Identification de Fissures en Elastodynamique Temporelle par la Méthode de Sensibilité Topologique

Identification de Fissures en Elastodynamique Temporelle par la Méthode de Sensibilité Topologique Cédric Bellis Marc Bonnet Laboratoire de Mécanique des Solides, UMR CNRS 7649 Ecole Polytechnique Journée GDR MSPC et GDR Ondes

Crack identification framework Crack Γ true embedded in reference linear elastic domain Ω(C, ρ) Elastodynamic measurements u obs = u true on S obs S N and t [0;T ] J(Γ,T )= T 0 Trial crack Γ defining Ω Γ = Ω\Γ [C : u Γ ](ξ,t) + f (ξ,t) = ρü Γ (ξ,t) in Ω Γ t ± [u ± Γ ](ξ,t) = C : u± Γ (ξ,t) n± (ξ) = 0 on Γ ± t[u Γ ](ξ,t) = t N (ξ,t) u Γ (ξ,t) = u D (ξ,t) u Γ (ξ,0) = u Γ (ξ,0) = 0 No unilateral contact condition or friction on Γ Crack: Traction-free surface Minimization of cost functional S obs ϕ[u Γ, ξ,t]ds ξ dt on S N on S D arg min J(Γ,T ) Γ true Γ e.g. ϕ[w, ξ,t] = 1 2 w uobs 2 (ξ,t) C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 2 / 25

Context Inverse scattering: non-destructive material testing, geo-imaging,... Boundary integral formulations: Nishimura, Bonnet, Rus, Gallego,... Reciprocity gap: Bui, Andrieux, Ben Abda, Constantinescu,... Non-linear optimization approaches Iteratively: resolution of forward scattering problem and evaluation of J Global search techniques: substantial computational cost Gradient-based optimization algorithms: require preliminary informations Development of non-iterative and qualitative reconstruction methods Construction of indicator functions for global search Seek qualitative informations in computationally efficient and robust framework Probe method: Ikehata Linear Sampling method: Colton, Cakoni, Guzina, Arens,... Topological Sensitivity method: Ammari, Kang, Bonnet, Rus, Guzina,... C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 3 / 25

Outline 1 Topological sensitivity analysis 2 Qualitative crack identification 3 Numerical examples 4 Acoustic case 5 Interface cracks C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 4 / 25

Outline 1 Topological sensitivity analysis 2 Qualitative crack identification 3 Numerical examples 4 Acoustic case 5 Interface cracks C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 5 / 25

Conceptual framework Shape sensitivity analysis from topological optimization Eschenauer, Sokolowski, Garreau,... Trial infinitesimal crack Γ ε (z) = z + ε Γ location or sampling point z radius ε, shape Γ orientation Asymptotic expansion for trial domain Ω ε (z) J(Γ ε (z),t ) = ε 0 J(,T ) + f (ε)t(z, Γ,T ) + o(f (ε)) Topological derivative T(z, Γ,T ): Indicator of local sensitivity Displacement u ε =u+v ε with incident field u and scattered v ε =o(ε) T f (ε)t(z, Γ,T ϕ ) = ε 0 u [u, ξ,t]v ε(ξ,t) ds ξ dt H. Ammari, H. Kang, Springer, 2004. 0 S obs C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 6 / 25

Leading contribution of v ε as ε 0 Sufficient regularity conditions for incident field τ u(ξ,τ) Lipschitz-continuous uniformly for ξ in a neighbourhood of z Differentiable in a neighbourhood of τ = t Limited high-frequency content of excitation C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 7 / 25

Leading contribution of v ε as ε 0 Sufficient regularity conditions for incident field τ u(ξ,τ) Lipschitz-continuous uniformly for ξ in a neighbourhood of z Differentiable in a neighbourhood of τ = t Limited high-frequency content of excitation Crack-opening displacement on Γ ε (z) using integral equations [[v ε ]](ξ,t)=ε[[v]]( ξ):σ[u](z,t)+o(ε) with ξ =(ξ z)/ε Canonical elastostatic solution V defined for Γ R 3 Scattered field on S obs v ε (ξ,t) = ε 3 U(ξ,t, z) A: u(z,t) + o(ε 3 ) Time-impulsive fundamental solution U(ξ,t, z) in Ω with homogeneous B.C. Polarization tensor A Time convolution at t 0 [a b](ξ,t)= t a(ξ,τ) b(ξ,t τ)dτ 0 C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 7 / 25

