Lamb modes and Born approximation for small shape defects inversion in elastic plates Éric Bonnetier1 Angèle Niclas2 and Laurent Seppecher3

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Lamb modes and Born approximation for small shape defects

inversion in elastic plates

Éric Bonnetier1, Angèle Niclas2,*, and Laurent Seppecher3

1Institut Fourier, Université Grenoble Alpes, France

2CMAP, École Polytechnique, France

3Institut Camille Jordan, École Centrale Lyon, France

*Corresponding author: angele.niclas@polytechnique.edu

Abstract

The aim of this work is to present theoretical tools to study wave propagation in elastic

waveguides and perform multi-frequency scattering inversion to reconstruct small shape defects

in a 2D and 3D elastic plate. Given surface multi-frequency waveﬁeld measurements, we use a

Born approximation to reconstruct localized defect in the geometry of the plate. To justify this

approximation, we introduce a rigorous framework to study the propagation of elastic waveﬁeld

generated by arbitrary sources. By studying the decreasing rate of the series of inhomogeneous

Lamb mode, we prove the well-posedness of the PDE that model elastic wave propagation in

2D and 3D planar waveguides. We also characterize the critical frequencies for which the Lamb

decomposition is not valid. Using these results, we generalize the shape reconstruction method

already developed for acoustic waveguide to 2D elastic waveguides and provide a stable recon-

struction method based on a mode-by-mode spacial Fourier inversion given by the scattered ﬁeld.

1 Introduction

This work is devoted to the reconstruction of small shape defects in a waveguide using multi-

frequency scattering data. It is an extension of the method exposed in [11] int the case of acoustic

waveguide to the case of elastic plates. If the scalar Helmholtz case is relevant to the non destructive

testing of pipes or optical ﬁbers (see [17]), applications in the elastic case concern the monitoring

of structural parts, airplane, ship, oﬀshore wind energy plants or bridges for instance (see [34]).

The main common point between acoustic and elastic waveguides is the existence of a modal

decomposition of the waveﬁeld in a sum of explicit guided modes. The acoustic modes form an or-

thonormal basis, a property not satisﬁed by their elastic counter-parts, called Lamb modes. Several

authors have looked into this feature. The books [28, 1] provide analytic expressions of Lamb modes

as well as dispersion relations for their wavenumbers. In [22, 25, 26] a new formulation is introduced,

the X/Yformulation, under which the family of Lamb mode turns out to be complete [3, 18, 8].

The associated bi-orthogonality relations [14] thus allow the use of the Lamb basis to decompose

any waveﬁeld that propagates in an elastic waveguide as a sum of Lamb modes.

However, a rigorous mathematical framework is still missing to study the propagation of an

elastic waveﬁeld generated by an arbitrary source term (see however [5, 6] in 2D). One main goal of

arXiv:2210.01899v1 [math.AP] 4 Oct 2022

the present article is to prove well-posedness of the system of PDE’s, that models 2D or 3D planar

elastic waveguides with internal and boundary source terms. To this end, we adapt the strategy

developed for acoustic waveguides in [11], which diﬀers from [5]. Under stronger assumptions on

the regularity of the source terms than those in [5], we present in Theorem 2 a constructive proof

of existence and regularity of a waveﬁeld propagating in a two dimensional elastic waveguide.

As it turns out, this result is not valid at some particular frequencies, which we call critical

frequencies, and that are characterized in the proof of Theorem 2. In particular, we establish in

Corollary 1 that the critical frequencies, for which the Lamb family is no longer complete, coincide

with the vanishing of the bi-orthogonality relation established by [14]. This result, up to our

knowledge, has not been proven before and may help understanding the mathematical analysis of

elastic waveguides.

Concerning the study of wave propagation in three-dimensional plates, most of the work that we

are aware of consists in adapting the 2D framework to situations with radial or axial symmetry (see

for instance [19, 28, 2, 33]). In [31], arbitrary source terms are considered, without mathematical

justiﬁcation however. Introducing the Helmholtz-Hodge decomposition of the waveﬁeld [9], we split

the three dimensional system of elasticity into a system of two independent equations. One of them

ﬁts into the scalar wave framework developed in [11], while the other can be rewritten using the

X/Yformulation. This provides a full expression for the decomposition of the waveﬁeld generated

by arbitrary source terms in dimension 3, see Theorem 3.

