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Ultrasonic imaging in highly heterogeneous backgrounds

This work formally investigates the differential evolution indicators as a tool for ultrasonic tracking of elastic transformation and fracturing in randomly heterogeneous solids. Within the framework of periodic sensing, it is assumed that the background at time to contains (i) a multiply connected set of viscoelastic, anisotropic, and piece-wise homogeneous inclusions, and (ii) a union of possibly disjoint fractures and pores. The support, material properties, and interfacial condition of scatterers in (i) and (ii) are unknown, while elastic constants of the matrix are provided. The domain undergoes progressive variations of arbitrary chemo-mechanical origins such that its geometric configuration and elastic properties at future times are distinct. At every sensing step to, t1, ..., multi-modal incidents are generated by a set of boundary excitations, and the resulting scattered fields are captured over the observation surface. The test data are then used to construct a sequence of wavefront densities by solving the spectral scattering equation. The incident fields affiliated with distinct pairs of obtained wavefronts are analyzed over the stationary and evolving scatterers for a suit of geometric and elastic evolution scenarios entailing both interfacial and volumetric transformations. The main theorem establishes the invariance of pertinent incident fields at the loci of static fractures and inclusions between a given pair of time steps, while certifying variation of the same fields over the modified regions. These results furnish a basis for theoretical justification of differential evolution indicators for imaging in complex composites which, in turn, enable the exclusive tomography of evolution in a background endowed with many unknown features.


Introduction
Many critical components in aerospace structures and energy systems are comprised of highly heterogeneous composites [41,11]. Examples include (a) single-and polycrystalline superalloys deployed in aeroengine and gas turbine blades [6], (b) interpenetrating phase metamaterials such as SiC-SiC composites used in accident-tolerant nuclear fuel claddings [1,46], and (c) multifunctional polymer matrix composites with a wide spectrum of applications thanks to their exceptional mechanical properties [25,47]. The topology and characteristics of such materials at micro-and meso-scales are often unknown, or only known to a limited extent because of variabilities in the manufacturing process [35,39] and/or aging [49]. In addition, mechanisms of deterioration via corrosion, fatigue, irradiation, and thermal cycling are yet to be fully understood. These processes, however, are responsible for continuous microstructural evolution leading to inevitable development of micro/macro cracks and volumetric damage zones which may result in the loss of functional performance in key components [8,45].
Ultrasonic imaging emerge as the preferred (or the only feasible) sensing modality in many applications. Laser ultrasonics, in particular, has come under the spotlight for enabling noncontact actuation and measurement [48,50,24]. This is crucial for high-fidelity, in-situ monitoring of complex components and (e.g., additive manufacturing) processes which may be achieved through careful integration of state-of-the-art instrumentation and advanced data analytic solutions capable of fast ultrasonic tomography of progressive variations in uncertain or unknown environments. Existing imaging solutions mostly rely on (a) significant assumptions on the nature of wave motion in the background, or (b) optimization-based full-waveform inversion. Expedited data processing in (a), which underpins the prevalent time-of-flight methods in laserultrasonic imaging [51,33], is made possible by leveraging the propagating modes of an isotropic elastic solid. This leads to significant errors in heterogeneous and anisotropic solids where multiple scattering generates remarkable wave dispersion and attenuation. On the other hand, inverse algorithms in (b) have so far been associated with tardy reconstructions due to their high computational cost. Lately, however, efforts are underway to address the latter by taking advantage of deep learning solutions such as physics-informed neural networks [22,21].
In applied mathematics, in parallel, over two decades of research in inverse scattering and transmission eigenvalues has given rise to a suit of rigorous algorithms for non-iterative waveform inversion [15,14,5,28,13]. Recently, sampling-based approaches to inverse scattering have been applied to laser-ultrasonic test data [38]. The results demonstrate superior reconstructions in terms of quality and resolution compared to conventional methods. In addition, introduction of the differential evolution indicators [4] as a tool for waveform tomography in unknown media showcase a unique opportunity to achieve the above-mentioned goal for real-time, laser-based monitoring. The differential indicators have so far been theoretically established for (i) acoustic imaging of obstacles in periodic and random media [40,3], and (ii) elastic-wave imaging of fractures in monolithic solids [44,42]. A rigorous justification of this imaging modality for its potentially dominant field of application i.e., ultrasonic tomography in complex compositeswhere the background features a random distribution of heterogeneities and discontinuities of unknown support and material characteristics which are subject to both interfacial and volumetric evolution, is still lacking. The present study is an effort toward establishing the differential evolution indicators in the general case of solids. In this vein, a special attention is paid to pose the forward scattering problem in a broad sense by taking advantage of contributions on the nature of transmission eigenvalues in elastodynamics [17,19,18,20,10,9,16]. In this setting, the main theorems provide a comprehensive analysis on the behavior of differential indicators for relevant evolution scenarios involving fracturing and elastic transformation.
