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FEW REMARKS ON THE USE OF LOVE WAVES IN NON DESTRUCTIVE TESTING
Philippe Destuynder, Caroline Fabre
To cite this version:
Philippe Destuynder, Caroline Fabre. FEW REMARKS ON THE USE OF LOVE WAVES IN NON
DESTRUCTIVE TESTING. 2014. �hal-01084471v2�
AIMS’ Journals
VolumeX, Number0X, XX200X pp.X–XX
FEW REMARKS ON THE USE OF LOVE WAVES IN NON DESTRUCTIVE TESTING
PHILIPPEDESTUYNDER Département d’ingénierie mathématique Conservatoire National des Arts et Métiers
292, rue saint Martin, 75003 Paris France
CAROLINEFABRE
Laboratoire de mathématiques d’Orsay UMR 8628 Université paris-sud
Orsay 91405 France
(Communicated by the associate editor name)
ABSTRACT. This paper concerns a theoretical study on the possibility of using Love waves for non destructive testing. A mathematical model is presented and analyzed. Several numerical tests are given in order to show the mechanical behaviour of this model.
1. Introduction. The detection of cracks at the interface between two materials, for in- stance in pipes with a coating, is often treated using ultrasonic waves or Foucault currents propagating transversally to the interface. The detection is based on the analysis of the return signal which is different depending if the wave meets a crack or not. The possibility of using Love waves is a new idea suggested by many authors (for instance one can consult [9]-[11]-[12]-[14]-[15]-[18]). Such waves could enable one to explore larger area from a single shot. But the signal processing which should be implied is much more complicated.
Therefore, a mathematical model can be useful in order to determine if there is a crack and where it is localized. One solution is to have a wave simulator model set on a finite part of the structure, but the boundary conditions should avoid artificial reflections. The goal of this paper is to make explicit these remarks and to suggest appropriate numerical transpar- ent boundary conditions for this challenge. Nevertheless, this study is only a discussion on a mathematical model which could be used in order to improve the understanding of Love waves for NDT. Concerning the details of the mathematical proofs and the application to an operational use, we refer the reader to the references. Let us consider a rectangular open set as the one drawn on figure1. There are two subsetsΩ−andΩ+corresponding to two different materials. The wave velocities are respectivelyc− andc+. For sake of clarity, it is assumed thatc− < c+. The upper and lower boundaries -sayΓ+andΓ−- are free edges and the lateral boundariesΓeandΓsare artificial cuts on which transparent boundary con- ditions are applied in order to simulate the propagation on an infinite strip. The separation line between the two media is denoted byΓi. A crack may exist somewhere on this line.
Its two extremities are positioned at pointsAandBwith abscissax1=xAandx1=xB. A mechanical excitation is produced on a subdomain ofΩ+orΩ−which is assumed to be remote from the crack tips. It can be an initial condition or a time dependent force denoted
2010Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.
Key words and phrases. Non destructive testing, Wave equation, Transparent boundary conditions, Fracture mechanics, Inverse problems.
1
FIGURE1. The open set used for the wave model
byf and such that:f(x, t) =g(t)η(x). The mathematical model that we discuss consists in findingu(x, t)solution for anytof the following system:
∂2u
∂t2 −div(c2∇u) =f inΩ,
∂u
∂ν = 0 onΓ−∪Γ+, and ∂u
∂t +c∂u
∂ν = 0 onΓe∪Γs, u(x,0) =u0(x), ∂u
∂t(x,0) =u1(x) inΩ.
(1)
The existence and uniqueness of a solution is not exactly the result contained in most books on partial differential equations, because of the dissipation term which appears on the boundariesΓeandΓs. The proof method consists in constructing a sequence of approx- imations of the model (1) using a Galerkin approximation with the variational formulation and the family of eigenfunctionswn, n≥1solution of:
−div(c2∇wn) =λnwninΩ, 0≤λ1≤. . . λn≤λn+1. . . ,
∂wn
∂ν = 0 onΓ+∪Γ−∪Γe∪Γs, Z
Ω
wn2(x)dx= 1.
(2)
The first termw1 is the constant function corresponding toλ1 = 0. One can define a sequence of approximate solutionsunto the equation (1) using the spaceVN span by the N-first eigenvectors solution of (2). This is obtained from the variational formulation using
test functions inVN. This solution can be written:
X
i=1,N
αn(t)wn(x), (3)
and one can prove that it is bounded in the spaceL∞(]0, T[;H1(Ω)). By extracting a subsequence, one proves the existence and then the uniqueness of a solution to (1). The time regularity is obtained by taking the time derivative of the model. The functional space used for the space dependence of the solution isH1(Ω)andL2(Ω)for its time derivative.
