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):  P ρ, C u = 0 in Ω × (0 , T )    u | ∂ Ω = for t ∈ [0 , T ] , f    u | t =0 = 0 , ( ∂ t u ) | t =0 = 0 in Ω where P ρ, C is the 3 × 3 non-constant coefficient system � � 3 ρ ( x ) ∂ 2 u � ∂ C ijkl ( x ) ∂u k P ρ, C u = − ∂t 2 ∂x j ∂x l j,k,l =1 with u = u a displacement vector field, ρ ∈ C ∞ density scalar field, C = ( C abcd ) ∈ C ∞ elasticity tensor with symmetries ( hyper- elasticity ): C abcd = C cbad = C abdc = C cdab . 1

Dynamic Inverse Problem Determine material parameters { ρ, C } from surface measurements: surface traction ⇒ resulting displacement Boundary data encoded by dynamic DN map (displacement-to- f : ∂ Ω × [0 , T ] → R 3 : traction map) f → Λ ρ, C ( f ), (Λ ρ, C f ) a = � C · ∇ U , ν � a = C abcd ( ∇ u ) cd ν B Uniqueness question = injectivity of Λ ρ, 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). 2

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 C = (det Dψ ) ψ ∗ C . ˜ ρ = (det Dψ ) ρ ◦ ψ, ˜ ρ, ˜ and consider the elastic object (Ω , ˜ C ) . Then, the DN maps agree: Λ ρ, C = Λ ˜ C . ρ, ˜ 3

Type of anisotropy determines the form of the elasticity tensor. Ex: isotropic elastic media, with λ, µ the Lam´ e parameters: � δ ac δ bd + δ ad δ bc � = λ ( x ) δ ab δ cd + µ ( x ) C abcd . iso 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). 4

If C strongly elliptic, i. e., ∃ c > 0 such that C abcd ( x ) V a W b V c W d ≥ c | V | 2 | W | 2 , V , W ∈ T ∗ x Ω ≈ R 3 , P ρ, C is well-posed in H s (Ω), s > 1 (symmetric hyperbolic). Study the inverse problem by studying propagation of singulari- ties by P ρ, C . Under certain conditions, the wave-front set WF( u ) of solutions u to IBVP determines the travel times and entry/exit directions of 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 ∗ 0 Ω \{ ( x 0 , ξ 0 ) | ∀ N, | � ( φ u )( r ξ ) | = O ( r − N ) , ξ ∈ V, r → ∞} . V neighborhood of ξ 0 and φ cut-off near x 0 . 5

Wave Propagation Generically, three distinct wave modes in elasticity. For given x ∈ Ω, ξ ∈ T ∗ x Ω, ∃ three eigenvectors v i ( x, ξ ) of the principal symbol σ o ( P ρ, C )( x, ξ, τ ) = − ρτ 2 I + C [ · , ξ, · , ξ ] with eigenvalue µ i = − ρ τ 2 + λ i ( x, ξ ), i = 1 , 2 , 3. λ i homogeneous of deg 2 in ξ . The polarization vector v i ( x, ξ ) gives the (approximate) direction � of displacement of i -th wave mode with speed λ i ( x, ξ ) at x in the direction ξ . The surface λ i ( x, s ) = 1 is called a slowness surface. Ex: in isotropic elastodynamics, two coincident shear wave modes and one longitudinal wave mode. 6

Propagation of singularities can occur only where the principal symbol of P ρ, C , σ 0 ( P ρ, C ), is degenerate ( P ρ, C does not have a parametrix ) , i.e., in the (bi)characteristic set of the operator P ρ, C in T ∗ ([0 , T ] × Ω) ≈ [0 , T ] × Ω × R 6 : Char( P ρ, C ) = { ( x, t, ξ, τ ) | Det σ o ( P ρ, C ) = 0 } . Integral curves ( t, x ( t ) , ξ ( t ) , τ ( x ( t ) , ξ ( t ))) of the Hamilton vector field H Det( σ o ( P ρ, 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( P ρ, C ) are the null bicharacteristics. 7

Propagation of singularities by hyperbolic operators: • If P = p ( x, t, D x , D t ) ∈ OPS m is a ΨDO of order m and Pu = f , then WF( u ) ⊂ WF( f ) ∪ Char( P ) . • If there are multiple eigenvalues µ i , H Det( σ o ( P ρ, C )) ≡ 0 in Char P . Moreover, waves may not be distinguished from their speed, as waves can have same speed when multiplicity changes. • Assume each µ i has constant multiplicity. Set Γ i = Char( µ i ( x, t, D x , D t )). Then: WF( u ) = ⋒ µ i � = µ j Γ i ∪ WF( f ) , ( ⋒ disjoint union) and WF( u ) ∩ Γ i is a union of bicharacteristics of µ i (Egorov’s Theorem). 8