Polarization tensor A Canonical elastostatic solution V defined for unit crack Γ in R 3 Depends on shape Γ and orientation of infinitesimal trial crack Canonical problems can be, at least, solved numerically Polarization tensor A = C : A σ : C using canonical solution A σ ijkl = 1 { } n i [[V kl ]] j ( ξ) + n j [[V kl ]] i ( ξ) ds ξ 2 where Vkl =V lk =(e k e l ):V Γ C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 8 / 25

Polarization tensor A Canonical elastostatic solution V defined for unit crack Γ in R 3 Depends on shape Γ and orientation of infinitesimal trial crack Canonical problems can be, at least, solved numerically Polarization tensor A = C : A σ : C using canonical solution A σ ijkl = 1 { } n i [[V kl ]] j ( ξ) + n j [[V kl ]] i ( ξ) ds ξ 2 where Vkl =V lk =(e k e l ):V Γ Penny-shaped crack: Γ is the unit disk Limit case of ellipsoidal cavity Analytical crack opening displacement [[V kl ]]( ξ)= 4(1 ν) [ ] 2 1 ξ πµ 2 2 ν σkl in e i +σnnn kl with i {1, 2} and σ kl jn = 1 2 e j (e k e l +e l e k ) n on Γ A σ = 8(1 ν) 3µ(2 ν) n ( 2I νn n ) n with identity tensor I C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 8 / 25

Adjoint-field formulation Topological sensitivity with leading contribution f (ε) = ε 3 (3D) T T(z, Γ,T ϕ )= u [u, ξ,t] { U(ξ,t, z) A: u(z,t) } ds ξ dt 0 S obs Limited due to reference to fundamental solution of domain Ω C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 9 / 25

Adjoint-field formulation Topological sensitivity with leading contribution f (ε) = ε 3 (3D) T T(z, Γ,T ϕ )= u [u, ξ,t] { U(ξ,t, z) A: u(z,t) } ds ξ dt 0 S obs Limited due to reference to fundamental solution of domain Ω Definition of the adjoint-field û in reference (defect-free) domain Ω(C, ρ) [C : û](ξ,t) = ρ û(ξ,t) t[û](ξ,t) = ϕ [u, ξ,t t] u t[û](ξ,t) = 0 û(ξ,t) = 0 û(ξ,0) = û(ξ,0) = 0 in Ω on Sobs on S N \S obs on S D Reciprocity identity between elastodynamic solutions u 1 and u 2 in domain O { } t[u1 ] u 2 t[u 2 ] u 1 (ξ,t) dsξ = 0 O with u 1 (ξ,0) = u 1 (ξ,0) = 0, u 2 (ξ,0) = u 2 (ξ,0) = 0 and no body forces in O C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 9 / 25

Adjoint-field formulation Reciprocity identify in Ω\Γ ε (z) with scattered v ε and adjoint û fields { } f (ε)t(z, Γ,T ) = t[v ε ] û t[û] v ε (ξ,t) dsξ ε 0 = ε 0 Γ + ε (z) Γ ε (z) Γ ε(z) t[û](ξ,t) [[v ε ]](ξ,t) ds ξ with reference normal n =n and t[û]=c : û n C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 10 / 25

Adjoint-field formulation Reciprocity identify in Ω\Γ ε (z) with scattered v ε and adjoint û fields { } f (ε)t(z, Γ,T ) = t[v ε ] û t[û] v ε (ξ,t) dsξ ε 0 = ε 0 Γ + ε (z) Γ ε (z) Γ ε(z) t[û](ξ,t) [[v ε ]](ξ,t) ds ξ with reference normal n =n and t[û]=c : û n Topological derivative with incident field u T(z, Γ,T ) = { û A: u}(z,t ) Incident and adjoint fields defined in reference (defect-free) domain Ω u(z,t) = U(z,t, ξ) t N (ξ,t) ds ξ T (z,t, ξ) u D (ξ,t) ds ξ S N S D û(z,t) = U(z,t, ξ) ϕ S u [u, ξ,t t] ds ξ obs Generic adjoint-field formulation with misfit function ϕ and observations u obs Choice a-priori of shape Γ and orientation of trial crack C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 10 / 25