For instance, one has the following result:
Theorem 1.1. Let us assume thatu0∈H1(Ω), u1∈L2(Ω)and:
f ∈L2(]0, T[×Ω).
Then there is a unique solutionuto (1) such that:
u∈C0([0, T];H1(Ω))∩C1([0, T];L2(Ω)).
Remark 1. It is worth noting that the convergence of the series (3) can’t be proved in the spaceH2(Ω). This can be justified by the fact thatun satisfies on the boundary ofΩthe condition:
∂un
∂ν = 0,
which is incompatible with the boundary condition which should be satisfied byuat any time onΓe∪Γs.
2. A more refined regularity analysis foru. Let us discuss locally what happens in the vicinity of pointsE(or/andF) (see figure2). On a close neighbourhoodOofE(same is
FIGURE2. The neighbourhoodOnear the pointE
true forF), the solutionucan be locally split into its symmetrical and unsymmetrical parts with respect to the coordinatex2as shown on figure2. They are respectively denoted by usandua. From a classical result, the functionusand its derivative with respect tox2are continous acrossΓi. Therefore one can prove from the local equation satisfied byus, that
it is locally in the spaceH2(O). The discontinuity of ∂u
∂x1
is therefore taken into account by the unsymmetrical term∂ua
∂x1
as it is explained hereafter. Because:
∂u
∂t +c∂u
∂ν = 0onΓe, the function ∂u
∂x1
is piecewise smooth as far as:
∂u
∂t ∈C0([0, T];H1/2(Γe))
which can’t be discontinuous. Since the velocityc is discontinuous, ∂u
∂x1
is also discon- tinuous. In fact, one can introduce the following function defined onO and which takes into account this discontinuity (c∓ =c−, c±=c+forx2 >0andc∓ =c+, c± =c− for x2<0):
q(x1, x2) = (c∓
c±)( c+−c−
π(c2++c2−))Log(1 + 2 cos(x2π)e−πx1+e−2x1π
1−2 cos(x2π)e−πx1+e−2x1π) (4) A primitive ofqwith respect tox1 is a obtained using the so-called Dilog function [22].
More precisely [5], let us set:
D(z) =X
n≥1
zn
n2 withz=e−π(x1−ix2). (5) The singular fonctionSinvolved in our problem is:
S(x1, x2) =Im(D(z)−D(−z))/2, andq(x1, x2) = ( c+−c−
π(c2++c2−))(c∓
c±
)∂S
∂x2
. (6) From a simple computation one can check that this function is locally equivalent toLog(r) (ris the distance fromE). Furthermore it can be truncated symmetrically aroundΓi (the interface) such that its integral overΩ is zero. S and its derivative with respect to the coordinatex2have been plotted on figure3.
FIGURE3. The singular functionSon left and∂S/∂x2on right
Theorem 2.1. LetQbe a sub-open set strictly included inΩ+∪Ω−. In other words,Q doesn’t cross the interfaceΓi. Assuming that the initial data are such thatu0 ∈ H2(Ω)
andH1(Ω), and that the right hand side satisfiesf ∈H1(]0, T[;L2(Ω), with support in Q, one has:
u=α(t)S(x) +uR(x, t), withuR(x, t)∈W1,∞(]0, T[;H1(Ω))∩W2,∞(]0, T[;L2(Ω)).
More generally, the following array using the same notations as above and the assumptions of theorem2.1, describes the regularity ofuin the different parts ofΩ:
Function→
Open set↓ | ∂u
∂t | ∂u
∂x1 | ∂u
∂x2
U ⊂⊂Ω± | L∞(]0, T[;H1(Ω)) | L∞(]0, T[;H1(U))| L∞(]0, T[;H1(U))
Q⊂⊂Γe±| L∞(]0, T[;H12(Γe))| L∞(]0, T[;H12(Q))| L∞(]0, T[;H12(Q))
P⊂⊂Γi | L∞(]0, T[;H12(Γi)) | L∞(]0, T[;H12(P))| L∞(]0, T[;H−12(Γi))
With another respect, the time regularity can be derived up to any order assuming regularity on the initial datau0etu1and the right hand sidef of the wave equation.