In the sequel, Section 2 provides the geometric and mechanistic description of the direct scattering problem. Section 3 specifies the fundamental properties of scattering operators. The main theorems in Section 5 establish the behavior of differential indicators in the case of composites. Section 6 is dedicated to a numerical implementation and discussion of the results.

Preliminaries
Consider periodic illumination of mechanical evolution in a randomly structured composite shown in Fig. (1). At the first sensing step t = t • , the specimen B ⊂ R 3 is comprised of (i) a linear, elastic, isotropic and homogeneous binder of mass density ρ ∈ R and Lamé parameters µ, λ ∈ R, (ii) a union of bounded inclusions D • ∪ D o • = D • ⊂ B with Lipschitz boundaries composed of penetrable D • and impenetrable D o • components where in the former case, the inclusions may be viscoelastic, anisotropic, and multiply connected, and (iii) a network of discontinuity surfaces Γ • ⊂ B characterized by the complex-valued and heterogeneous interfacial stiffness matrix K • (ξ) where ξ ∈ Γ • is the position vector. The specimen may be exposed to irradiation or chemical reactions as common producers of interfacial damage [31], and/or subject to thermal cycling, fatigue, and shock-waves which are mostly responsible for volumetric degradation [32] so that at any future sensing steps For further clarity, let ρ κ = ρ κ (ξ) > 0 and C κ = C κ (ξ), κ = •, 1, . . ., designate the mass density and (complex-valued) viscoelasticity tensor associated with the penetrable obstacles D κ at t κ . Here, (C κ , ρ κ ) are understood in a piecewise-constant sense i.e., D κ can be decomposed into N κ open, simply connected, and non-overlapping subsets D n κ ⊂ D κ (of Lipschitz boundaries) where both ρ κ and C κ are constants ∀ξ ∈ D n κ , and D κ = N κ n =1 D n κ . It should be noted that ∀t κ , D κ and D o κ are assumed to be disjoint i.e., D κ ∩ D o κ = ∅. In addition, the support of Γ κ may be decomposed into N κ smooth open subsets Γ n ⊂ Γ κ , n = 1, . . . , N κ , such that Γ κ = Nκ n =1 Γ n . The support of Γ n may be arbitrarily extended to a closed Lipschitz surface ∂D n of a bounded simply connected domain D n , so that the unit normal vector n to Γ n coincides with the outward normal vector to ∂D n . We assume that Γ n is an open set (relative to ∂D n ) with a positive surface measure.
In this setting, let E i ∪ E o i andΓ i ∪Γ i respectively specify the support of volumetric and interfacial evolution within [t i−1 , t i ] for i ∈ N such that At t = t • , the specimen's external boundary ∂B and the binder's constants (λ, µ, ρ) are known which may be used to define a baseline model of the background. On the other hand, the support of pre-existing scatterers D • ∪ Γ • and their designated properties (C • , ρ • , K • )(ξ) are unknown. Given sequential sensory data at t i−1 and t i for i ∈ N, the objective is to reconstruct the support of volumetric and interfacial evolution E i ∪ E o i ∪Γ i ∪Γ i defined by (1).
