Inverse scattering in acoustics and elasticity using high-order topological derivatives Marc Bonnet 1 1 POems, (ENSTA, CNRS, INRIA, Universit´ e Paris-Saclay), Palaiseau, France mbonnet@ensta.fr Workshop ”Analysis and Numerics of Acoustic and Electromagnetic Problems”, Oct. 21, 2016 Marc Bonnet 1 Inverse scattering with high-order topological derivatives 1 / 42
Outline 1. Identification using topological derivative: overview 2. Higher-order topological expansion (acoustics) 3. Higher-order topological expansion (elastodynamics – with R. Cornaggia) Marc Bonnet 1 Inverse scattering with high-order topological derivatives 2 / 42
Model problem: acoustic transmission problem Scattering of an acoustic wave u by a penetrable obstacle B ( ρ ⋆ , c ⋆ ) embedded in an acoustic medium Ω ( ρ, c ). � ∆ + k 2 � � β ∆ + η k 2 � u B = 0 in Ω \ B , u B = 0 in B , ∂ n u B = i ρω V D on S , u B | + = u B | − and ∂ n u B | + = η∂ n u B | − on ∂ B . u obs β := ρ/ρ ⋆ u (mass density ratio) , ? B ( ρ ⋆ , c ⋆ ) η := ( ρ c 2 ) / ( ρ c 2 ) ⋆ (bulk modulus ratio) Ω( ρ, c ) Problem considered: identification of B by means of small-inclusion asymptotics. Marc Bonnet 1 Inverse scattering with high-order topological derivatives 3 / 42
Wave-based identification u obs u ? B ( ρ ⋆ , c ⋆ ) Ω( ρ, c ) Standard approach (e.g. Full Waveform Inversion): minimize a cost functional (e.g. � � � output least-squares) � � u [ B ′ , ρ ′ , c ′ ] − u obs � J LS ( B ′ , ρ ′ , c ′ ) := 1 2 � � ( B ⋆ , ρ ⋆ , c ⋆ ) = arg min dΓ � 2 B ′ ,ρ ′ , c ′ M Entails repeated evaluations of forward solutions u [ B ′ , ρ ′ , c ′ ] (with varying B ′ , ρ ′ , c ′ ) Impetus for development of alternative identification methods: ◮ Linear sampling method (Colton, Kirsch ’96), factorization method (Kirsch’98) Mathematically justified; require abundant data, qualitative ◮ Topological derivative (TD): (very) partial mathematical justification so far; any data, qualitative ◮ High-order topological derivative (this talk): quantitative enhancement of TD, any data Marc Bonnet 1 Inverse scattering with high-order topological derivatives 4 / 42
Identification problem and cost functional Experimental configuration fur true (unknown) inhomogeneity B ⋆ ( ρ ⋆ , c ⋆ ) Simulation for trial inhomogeneity B ′ ( ρ ′ , c ′ ) u u ′ B B ′ ( ρ ′ , c ′ ) Ω( ρ, c ) The discrepancy between B ⋆ and B ′ is evaluated by the cost functional � � � u B ′ − u obs � J ( B ′ ) := J ( u B ′ ) = 1 � 2 dΓ u B ′ := u [ B ′ , ρ ′ , c ′ ] 2 Γ Usual idea: minimize J ( B ′ ) w.r.t. (some of the characteristics of) the trial obstacle B ′ . Note: quadratic cost functional considered for definiteness in this talk, but more general choices possibble as well. Marc Bonnet 1 Inverse scattering with high-order topological derivatives 5 / 42
Identification using topological derivative: overview 1. Identification using topological derivative: overview 2. Higher-order topological expansion (acoustics) Numerical example 3. Higher-order topological expansion (elastodynamics – with R. Cornaggia) Numerical example Marc Bonnet 1 Inverse scattering with high-order topological derivatives 6 / 42
Identification using topological derivative: overview Cost functional for a small trial defect B a Experimental configuration fur true (unknown) inhomogeneity B ⋆ ( ρ ⋆ , c ⋆ ) Simulation for small trial inhomogeneity B a ( ρ ′ , c ′ ) u u a a z B a ( ρ ′ , c ′ ) Ω( ρ, c ) B a = z + a B Introducing the scattered field v a := u a − u , we have � � � u + v a − u obs � J ( B a ) := J ( u a ) = 1 � 2 dΓ 2 Γ = J ( u ) + J ′ ( u ; v a ) + 1 2 J ′′ ( u ; v a ) Marc Bonnet 1 Inverse scattering with high-order topological derivatives 7 / 42
Identification using topological derivative: overview Topological derivative T 3 The leading-order expansion of J ( u a ) is already known as: J ( u a ) = J ( u ) + a 3 T 3 ( z , B ) + o ( a 3 ) T 3 is called the topological derivative (or sensitivity , or gradient ) of J . [Sokolowski, Zochowski, 1999; Garreau, Guillaume, Masmoudi, 2001; Guzina, B 2004; Amstutz, Takahashi, Wexler 2008] ... Heuristic: locations z where T 3 ( z ) takes the most negative values are good candidates for the real defect location. Many empirical validations of this heuristic even for macroscopic defects ◮ using synthetic data (references above and many more) ◮ using experimental data [Tokmashev, Tixier, Guzina 2013]. Heuristic proved in some cases: ◮ β = 0 and “moderate” scatterers [Bellis, B, Cakoni 2013], ◮ small obstacles [Ammari et al. 2012], High-frequency behavior investigated [Guzina, Pourahmadian 2015] ( T 3 tends to emphasize the boundaries of the obstacles). Marc Bonnet 1 Inverse scattering with high-order topological derivatives 8 / 42