Remark 2. Let us consider a vertical line -sayΓv(corresponding to0< x1 < L). Along Γ+v (x2>0)orΓ−v (x2<0)the function ∂u
∂x2
belongs to the space:
L∞(]0, T[;H3/2(Γ+v ∪Γ−v)) (it is discontinuous acrossΓi).
Hence there is a finite limit forx2= 0+andx2= 0−. This property is no more true along the boundaryΓe∪Γsbecause of the singularitySgiven at (6). This remark is meaningful in our discussion.
For instance, one has:
∂u
∂t ∈L∞(]0, T[;H1/2(Γe∪Γs)) and therefore:
c∂u
∂ν|Γe∪Γs
∈L∞(]0, T[;H1/2(Γe∪Γs)).
Butcis discontinuous acrossΓiand therefore one has:
∂u
∂ν|Γe∪Γs
∈/ L∞(]0, T[;H1/2(Γe∪Γs)),
but (assuming an additional regularity on the initial data and on the right hand side of (1)), one has (and even more):
∂u
∂ν|Γe∪Γs
∈L∞(]0, T[;L2(Γe∪Γs)).
2.1. Effect of the transparent boundary condition onu. For sake of simplicity in this subsection, let us assume thatf = 0. But this is just a facility, not a restriction. The mechanical energy of the structure is defined by:
E(t) =1 2
Z
Ω|∂u
∂t|2(x, t)dx+1 2 Z
Ω
c2|∇u|2(x, t)dx.
The mechanical energy is decreasing with respect to the time variable because there are only initial conditions and due to the transparent boundary condition which acts as a damp- ing. Let us check it. One has, by multiplying the wave equation by∂u
∂t and integrating over Ω×]0, t[:
E(t) + Z t
0
Z
Γe∪Γs
c|∂u
∂t|2(x2, t)dx2dt≤E(0) ∀t >0. (7) This proves two properties:
1. the functiont→E(t)which is continuous, is decreasing whent→ ∞. Hence there is a lower limit -sayE∞- such that lim
t→∞E(t) =E∞; 2. There exists a constantcindependent on time such that:
∀t >0, ||∂u
∂t||0,2,Ω≤c, ||∇u||1,Ω≤c.
Let us point out that the estimate (7) proves a somewhat hidden regularity concerning the term ∂u
∂t on the boundaryΓe∪Γs. Let us introduce the sequence of functions ut0(x, t) =u(x, t0+t). The previous estimate enables one to extract a subsequence which converges to a the term denoted byu∗ (with respect to the time t0 and the uniqueness of the limit will prove that all the sequence converges) such that:
ut0(x, t)→t0→∞u∗ inL∞(]0, T[;H1(Ω)) weakly,
∂ut0
∂t (x, t)→t0→∞
∂u∗
∂t inL∞(]0, T[;L2(Ω)) weakly,
∂ut0
∂t (x, t)→t0→∞
∂u∗
∂t inL2(]0, T[×{Γe∪Γs})) weakly.
(8)
From the convergence ofE(t)toE∞one can state that:
∀ε >0, ∃tε>0, ∀t≥tε: E(t)≤E∞+ε. (9) Therefore:
∀t≥tε, E∞≤lim inf
t→∞ E(t) + Z ∞
tε
Z
Γe∪Γs
c|∂u
∂t(x2, t)|2dx2dt≤E(t0)≤E∞+ε.
This implies that:
∀ε >0, ∃tε∀t1≥tε, Z t1
tε
Z
Γe∪Γs
c|∂u
∂t(x2, t)|2dx2dt≤ε, and finally, because of the lower semi-continuity of a continuous convex function:
∂u∗
∂t = 0on]0, T[×{Γe∪Γs}. (10)
From the equations satisfied byu, one obtains thatu∗is solution of the limit model :
∂2u∗
∂t2 −div(c2∇u∗) = 0in]0, T[×Ω,
∂u∗
∂ν = 0, on]0, T[×Γ0,∂u∗
∂t = 0and ∂u∗
∂ν = 0on]0, T[×Γe∪Γs.
(11)
The Holmgrem theorem [13] enables one to conclude that the functionu∗ (in a first step one proves that the time derivative ofu∗ is zero) is a constant with respect to space and time variables on]0, T[×Ω. Such a constant doesn’t generate wave. Hence whent→ ∞ the waves disappears. One can get rid of this constant as follows. By integrating the wave equation (1) on]0, t[×Ω, one obtains:
∀t >0, ∂
∂t( Z
Ω
u(x, t)dx1dx2) + Z
Γe∪Γs
cu(x, t)dx2) = Z
Ω
u1(x)dx1dx2+ Z
Γe∪Γs
cu0(x)dx2. If the initial conditions satisfy:
Z
Ω
u1+ Z
Γe∪Γs
cu0= 0, one deduces that whent→ ∞and becauseu∗=constant:
Z
Γe∪Γs
cu∗= 0 => u∗= 0.
