Introduction à la méthode de sensibilité topologique pour l’identification de défauts. Aspects théoriques et numériques

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Introduction ` a la m´ ethode de sensibilit´ e topologique pour l’identification de d´ efauts.

Aspects th´ eoriques et num´ eriques

Marc Bonnet¹

1UMA (Dept. of Appl. Math.), POems, UMR 7231 CNRS-INRIA-ENSTA ENSTA, PARIS, France

mbonnet@ensta.fr

Formation Probl`emes inverses - Contrˆole des EDP, Amiens, 5-6 septembre 2011

Marc Bonnet (POems, ENSTA) Sensibilit´e topologique et identification 1 / 148

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Plan

1. Introduction

2. Time-independent (Laplace, elasticity)

3. Waves, frequency domain (Helmholtz, elastodynamics) 4. Waves, time domain

5. Crack identification

6. Higher-order topological sensitivity 7. Further reading

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To-do list

◮ champ total dans lippmann-schwinger

◮ estimations au moyen deδ^a

◮ separer inner and outer expansions

◮ definir¯v mais pas¯u

◮ define polarization tensor from far-field representation

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1. Introduction

2. Time-independent (Laplace, elasticity) Scalar (conductivity) problems

Elasticity

Numerical example

Energy-based cost functional

3. Waves, frequency domain (Helmholtz, elastodynamics) Helmholtz

Elastodynamics

Heuristics and partial justification for limiting situation Numerical examples

4. Waves, time domain Numerical examples

Experimental studies (Dominguez and Gibiat, Tixier and Guzina) 5. Crack identification

Numerical examples

6. Higher-order topological sensitivity Numerical examples

7. Further reading

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Forward and inverse problems

Ω^true= Ω\B^true

S

q S_obs

Sobs

Γ^true B^true

JLS(ΩΓ) = 1 2

Z

Sobs

|uΓ−uobs(ξ)|² dΩξ

J(ΩΓ) = Z

Sobs

ϕ(uΓ(ξ),ξ) dΩξ

where uobs(ξ) =utrue(ξ)|Sobs+ (noise)

◮ Insulated or conductive inhomogeneities (heat transfer, electrostatics)

◮ Sound-hard or penetrable objects (linear acoustics)

◮ Cavities, inclusions or cracks (elasticity)

◮ Electromagnetic inverse scattering ...

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Background

◮ Extensive literature on (acoustic, electromagnetic, elastic) inverse scattering e.g. books by Colton & Kress (93), Ramm (92), Cakoni & Colton (06), Kirsch (96), Kirsch and Grinberg (08)

◮ Iterative solution techniques (optimization-based, Newton): facilitated by sensitivity formulae, e.g.:

δJ = Z

Γ

b(u,û;C, ρ)vndS (shape sensitivity) withû: adjoint solutionsuch thatσ[û].n=∂uϕ

vn: normal perturbation of unknown surface Γ

Extension to combined shape-material sensitivity: b(u,u;ˆ C, ρ,∆C,∆ρ)

◮ Combined optimization and level-set approaches (Dorn and Lesselier 06, topical review)

MBEng. Anal. with Bound. Elem.(1995)

S. Nintcheu Fata, B. Guzina, MB,Comput. Mech.(2003) MB and B. Guzina,J. Comput. Phys. (2008)

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Optimization-based identification: example (inclusion, elastodynamics)

0 10 20 30 40 50

Iteration k 0

0.2 0.4 0.6 0.8 1

Parameter

c₁/c₁^true c₂/c₂^true c3/c

3 true J/J₀ Centroidal coordinates

0 10 20 30 40 50

Iteration k 0

0.2 0.4 0.6 0.8 1 1.2

Parameter

α1/α 1 true α₂/α₂^true α₃/α₃^true J/J₀ Semi-axes

0 10 20 30 40 50

Iteration k -2

-1 0 1 2 3 4

Parameter

µ/µ^true ν/ν^true ρ/ρ^true J/J0

Material properties

^

−3

−1 −2

0 ξ₁/d

−2

−1 ξ₃/d 1.5

2 2.5

3 3.5

0 1 ξ2/d

Initial

True 0 True Initial

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

µ/µ

M. Bonnet, B. B. Guzina,J. Comput. Phys (2009)

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Background

◮ Iterative solution techniques:

Computationally expensive: ≥1forward solution(s) per iteration;

Success strongly depends on choice of initial guess.

◮ Non-iterative, fast preliminary ‘probing’ (or ‘sampling’) methods (Potthast 06, IP topical review):

(a) Linear sampling (Colton, Kirsch 96)

Extension to near-field elastodynamic data (Guzina, Nintcheu Fata 04);

(b) Topological sensitivity

Mathematical studies (Sokolowski-Zochowski 97, Guillaume-Sid Idris 04) Asymptotic methods (Maz’ya-Nazarov-Plamenevskij 91, Il’in 92,

Vogelius-Volkov 00, Ammari-Kang 04...)

Initially proposed for topological optimization (Eschenauer et al. 1994);

Inverse problems (Guzina, MB 04, Feijoo 04, Masmoudi 05, MB 06...)