Outline 1 Topological sensitivity analysis 2 Qualitative crack identification 3 Numerical examples 4 Acoustic case 5 Interface cracks C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 11 / 25

Methodology Topological derivative: Local sensitivity to introduction of infinitesimal guess T(z, Γ,T ) = { û A: u}(z,t ) Analysis rigorous in the limit ε 0 of an infinitesimal crack T(z, Γ,T ) 0 Infinitesimal trial crack at z improve the fit with u obs Heuristic: Pronounced negative values coincide with finite cracks locations { } Γ eq (α) = z Ω : T(z, Γ,T ) α min T(z, Γ,T ) < 0 z Ω Topological sensitivity as a defect indicator function Non-iterative: Require free and adjoint fields in reference configuration Ω Local indicator at the cost of two forward solutions Qualitative approach: Location, number of cracks Global nature: Probing of domain Ω Adjoint solution contains entire experimental informations C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 12 / 25

Methodology Topological derivative formula featuring infinitesimal penny-shaped crack Identification of local orientations T(z, Γ,T ) T(z, n,t ) If prior information available: n = n true Alternatively: Minimization n min (z)=arg min T(z, n,t ) over unit sphere S n S C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 13 / 25

Methodology Topological derivative formula featuring infinitesimal penny-shaped crack Identification of local orientations T(z, Γ,T ) T(z, n,t ) If prior information available: n = n true Alternatively: Minimization n min (z)=arg min T(z, n,t ) over unit sphere S n S Reconstruction of equivalent crack geometry { } Γ eq (α) = z Ω : min T(z, n,t ) α min min T(z, n,t ) < 0 n S z Ω n S C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 13 / 25

Methodology Topological derivative formula featuring infinitesimal penny-shaped crack Identification of local orientations T(z, Γ,T ) T(z, n,t ) If prior information available: n = n true Alternatively: Minimization n min (z)=arg min T(z, n,t ) over unit sphere S n S Reconstruction of equivalent crack geometry { } Γ eq (α) = z Ω : min T(z, n,t ) α min min T(z, n,t ) < 0 n S z Ω n S Polarization tensor A{ is a function of even powers of n } S min (α, z) = n S min (z) : T(z, n,t ) < α min min T(z, n,t ) z Ω n S Average equivalent optimal normal n min eq (α, z) = n ds ξ S min (α,z) C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 13 / 25

Outline 1 Topological sensitivity analysis 2 Qualitative crack identification 3 Numerical examples 4 Acoustic case 5 Interface cracks C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 14 / 25

FEM-based simulations FEM-based time domain crack identification using synthetic data Unconditionally stable Newmark algorithm β = 1/4, γ = 1/2 No prescribed Dirichlet condition S D = Prescribed normal component of traction vector t N (ξ,t) on S N Measurements on discretized surface Sh obs Ω Γ,h = Ω h Unique numerical experiment Discretized least-squares cost functional n obs n T 1 J h (Ω Γ,h,T ) = 2 u Γ,h(ξ i,t j ) u obs h (ξ i,t j ) 2 i=1 j=0 Discrete version of time convolution k [v h w h ](ξ i, t k ) t v h (ξ i, t j )w h (ξ i, t k t j ) (0 k n T ). j=0 Corresponding adjoint state in Ω h defined by nodal forces over S obs h ˆF h (ξ i,t j ) = u h (ξ i,t T t j ) u obs h (ξ i,t T t j ) (1 i n obs, 0 j n T ) C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 15 / 25

Numerical examples: Penny-shaped crack Unit cubic mesh with 27840 nodes Time-step compressional loading on top face Adimensionalization w.r.t. longitudinal wave velocity Observation on whole domain boundary T(z, n min,t ) 6 0 Γeq (α) n min eq (α, z) θ=0 α = 0.7 θ = π/4 α = 0.75 C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 16 / 25