Decoupling system of elastodynamics For inhomogeneous, anistropic elastic media, P ρ, C has multiple eigenvalues of non-constant multiplicity. Generically P ρ, C can be conjugated to normal form via ellip- tic FIOs, but conjugation is not explicit (H¨ ormander, Braam- Duistermaat, Nolan-Uhlmann). Diagonalize principal symbol, but eigenvectors not smooth when slowness surfaces intersect ⇒ Consider case with a multiplicity-one eigenvalue, e.g. λ 3 ( 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. 9

• Extend ρ , C to R 3 . Consider Cauchy Problem on R 3 × [0 , T ]. • “Boundary” is at t = 0. ∈ Ω, ξ � = 0, ∃ U x 0 ,ξ 0 ⊂ Theorem 1 (M-Rachele). For all x 0 / T ∗ R 3 \ 0 and microlocally invertible Q ( x, t, D ) ∈ OPS 0 1 , 0 ( R 3 ), smooth in t ∈ [0 , T ] such that:    A ( x, t, D ) 0 Q − 1 P ρ, C Q = I ∂ 2 mod OPS −∞ �  − t 0 a ( x, t, D ) in C ( t ) U , t ∈ [0 , T ], with A ( x, t, D ) ∈ OPS 2 1 , 0 ( R 3 ) 2 × 2 symmetric, a ( x, t, D ) ∈ OPS 2 1 , 0 ( R 3 ) scalar with a 0 2 ( x, ξ ) = λ 3 ( x, ξ ). C ( t ) canonical relation induced by H λ 3 . 10

Travel times in Ω for λ 3 determined by DN map ρ, ˜ • Two versions of elastic body B = (Ω , ρ, C ), ˜ B = (Ω , ˜ C ) with disjoint mode. ρ and C = ˜ • Assume ρ = ˜ C to infinite order at ∂ Ω ⇒ Extend parameters to R 3 so that ρ = ˜ ρ , C = ˜ C in Ω c . • For initial data φ 0 ∈ H s , φ 1 ∈ H s − 1 , s > 2, Supp φ i ⊂ Ω c , U to the Cauchy problems in R 3 for B and i = 0 , 1, solutions U , ˜ U in Ω c weak solution). B agree in Ω c . ( v = U in Ω, v = ˜ ˜ • Fix x 0 / ∈ Ω and ξ 0 � = 0. Partially decouple elastodynamics system as in Theorem 1. W , ˜ W solutions to decoupled systems. • Choose initial data for decoupled system W 0 = ˜ W 0 , W 1 = ˜ W 1 with W 0 ( x ) = h ( x )(0 , 0 , 1) , W 1 ( x ) = 0. 11

• Construct h with wave-front set on a single ray : WF( h ) = { ( x 0 , αξ 0 ) : α > 0 } WF( ˜ W ) = WF( ˜ ⇒ WF( W ) = WF( W 3 ), W 3 ). • Pick U in Theorem 1 with WF( W 0 ) ⊂ U ⇒ WF( W ) ⊂ C ( t ) U , ∀ t ≥ 0. • W 3 , ˜ W 3 solve strictly hyperbolic equations ( L, ˜ L lower order): ∂ tt ˜ W 3 = ˜ λ 3 ( x, D ) ˜ W 3 + ˜ L ˜ ∂ tt W 3 = λ 3 ( x, D ) W 3 + LW 3 , W 3 , ⇒ Wave-front set contains only a few null bicharacteristics: WF( W ) = WF( W 3 ) = C ( t )( x 0 , ξ 0 ) = γ + ⋒ γ − γ ± the forward (backward) null bicharacteristic through ( x 0 , ξ 0 ) 12

• 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: P ρ, C v = 0 , mod C ∞ , - v = U − Q ( x, t, D ) φ ( x, t, D ) W solves - v (0) = U 0 − Q ( x, 0 , D ) φ ( x, 0 , D ) W 0 so that WF( v (0) = ∅ ) WF( ˜ W ) = WF(˜ ⇒ WF( W ) = WF( U ), U ) U in Ω c ⇒ WF( U )= WF(˜ U ) in Ω c • U = ˜ Forward, backward bicharacteristics can be distinguished ⇒ entry/exit points, directions, & travel γ ± in Ω c . γ ± = ˜ ⇒ times in Ω same for λ 3 and ˜ λ 3 . 13

Transverse isotropy Specialize to media with geodesic wave propagation (GWP): eigenvalues of σ o ( P ρ, C ) are given by λ i ( x, ξ, τ ) = ( ρ ( x ) τ 2 − ξ T g − 1 1 ( x ) ξ ) , where g i , i = 1 , 2 , 3, Riemannian metric (ellipsoidal slowness sur- faces). Characteristics are geodesic segements. Only two classes of transversely isotropic media with GWP: (GWP1) µ L + C = 0, and B ≤ µ L ≤ A + µ T or A + µ T ≤ µ L ≤ B ; ( µ L + C ) 2 = ( A + µ T − µ L )( B − µ L ). (GWP2) where A, B, C depends on shear and Young’s moduli, and Pois- son’s ratios. 14