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Linear sampling

Sobs

S

B f

◮ Scattering operator:

uobs(x) = [R(B)f](x) (x∈Sobs)

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Linear sampling

z Sobs

S

B g

◮ Linear sampling equation (withG: Green’s tensor):

∃?g(ξ), kG(z;x).e−[R(B)g^ε_z](x)k ≤ε (x∈S_obs) z∈B =⇒ lim

z→∂Bkg^ε_zkL²(S)=∞ z6∈B =⇒ lim

ε→0kg^ε_zkL²(S)=∞

◮ Defect indicator function: z7→1/kg^ε_zk_L²_(S)

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Topological sensitivity

Sobs

S

B f

Ba(z) z z

◮ Small-obstacle expansion of cost function:

J(Ba(z)) =J(∅) +δ(a)T(z) +o(δ(a)) whereδ(a)quantifies the perturbation behavior abouta= 0;

T(z)is thetopological derivativeofJ; zacts as a sampling point.

◮ Heuristic idea: find regions inΩwhere field T ismost negative.

◮ Proposed defect indicator function:

z7→ T(z)

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Concept of topological sensitivity

◮ Sensitivity analysis toolinitially proposed for topological optimization (Eschenaueret al94; Sokolowski, Zochowski 99; Garreauet al01; Allaireet al05. . . )

Allaire, de Gournay, Jouve, Toader (2005)

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Example (Topological optimisation)

0 20 40 60 80 100

Iterations

4 4.5 5 5.5 6

Compliance

Th`ese X-EADS “Optimisation g´eometrique et topologique du drapage des composites” (2010–

2013, G. Delgado, dir. G. Allaire) — Algorithm combining topological and shape derivatives

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1. Introduction

Elasticity

Numerical example

Elastodynamics

Numerical examples

7. Further reading

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1. Introduction

Elasticity

Numerical example

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Background problem (BP)

◮ Media endowed with scalar (thermal, electrostatic...) conductivityκ Generalization to anisotropic (tensorial) conductivity not addressed, but relatively straightforward

◮ Primary field variableuis scalar (temperature, electrostatic potential...) Background problem (κ: background conductivity): defines the physical state in the absence of inclusion

div (κ∇u) +f = 0 inΩ, κ∂nu=g onSN, u=u^Don SD

i.e., in weak form:

Findu∈W(u^D), hu,wi^κ_Ω=F(w), ∀w∈W0

withW(u^D) =

w∈H¹(Ω), w=u^DonSD . W0=W(0)and hu,wi^κ_Ω:=

Z

Ω

κ∇u·∇w dV, F(w) :=

Z

Ω

fw dV+ Z

SN

gwdS

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Generic transmission problem (TP)

◮ Generic (possibly multiple) inclusionB⊂Ωand conductivityκB: κB = [1−χ(B)]κ+χ(B)κ^⋆=κ+χ(B)∆κ

◮ Transmission problem:

div (κB∇uB) +f = 0inΩ, κ∂nu=g on SN, u=u^DonSD

i.e., in weak form:

FinduB∈W(u^D), huB,wi^κ_Ω+huB,wi^∆κ_B =F(w), ∀w∈W0

◮ Both formulations implicitly enforce the transmission conditions u_B |+=u_B |− and κ∂nu_B |+=κ^⋆∂nu_B |− on∂B

◮ TP in terms of solution perturbationvB :=uB−u:

FindvB∈W0, hvB,wi^κ_Ω+hvB,wi^∆κ_B =−hu,wi^∆κ_B , ∀w∈W0

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Free-space transmission problem (FSTP)

◮ Auxiliary TP for an inclusionBembedded in an infinite mediumΩ =R³ subjected to a given remote uniform gradientE∈R³ through the background solutionu(ξ) =E·ξ=:ψ[E](ξ), i.e.:

( div (κB∇uB[E]) = 0inR³,

uB[E](ξ)−ψ[E](ξ) =O(kξk⁻²)askξk → ∞ withκB:=κ+χ(B)∆κ.

◮ Weak formulation for the solution perturbationvB:=uB−ψ[E]:

FindvB∈H¹(R³), hvB,wi^κR³ =−hψ[E],wi^∆κB , ∀w∈H¹(R³)

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Cost functional

Consider generic cost functionals of the form J(κB) =J(uB) with J(w) :=

Z

Ω

ϕV(·,w) dV + Z

∂Ω

ϕS(·,w) dS with twice-differentiable densitiesw 7→ϕV(·,w)andw 7→ϕS(·,w)

◮ Example 1: weighted least-squares output residuals (identification based on overdetermined datag,u^obs onSN)

J(w) = 1 2

Z

SN

Q|w−u^obs|²dS

◮ Example 2: potential energy (expressed as boundary integrals) J(w) = 1

2 Z

SD

(κ∂nw)u^DdS−1 2

Z

SN

gwdS

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Concept of topological sensitivity

SN

SD

Ba

Ω⊂R^D

(κ) Ba z

(κ^⋆)

Ba(z) =z+aB

Definition (Topological derivative)

Assume thatJ(u^a)can be expanded in the form

J(κa) =J(κ) +δ(a)T(z) +o(δ(a))

whereδ(a)is such thatδ(a) =o(1)asa→0 and characterizes the small-inclusion behavior ofJ(κB).

Then, the coefficientT(z)is called the topological derivative ofJ atz∈Ω.

◮ T(z)also depends on shapeB and conductivity contrastβ:=κ^⋆/κ;

◮ Terminology forT varies, with “gradient” or “sensitivity” sometimes used instead of “derivative”.

