Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
4.6 Some problems of detection of defects
−v=div(kχ(D)∇u0) in, v=0 on∂,
with the additional Neumann data∂νv=hon∂. The corresponding linearized inverse problem is easier to analyse. In particular, whenk is a known constant andu0is a linear function one can show uniqueness ofDwhich is convex in the direction of∇u0. However, the remaindervδis noto(δ). A more detailed analysis shows that the genuine linearization is more complicated. To explain difficulities we mention the recent result of Beretta, Francini, and Vogelius [BerFV].
Let σ be an orientable hypersurface in Rn of classC3 withσ ⊂ with a fixedC2-smooth unit normalν. Leta0 be a constant,aδ =kχ(Dδ) where Dδ = {y:di st(y, σ)< δ}. Letb0=bδ =0,c0=cδ =0. Let (τ1, . . . , τn−1, ν) be an orthonormal basis inRn, withτ1, . . . , τn−1tangent toσ. We defineA(;σ,a0,k) as a symmetric matrix with eigenvectorsτ1, . . . , τn−1, νand corresponding eigenvalues 1, . . . ,1,a0/(a0+k). One can assume that A∈C2(σ). It was shown in [BerFV]
that
u(x)−u0(x)=2δ
σk A(y;σ)∇u0(y)ã ∇yG(x,y)dσ(y)+r(x;δ) where|r(;δ)| ≤Cδ1+κg(1/2)(∂),κ ∈(0,1) depends onn andC depends on n, ,a0/(a0+k), σ. Most likely the result of the linearization depends on ap- proximating domains Dδ, which are chosen to be “uniformly thin” in [BerFV].
While the methods of section 4.4 are quite sharp and powerful they are restricted to the plane case. In any dimension linearization at fixed domains can be obtained by using the theory of domain derivatives exposed, for example, in the book of Sokolowski and Zolesio [SoZ].
4.6 Some problems of detection of defects
Locating defects inside of material, in particular, locating cracks, is a fundamental importance in (airplane and automotive) engineering. The mathematical study of this problem originated in the paper of Friedman and Vogelius [FV]. We will formulate some recent results.
Letuσbe a solution to the following boundary value problem:
(+k2)u =0 in\σ, ∂νu=g1on∂;
u =Conσ (insulating crack) or∂νu =0 onσ (conducting crack).
(4.6.1)
Hereis a bounded Lipschitz domain inRn,n=2,3;Cis a (unknown) constant at each connected component ofσ, andσis a Lipschitz curve inwith connected
\σinR2or a Lipschitz surface inR3. Whenk=0, by minimizing the Dirichlet integral or by using conformal mapping of\σonto an annulus, one can secure the existence and uniqueness ofu∈ H(1)(\σ), provided that natural orthogonality relations are satisfied (the integrals ofg1,uover∂are zero). By standard elliptic theory this solvability result will hold for all realkexcept of some set accumulating to infinity.
First, we considerburiedcracks; i.e., ¯σ ⊂.
Whenk=0, in the plane case there is a quite complete result on uniqueness due to Alessandrini and Valenzuela [A1V] and to Kim and Seo [KiS]. Let1, . . . , 3
be three connected curves with disjoint interiors whose union is∂. Letφj be L2(∂)-functions,φj ≥0, and suppφj ⊂j and let the integral ofφj over∂ be 1. Define g1,j =φ3−φj,j =1,2. In the next Theorem a buried crack σm
is assumed to be the union of finitely many connected nonintersecting Lipschitz curves. Letuj mbe the solution to the boundary value problem (4.6.1) withσ =σm
andg1=g1,j,j,m=1,2.
Theorem 4.6.1. Let k=0. Letbe a simply connected bounded domain inR2 with C2-piecewise smooth Lipschitz boundary.
If for solutions uj mto the boundary value problem(4.6.1)one has uj1=uj2on⊂∂, j =1,2,
(4.6.2) thenσ1 =σ2.