Experiments. The domain B is subject to periodic inspections at time steps t κ = {t • , t 1 , . . .}. At every t κ , the specimen is excited by a combination of ultrasonic sources on its external boundary ∂B t so that the corresponding incident field u f ∈ H 1 (B) 3 in the baseline model is governed by Here, the illumination frequency ω > 0 is selected such that the shear wavelength λ s = 2π µ /(ρω 2 ) is sufficiently smaller than the characteristic length scale of the sought-for objects; n is the unit outward normal to the sample's boundary ∂B; τ (ξ) represents the external traction on the Neumann part of the boundary ∂B t ⊂ ∂B which includes the source input. It is assumed that supp(∂B \ ∂B t ) = ∅, i.e., a set of fixed boundary points of zero surface measure prevent the rigid body motion. Henceforth, the homogeneous Dirichlet part of the boundary will be implicitly indicated. At every sensing step t κ , the interaction of u f with the hidden scatterers Γ κ ∪ D κ gives rise to the total field (u κ , Here, wherein n is the unit outward normal to ∂B and ∂D κ . In addition, u (resp. t[u] ) denotes the jump in displacement u (resp. traction t[u]) across Γ κ such that Keep in mind that the unit normal vector n to Γ κ is specified earlier. Also, the second of (4) should be understood as a shorthand for the set of N κ governing equations over the respective homogeneous regions D n κ (n = 1, . . . , N κ ), supplemented by the continuity conditions for displacement and traction across ∂D n κ as applicable.

Dimensional platform.
In what follows, all quantities are rendered dimensionless by taking ρ, µ, and • -denoting the minimum length scale attributed to the hidden scatterers, as the respective reference scales for mass density, elastic modulus, and length -which amounts to setting ρ = µ = • = 1 [7].
Function spaces. It is known that stress singularities at the branch points of multiple intersecting fractures in an isotropic and homogeneous background is weaker than the classical crack-tip singularity [34,23]. The latter is also the case for delamination cracks propagating along bi-material interfaces [37]. High-order singularities may occur when a crack tip meets a bi-material interface in an angle [27,36], in which case it is shown that the contact laws in the vicinity of the crack tip may be modified such that the usual asymptotic forms for stress still applies [2]. In light of this, it will be assumed that u ∈H Here, D = Nκ n=1 D n is a multiply connected Lipschitz domain of bounded support such that Γ κ ⊂ ∂D. Recall that D n is an arbitrary extension of Γ n defined in the above. On invoking H −1/2 (Γ κ ) 3 andH −1/2 (Γ κ ) 3 as the respective dual spaces ofH 1/2 (Γ κ ) 3 and H 1/2 (Γ κ ) 3 , it follows thatH In this setting, t[u] ∈ H −1/2 (Γ κ ) 3 . For future reference, let us also define Wellposedness. Under Assumptions 2.1 and 2.2, observe that the direct scattering problem (5) is of Fredholm type, and thus, its wellposedness may be established by drawing from the unique continuation principles. See Appendix A in the supplementary material for details.
Scattering signatures. By deploying Betti's reciprocal theorem, one obtains the following integral representation for the scattered fields v κ : . .}, on the specimen's boundary.

Scattering operators
Let us define the scattering operator Λ κ : where v κ solves (5). Then, there exists the factorization such that S κ : where wherein t[v * ] : = n · C : ∇v * . This may be observed by (i) premultiplying the first of (15) byū f , and (ii) postmultiplying the conjugated first of (3) by v * . Integration by parts over B\Γ κ ∪ ∂D κ followed by application of the contact condition over Γ κ ∪ ∂D κ and summation of the results yield extend L 2 inner products. In this setting, the middle operator T κ : 4 Properties of operators Then for any j ( resp. m), it is assumed that ω > 0 is not a "Neumann" eigenfrequency of the Navier equation in D o κ,j ( resp. D m ), i.e., as long as vanish identically in D o κ,j and D m respectively. If D m is bounded, the real eigenfrequencies of (18) form a discrete set [9,10].
Lemma 4.1. In light of the unique continuation principle, the operator S κ :

Lemma 4.2. Let
where the latter denotes the closure of the range of S κ .
Proof. Observe that the restriction of To prove the claim, it is then sufficient to establish that S * κ : S(H ∆ ) → L 2 (∂B t ) 3 given by (5) that v κ has trivial Dirichlet and Neumann traces on ∂B t , and thus, the unique continuation principle reads v κ = 0 in B \Γ κ ∪ D κ . From (a) properties of the layer potentials i.e., φ = u κ and ψ = v κ , (b) fifth of (5) which reads t f = 0 on ∂D o κ , and (c) Assumption 4.1 indicating that ω > 0 is not a Neumann eigenfrequency of the Navier equation affiliated with any simply-connected subset of D o κ , one may conclude that ψ = φ = 0. Now, on denoting From the regularity of volume potentials, one may infer that ∀v ∈ Also, on recalling ∇ · C : Combining (22) and (23) reads (φ, Φ, ϕ) = 0 which completes the proof. 3 is compact and has a dense range.
