Determination of material properties from boundary measurements in - - PowerPoint PPT Presentation

Determination of material properties from boundary measurements in anisotropic elastic media

Anna L. Mazzucato - Penn State University Joint with Lizabeth Rachele, RPI AIP, Vienna July 20, 2009

Classical elastodynamics Direct problem is to solve initial-boundary-value problem (IVBP):

      

Pρ,Cu = in Ω × (0, T) u|∂Ω = f for t ∈ [0, T] u|t=0 = 0, (∂tu)|t=0 = 0 in Ω , where Pρ,C is the 3 × 3 non-constant coefficient system Pρ,Cu = ρ(x)∂2u ∂t2 −

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• j,k,l=1

∂ ∂xj

• Cijkl(x)∂uk

∂xl

• with u = ua displacement vector field, ρ ∈ C∞ density scalar

field, C = (Cabcd) ∈ C∞ elasticity tensor with symmetries (hyper- elasticity):

Cabcd = Ccbad = Cabdc = Ccdab.

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Dynamic Inverse Problem Determine material parameters {ρ, C} from surface measurements: surface traction ⇒ resulting displacement Boundary data encoded by dynamic DN map (displacement-to- traction map) f → Λρ,C(f), f : ∂Ω × [0, T] → R3: (Λρ,Cf)a = C · ∇U, νa = Cabcd (∇u)cdνB Uniqueness question = injectivity of Λρ,C w.r.t. ρ and C: 22 unknown parameters. Inverse problem has applications to imaging in Elastography (J. McLaughlin, F. Natterer, J. Greenleaf), seismology (B. Symes,

• M. DeHoop), crack detection (Nakamura-Uhlmann-Wang).

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Study non-uniqueness with respect to change of coordinates ψ : Ω → Ω fixing boundary, ψ⌊∂Ω= Id: “Natural Obstruction”. Well-known approach for the wave equation in anisotropic media ( Belishev, Lassas, Sharafudtinov, Sylvester, Romanov, Uhlmann). It requires a covariant formulation of elasticity and frame-free representation of C. Theorem 1 (M. & Rachele). Let (Ω, ρ, C) be an elastic object. Set ˜ ρ = (det Dψ) ρ ◦ ψ, ˜

C = (det Dψ) ψ∗C.

and consider the elastic object (Ω, ˜ ρ, ˜

C). Then, the DN maps

agree: Λρ,C = Λ˜

ρ,˜

C.

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Type of anisotropy determines the form of the elasticity tensor. Ex: isotropic elastic media, with λ, µ the Lam´ e parameters:

Cabcd

iso

= λ(x)δabδcd + µ(x)

• δacδbd + δadδbc

. For isotropic hyperleastic media, the DN map uniquely deter- mines the density and Lam´ e parameters (Rachele, Hansen-Uhlmann in the presence of caustics and residual stress). Study next simplest case: transversely isotropic media, isotropic at each point x in the plane orthogonal to unit vector k(x), the fibre direction. Principally fibred materials such as biological tissues, hexagonal crystals are transversely isotropic. In transversely isotropic elastostatics, C can be recovered asymp- totically from the DN map via layer stripping up to coordinate changes (Nakamura-Tanuma-Uhlmann).

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If C strongly elliptic, i. e., ∃ c > 0 such that

Cabcd(x)V aW bV cW d ≥ c|V|2|W|2, V, W ∈ T ∗

xΩ ≈ R3,

Pρ,C is well-posed in Hs(Ω), s > 1 (symmetric hyperbolic). Study the inverse problem by studying propagation of singulari- ties by Pρ,C. Under certain conditions, the wave-front set WF(u) of solutions u to IBVP determines the travel times and entry/exit directions

• f elastic waves through interior.

Wave-front set of a distribution u = “set of points x and direc- tions ξ along which u is not smooth”: WF(u) = T ∗

0Ω\{(x0, ξ0) | ∀N, |

(φ u)(r ξ)| = O(r−N), ξ ∈ V, r → ∞}. V neighborhood of ξ0 and φ cut-off near x0.

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Wave Propagation Generically, three distinct wave modes in elasticity. For given x ∈ Ω, ξ ∈ T ∗

xΩ, ∃ three eigenvectors vi(x, ξ) of the

principal symbol σo(Pρ,C)(x, ξ, τ) = −ρτ2I + C[·, ξ, ·, ξ] with eigenvalue µi = −ρ τ2+λi(x, ξ), i = 1, 2, 3. λi homogeneous

• f deg 2 in ξ.