We will not prove this result, referring instead to the above-mentioned papers.
We observe only that in those proofs based on study of level-lines solutions of elliptic equations, it is used that due to the particular choice of the Neumann boundary data, solutions have no critical points inside, and that level curves corresponding to two boundary data form a global coordinates in.
We emphasize that one boundary measurement (g1 =g1,1) is not sufficient. A simple counterexample ( Friedman and Vogelius [FV]) is given by level curves of a harmonic function in. In particular, whenu(x)=x1all interval of vertical straight lines can be viewed as insulating cracks and all intervals of horizontal lines as conducting cracks. A conditional logarithmic type stability estimate for σ ∈C1+λ in terms of differences of the boundary datauσ(1)−uσ(2) and under additional natural a priori constraints onσ (and) was obtained by Alessandrini and more generality by Rondi [Ro].
An useful orthogonality relation that helps to identify linear cracks inR2 and planar cracks inR3 was found and exploited by Andrieux and Ben Abda [AnB].
We illustrate this relation in the following exericise.
Exercise 4.6.2. Show that for anyH(1)() solutionvto the Helmholtz equation v+k2v=0 inand a solutionu∈ H(1)() to the boundary value problem
4.6. Some problems of detection of defects 121 (4.6.1) (for insulating cracks) one has the following integral relation
σ[u]∂νvdσ =
∂(vg1−u∂νv)d
where [u] (“opening gap”) is the difference of limits ofuonσfrom the “negative”
and the “positive” sides ofσ determined by the unit normalν.
To derive the identity of this exercise surroundσ by “thin” domains and pass to the limit when “thinness” goes to zero.
The integral relation from Exercise 4.6.2 can be used to evaluate size of σ by taking asvparticular explicit solutions to the Helmholtz equation in . For example, whenk=0 one can exploit the coordinate harmonic functionsv(x)=xj
Of course, when the “opening gap” is zero this relation is useless, so it is an important (and largely open) problem to find sets of boundary datag1so that one of this sets maximises this gap.
∂σ (end points in the plane case) are expected to be branch points of solutions to the equation (4.6.1), moreover, the solution near these points in a generic case should have special singular behavior (for example,∇u(x) near crack tipx0is most likelyc|x−x0|−1/2+ ã ã ãwhere. . .is bounded terms). For an effective solution of the inverse problem it would be very helpful to use the boundary datag1which maximizec. Again, finding such boundary data is an quite interesting, but open problem. In the plane case for k=0 one can probably utilize index (winding number) of gradient of a harmonic function as in [AlIP], [AlM].
A iterative computational algorithm to find an interval σ was suggested and tested by Santosa and Vogelius [SaV2], who actually used results of many boundary measurements with the boundary data updated on each iteration of their method.
We observe that the methods of all mentioned papers do not work for the Helmholtz equation, which models prospecting cracks by (ultrasonic or elastic) waves, and uniqueness results are unknown in the case of single or finitely many measurements. Kress [Kres] proved uniqueness ofσ from many boundary mea- surements (i.e., from the operatorg1→g0on) in the plane case by the methods described in Section 3.3. He also designed an effective numerical algorithm.
In the three-dimensional case quite complete uniqueness results for insulat- ing cracks with two boundary measurements were reported by Alessandrini and DiBenedetto [A1D]. They also proved uniqueness of planar conducting cracks.
Eller [Ell] proved uniqueness of a general conducting crack in R3 from many boundary measurements.
Another type of cracks quite important in applications issurfacecracksσ con- sisting of finitely many connected Lipschitz components intersecting∂. Elcrat, Isakov, and Neculoiu [ElcIN] proved uniqueness of such insulated cracks from one boundary measurement in a quite general situation, in particular for finitely multiconnected domains, and suggested an efficient numerical algorithm based on Schwarz-Christoffel transformations. We will formulate one result from their paper and outline its proof.