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Small-inclusion asymptotic behavior of

u^a

Established using a domain integral equation (DIE) formulation of supportBa

◮ The Green’s functionG is defined by div κ∇G(·,x)

+δ(· −x) = 0 in Ω G(·,x) = 0 on S_D κ∂nG(·,x) = 0 on SN







(x∈Ω)

and verifies

hG(·,x),wi^κ_Ω=w(x) x∈Ω, ∀w∈W0∩C0(Ω)

◮ SplitG into singular and nonsingular parts:

G(ξ,x) =G(ξ−x) +GC(ξ,x)

whereG(r) = 1/(4πkrk)is the (singular) full-space fundamental solution and the complementary Green’s functionGC is bounded atξ=x (and in factC^∞ forξ,x∈Ωby virtue of solving a BVP with regular boundary data).

◮ Note thatG and∇G are homogeneous of degree -1 and -2, respectively:

G(λr) =|λ|⁻¹G(r), ∇G(λr) =|λ|⁻²sgn(λ)∇G(r)

(r∈R³\ {0}, λ∈R\ {0})

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Small-inclusion asymptotic behavior of

u^a

◮ Domain integral equation (DIE) for the TP:

combine weak formulations for the TP (withw =G(·,x)) and the Green’s function (withw =v^a):

(hv^a,G(·,x)i^κ_Ω+hv^a,G(·,x)i^∆κ_B_a =−hu,wi^∆κ_B_a

hG(·,x),v^ai^κ_Ω=v^a(x) (x∈Ω) to obtain

La[v^a](x) =−hu,G(·,x)i^∆κ_B_a

x∈Ba (integral equation forv^a) x∈Ω\B¯a (integral representation forv^a) with the linear operatorLa defined by La[w](x) :=w(x) +hw,G(·,x)i^∆κ_B_a i.e. (in expanded form):

La[w](x) =w(x) + Z

Ba

∆κ∇w·∇G(·,x) dV

◮ DIE for the FSTP:(similarly)

LB[vB](x) =−hψ[E],G(·,x)i^∆κB with LB[w](x) :=w(x) +hw,G(·,x)i^∆κB

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Small-inclusion asymptotic behavior of

u^a

◮ Perturbation due to small inclusion governed by domain integralof supportBa

La[v^a](x) =−hu,G(·,x)i^∆κ_B_a Facilitates investigation of limiting behavior asa→0

◮ Introduce normalized coordinates

(a) (ξ,x) =z+a(¯ξ,¯x), (b) dVξ =a³d ¯Vξ¯ (ξ∈B_a,¯ξ∈ B)

Asymptotic behavior of v^a asa→0: InsideB_a, one has (inner expansion) v^a(x) =avB[∇u(z)](¯x) +o(a) ξ∈Ba,ξ¯∈ B

Moreover, at any fixed locationx6=z (outer expansion), one has v^a(x) =−a³∇₁G(z,x)·A·∇u(z) +o(a³) x6=z

whereA=A(κ, κ^⋆,B)is the (second-order) polarization tensor, defined by A·E=

Z

B

∆κ∇uB[E] dV = Z

B

∆κ(E+∇vB[E]) dV ∀E∈R³ (uB[E] =ψ[E] +vB[E]: solution to the FSTP)

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Small-inclusion asymptotic behavior of

u^a

(proof sketch 1/2)

Proof based on seeking and exploiting the limiting form of the DIE, and using scaled coordinates.

(i) Asymptotic behavior of the Green’s function:

∇₁G(ξ,x) =a⁻²∇G(¯ξ−¯x) +∇₁GC(z,z) +o(1) (ii) Defining¯v^a(¯ξ) :=v^a(z+a¯ξ), ¯u(¯ξ) :=u(z+a¯ξ), one finds

∇v^a(ξ) =a⁻¹∇v¯^a(¯ξ), ∇¯u(¯ξ) =∇u(z) +o(1) (iii) Substitute above expansions and rescale coordinates in DIE:

La[vâ](x) =LB[¯vâ](¯x) +aO(k∇v¯âk2,B) hu,G(·,x)i^∆κ_B_a =ahψ[∇u(z)],G(·,¯x)i^∆κB +o(a) (iv) Make ansatz ¯vâ(¯ξ) =aV(¯ξ) +o(a)and isolate lowest-order (ina)

contributions in DIE, which correspond to the FSTP:

LB[V](¯ξ) =−hψ[∇u(z)],G(·,¯x)i^∆κ_B Hence: V(¯ξ) =vB[∇u(z)](¯ξ) ¯ξ∈ B

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Small-inclusion asymptotic behavior of

u^a

(proof sketch 2/2)

(v) Now, considerx6=zandasmall enough. In that case, one has

∇₁G(ξ,x) =∇₁G(z,x) +o(1), ξ∈Ba,x6∈Ba. (vi) Substitute¯v^a(¯ξ) =aV(¯ξ) +o(a)and the above into the integral

representation outsideBa, to obtain v^a(x) =−a³∇₁G(z,x)·

Z

B

∆κ∇vB[∇u(z)](¯ξ) d ¯Vξ¯+o(a³)

=−a³∇₁G(z,x)·A·∇u(z) +o(a³)

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Topological derivative of cost functionals

(i) Expand cost functional w.r.t. v^a:

J(κa) =J(κ) +J^′(u;vâ) +o(kvâk2,Ω+kvâk2,∂Ω) with J^′(u;w) :=

Z

Ω

∇₂ϕV(·,u)w dV+ Z

SN

∇₂ϕS(·,u)wdS (ii) Interpret J^′(u;v^a)as component of a weak formulation with test function v^a

and define theadjoint solutionˆuby

Find ˆu∈W0, hˆu,wi^κ_Ω=J^′(u;w), ∀w∈W0

(iii) Reciprocity between transmission and adjoint problems:

hvâ,uiˆ ^κ_Ω+hvâ,ûi^∆κ_B_a =−hu,uiˆ ^∆κ_B_a , hû,vâi^κ_Ω=J^′(u;vâ) to obtain J^′(u;vâ) =−hû,ui^∆κ_B_a − hû,vâi^∆κ_B_a =−hû,uâi^∆κ_B_a (iv) Exploit known limiting behavior ofvâ in Ba (inner expansion), yielding