Numerical examples: Helicoidal crack Elongated thin pipe with 40650 nodes Torsion and normal compression on narrow band Gaussian distribution of the loading Observation on external domain boundary T(z, n min,t ) 6 0 Γeq (α) n min eq (α, z) θ = π/20 α = 0.85 θ = π/2 α = 0.75 C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 17 / 25

Outline 1 Topological sensitivity analysis 2 Qualitative crack identification 3 Numerical examples 4 Acoustic case 5 Interface cracks C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 18 / 25

Acoustic case Dynamical acoustic solicitations: Γ with homogeneous Neumann condition u ± Γ (ξ,t) n± (ξ) = 0 on Γ ± Rigid thin screen Definition of adjoint-field û û(ξ,t) n(ξ) = ϕ u [u, ξ,t t] on Sobs and other B.C. and I.C. set to zero Scalar dynamical reciprocity relation Scattered field behavior using solutions of stationary canonical problems Acoustic topological derivative T(z, Γ,T ) = { û B u}(z,t ) ; η(ε) = ε 3 with adjoint û and incident u acoustic pressure fields Polarization tensor for penny-shaped screen: B = 8 3 n n Analytical minimization of topological derivative w.r.t. normal n Amstutz, Dominguez, Masmoudi,... C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 19 / 25

Numerical examples: Penny-shaped crack Unit cubic mesh with 27840 nodes Time-step uniform normal gradient imposed on top Adimensionalization w.r.t. sound speed Observation on whole domain boundary T(z, n min,t ) 6 0 Γeq (α) n min (z) θ=0 α = 0.9 θ = π/4 α = 0.8 C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 20 / 25

Numerical examples: Helicoidal crack Elongated thin pipe with 40650 nodes Uniform normal gradient on narrow band Gaussian distribution of the loading Observation on external domain boundary T(z, n min,t ) 6 0 Γeq (α) n min (z) θ = π/20 α = 0.85 θ = π/2 α = 0.75 C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 21 / 25

Outline 1 Topological sensitivity analysis 2 Qualitative crack identification 3 Numerical examples 4 Acoustic case 5 Interface cracks C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 22 / 25

Interface crack Bi-material domain Ω = Ω + Ω with interface S I Interface crack Γ S I Number of applications Delamination cracks in composite materials Fatigue behaviors of laminates Matrix-matrix or fibre-matrix interface debonding... Specificities: Interface location and orientation are known beforehand C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 23 / 25

Interface crack Bi-material domain Ω = Ω + Ω with interface S I Interface crack Γ S I Number of applications Delamination cracks in composite materials Fatigue behaviors of laminates Matrix-matrix or fibre-matrix interface debonding... Specificities: Interface location and orientation are known beforehand Derivation of specific polarization tensor for the elastic case A σ ± = 8 π 2 κ(1 + κ 2 )(α 2 β 2 { ) 3 (α δ)πκ(1 + κ 2 ) + β n 2I + (α δ)πκ(1 + } κ2 ) β n n n β with constants α, β, δ, κ depending on isotropic elastic material constants ν ±, µ ± C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 23 / 25

Numerical examples FEM-based time domain 3D simulations in stiff/soft bi-material domain Gaussian time distribution of compressional loading on top face Adimensionalization w.r.t. longitudinal wave velocity, T = 1 Observation on top face Topological derivative T(z, n true,t ) 0 at interface Extension to multilayered domains Study of other type of interface, e.g. fiber reinforced composites C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 24 / 25

Conlusion Elastic or acoustic topological sensitivity provides non-iterative indicator Qualitative geometrical identification (locations, number of cracks,... ) Qualitative reconstruction of cracks local orientations Method easily implementable in classical computational platforms Quantitative approach enabled by higher-order expansions Perspectives and future works Configurations with heterogeneous media Explore error in constitutive relation Apply to real experimental data... C. Bellis, M. Bonnet Méthode de Sensibilité Topologique GDR MSPC - Ondes 25 / 25