The polarization vector vi(x, ξ) gives the (approximate) direction

• f displacement of i-th wave mode with speed
• λi(x, ξ) at x in

the direction ξ. The surface λi(x, s) = 1 is called a slowness surface. Ex: in isotropic elastodynamics, two coincident shear wave modes and one longitudinal wave mode.

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Propagation of singularities can occur only where the principal symbol of Pρ,C, σ0(Pρ,C), is degenerate (Pρ,C does not have a parametrix) , i.e., in the (bi)characteristic set of the operator Pρ,C in T ∗([0, T] × Ω) ≈ [0, T] × Ω × R6: Char(Pρ,C) = {(x, t, ξ, τ) | Detσo(Pρ,C) = 0}. Integral curves (t, x(t), ξ(t), τ(x(t), ξ(t))) of the Hamilton vector field HDet(σo(Pρ,C)) are the bicharacteristics curves. The characteristic curves (t, x(t)) are the projection of the bichar- acteristics from T ∗([0, T] × Ω) to [0, T] × Ω. Bicharacteristics in Char(Pρ,C) are the null bicharacteristics.

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Propagation of singularities by hyperbolic operators:

• If P = p(x, t, Dx, Dt) ∈ OPSm is a ΨDO of order m and

Pu = f, then WF(u) ⊂ WF(f) ∪ Char(P).

• If there are multiple eigenvalues µi, HDet(σo(Pρ,C)) ≡ 0 in CharP.

Moreover, waves may not be distinguished from their speed, as waves can have same speed when multiplicity changes.

• Assume each µi has constant multiplicity.

Set Γi = Char(µi(x, t, Dx, Dt)). Then: WF(u) = ⋒µi=µjΓi ∪ WF(f), (⋒ disjoint union) and WF(u)∩Γi is a union of bicharacteristics of µi (Egorov’s Theorem).

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Decoupling system of elastodynamics For inhomogeneous, anistropic elastic media, Pρ,C has multiple eigenvalues of non-constant multiplicity. Generically Pρ,C can be conjugated to normal form via ellip- tic FIOs, but conjugation is not explicit (H¨

• rmander, Braam-

Duistermaat, Nolan-Uhlmann). Diagonalize principal symbol, but eigenvectors not smooth when slowness surfaces intersect ⇒ Consider case with a multiplicity-one eigenvalue, e.g. λ3 ( say corresponding wave mode is disjoint). Partially decouple disjoint mode from system (Stolk-DeHoop). Based on result of M. Taylor for reflection of singularities at

• boundary. Can decouple at every order.

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• Extend ρ, C to R3. Consider Cauchy Problem on R3 × [0, T].
• “Boundary” is at t = 0.

Theorem 1 (M-Rachele). For all x0 / ∈ Ω, ξ = 0, ∃ Ux0,ξ0 ⊂ T ∗R3\0 and microlocally invertible Q(x, t, D) ∈ OPS0

1,0(R3), smooth

in t ∈ [0, T] such that: Q−1

• Pρ,C Q = I ∂2

t

−

 A(x, t, D)

a(x, t, D)

 

mod OPS−∞ in C(t)U, t ∈ [0, T], with A(x, t, D) ∈ OPS2

1,0(R3) 2×2 symmetric,

a(x, t, D) ∈ OPS2

1,0(R3) scalar with a0 2(x, ξ) = λ3(x, ξ).

C(t) canonical relation induced by Hλ3.

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Travel times in Ω for λ3 determined by DN map

• Two versions of elastic body B = (Ω, ρ, C), ˜

B = (Ω, ˜ ρ, ˜

C) with

disjoint mode.

• Assume ρ = ˜

ρ and C = ˜

C to infinite order at ∂Ω ⇒ Extend

parameters to R3 so that ρ = ˜ ρ, C = ˜

C in Ωc.

• For initial data φ0 ∈ Hs, φ1 ∈ Hs−1, s > 2, Supp φi ⊂ Ωc,

i = 0, 1, solutions U, ˜ U to the Cauchy problems in R3 for B and ˜ B agree in Ωc. (v = U in Ω, v = ˜ U in Ωc weak solution).