Equipped with these results, we can generalize the shape reconstruction method presented in

[11] to the case of elastic plates, so as to determine possible defects (bumps or dips) in the geometry

of a plate, from multi-frequency measurements. We use the very same procedure as in the acoustic

case : after mapping the perturbed plate to a straight conﬁguration, we simplify the resulting

system of equations using the Born approximation. The scattered waveﬁeld generated by a known

incident waveﬁeld in the original geometry, gives rise in the straightened plate to a boundary source

term, that depends on the shape defect. Using measurements of the scattered ﬁeld on the surface

of the plate at diﬀerent frequencies, we can reconstruct in a stable way the shape defect (provided

the latter is small enough). Numerical reconstructions are presented in the last part of the article,

which show the eﬃciency of the method.

The paper is organized as follows. In section 2, we study the forward source problem in a two

dimensional waveguide and introduce all the tools needed to use Lamb waves as a modal basis. In

section 3, we generalize the results of section 2 to the forward source problem in three dimensional

plates. Section 4 is devoted to the reconstruction of shape defects in two dimensional plates,

generalizing the method presented in [11]. Finally, in section 5 we show numerical illustrations of

the propagation of waves in two and three dimensional plates as well as reconstructions of diﬀerent

shape defects.

2 Forward source problem in a regular 2D waveguide

In this section, we present a complete study of the forward elastic source problem in a two-

dimensional regular waveguide. We use the X/Yformulation developed in [25, 26] which allows a

modal decomposition of any elastic waveﬁeld using Lamb modes. Most of the results presented here

are already known, and can be found in [26, 28, 1]. Our main contribution is to provide a rigorous

proof of well-posedness for the direct problem and of the fact that its solutions can be represented

in terms of Lamb modes (Theorem 2). We also follow the suggestions in [18] to deﬁne the set of

critical frequencies and critical wavenumbers in Deﬁnition 3, and we prove in Corollary 1 that it

coincides with the set of frequencies for which the components Xnand Ynof the eigenmodes are

orthogonal for some n.

2.1 Lamb modes and critical frequencies

We consider a 2D inﬁnite, straight, elastic waveguide Ω = {(x, z)∈R×(−h, h)}of width 2h > 0.

The displacement ﬁeld is denoted by u= (u, v). Given a frequency ω∈R, and given (λ, µ)the

Lamé parameters of the elastic waveguide, the waveﬁeld usatisﬁes

∇ · σ(u) + ω2u=−fin Ω,(1)

where f= (f1, f2)is a given source term, and where the stress tensor σ(u)is deﬁned by

σ(u) = (λ+ 2µ)∂xu+λ∂zv µ∂zu+µ∂xv

µ∂zu+µ∂xv λ∂xu+ (λ+ 2µ)∂zv!:= s t

t r !.(2)

In this work, we assume that a Neumann boundary condition is imposed on both sides of the plate

σ(u)·ν=btop on ∂Ωtop,σ(u)·ν=bbot on ∂Ωbot,(3)

where btop = (btop

1, bbot

2)and bbot = (bbot

1, bbot

2)are given boundary source terms. This condition

could easily be replaced by a Dirichlet or a Robin condition without much changes in the following

analysis. The setting is represented in Figure 1.

fΩ

−h

btop

bbot

Figure 1: Parametrization of a two dimensional plate Ω. Elastic waveﬁelds are generated using an

internal source term f, and boundary source terms btop and bbot.

In [22] this equation is analyzed in an operator form Z=L(Z)where Z= (u, t, s, v). This idea

was then adapted in [25] to formalize the so-called X/Yformulation. We introduce the variables

X= (u, t),Y= (−s, v),(4)

with which the elasticity equation can be rewritten as follows:

Proposition 1. The system (1), with the Neumann boundary conditions (3), is equivalent to

∂x X

Y!=L(X,Y) + 





−f2−btop

2δz=h−bbot

2δz=−h

f1+btop

1δz=h+btop

2δz=−h





in Ω,(5)

with the boundary condition B1(X) = B2(Y)=0, where L(X,Y)=(F(Y); G(X)) and F,G,B1

and B2are diﬀerential matrix operators deﬁned by

F=



−1

λ+ 2µ−λ

λ+ 2µ∂z

λ+ 2µ∂z−ω2−4µ(λ+µ)

λ+ 2µ∂2





, G =

ω2∂z

−∂z

µ

,(6)

B1(X) = X·ez, B2(Y) = −λ

λ+ 2µY·ex+4µ(λ+µ)

λ+ 2µ∂zY·ez.(7)

The proof of this proposition follows the same steps as that presented in Appendix A of [26]. In

this formulation, the operators Fand Gonly depend on z, and are deﬁned on one section of the

waveguide, while derivatives with respect to xonly appear in the left-hand side of (5). We consider

the space

H0:= n(X,Y)∈(H2(−h, h))4|B1(X)(±h) = B2(Y)(±h)=0o,(8)

and the operator

L:H0→(L2(−h, h))4

(X,Y)7→ (F(Y), G(X)) .(9)

Our goal is to diagonalize this operator and, to this end, we introduce the Lamb modes:

Deﬁnition 1. A Lamb mode (X,Y)∈H0, associated to the wavenumber k∈C, is a non-trivial

solution of L(X,Y) = ik(X,Y).

