased in order to ensure an discretizationof 8 “nodes” per wavelength over the thickness. This is necessary as the wavelength of theA0-mode is almost equalto the plate thickness. Again an analytical reference solution is taken in order to evaluate the quality of the results.Fig. 11 displays the time history plot saved at pointsA andB for the semi-analytical [5] and SEM solutions. As canbe seen a good agreement between numerical and SAFE solution is obtained using the introduced guideline. Bothgroup velocity as well as the signal amplitudes are captured. p-FEM and N-FEM exhibit the same behaviour and theirresults are not shown, as no new insight can be gained. Considering the time history plot at pointB also the effects ofdispersion are clearly demonstrated. In the chosen frequency range theA0-mode is almost not dispersive whereas thewave form of theA1-mode is notably distorted compared to the initially introduced excitation signal, cf. Fig. 4.

-15

-10

-5

0

5

10

15

0.2 0.25 0.3 0.35 0.4 0.45

u 2[1

0−5

m]

t [10−4 s]

SEMSAFE

(a) PointA.

-15

-10

-5

0

5

10

15

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

u 2[1

0−5

m]

t [10−4 s]

SEMSAFE

(b) PointB.

Figure 11: Time history of the displacement field at the measurement pointsA and B in thickness direction (u2) -SEM.A0- andA1-mode are excited

Using this example it has been demonstrated that also higher order Lamb wave modes can be captured if thediscretization is chosen according the method proposed in this article. In contrast to conventional lower order FEMthe various higher order approaches are also an effective tool when a frequency range is to be studied where more thanthe two basic modes exist.

7. Wave propagation in a three-dimensional plate with a conical hole

To verify the results gained from the two-dimensional benchmark test in section 5 the Lamb wave propagationin a three-dimensional plate with a conical hole is studied. This model is seen as an example of a more complexgeometry and allows for both reflections at the boundaries of the structure and a mode conversion at the defect. The

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 20

more advanced test structure is to show that the guideline for an optimal discretization obtained in the previous sectioncan easily be generalized even to three-dimensional problems involving more complex parts, like conical holes. Thematerial and geometrical data as well as the simulation parameters, used for this example, are given in Fig. 12. TheLamb waves are excited using collocated point force normal to the plate surface. As before they are driven in-phasein order to facilitate a mono-modal excitation of theS0-mode. Thus, a mode conversion can be observed at the hole.

Figure 12: Three-dimensional test structure (plate with circular hole). Dimensions:a = 300mm, b = 200mm,c = 50mm, d = 200mm, ra = 10mm, ri = 9mm, h = 2mm(thickness of the plate); Material properties: aluminum(see Table 2); Excitation: point force at the origin of coordinates,F = 107 N, Hann-window modulated sine-burst,f = 175kHz, n = 3 (see Eq. (23)).

Since no analytical solution is available for this benchmark problem the results of the different higher order FE-approaches are compared to an ABAQUS reference solution deploying 20-noded hexahedral finite elements (C3D20)on a very fine grid (ndo f ≈ 15 ∙ 106). This corresponds to more than 50 nodes per wavelength to ensure a convergedsolution with high accuracy. The symmetry of the structure is exploited to save numerical effort. Only one half of theplate needs to be discretized. Atx2 = 0 symmetry boundary conditions are applied to the numerical model.In Fig. 13 snapshots of theu3-displacement of the three-dimensional wave propagation in the plate are shown. Theconversion of theS0- into theA0-mode is to be seen clearly as well as their interaction with the introduced discon-tinuity. In Fig. 13a theS0-mode reaches the hole, partially reflects and is converted into theA0-mode, since thisdiscontinuity is ani-symmetric with respect to the middle plane of the plate [5]. In the second snapshot, see Fig. 13b,the reflections of the symmetric mode at the boundary,x1 = 300mmcan be observed.

−0.1

−0.05

0.0

0.05

0.1

x 2[m

]

0.0 0.05 0.1 0.15 0.2 0.25 0.3x1 [m]

(a) t = 2.73 ∙ 10−5 s

−0.1

−0.05

0.0

0.05

0.1

x 2[m

]

0.0 0.05 0.1 0.15 0.2 0.25 0.3x1 [m]

(b) t = 7.32 ∙ 10−5 s

Figure 13: Snapshots of Lamb wave propagation (u3) in a plate with a conical hole at different points in time.