A similar analysis can be derived iff 6= 0but with some conditions on this term.
3. The Fourier transform for the operational use. For sake of simplicity in the nota- tions, it is assumed in this section that the initial conditions onuare homogeneous and that u= 0fort < 0.The only external force applied isf at the right hand side of the wave equation. Furthermore, it is assumed to decrease sufficiently fast whent→ ∞in order to make sense to the following. Let us set:
ˆ
u(ω, x) = Z ∞
0
e−iωtu(x, t)dt=ur+iui, (real and imaginary parts of u).ˆ (12) One has:
−ω2uˆ−div(c2∇u) = ˆˆ f inΩ,
∂uˆ
∂ν = 0, onΓ0, iωˆu+c∂uˆ
∂ν = 0onΓe∪Γs.
(13)
By splitting these equations into the real and imaginary parts, one deduces that:
−ω2ur−div(c2∇ur) =frinΩ,
∂ur
∂ν = 0, onΓ0, ωui=c∂ur
∂ν onΓe∪Γs.
(14)
−ω2ui−div(c2∇ui) =fiinΩ,
∂ui
∂ν = 0, onΓ0, ωur+c∂ui
∂ν = 0 onΓe∪Γs.
(15)
3.1. Property of the Fourier model. Let us consideruˆthe Fourier transform ofudefined at (12); (uis decreasing to zero whent→ ∞because of the transparent boundary condi- tion which acts as a damping as shown in the previous section as soon as a compatibility condition is satisfied by the data): One can prove the following result.
Theorem 3.1. Let us denote by(λn, wn)the eigenvalues and eigenfunctions solution of the model (2). Let us setΛ = {√
λn, n∈N∗}. For anyω ∈R\Λ, the coupled system (15) and (14) has a unique solution.
Proof Let us introduce the bilinear forma(..)defined on the spaceH1(Ω)×H1(Ω) and the linear forml(.)onH1(Ω)by:
∀u˜= (ur, ui),v˜= (vr, vi)∈H1(Ω)×H1(Ω), a(˜u,˜v) =−ω2
Z
Ω
urvr+uivi
+ Z
Ω
c2(∇ur.∇vr+∇ui.∇vi) +ω Z
Γe∪Γs
c(urvi−uivr),
l(˜v) = Z
Ω
(frvr+fivi)(x, ω)dx.
(16)
One has:
a(˜v,˜v) =−ω2 Z
Ω
(vr2+vi2) + Z
Ω
c2(|∇vr|2+|∇vi|2).
Therefore the bilinear forma(., .)is Garding-coercive [16]. This enables one to apply the Fredholm alternative [17] ensuring that the model (14)-(15) which is equivalent to find
˜
u∈H1(Ω)×H1(Ω)such that:
∀˜v, a(˜u,v) =˜ l(˜v), (17) has a unique solution as soon asω6=√
λn, ∀n∈N. ✷
From (17), it can be deduced that if one uses Shannon wavelet [21], the solutionuˆis zero excepted for the frequencies in the bandwidth concerned.
4. Local behaviour of the solution(ur, ui)near the crack tip. Let us assume that there is a crack on the interfaceΓi as shown on figure1. The two extremities are denoted by AandB and their abscissa onΓi are respectivelyx1 = aandx1 = b. From classical results (see [?], [3]) one can write in the vicinity of each crack tip ((RA, θA), -respectively (rB, θb)- are the polar coordinates fromA-respectivelyB- the reference axis for the polar angle being the lineΓias shown on figure4):
FIGURE4. Local polar coordinates
On a neighbourhoodVAofA, one can write: :
u(x, t) =KA(t)SA(x) +uRA, uRA∈L∞(]0, T[;H2(VA)), withSA(x) =
√rA
c sin(θA
2 ), c=c+ifθA>0, c=c−ifθA<0, and on a neighbourhoodVBofB :
u(x, t) =KB(t)SB(x) +uRB, uRB ∈L∞(]0, T[;H2(VB)), withSB(x) =
√rB
c sin(θA
2 ), c=c+ifθB >0, c=c−ifθB<0.