The next Proposition provides the analytical expressions of these modes. The proof can be

found in [1, 28].

Proposition 2. The set of wavenumbers k∈Cassociated to Lamb modes is countable, and every

such wavenumber ksatisﬁes the symmetric Rayleigh-Lamb equation

p2=ω2

λ+ 2µ−k2, q2=ω2

µ−k2,q2−k22=−4k2pq tan(ph)

tan(qh),(10)

or the antisymmetric Rayleigh-Lamb equation

p2=ω2

λ+ 2µ−k2, q2=ω2

µ−k2,q2−k22=−4k2pq tan(qh)

tan(ph).(11)

If ksatisﬁes (10), the associated Lamb mode is called symmetric and is proportional to

(X(z),Y(z)) =







u(z)

t(z)

−s(z)

v(z)





=





ik(q2−k2) sin(qh) cos(pz)−2ikpq sin(ph) cos(qz)

2ikµ(q2−k2)p(−sin(qh) sin(pz) + sin(ph) sin(qz))

(q2−k2)((λ+ 2µ)k2+λp2) sin(qh) cos(pz)−4µpqk2sin(ph) cos(qz)

−p(q2−k2) sin(qh) sin(pz)−2k2psin(ph) sin(qz)





.(12)

If ksatisﬁes (11), the associated Lamb mode is called anti-symmetric and is proportional to

(X(z),Y(z)) =







u(z)

t(z)

−s(z)

v(z)





=





ik(q2−k2) cos(qh) sin(pz)−2ikpq cos(ph) sin(qz)

2ikµ(q2−k2)p(cos(qh) cos(pz)−cos(ph) cos(qz))

(q2−k2)((λ+ 2µ)k2+λp2) cos(qh) sin(pz)−4µpqk2cos(ph) sin(qz)

p(q2−k2) cos(qh) cos(pz)+2k2pcos(ph) cos(qz)





.(13)

Remark 1. We see on the above expressions that pand qare deﬁned up to a multiplication by

−1. However, since Lamb modes are deﬁned up to a multiplicative constant, the choice of the sign

of por qdoes not change the associated value of kor the associated Lamb mode.

We notice that if kis a solution of the Rayleigh-Lamb equation then −kand ¯

kare also solutions.

Figure 2 depicts diﬀerent wavenumbers kwhere Real(k)≥0and Imag(k)≥0, in terms of the

frequency ω.

02468

Imag(k)h

Real(k)h

ωh

propagative inhomogeneous evanescent critical

Figure 2: Solutions of the symmetric Rayleigh-Lamb equation (12) in the space Imag(k)≥0,

Real(k)≥0with µ= 0.25 and λ= 0.31. Solutions on the full space can be obtained by axial

symmetries. Propagative, evanescent and inhomogeneous modes are represented by diﬀerent colors.

Critical points are represented by red dots.

We can distinguish three diﬀerent types of modes (represented in diﬀerent colors in the above

Figure) as in [19]:

Deﬁnition 2. There are three types of Lamb modes:

•If k∈R, the mode oscillates in the waveguide without energy decay and is called propagative.

•If k∈iR, the mode decays exponentially to zero as |x|→∞, and is called evanescent.

•If Real(k)6= 0 and Imag(k)6= 0, the mode oscillates quickly toward zero and is called inho-

mogeneous.

The completeness of Lamb modes depends on whether the frequency ωis critical as deﬁned

below:

Deﬁnition 3. A frequency ωand a wavenumber kare said to be critical if they satisfy k= 0 or

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摘要：

LambmodesandBornapproximationforsmallshapedefectsinversioninelasticplatesÉricBonnetier1,AngèleNiclas2,*,andLaurentSeppecher31InstitutFourier,UniversitéGrenobleAlpes,France2CMAP,ÉcolePolytechnique,France3InstitutCamilleJordan,ÉcoleCentraleLyon,France*Correspondingauthor:angele.niclas@polytechnique.ed...

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Lamb modes and Born approximation for small shape defects inversion in elastic plates Éric Bonnetier1 Angèle Niclas2 and Laurent Seppecher3

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