J^′(u;vâ) =−hû,uâi^∆κ_B_a =−a³∇u(z)·A·ˆ ∇u(z) +o(a³) Topological expansion ofJ: The topological expansion ofJ is given by

J(κa) =J(κ) +a³T(z) +o(a³), T(z) =−∇ˆu(z)·A·∇u(z)

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Remarks on topological derivative and polarization tensor

(i) Ais symmetric and has the same sign as∆κ(Cedyo-Fengya, Volkov,

Vogelius, 1998);

(ii) By superposition:

V(¯x) =vB[∇u(z)](¯x) = X3

i=1

ei·∇u(z)

vB[ei](¯x) One thus needs only compute thevB[ei], which are independent onz.

(iii) Spherical inclusion(i.e. Bis unit sphere): FSTP solution is given by vB[E](¯x) = 3

2 +βE·¯x withβ:=κ^⋆/κ (¯x∈ B).

Hence:

A(κ, κ^⋆,B) = 4πκ(β−1) β+ 2 I ;

(iv) Analytical FSTP solutions available for other shapes (e.g. ellipsoids);

otherwise, solve numerically (once and for all) the normalized DIE with E=ei (i= 1,2,3);

(v) Special caseβ= 0: insulated(i.e. impenetrable) inclusion.

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Remarks on topological derivative and polarization tensor

(vi) Similar analysis available for (penetrable or insulated) inclusions in 2D

media, yielding cost function asymptotics of the form J(κa) =J(κ) +a²T(z) +o(a²)

(vii) If homogeneous Dirichlet conditions are instead considered on∂Ba, one finds cost function asymptotics of the form (e.g. Guillaume, Sid Idris, 2002)

J(κa) =J(κ) +aT(z) +o(a) (three-dimensional case)

=J(κ)− 1

lnaT(z) +o−1 lna

(two-dimensional case) (viii) Cost function example 1 (output least squares): adjoint loads are (as usual)

measurement residuals, since J(uB) =1

2 Z

SN

Q|uB−u^obs|²dS =⇒ J^′(u;w) = Z

SN

Q(w−u^obs)wdS (ix) Cost function example 2 (potential energy, withu^D= 0):

J(uB) =−1 2

Z

SN

guBdS =⇒ J^′(u;w) =−1 2

Z

SN

gwdS

=⇒ uˆ=−u/2 =⇒ T(z) = (1/2)∇u·A·∇u Use sign properties ofAfor interpretation

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1. Introduction

Elasticity

Numerical example

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Background solution

Background solution (C: background elasticity tensor):

div (C:ε[u]) +f=0in Ω, t[u] =gonSN, u=u^DonSD

where

ε[u] = 1

2(∇u+∇u^T) (linearized strain tensor), t[u] = (C:ε[u])·n (traction vector)

In weak form:

Findu∈W(u^D), hu,wi^C_Ω=F(w), ∀w∈W₀ withW(u^D) :=

v∈H¹(Ω;R³),v=u^DonSD . W0=W(0)and hu,wi^C_D :=

Z

D

ε[u] :C:ε[w] dV = Z

D

∇u:C:∇wdV, F(w) =

Z

Ω

f·wdV+ Z

SN

g·wdS

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Generic transmission problem (TP)

◮ Generic TP:(possibly multiple) inclusionB⊂Ωand elasticity tensor C_B = [1−χ(B)]C+χ(B)C^⋆=C+χ(B)∆C

div (C_B:ε[uB]) +f=0in Ω, t[uB] =gonSN, uB =u^DonSD

i.e., in weak form:

Findua∈W(u^D), hua,wi^C_Ω+hua,wi^∆C_B_a =F(w), ∀w∈W0

Both formulations implicitly enforce the perfect-bonding conditions ua|+=ua|− and t

ua|+

=t^⋆ ua|−

on∂B TP in terms of solution perturbationvB :=uB−u:

FindvB ∈W0, hvB,wi^C_Ω+hvB,wi^∆C_B =−hu,wi^∆C_B , ∀w∈W0

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Free-space transmission problem (FSTP)

◮ Auxiliary TP for an inclusionBembedded in an infinite mediumΩ =R³ subjected to a given remote uniform strainE∈R^3,3 through the background solutionu(ξ) =E·ξ=:ψ[E](ξ), i.e.:

( div (C_B:∇uB[E]) =0 inR³, uB[E](ξ)−ψ[E](ξ) =O(kξk⁻²) askξk → ∞ withC_B:=C+χ(B)∆C.

◮ Weak formulation for the solution perturbationvB:=uB−ψ[E]:

FindvB∈H¹(R³;R³), hvB[E],wi^CR³ =−hψ[E],wi^∆CB , ∀w∈H¹(R³;R³)

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Cost functional

Cost functional:

J(C_B) =J(uB) with J(w) :=

Z

Ω

ϕV(·,w) dV + Z

∂Ω

ϕS(·,w) dS with twice-differentiable densitiesw7→ϕV(·,w)andw7→ϕS(·,w)

Topological derivative ofJ: sought by considering the limiting behavior of j(a) :=J(C_a) =J(ua).