(18)
The coefficientsKAandKBare called the stress intensity factors and they depend on the time variable. Concerning the Fourier transform ofu, the previous result becomes for instance around the extremityA:
ˆ
u(x, ω) = ˆKA(ω)SA(x) + ˆuRA, (19) with:
KˆA(ω) = Z ∞
0
e−iωtKA(t)dt anduˆRA(x, ω) = Z ∞
0
e−iωtu(x, t)dt.
The same result is also true for the extremityBof the crack tip.
5. An energy release rate for crack detection. Let us now introduce the so-called mul- tiplier method for partial differential equations which in fact is nothing else but a domain derivative of the elastic energy with respect to a displacement of the domainΩ(see [8]).
Several integrations by parts are used. Multiplying (15)-(14) by ∂ur
∂x1
or ∂ui
∂x1
and by inte- grating overΩδ, one obtains the following relations (ν1is+1onΓsand−1onΓs:
− Z
Γe∪Γs
[ω2(u2r+u2i)ν1+c2 2(|∂ur
∂x2|2+|∂ui
∂x2|2)]ν1− Z
Ω
(fr
∂ur
∂x1
+fi
∂ui
∂x1
)
=G(ω) = π
4(|KˆB|2− |KˆA|2)
(20)
Let us discuss briefly this relation. This is an important point in the detection criterion of the crack located on the interface lineΓi.
1. There is a formula which gives the expression of the stress intensity factors with respect to the solutionuˆin a neighbourhood of the crack tip. Let us recall it ([8]).
First of all we denote by∂VAthe boundary of the neighbourhood ofA as shown on figure4. Let us denote byKˆArthe stress intensity factor at pointAforur. If ν′ = (ν1′, ν2′)is the unit normal to∂VAinsideVA, one has (a similar result is also true in the vicinity of pointB):
π
4|KˆAr|2= Z
∂VA
[−ω2
2 u2rν1′ −c2∂ur
∂ν′
∂ur
∂x1
+c2
2|∇ur|2ν1′]. (21) This formula enables one to compute accurately the stress intensity factorKAr. 2. If there is no crack (KˆAr = ˆKBr = 0) one has alwaysG= 0;
3. Let us notice that if:
G(ω) = 0, ∀ω∈[ω1, ω2] then (it is assumed thatω16=ω2):
|KˆA(ω)|=|KˆB(ω)|, ∀ω∈[ω1, ω2].
4. If G(ω) = 0 on [ω1, ω2],and if the functionf is correctly chosen, one can prove that there is no crack. It looks like a controllability result. Let us give few ideas on the justification of this property. For sake of simplicity we only considerurassuming for instance thatui = 0and we restrict the discussion to the casec+=c− in order to avoid the two singularities appearing atEandF (see [7] for the complete case).
In fact one can use the characterization of the stress intensity factor from the dual singular functions (one for each crack tip) [10] -saySA∗ andSB∗- which leads to the expressions:
KAr= Z
Ω
[frSA∗ +ω2urSB∗]dx
− Z
Ω
div(c2∇SAr)SA∗
, KBr= Z
Ω
[frSB∗ +ω2urSB∗]dx
− Z
Ω
div(c2∇SBr)SB∗
. (22)
In fact, one can use the eigenvector basis{wn}defined at (2) in order to represent the solutionur. Hence one has for anyω /∈Λ ={λn}(see (2)):
ur(x, ω) =X
n≥1
ξn(ω)wn(x) with ξn(ω) = Z
Ω
frwn
λn−ω2. Thus, setting:
δA= 1
− Z
Ω
div(c2∇SA)SA∗
, δB = 1
− Z
Ω
div(c2∇SB)SB∗ ,
one has:
KAr(ω)=δA
X
n≥1
λn
Z
Ω
frwn
Z
Ω
SA∗wn
λn−ω2 KBr(ω)=δB
X
n≥1
λn
Z
Ω
frwn
Z
Ω
SB∗wn
λn−ω2 .
Let us consider thatfr = χ[ω1,ω2]d(x). Let us introduce a function of the space L2(Ω)-sayh(x, ω)- such that;
h(x, ω) =X
n≥1
[ λn
λn−ω2 Z
Ω
d(x)wn(x)dx]wn(x).
For instance
h(x,0) =d(x).
If :
∀ω∈[ω1, ω2], |KAr|=|KBr|,
one has, assuming thatfˆ(x, ω) =χ[ω1,ω2]d(x), one can deduce:
Z
Ω
h(ω)SA∗ =± Z
Ω
h(ω)SB∗ = 0.