Let be a bounded domain inR2 with the C2-piecewise smooth Lipschitz boundary∂. Let0be the connected component of∂that is also the boundary of the unbounded component ofR2\. We break0into three components,, 1, and 2. We will consider cracksσsatisfying the same conditions as in Theorem 4.6.1, but in contrast assuming that any component ofσhas an endpoint on∂\(∪1).
Also, in the boundary value problem (4.6.1) we will replace the Neumann condition on∂by the Dirichlet conditionu=gon∂.
Theorem 4.6.3. Let the Dirichlet data g0≥0on∂be not identically zero, and suppg0⊂1.
Then the additional Neumann data∂νu onuniquely determineσ.
PROOF. Let us assume the opposite. Let σ1, σ2 be two collections of Lipschitz curvesσ11, . . . , σ1k, σ21, . . . , σ2mgenerating the same Neumann data on.
Let\(σ1∪σ2) be not connected. Take the connected component0of this set whose boundary contains. Since the two solutionsu1,u2forσ1, σ2have the same Cauchy data on, by uniqueness in the Cauchy problem for the Laplace equation they are equal on0. We consider1=(\σ1)\0. We haveu1=0, and we will show thatu1=0 on∂1, provided that1is disjoint from∂1.
Letx∈∂1. Ifx∈∂(\σ1), thenu1(x)=0. If not, then nearxthere are points of0whereu1=u2; sou1(x)=0, becausex∈σ2whereu2=0.
By the maximum principle,u=0 in1. Due to connectedness of\σwe have u=0 in\σ, which contradicts our assumption aboutg.
If 1 intersects ∂1, then it does not intersect ∂0, and we can repeat the previous argument with0instead of1.
This contradiction shows that the set\(σ1∪σ2) is connected. By uniqueness in the Cauchy problemu1 =u2, soσ1must coincide withσ2becauseuj =0 on σjanduj >0 on\σjby the maximum principle.
The proof is complete.
This proof is valid in three-dimensional space. InR2, by using harmonic con- jugates, one can obtain uniqueness for conducting surface cracks.
A solutionuto the above direct problem can be represented by a single or double layer potential with some density distributed overσ. When this density is given, there are uniqueness and (logarithmic type) stability results forσ. We refer to the book [Is4] and to the paper by Beretta and Vessella [BerV], where this problem is related to the inverse problem of cardiology.
The boundary value problem (4.6.1) is a simplest mathematical model of a crack.
A way to derive it is to approximateσby slender domainsDδsurrounding crack, prescribing a natural boundary condition on∂Dδand finding the limit of solutions for Dδ asδgoes to 0. In section 4.5 it was mentioned that limits of solutions can be different from what one expects. It is obvious that this limit depends on the choice of approximationsDδ and it is not obvious that the choice in section 4.5 was natural. Indeed, probably this approximation process repeats dynamics of the growth of crack, so it is hard to believe thatDδmust be a “uniform” layer used in
4.6. Some problems of detection of defects 123 section 4.5. However, it is not obvious that uniform layers are the most appropriate ones. Due to these ambiguities a more appropriate model of a stationary wave in a domainwith crackσ is given by the boundary value problem
(A+k2)u=0 in\σ, ∂ν(a)u=g1on∂ (4.6.3)
with the general transmission conditions
u+=b0u−, ∂νu+=b1∂νu−+b00u+onσ (4.6.4)
whereu+,u− are limits ofu from different sides ofσ andb0,b1,b00 are some coefficients which are to be found together with σ from certain collection of boundary measurements. Here A is a general elliptic operator of second order.
While this general formulation is physically motivated, the particular case ofA= is far from understanding. In particular, it is not known whether the Neumann-to Dirichlet operator ( i.e. all possible Cauchy data for solutionsuto (4.6.3), (4.6.4)) uniquely determinesσand coefficients of the transmission condition.
Few large cracks considered above hypothetically result from a collection of many microcracks. Evaluation of amount of these microcracks is quite important engineering problem. However, there are only first mathematical results available.