Identification using topological derivative: overview TD as topology optimization tool 6 Compliance 5.5 5 4.5 4 0 20 40 60 80 100 Iterations PhD G. Delgado (2014) — Topology optimization combining topological and shape derivatives Marc Bonnet 1 Inverse scattering with high-order topological derivatives 9 / 42
Identification using topological derivative: overview TD as imaging functional for qualitative flaw identification FEM-based computation of T 3 , transient wave equation, 2D: simultaneous identification of a multiple scatterer α = 0 . 5 Bellis, B, Int. J. Solids Struct. (2010) Marc Bonnet 1 Inverse scattering with high-order topological derivatives 10 / 42
Identification using topological derivative: overview TD as imaging functional for qualitative flaw identification S piezo S piezo S piezo (a) (b) 2 3 4 B hole 9 cm S piezo S piezo 10 cm 1 5 B slit 0 . 6 cm 14 S obs 1 . 5 cm S D 5 13 obs ⊂ S S N S N 63 4 1 66 Figure 4. Testing configuration: (a) photograph of the damaged plate, and (b) boundary conditions and spatial arrangement of the LDV scan points for five individual source locations ( S piezo , k = 1 , 5). Figure 2. Three-dimensional motion sensing via laser Doppler vibrometer (LDV) system. k Topological derivative z �→ T ( z ) for � T � J ( u D ) = 1 | u D − u obs | 2 d S d t 2 0 S R. Tokmashev, A. Tixier, B. Guzina, Inverse Problems (2013) Marc Bonnet 1 Inverse scattering with high-order topological derivatives 11 / 42
b b b b b b b b b b b b b b b b b b b b b b b b b Identification using topological derivative: overview T 3 : Full and partial aperture (time-harmonic elastodynamics) Full aperture: Partial aperture: Γ = � x n Ω test Ω test B true Γ u ( x ) B true u ( x ) [PhD thesis R. Cornaggia, 2016] Marc Bonnet 1 Inverse scattering with high-order topological derivatives 12 / 42
Higher-order topological expansion (acoustics) 1. Identification using topological derivative: overview 2. Higher-order topological expansion (acoustics) Numerical example 3. Higher-order topological expansion (elastodynamics – with R. Cornaggia) Numerical example Marc Bonnet 1 Inverse scattering with high-order topological derivatives 13 / 42
Higher-order topological expansion (acoustics) Higher-order topological expansion: motivation Computation of topological derivative of J Non-iterative; Computationally faster than a minimization-based inversion algorithm; (2 forward solutions governed by same linear field equations) Yields qualitative results, since the approximation J ( B a ) ≈ J ( u ) + a 3 T 3 ( z ) (Ω ⊂ R D ) cannot be minimized Inaccurate localization when using partial-aperture measurements Higher-order topological expansion − → polynomial (in a ) approximation of cost function J : Much faster to compute than J ; Lends itself to minimization w.r.t. defect size a Provide usable results with sparser data Marc Bonnet 1 Inverse scattering with high-order topological derivatives 14 / 42
Higher-order topological expansion (acoustics) Expansion of the cost functional Expansion of the cost functional: J ( a ) = J ( u a ) = J ( u ) + J ′ ( u ; v a ) + 1 2 J ′′ ( u ; v a ); Known behavior of scattered field: v a ( x ) = a 3 W ( x ; z ) + O ( a 4 ) for x �∈ B a , hence: J ( u a ) = J ( u ) + J ′ ( u ; v a ) + a 6 � � 1 2 J ′′ ( u ; W ) + o (1) To retain the contribution of the last term in the expansion, we seek J ( u a ) = J ( u ) + a 3 T 3 ( z ) + a 4 T 4 ( z ) + a 5 T 5 ( z ) + a 6 T 6 ( z ) + o ( a 6 ) ⇒ J ′ ( u ; v a ) is to be expanded to order O ( a 6 ). = � � a 3 T 3 ( z ) + a 4 T 4 ( z ) + a 5 T 5 ( z ) + a 6 T 6 ( z ) Identification approach: ( a est , z est ) = arg min ( a , z ) Quantitative evaluation of size a . Improved localization when using partial-aperture measurements. Previous work: [B 2008] (3D acoustics, sound-hard obstacles), [B 2010] (2D electrostatics, inclusions), [Silva et al., 2010] (2D elastostatics, holes), in all cases with expansions derived but not justified. Marc Bonnet 1 Inverse scattering with high-order topological derivatives 15 / 42
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