The expressions ofSA∗ andSB∗ are known (for instance using the polar coordinate around the pointAone has [10]:
SA∗(rA, θA) = 1 c√rA
sin(θA
2 ).
Therefore, ifdis chosen sucht that one can conclude that this is impossible for any position ofAandB, one can claim that there is no crack.
5. Let us denote byGobsthe quantityGbut computed from an experimental measure- ment (or an estimation of it). With another respect, for a given position of the crack (iexA andxB given) one can compute the numerical expression ofG(ω, xA, xB) and compare it toGobs. Therefore, one can define an observability criterion of a crack by:
CN D(xA, xB) = Z ω2
ω1
|G(ω, xA, xB)−Gobs|)2dω. (23) And the localisation of the crack can be handle through the following optimisation problem forxCgiven between on]0, L[and(xA, xB)∈R2being the control vari- ables:
0≤xA≤xminC≤xB≤LCN D(xA, xB) +ε[(xA−xC)2+ (xB−xC)2]. (24) The last term is introduced in order to select the solution for which the middle of the crack is the nearest of the point with abscissax1 =xC andεis a small parameter (inducing a Tikhonov regularisation because of thea priorinon uniqueness of the solution [20]). The continuity ofCN Dwith respect to(xA, xB)and its positiveness ensure the existence of a solution (but not the uniqueness). In fact, the main point is to detect the presence of a crack. Its exact position is a second step much more complicated from the computational point of view. It can performed numerically by usinf an adjoint state which leads to the expression of the gradient o the criterion to be minimized. This will be discussed in future works.
6. The numerical simulation in time. The Fourier transform can be computed directly or derived from a time simulation which is the choice we did. In fact we choose to solve numerically in time in order to have a full description of the mechanical phenomenon.
Furthermore, in order to restrict the frequency range, we use a pseudo-Shannon wavelet setting:
f(x, t) =d(x)[sin(at)
πt −sin(bt)
πt ], a > b, (25)
whered(x) is a function the support of which is included in the soft media and which interacts with Love waves as shown on the left of figure5. On the right of figure5, we have plotted the solution representing the wave propagation for two materials with the velocities c+ = 2c−and the soft media corresponds to a a small strip (thickness=.1Land localized downwards on the figure))
FIGURE5. Right hand side (left); solution withc+= 2c−(right)
In order to compare the influence of the soft media we have drawn on figure6two cases of ratio betweenc+ andc−. One can see thetrapping effect in the softest media (Love waves) which increases with the ratioc+/c−compared to the figure5on the right.
FIGURE6. Left :c=c− Right:c+= 4c−
The Fourier transform offis the product of the characteristic functionχ[−b,−a[∪[a,b](ω) of the segment±[a, b]byq(x). In particular one hasfi = 0. The Fourier transform ofu is therefore computed by a symmetrisation with respect to the time (the initial conditions are zero). The values ofaandbare chosen such that the segment[a, b]contains Love wave frequencies. The figures7represents the wave propagation for two examples of frequencies ranges. The terms implying ∂u
∂x2
on the boundariesΓeandΓswhich are plotted on figures8 (without crack) and9(with a crack), show the effects of the artificial singularities appearing at pointsE andF compared to what happens inside the open setΩ. These terms are the most important in the computation of the detection criterionGexplicited at (20).
FIGURE 7. Without crack Solution of (1): low frequencies (left),
∂u/∂x2 (right); f is given at (25): a = 50b = 10. Comparing with figure8one can see that the energy is less localized in the softest media than with high frequencies.
FIGURE8. Without crack∂u/∂x2on the entryΓe(left) on the exitΓs
(right), (c+ = 2c−the initial data are zero,f is given at (25): a= 260 b= 360.
FIGURE9. With a crack on the interface between the two materials The left figure represents the wave equation solution. The right one is
∂u/∂x2 onΓeand on Γs. Same conditions as on figure8but with a crack betweenx1=.4Landx1=.6L. One can see the influence of the crack on this quantity and therefore on the criterion CND defined at (23)
7. The Fix method for the singularities at the pointsEandF. In order to improve a finite element scheme for solving a PDEs, Fix [19] has suggested in fracture mechanics modelling, to add the singular functionS to the classical f.e.m. spaceVh spanned by first order polynomials (using here quadrangular elements). This leads to an approximate solution denoted byuhand such that:
uh(x, t) =αh(t)S(x) +uhr(x, t), uhr(x, t)∈Vh. (26) Our goal is to discuss this method in our framework. In particular, the error estimate be- tweenα(t)andαh(t)is analyzed in terms of error with respec to the mesh sizesay−h.