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Small-inclusion asymptotic behavior of

u^a

◮ The elastostatic Green’s tensor is defined by div C:ε[G(·,x)]

+δ(· −x)I=0 in Ω G(·,x) =0 onSD

t[G(·,x)] =0 onSN







(x∈Ω)

and verifies

hG(·,x),wi^C_Ω=w(x) x∈Ω, ∀w∈W₀∩C0(Ω)

G(ξ,x) =G(ξ−x) +G_C(ξ,x)

whereG(r)is the (singular) full-space (Kelvin) fundamental solution and the complementary Green’s tensorGC is bounded atξ=x (and in factC^∞for ξ,x∈Ωby virtue of solving a BVP with regular boundary data).

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Small-inclusion asymptotic behavior of

u^a

◮ Applying 3-D Fourier transform, one has in fact G(r) = 1

(2π)³ Z

R³

exp(iη·r)N(η) dV(η) (r∈R³\ {0}) withN(η) =K⁻¹(η)in terms of theacoustic tensorK(η), defined by Kik(η) =Cijkℓηjηℓ

◮ This implies that∇G is homogeneous of degree -2:

∇G(λr) =|λ|⁻³λ∇G(r) =|λ|⁻²sgn(λ)∇G(r)

(r∈R³\ {0}, λ∈R\ {0})

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Small-inclusion asymptotic behavior of

u^a

◮ Domain integral equation (DIE) for the TP:

combine weak formulations for the TP (withw =G(·,x)) and the Green’s function (withw =v^a):

(hvB,G(·,x)i^C_Ω+hvB,G(·,x)i^∆C_B_a =−hu,G(·,x)i^∆C_B_a

hG(·,x),vai^C_Ω=va(x) (x∈Ω) to obtain

La[va](x) =−hu,G(·,x)i^∆C_B_a

x∈Ba (integral equation forv^a) x∈Ω\B¯a (integral representation forv^a) with the linear operatorLa defined by La[w](x) =w(x) +hw,G(·,x)i^∆C_B_a i.e. (in expanded form):

La[w](x) =w(x) + Z

Ba

∇w: ∆C:∇G(·,x) dV

◮ DIE for the FSTP:(similarly)

LB[vB](x) =−hψ[E],G(·,x)i^∆CB with LB[w](x) :=w(x) +hw,G(·,x)i^∆CB

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Small-inclusion asymptotic behavior of

u^a

Asymptotic behavior of va asa→0: InsideBa, one has (inner expansion) va(x) =avB[∇u(z)](¯x) +o(a) x∈B_a,¯x∈ B

Moreover, at any fixed locationx6=z (outer expansion), one has va(x) =−a³∇₁G(z,x) :A:∇u(z) +o(a³) (x6=z)

whereA=A(C,C^⋆,B)is the (fourth-order) elastic moment tensor, defined by A:E=

Z

B

∆C:∇uB[E] dV = Z

B

∆C: (E+∇vB[E]) dV ∀E∈R^3,3 (uB[E] =ψ[E] +vB[E]: solution to the FSTP)

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Small-inclusion asymptotic behavior of

ua

(proof sketch 1/2)

Proof based on seeking and exploiting the limiting form of the DIE, and using scaled coordinates.

(i) Asymptotic behavior of the Green’s tensor:

∇₁G(ξ,x) =a⁻²∇G(¯ξ−¯x) +∇₁GC(z,z) +o(1) (ii) Defining¯va(¯ξ) :=va(z+a¯ξ),¯u(¯ξ) :=u(z+a¯ξ), one finds

∇¯va(¯ξ) =a⁻¹∇va(ξ), ∇¯u(¯ξ) =∇u(z) +o(1) (iii) Substitute above expansions and rescale coordinates in DIE:

La[va](x) =LB[¯va](¯x) +aO(k∇¯vak2,B) hu,G(·,x)i^∆C_B_a =ahψ[∇u(z)],G(·,¯x)i^∆CB +o(a) (iv) Make ansatz ¯va(¯ξ) =aV(¯ξ) +o(a)and isolate lowest-order (ina)

contributions in DIE, which correspond to the FSTP:

LB[V](¯ξ) =−hψ[∇u(z)],G(·,¯x)i^∆C_B Hence: V(¯ξ) =vB[∇u(z)](¯ξ) ξ¯∈ B

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Small-inclusion asymptotic behavior of

ua

(proof sketch 2/2)

(v) Now, considerx6=zandasmall enough. In that case, one has

∇₁G(ξ,x) =∇₁G(z,x) +o(1), ξ∈Ba,x6∈Ba. (vi) Substitute¯va(¯ξ) =aV(¯ξ) +o(a)and the above into the integral

representation outsideBa, to obtain va(x) =−a³∇₁G(z,x) :

Z

B

∆C:∇uB[∇u(z)](¯ξ) d ¯Vξ¯+o(a³)

=−a³∇₁G(z,x) :A:∇u(z) +o(a³)

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Topological derivative of cost functionals

(i) Expand cost functional w.r.t. va:

J(C_a) =J(C) +J^′(u;va) +o(kvak2,Ω+kvak2,∂Ω) with J^′(u;w) :=

Z

Ω

∇₂ϕV(·,u)·wdV + Z

SN

∇₂ϕS(·,u)·wdS (ii) Interpret J^′(u;va)as component of a weak formulation with test functionva

and define theadjoint solutionˆuby

Find ˆu∈W0, hˆu,wi^C_Ω=J^′(u;w), ∀w∈W0

hva,uiˆ ^C_Ω+hva,ûi^∆C_B_a =−hu,uiˆ ^∆C_B_a , hû,vai^C_Ω=J^′(u;va) to obtain J^′(u;va) =−hû,ui^∆C_B_a − hû,vai^∆C_B_a =−hû,uai^∆C_B_a (iv) Exploit known limiting behavior ofvâ in Ba (inner expansion), yielding

J^′(u;va) =−hˆu,uai^∆C_B_a =−a³∇ˆu(z) :A:∇u(z) +o(a³) Topological expansion ofJ: The topological expansion ofJ is given by

J(C_a) =J(C) +a³T(z) +o(a³), T(z) =−∇u(z) :A:∇u(z)ˆ

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Elastic moment tensor (anisotropic elastic inclusion)

1. The EMTAhas major and minor symmetries: for any (E,E^′)∈R^3,3×R^3,3, one has

E^′:A:E=E:A:E^′ (major symmetry) E^′:A:E=E^′:A:E^T =E^′^T:A:E (minor symmetry) 2. By superposition, one has

VB[∇u(z)] = X

1≤i≤j≤3

E^ij:∇u(z) VB[E^ij]

with E^ij=ei⊗ei (i=j), E^ij= 2^−1/2(ei⊗ej+ej⊗ei) (i6=j) One thus needs only compute thevB[Eij](6 auxiliary solutions independent onz) once and for all.

3. C^⋆=0, i.e. ∆C=−C, corresponds to atraction-free cavity.

Thus, small-cavity asymptotics is special case of small-inclusion asymptotics.

Beretta, Bonnetier, Francini, Mazzucato (2011); B, Delgado (in preparation, 2011) Isotropic case: Ammari, Kang, Nakamura, Tanuma (2002); Guzina, Chikichev (2006)

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Elastic moment tensor (anisotropic elastic inclusion)

4. LetBbe an ellipsoid (of principal semiaxesa1,a2,a3). The Eshelby tensor S(a1,a₂,a₃)ofBis given by (Mura, 1987):

Sijmn(a1,a2,a3) = 1 8πCpqmn

Z 1

−1

du Z 2π

0

Gipjq(η(u, θ)) +Gjpiq(η(u, θ)) dθ

Gijkℓ(η) :=ηkηℓNij(η), η(u, θ) :=p

1−u²[a⁻¹₁ cosθe1+a⁻¹₂ sinθe2] +a⁻¹₃ ue3

Then,Ahas the explicit expression (valid for arbitrary anisotropicC andC^⋆) A(C,C^⋆,a1,a2,a3) =C:

C+ ∆C:S(a1,a2,a3)−1

: ∆C

S(a1,a2,a3)known in closed form forisotropicC (elliptic integrals involved in the case of ellipsoids).

5. For ellipsoidal cavities, one has A(C,a1,a2,a3) =

S(a1,a2,a3)−I−1

:C

6. Closed-form expression for spherical cavity (i.e. whenBis unit sphere):

A=−2µh 3(1 +ν)

2(1−2ν)²J +15(1−ν) 7−5ν Ki

(3J =I⊗I, K=I−J)

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1. Introduction

Elasticity

Numerical example

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Numerical example

Ba Ω

ΓD g

ΓN

L= 2 H= 1

α= arctan(0.5) O

z= (1/3,2/3)

J(Ba) =−1 2

Z

SN

g·uadS (compliance)

Isotropic tests: C=





1.34 0.57 0.

0.57 1.34 0.

0. 0. 0.38



, C^⋆= 10⁻⁹C

Anisotropic tests: C=





1. 0.5 0.

0.5 2. 0.

0. 0. 0.04



, C^⋆=





3. 0.4 0.

0.4 1.5 0.

0. 0. 0.03



.

B, Delgado (in preparation, 2011)

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Numerical example

0 0.02 0.04 0.06 0.08 0.1 0.12

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Characteristic size a

Relative error e(a)

Anisotropic, a1/a2=2, angle = pi/4

Isotropic, a1/a2=1

Anisotropic, a1/a2=2, angle = 0 Isotropic, a1/a2=2, angle = 0 Isotropic, a1/a2=2, angle = pi/4

e(a) = |J(C_a)−J(C)−a²T(z)|

|a²T(z)|

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1. Introduction

Elasticity

Numerical example

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Energy-based cost functional

◮ ’Neumann’ and ’Dirichlet’ background fields:

Findu^N∈W^N(u^D), hu^N,wi^C_Ω=Fno(w) +Fo(w), ∀w∈W₀^N. Findu^D∈W^D(u^D), hu^D,wi^C_Ω=F_no(w), ∀w∈W₀^D. whereW^N(u^D) =W(u^D),W₀^N=W0and

W^D(u^D) :=

v∈H¹(Ω;R³),v=u^DonSD, v=uobson So , W₀^D:=W^D(0)

◮ ’Neumann’ and ’Dirichlet’ fields for small trial inclusion:

Findv^N_a ∈W₀^N, hv^N_a,wi^C_Ω+hv^N_a,wi^∆C_B_a =−hu^N,wi^∆C_B_a, ∀w∈W₀^N. Findv^D_a ∈W₀^D, hv^D_a,wi^C_Ω+hv^D_a,wi^∆C_B_a =−hu^D,wi^∆C_B_a , ∀w∈W₀^D.