To illustrate difficulties and features of this problem we briefly decribe the findinds of Bryan and Vogelius [BrV] for a very particular case of a periodic array of simplest possible cracks in the plane.
Letσbe an interval in the unit squareY =(0,1)×(0,1) in the planeR2. Letσ has the end points (s,s),(1−s,1−s),0<s<1/2. Letbe a bounded domain inR2 withC2-boundary. We denote by σ(ε) the union ofε-scaled translations ε((n1,n2)+σ) of σ over all integer n1,n2. Let (ε)=\σ(ε). An electric potentialu(;ε) in the domain(ε) with periodic array of small insulating cracks σ(ε) solves the following Neumann problem for the Laplace equation:
u(;ε)=0 in(ε), ∂νu(;ε)=0 onσ(ε)∩, ∂νu(;ε)=g1on∂, where we assume that the total flux ( the integral of g1 over ∂) is zero and for uniqueness ofu(;ε) we request one of standard normalization conditions, for example assuming that the integral ofu(;ε) overis zero. Because of expected complex behavior ofu(;ε) asεgoes to zero one would be interested in finding the limit (in some sense) ofu(;ε). The most suitable available technique is the so- called homogenization, also used in [BrV]. Homogenization is rather complicated procedure and it is not quite clear how to effectively utilize it to solve inverse problems.
To continue with results of [BrV] we introduce the (y1,y2)-periodic ( with periods (1,0),(0,1)) solutionχk(y) to the boundary value problem
χk=0 inR2\σ(1), ∂νχk= −νkonσ(1)
which is unique up to an additive constant. Letabe the positive symmetric matrix aj k=
Y
(δj k+∂χj/∂yk)
and letube the normalized solution to the Neumann problem di v(a∇u)=0, in, ∂ν(a)u =g1 on∂.
One of typical results obtained by homogenization methods (use of two-scale test oscillating test function) claims that in certain senseu(;ε) converges touasεgoes to 0.
Another related problem is about prospecting corrosion on inaccessible parts of surfaces. A model of electric prospecting is described by the elliptic boundary value problem
u=0 inγ,
u =0, when b=0
∂νu+bu=g1on∂,
g1=0, whenb=0
,
whereγis the domain{x: 0<x1<1, . . . ,0<xn−1 <1,0<xn < γ(x)}and 0is the part∂∩ {xn=0}of its boundary. It is assumed thatg1=0 on∂\0, 0≤b, and b=0 outside ∂∩ {xn=γ(x)} (the possibly corroded boundary part). One is looking for functionsγandb(or a more general nonlinear boundary condition) from the additional Dirichlet data
u =g0 on0.
Wheng1 is not identically zero andb=0 is given, the uniqueness ofγ can be shown as for obstacles in Section 6.3. One boundary measurement is obviously not sufficient to determine bothγ andb, so it makes sense to consider two mea- surements (or even the local Neumann-to-Dirichlet map on0 especially when one tries to determine a nonlinear boundary condition (b=b(x,u)). While this inverse corrosion problem is mathematically largely open, there are preliminary uniqueness theorems and numerical reconstruction algorithms.
We will conclude with a quite explicit counterexample of Alessandrini [Al4]
which shows exponential instability of the inverse problem about determinationγ from one set of remote data on0
COUNTEREXAMPLE4.6.4. Let Sm be the strip inR2 bounded by the x1-axis 0
and the curveγmgiven by the parametric equationsx1=t+1/(2m)si nmt,x2 = 1+1/(2m)(1−e−2m)/(1+e2m)cosmt,t ∈R,m=1,2, . . .. Let S0 be the strip bounded by the curves 0 and γ = {x2=1}. For the (Hausdorff) distance di st(γm, γ) between the curvesγmandγ we have
1/(4m)≤di st(γm, γ)≤1/(2m).
These inequalities can be derived by using the definition of the Hausdorff distance and minimization and maximization of functions of one variable. We will consider