Finally, this suggests to substract the termαh(t)S(x)to the approximate solutionuhin order to try to eliminate the effect of this non-physical singularity in the computation of the criterionCN D(xA, xB).
For sake of simplicity, we discuss the case of the following model (the functionsf and gare given):
−div(c2∇u) =f inΩ assuming that:
Z
Ω
f + Z
Γe∪Γi
gv= 0,
∂u
∂ν = 0onΓ1, ∂u
∂ν =g onΓe∪Γs, Z
Ω
u= 0
(27)
The variational formulation consists in findingu∈ V0 ={v ∈H1(Ω) and Z
Ω
v = 0} such that:
∀v∈V0, Z
Ω
c2∇u.∇v= Z
Ω
f v+ Z
Γe∪Γs
gv. (28)
Let us denote byVhthe approximation space spanned by the first degree Lagrange finite element with the classical assumptions.V0his the subspace of functionsv∈Vhsatisfying
the additional condition Z
Ω
v= 0. Setting:
Vsh={R×S} ⊕Vh, and V0sh ={v∈Vsh, Z
Ω
v= 0}. (29) The upgrade approximate model consists in findinguh∈V0shsuch that:
∀v∈V0sh, Z
Ω
c2∇uh.∇v= Z
Ω
f v+ Z
Γe∪Γs
gv. (30)
Theorem 7.1. Letu∈V0anduh∈V0sh be the solutions of (28) and (30). Let us set:
u=αS+ur, uh=αhS+urhwhereur∈H2(Ω)∩V0, urh∈V0h.
The family of meshes is assumed to satisfy the classical regularity assumptions (see Ciarlet [4]or Raviart-Thomas[17]). Then there exists a constantcindependent on the mesh size hsuch that:
||u−uh||1,2,Ω≤ch|ur|2,2,Ω and
|α−αh| ≤ c
Log(h)|ur|2,2,Ω.
Indications on the proof following the method used in Amara-Destuynder-Djaoua[1].
Fom the variational formulation characterizinguat (28), one gets: Let us set (see figure8):
Kh =]0, h[×]−h, h[, which are the two elements near the singular pointEfor instance.
Furthermore, we introduce the projection -say P- fromV0 ontoV0h with respect to the scalar product induced by the bilinear forma(., .). One has for instance (constant functions disappear in the expression of the bilinear forma(., .)):
∀v∈V0h, a(S−P S, v) = 0.
Let us define byπthe Lagrange interpolation operator fromV ∩C0(Ω)intoVh. From the relations satisfied buuanduhone has, using classical error estimates (see Strang-Fix [19]
and P.A. Raviart-J.M. Thomas [17] ):
∀v∈V0sh, a(uh−u, v) = 0 =>q
a(uh−u, uh−u)≤p
a(u−πu, u−πu)≤ch|ur|2,2,Ω. And by separating the term inV0hfrom the singular functionS, one obtains:
∀v∈V0sh, (αh−α)a(S, v) +a(uhr−ur, v) = 0, ∀v∈V0h, a(S−P S, v) = 0.
Setting:
v=S−P S, one deduces that (with a new constantc):
|αh−α|= ch
pa(S−P S, S−P S)
≤ch v u u u t
1
z=ax1+bxinf2+cx1x2
Z
Kh
c2|∂(S−z)
∂x2 |2 .
The computation of the previousinfimumusing a symbolic software (Maple) enables one to complete the proof , becauseSis known analytically. ✷ Remark 3. The extension to the stationary case (ω 6= 0) is based on the same method as the one described quickly here. Once,αhknown, the idea is to substractαhS fromuhin order to avoid the effect of the singularitySin the computation of the detection criterionG defined at (20). The effect is to smooth the behaviour of the solution of the wave equation near the singular points.
Remark 4. This theoretical result suggests to useuhr in the CND criterion for detecting the crack. It appears has a way for eliminating the artifical singularityS even if the error estimate onα−αhis very coarse.
Another possibility is to project the finite element solution (without adding the singularity) orthogonally to the singularityS. This is done by setting for instance:
R0uh=uh− Z
Ω
Suh Z
Ω
S2
S or R1uh=uh− Z
Ω
c2∇uh.∇S Z
Ω
c2|∇S|2 S.
We have plotteduhandR0uhon figure10the results obtained by this simple method in the computation of the term ∂u
∂x2
. But up to now this is just an idea without mathematical foundations.