◮ Energy discrepancy functional:

E(C_a) =E(u^N_a,u^D_a,C_a) = 1

2hu^N_a−u^D_a,u^N_a−u^D_ai^C_Ω^a

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Energy-based cost functional

Topological derivative ofE: it is given by TE(z) = 1

2ε[u^N](z) :A:ε[u^N](z)−1

2ε[u^D](z) :A:ε[u^D](z)

◮ Energy-based cost functionals are very useful for identification of (possibly heterogeneous) material parameters, see e.g. Kohn, Vogelius (1987), Kohn, McKenney (1989), Constantinescu (1995), Chavent, Kunisch, Roberts (1996).

◮ Specific treatment needed to findTE, because (i)ϕV(·,∇w,C_B)rather than ϕV(·,w), and (ii) va=O(a)butva =O(1)insideBa (so care needed in doing expansions abouta= 0).

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1. Introduction

Elasticity

Numerical example

Elastodynamics

Numerical examples

7. Further reading

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1. Introduction

Elastodynamics

4. Waves, time domain 5. Crack identification

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Acoustic transmission problem

Scattering of an acoustic wave by a penetrable inclusionB (ρ^⋆,c^⋆) embedded in an acoustic mediumΩ(ρ,c).







β:=ρ/ρ^⋆ mass density ratio γ:=c/c^⋆ velocity ratio η:= (ρc²)/(ρc²)^⋆ bulk modulus ratio Letm:= 1 + (β−1)χ(B)andn:= 1 + (η−1)χ(B).

◮ Field equation (assuming homogeneous inclusion and background) on acoustic pressurep:

div (m∇uB) +nk²u_B+f = 0 (in Ω)

◮ Transmission conditions implied (continuity of pressure and normal velocity across∂B)

uB |+=uB |− and ρ⁻¹∂nuB |+= (ρ^⋆)⁻¹∂nuB |− on∂B

Martin, P.A. (2003), Acoustic scattering by inhomogeneous obstacles. SIAM J. Appl. Math.

64:297-308

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Weak formulation of TP

Notations: LetW(u^D) :=

w∈H¹(Ω), w=u^DonSD . W0=W(0)and hu,wi^m_D :=

Z

D

m∇u·∇w dV (u,w)ⁿ_D:=

Z

D

nuw dV hu,wiD :=

Z

D

∇u·∇w dV (u,w)D:=

Z

D

uwdV F(w) =

Z

Ω

fw dV+ Z

SN

gwdS

◮ Background solution(weak formulation):

Findu∈W(u^D) hu,wiΩ−k²(u,w)Ω=F(w), ∀w∈W0

For unbounded-media idealizations (Ω =R³),uis an arbitrary solution to (∆ +k²)u= 0(e.g. a plane wave), in particular not constrained by radiation conditions.

◮ TP(weak formulation), in terms of the solution perturbation (scattered field) vB :=uB−u:

FindvB∈W0, hvB,wi^m_Ω^B−k²(vB,w)ⁿ_Ω^B =−hu,wi^∆m_B +k²(u,w)^∆n_B , ∀w∈W0

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Cost functional

J(mB,nB) =J(uB) with J(w) :=

Z

Ω

ϕV(·,w) dV+ Z

∂Ω

ϕS(·,w) dS with twice-differentiable real-valued densitiesw 7→ϕV(·,w)andw 7→ϕS(·,w).

Example (output least-squares for defect identification):

ϕV(·,w) = 0, ϕS(·,w) =1

2(w−u^obs)Q(·)(w−u^obs)

Topological derivative ofJ: sought by considering the limiting behavior of j(a) :=J(ma,n_a) =J(u^a)

withma:= 1 + (β−1)χ(Ba)andna:= 1 + (η−1)χ(Ba)

Ba(z)

z p

SN SD

Sobs

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Small-inclusion asymptotic behavior of

u^a

◮ Green’s function: defined, forx∈Ω, by (∆ +k²)G(·,x;k) +δ(· −x) = 0 (in Ω),

G(·,x;k) = 0 (onS_D), ∂nG(·,x;k) = 0 (onS_N)

G(ξ,x;k) =G(ξ−x;k) +GC(ξ,x;k)

whereG(r;k)is the (singular) Helmholtz full-space fundamental solution:

G(r;k) = eⁱ^kr

4πr (r =krk)

◮ Moreover, one has

G(r;k) =G(r) +GC(r;k)

whereG(r) = 1/(4πr)is the Laplace fundamental solution andGC(r;k)is nonsingular forr=0

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Small-inclusion asymptotic behavior of

u^a Domain integral equation (DIE) for the TP:

Combining weak formulations for the TP (withw =G(·,x;k)) and the Green’s function (withw =v^a) yields a DIE of Lippmann-Schwinger type:

L^k_a[v^a](x) =−hu,G(·,x;k)i^∆m_B_a +k²(u,G(·,x;k))^∆n_B_a

x∈Ba (integral equation forv^a) x∈Ω\B¯a (integral representation forv^a) with the linear operatorL^k_a defined by

L^k_a[w](x) =w(x) +hw,G(·,x;k)i^∆m_B_a −k²(w,G(·,x;k))^∆n_B_a i.e. (in expanded form):

La[w](x) =w(x) + Z

Ba

(β−1)∇w·∇G(·,x;k)−(η−1)k²wG(·,x;k) dV

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Small-inclusion asymptotic behavior of

u^a

L^k_a[v^a](x) =−hu,G(·,x)i^∆m_B_a +k²(u,G(·,x))^∆n_B_a (LS)

◮ Coordinate scaling:

ξ=z+a¯ξ, x=z+a¯x, dVξ =a³d ¯Vξ¯, ∇_ξ=a⁻¹∇ξ¯

◮ Scaling properties ofG(ξ,x;k)(by homogeneity properties ofG(ξ−x)):

G(ξ,x;k) =a⁻¹G(¯ξ−¯x) +O(1)

∇₁G(ξ,x;k) =a⁻²∇G(¯ξ−¯x) +O(1)

◮ Leading contribution in (LS) using ansatz v^a(ξ) =aV(¯ξ) +o(a) inBa : L^k_a[v^a](x) =aLB[V](¯x) +o(a¹)

hu,G(·,x)i^∆m_B_a −k²(u,G(·,x))^∆n_B_a =ahψ[∇u(z)],G(·,¯x)i^∆mB +o(a¹) Hence, retaining only theO(a)terms,V solves again thestatic FSTP for the normalized inclusion:

LB[V](¯x) =−hψ[∇u(z)],G(·,¯x)i^∆m_B (¯x∈ B)

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Small-inclusion asymptotic behavior of

u^a InsideBa, one has (inner expansion)

v^a(x) =avB[∇u(z)](¯x) +o(a) x∈Ba,¯x∈ B wherevB[E]is again the solution of thestaticFSTP.

Moreover, at any fixed locationx6=z (outer expansion), one has v^a(x) =−a³

∇₁G(·,x;k)·A·∇u−k²(η−1)|B|G(·,x;k)u

(z) +o(a³) (x6=z) whereA=A(β,B)is again the polarization tensor associated with the normalized staticFSTP.

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Topological derivative of cost functionals

(i) Expand cost functional w.r.t. v^a:

J(ma,na) =J(1,1) +J^′(u;vâ) +o(kvâk2,Ω+kvâk2,∂Ω)

(ii) Interpret J^′(u;vâ)as component of a weak formulation with test function vâ and define theadjoint solutionûby

Find û∈W0, hû,wiΩ−k²(û,w)Ω=J^′(u;w), ∀w∈W0

hvâ,ûi^m_Ωâ−k²(vâ,u)ˆ ⁿ_Ωâ =−hu,ûi^∆m_B_a +k²(u,u)ˆ ^∆n_B_a hû,vâiΩ−k²(û,vâ)Ω=J^′(u;vâ)

to obtain J^′(u;vâ) =−hû,uâi^∆m_B_a +k²(û,uâ)^∆n_B_a

(iv) Exploit known limiting behavior ofv^a in Ba (inner expansion), yielding J^′(u;v^a) =−a³

∇u(z)·A·ˆ ∇u(z)−k²(η−1)|B|ˆu(z)u(z)

+o(a³) Topological derivative of J:

T(z) =−∇ˆu(z)·A·∇u(z) +k²(η−1)|B|ˆu(z)u(z)

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Comments

SinceA(β,B)is associated with the normalizedstaticFSTP, the following hold:

◮ Ais symmetric;

◮ By superposition:

V(¯x) =vB[∇u(z)](¯x) = X3

i=1

ei·∇u(z)

vB[ei](¯x) One thus needs only compute thevB[ei], which are independent onz.

◮ Spherical vanishing inclusion(i.e. Bis unit sphere): The FSTP solution is given by

vB[E](¯x) = 3

2 +βE·¯x withβ :=κ^⋆/κ (¯x∈ B).

Hence:

A(β,B) = 4π(1−β) 2 +β I;

◮ FSTP solutions analytically available for other shapes (e.g. ellipsoids);

otherwise, solve numerically (and once and for all) the normalized DIE with E=ei (i= 1,2,3);

◮ The special caseβ= 0, i.e. ρ^⋆=∞, corresponds to theimpenetrable inclusion.

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1. Introduction

Elastodynamics

4. Waves, time domain 5. Crack identification

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Background solution

◮ Background solution (C, ρ: background elasticity tensor and mass density):

div (C:ε[u]) +ρω²u+f=0in Ω, t[u] =gon SN, u=u^DonSD

In weak form:

Findu∈W(u^D), hu,wi^C_Ω−ω²(u,w)^ρ_Ω=F(w), ∀w∈W0

withW(u^D) :=

v∈H¹(R³;C³),v=u^DonSD . W0=W(0)and hu,wi^C_D :=

Z

D

ε[u] :C:ε[w] dV = Z

D

∇u:C:∇wdV, (u,w)^ρ_Ω:=

Z

D

ρu·wdV F(w) =

Z

Ω

f·wdV + Z

SN

g·wdS

◮ For unbounded-media idealizations,uis an arbitrary solution to the Navier PDE (e.g. a plane wave), which in particular is not constrained by radiation conditions.

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Transmission problem (TP)

(possibly multiple) inclusionB⊂Ωwith

C_B = [1−χ(B)]C+χ(B)C^⋆ =C+χ(B)∆C (elasticity tensor) ρB = [1−χ(B)]ρ+χ(B)ρ^⋆ =ρ+χ(B)∆ρ (mass density) TP:

div (C_B:ε[uB]) +ρBω²u+f=0in Ω, t[uB] =g onS_N, uB =u^DonS_D i.e., in weak form and in terms of the solution perturbationvB :=uB−u:

FindvB∈W0, hvB,wi^C_Ω^B−ω²(vB,w)^ρ_Ω^B =−hu,wi^∆C_B +ω²(u,w)^∆ρ_B , ∀w∈W0

This weak form is also applicable to scattering in unbounded media.