The plate is also modelled deploying the different optimal polynomial degree templates for the proposed higher orderfinite element schemes

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 21

• SEM: px1 = 3, px2 = 3, px3 = 4,

• p-FEM: px1 = 3, px2 = 3, px3 = 4,

• N-FEM: px1 = 3, px2 = 3, px3 = 4,

and the optimized number of “nodes” per wavelength regarding the first anti-symmetric Lamb wave mode (A0)

• SEM: χA0 = 7 (see Fig. 6b),

• p-FEM: χλA0= 7 (see Fig. 6b),

• N-FEM: χλA0= 5 (see Fig. 6b).

The time history plots of the displacement fields at pointsP1 (x1 = 200mm, x2 = 0mm, x3 = 2mm) and P2

(x1 = 50mm, x2 = −10mm, x3 = 2mm) , obtained using SEM, p-FEM as well as N-FEM are given Fig. 14 -16. As can be seen there is an excellent agreement between the proposed higher order finite element schemes andresults obtained using the commercial software package ABAQUS. The relative error in group velocity between thereference solution and the three different higher order approaches is less than 0.1% regarding the proposed discretiza-tion. Hence, it is shown that the findings gathered from the two-dimensional benchmark test can be generalized andprovide a valuable guideline for an efficient discretization of numerical models being concerned with the simulationof Lamb wave propagation.However, we have to take into account that the proposed optimal discretization is determined using a fairly regulargrid. If the mesh is distorted slightly more “nodes” per wavelength are needed to obtain the desired accuracy. Sincehigher order finite elements do not drastically suffer from locking effects the required increase in degrees-of-freedomfor a distorted grid is quite low.It can be noted that the exact description of the hole geometry offered by isogeometric finite elements does not im-prove the results significantly. In the case of SEM and p-FEM a sub-parametric approach is chosen. To approximatethe geometry of the structure standard 20-noded finite element discretization is used. For the considered mesh densitythe error caused by the mapping from the local to the global coordinate system is negligible. This was expected asthe wavelengths of the Lamb wave modes at the chosen frequency are significantly larger than the inaccuracy of thegeometrical approximation. Therefore, the interaction between the waves and the hole is not noticeably disturbed.

-5-4-3-2-10123456

0 1 2 3 4 5 6 7 8 9 10

u 3[1

0−4

m]

t [10−5 s]

SEMABAQUS

(a) Comparison of the ABAQUS reference solution at pointP1 to theSEM results, obtained using the optimal discretization.

-2.5-2

-1.5-1

-0.50

0.51

1.52

2.5

0 1 2 3 4 5 6 7 8 9 10

u 3[1

0−4

m]

t [10−5 s]

SEMABAQUS

(b) Comparison of the ABAQUS reference solution at pointP2 to theSEM results, obtained using the optimal discretization.

Figure 14: Time history of the displacement field at the measurement points in thickness direction (u3) - SEM.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 22

-5-4-3-2-10123456

0 1 2 3 4 5 6 7 8 9 10

u 3[1

0−4

m]

t [10−5 s]

p-FEMABAQUS

(a) Comparison of the ABAQUS reference solution at pointP1 to thep-FEM results, obtained using the optimal discretization.

-2.5-2

-1.5-1

-0.50

0.51

1.52

2.5

0 1 2 3 4 5 6 7 8 9 10

u 3[1

0−4

m]

t [10−5 s]

p-FEMABAQUS

(b) Comparison of the ABAQUS reference solution at pointP2 to thep-FEM results, obtained using the optimal discretization.

Figure 15: Time history of the displacement field at the measurement points in thickness direction (u3) - p-FEM.

-5-4-3-2-10123456

0 1 2 3 4 5 6 7 8 9 10

u 3[1

0−4

m]

t [10−5 s]

N-FEMABAQUS

(a) Comparison of the ABAQUS reference solution at pointP1 to theN-FEM results, obtained using the optimal discretization.

-2.5-2

-1.5-1

-0.50

0.51

1.52

2.5

0 1 2 3 4 5 6 7 8 9 10

u 3[1

0−4

m]

t [10−5 s]

N-FEMABAQUS

(b) Comparison of the ABAQUS reference solution at pointP2 to theN-FEM results, obtained using the optimal discretization.

Figure 16: Time history of the displacement field at the measurement points in thickness direction (u3) - N-FEM.