FIGURE10. Comparison betwwenuh(left) andRuh(right) insideΩat the same time
8. Conclusion and future work. The non destructive testing is a challenge which requires efficient and reliable methods. The large amount of structures which should be tested requires new methods which enable to perform this duty faster than classical ones. The researches in this field are very active and a lot of problems are still to be solved from both the experimental aspect and the signal processing algorithms. In this paper, we have only suggested few remarks based on mathematical modelling but which are to be continued in order to confirm the operational possibilities of the tool considered here.
Acknowledgments. This work is developed in the context of the French-English GDR 2501:Wave propagation in complex media for quantitative and non destructive evaluation.
The support of this research group from both scientific exchanges and experimental point of view has been particularly useful in our study.
References
[1] M. Amara, Ph. Destuynder and M. Djaoua, On a finite element schem for plane crack problems, Numer.
Meth. in Frac. Mech., D.R.J. Owen and A.R. Luxmoore, Pinridge Press, Swansea, pp. 41-50, (1980).
[2] H. Brezis,Analyse fonctionnelle, Masson, (1984).
[3] H.D. Bui,Mécanique de la rupture fragile, Masson, (1979).
[4] P.G. Ciarlet,The finite element mehod for elliptic problems, Elsevier, Amsterdam, (1978).
[5] Ph. Destuynder and C. Fabre, Singularities occuring in multimaterials with transparent boundary conditions, research research report CNAM (2014).
[6] Ph Destuynder and C. Fabre,Numerical analysis of singularities appearing in bimaterials with transparent boundary conditions, research research report CNAM, (2015).
[7] Ph Destuynder and C. Fabre,On the detection of cracks in linear elasticity research report CNAM, (2015).
[8] Ph. Destuynder and M. Djaoua, Sur une interpretation mathématique de l’intégrale de Rice en mécanique de la rupture fragile, Mathematical Methods in the Applied Sciences Volume 3, Issue 1, pages 70-87, (1981) [9] G. Diot, A. Kouadri-David, L. Dubourg, J. Flifla, S. Guegan, E. Ragneau, Mesures de défauts par ultrasons
laser dans des soudures d’alliage d’aluminium, Publications du CETIM, (2014).
[10] M. Dobrowolski,Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationsschrift, Bonn (1981).
[11] J-C. Dumont-Fillon, Contrôle non destructif par les ondes de Love et Lamb, Editions Techniques de l’ingénieur, (2012).
[12] A.Galvagni and P.Cawley, The reflection of guided waves from simple supports in pipes, Journal of the Acoustical Society of America, (129), 1869-1880, (2011).
[13] E. Holmgrem, Eric Holmgren,Uber Systeme von linearen partiellen Differentialgleichungen,¨ Ofversigt af¨ Kongl, Vetenskaps-Academien F¨orhandlinger, 58, pp. 91-103, (1901).
[14] M.J.S. Lowe, Characteristics of the reflection of Lamb waves from defects in plates and pipes, Review of Progress in Quantitative NDE, Vol. 17, DO Thompson and DE Chimenti (eds), Plenum Pr ess, New-York, pp113-120, (2002).
[15] P. M. Marty, Modelling of ultrasonic guided wave field generated by piezoelectric transducers, Thesis at Imperial college of science, technology and medecine, university of London, (2002), http://www3.imperial.ac.uk/pls/portallive/docs/1/50545711.PDF
[16] J. Necas,Les méthodes directes en théorie des équations elliptiques, Masson, Paris, (1965).
[17] P.A. Raviart and J.M. Thomas,Approximation des équations aux dérivées partielles, Masson, Paris, (1986).
[18] R.Ribichini, F.Cegla, P.Nagy and P.Cawley, Study and comparison of different EMAT configurations for SH wave inspection, IEEE Trans.UFFC, (58), 2571-2581, (2011).
[19] Gilbert Strang and George Fix,Analysis of the Finite Elements Method, Prentice Hall; Edition : First Edition (1973).
[20] A. N. Tikhonov, http://fr.wikipedia.org/wiki/R%C3%Régularisation Tychonoff (1943).
[21] Wikipedia http://en.wikipedia.org/wiki/Shannon wavelet
[22] D. Zagler, The Dilog function,http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-30308- 4 1/fulltext.pdf (2007).
Received xxxx 20xx; revised xxxx 20xx.
E-mail address:philippe.destuynder@cnam.fr E-mail address:caroline.fabre@u-psud.fr