• Fix x0 /

∈ Ω and ξ0 = 0. Partially decouple elastodynamics system as in Theorem 1. W, ˜ W solutions to decoupled systems.

• Choose initial data for decoupled system W0 = ˜

W0, W1 = ˜ W1 with W0(x) = h(x)(0, 0, 1), W1(x) = 0.

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• Construct h with wave-front set on a single ray:

WF(h) = {(x0, αξ0) : α > 0} ⇒ WF(W) = WF(W3), WF( ˜ W) = WF( ˜ W3).

• Pick U in Theorem 1 with WF(W0) ⊂ U ⇒ WF(W) ⊂ C(t)U, ∀t ≥ 0.
• W3, ˜

W3 solve strictly hyperbolic equations (L, ˜ L lower order): ∂ttW3 = λ3(x, D)W3 + LW3, ∂tt ˜ W3 = ˜ λ3(x, D) ˜ W3 + ˜ L ˜ W3, ⇒ Wave-front set contains only a few null bicharacteristics: WF(W) = WF(W 3) = C(t)(x0, ξ0) = γ+ ⋒ γ− γ± the forward (backward) null bicharacteristic through (x0, ξ0)

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• Q(x, t, D) invertible on C(t)U ⇒

WF(W) = WF(Q(x, t, D)φ(x, t, D)W), where φ(x, t, D) microlocal cut-off near C(t)U.

• No contribution to WF(U) from outside C(t)U, as:
• v = U − Q(x, t, D)φ(x, t, D)W solves

Pρ,Cv = 0, mod C∞,

• v(0) = U0 − Q(x, 0, D)φ(x, 0, D)W0 so that WF(v(0) = ∅)

⇒ WF(W) = WF(U), WF( ˜ W) = WF(˜ U)

• U = ˜

U in Ωc ⇒ WF(U)= WF(˜ U) in Ωc Forward, backward bicharacteristics can be distinguished ⇒ γ± = ˜ γ± in Ωc. ⇒ entry/exit points, directions, & travel times in Ω same for λ3 and ˜ λ3.

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Transverse isotropy Specialize to media with geodesic wave propagation (GWP): eigenvalues of σo(Pρ,C) are given by λi(x, ξ, τ) = (ρ(x) τ2 − ξT g−1

1 (x) ξ),

where gi, i = 1, 2, 3, Riemannian metric (ellipsoidal slowness sur- faces). Characteristics are geodesic segements. Only two classes of transversely isotropic media with GWP: (GWP1) µL+C = 0, and B ≤ µL ≤ A+µT or A+µT ≤ µL ≤ B; (GWP2) (µL + C)2 = (A + µT − µL)(B − µL). where A, B, C depends on shear and Young’s moduli, and Pois- son’s ratios.

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Transv isotropic media with GWP and a disjoint mode Multiplicity of eigenvalues λi changes when slowness surfaces intersect (may give rise to conical refraction). Only two types of intersections:

• two coincident modes (constant multiplicity).
• tangential intersection along fiber direction (nonconstant mul-

tiplicity). There can be triple-point intersections. Under explicit mild conditions on parameters, the light cone of λ3 is always disjoint from the others. From calculated tables of elastic constants, disjoint-mode con- dition is generic. GWP is instead restrictive .

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Table 1: Cases of GWP with a disjoint slowness surface where a = 1 √A + µT , b = 1 √ B , l = 1 √µL , t = 1 √µT CM1

k k

l b a t

k k

b l t a

B < µL < µT < A + µT µT < A + µT < µL < B CM2

k k

l a t b

k k

l a b

k k

l a b t

µL > µT µL = µT µL < µT and µL < min{A+µT, B}

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• Consider transversely isotropy with GWP and a disjoint mode.
• If Ω strictly convex, then travel times determine the boundary

distance function dg = distance between boundary points along minimal geodesic (Guillemin).

• Under additional conditions (e.g.

curvature bounds), Ω is boundary rigid (Lassas, Romanov, Gromov, Stefanov, Uhlmann): dg3 = d˜

g3 ⇒ g3 = ψ∗˜

g3 for some diffeo ψ fixing the boundary of Ω.

• From knowledge of g3, 2 out of 5 independent parameters can

be reconstructed (use spectral representation of C).

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