8. Conclusion

In the current paper three different higher order finite element schemes have been compared with respect to theirsuitability concerning Lamb wave propagation analysis. The quality of their solutions has been assessed deployingan analytical approach. The results of the benchmark test have illustrated that higher order finite element methodsare an efficient numerical tool and are essential when dealing with ultrasonic guided waves in thin-walled structures.The accuracy which can be achieved deploying these approaches is much improved compared to conventional finiteelement methods. Furthermore, it is seen that the number of degrees-of-freedom can be reduced significantly whenhigher order schemes are applied.Two benchmark tests have been performed to determine the optimal polynomial degree templates for three differenthigher order FE-approaches, namely SEM, p-FEM and N-FEM. The results are compared to an analytical solutionand an error threshold of 1 % difference in the time-of-flight is chosen to evaluate their performances. It has been

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C. Willberg et al./ Computer Methods in Applied Mechanics and Engineering 00 (2012) 1–25 23

found that an polynomial degree distribution ofpx1 = 3 andpx2 = 4 is required for all investigated finite elementapproaches, namely SEM, p-FEM and NFEM. These findings are based on a minimal memory storage criterion fora specific relative error in time-of-flight.Spectral elements and p-elements exhibit similar convergence properties. Overall, p-elements promise the best accu-racy for px2 ≤ 4. Both spectral and isogeometric finite elements reach an equal quality of the results forpx2 ≥ 5.Furthermore, isogeometric elements offer the highest convergence rates. These results are in agreement with the find-ings published by Hughes et al. [63] and Schillinger et al. [64]. An undisputable advantage of spectral elements is thatbecause of the physical meaning of the degrees-of-freedom the pre- and post-processing is much more convenient.Additionally, mass-lumping techniques have already been developed and thus, explicit time integration schemes likecentral difference methods can easily be adapted for spectral elements. In summary, every scheme has its positive fea-tures and drawbacks. So far, if we neglect the computational time and study the results obtained so far isogeometricelements would have to be recommended due to their high convergence rates.After having determined the optimal discretization parameters, the proposed higher order FE-schemes are comparedto conventional linear finite elements and the wave propagation in a three-dimensional plate with a circular hole isstudied. As expected the results highlight the superiority of higher order approaches. The convergence behaviouris more rapid and considerably less degrees-of-freedom are needed for accurate results. For the three-dimensionalexample the proposed guideline has been applied to a more complex geometry and it has been found that the resultsare in good agreement with an ABAQUS reference solution, computed on a very fine grid. Both SEM and p-FEMutilize a sub-parametric approach to approximate the geometry, whereas isogeometric elements are able to describe itexactly. However, the discrepancy in the results is negligible. From this fact one cannot infer that the exact representa-tion of the geometry is insignificant. For arbitrary structures an exact representation can be important. As mentionedbefore, this property is inherent to isogeometric finite elements and is to be achieved for spectral and p-version finiteelements deploying the blending-function method [65, 66, 67, 68, 69, 70, 71]. Hence, for three-dimensional complexstructures the savings offered by using higher order Finite-Element-Method are even considerably higher. Here nomesh refinement is needed to describe small features of the geometry.In further studies the influence of mass lumping and the integration schemes will be investigated. This is a mainfocus in order to achieve an efficient numerical simulation tool with respect to computational time and memory re-quirements. In addition, the influence of the inter element continuity is another topic that will be studied. Whenscrutinizing isogeometric finite elements the order of continuity is another variable, which has an influence on theaccuracy of the solution. In general, it can be concluded that higher order finite elements are in any case significantlymore accurate than conventional lower order approaches and hence are a viable option computing ultrasonic guidedwave propagation in thin-walled structures.

AcknowledgmentsThe authors are indebted to the anonymous referees for their valuable hints and suggestions, which considerablyimproved the first version of this contribution. The authors like to thank the German Research Foundation (DFG)and all partners for their support (GA 480/13-1). S. Duczek is supported by the postgraduate program of the GermanFederal State of Saxony-Anhalt. This support is also gratefully acknowledged.

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Highlights

Higher order FE-methods are compared to analyze their potential for wave propagation.

We examine their convergence behavior and compare them to analytical results.

First, the optimal polynomial degree in longitudinal direction is determined.

Second, the optimal polynomial degree in transversal direction is determined.

The advantage of higher order schemes in comparison to commercial FEM is stressed.