Proof. The compactness of S * κ is established by the smooth kernels in its integral form (19), and its dense range results from the injectivity of S κ per Lemma 4.1. wherein It is assumed that ∀t κ , ω is such that (24) remains wellposed i.e., for (f κ , g κ ) = 0, the ITP does not admit a nontrivial solution (u κ , w κ ). For Γ κ = ∅, the elastodynamics ITP, with varying restrictions on C and C κ (ξ), is analyzed by [17,10,9,16] following the variational method introduced by [29]. In the most general case, under Assumptions 2.2 and 2.3, (24) with Γ κ = ∅ is well-posed when ω does not belong to (at most) a countable set of transmission eigenvalues [10]. The latter is also concluded in a recent study of acoustic ITP for penetrable obstacles with sound-hard cracks [3]. Similar analysis could be applied to (24) which is beyond the scope of the present study. (17) is bounded and satisfies ∀ Ξ ∈ S(H ∆ ) : Ξ = 0. Consequently, T κ is also injective provided that ω is not a transmission eigenvalue of (24) per Assumption 4.2.
Proof. Compactness of V κ follows immediately from Lemmas 4.3 and 4.4 which respectively establish the compactness of S * κ and the boundedness of T κ . To demonstrate the injectivity of V κ , let on ∂B t , one may observe that v κ satisfying (5) has trivial Dirichlet and Neumann traces on ∂B t so that by the unique continuation principle v κ = 0 in B − κ . Now, from (a) second and fifth of (5), and (b) Assumption 4.1 indicating that ω > 0 is not a Neumann eigenfrequency of the Navier equation affiliated with any simply-connected subset of D o κ , one may conclude that ψ = φ = 0. Next, let us define w κ such that w κ =û + v κ in D κ , then the pair (û, w κ ) satisfies the interior transmission problem (24) with trivial boundary potentials i.e., (f κ , g κ ) = 0. Since ω is not a transmission eigenvalue as per Assumption 4.2, on may deduce thatû = 0 in D κ \ Γ κ . Subsequently, ∇û = 0 in D κ \ Γ κ , and t[û] = 0 on Γ κ ∩ D κ which proves the injectivity of V κ .
Denseness of the range of V κ may be established by showing that the adjoint operator V * κ is injective. In this vein, from definition, V * κ : where where t[u τ ] : = n · C : ∇u τ , and This may be observed by (i) premultiplying the first of (5) by u τ , and (ii) postmultiplying the conjugated first of (28) by v κ . Integration by parts over B − κ ∪ D κ \Γ κ followed by application of the contact condition over Γ κ ∪ ∂D κ and summation of the results yield substantiating (28) as the adjoint operator. Next, let τ ∈ L 2 (∂B t ) 3 and assume that V * κ (τ) = 0, i.e., In this setting, the unique continuation reads u τ = 0, and thus, τ = 0 which concludes the proof. • • For some constants α, η > 0, Proof. We adopt a contradiction argument as follows. Suppose (29) does not hold, then one may find a sequence (Ξ n ) n∈N ⊂ S(H ∆ ) such that Denote by v n ∈ H 1 (B − κ ∪ D κ \Γ κ ) 3 the solution to (5) with Elliptic regularity implies that v n H 2 (B − κ ) 3 is bounded uniformly with respect to n. Then, up to changing the initial sequence, one may assume that Ξ n weakly converges to some Ξ in wherein ∇ · C : ∇u + ρ ω 2 u = 0 in D κ . In this setting, (26) along with | T κ Ξ n , Ξ n | → 0 imply that v n → 0 on Γ κ and ∇(u n + v n ) → 0 in D κ \ Γ κ , whereby the first of (5) reads that u n + v n → 0 in D κ \ Γ κ . One may then deduce from the unique continuation principle that the total field u n + v n also vanishes in B − κ . In which case, (9) indicates that v n → 0, and thus v = 0, on ∂B t . This implies that (a) ψ = φ = 0 by virtue of the second and fifth of (5) and the unique continuation, and (b) u = v = 0 which follows from Assumption 4.2.
Next, on recalling (16) and (17), (30) may be recast as where u n φ solves In addition, the variational form (61) with Since | T κ Ξ n , Ξ n | → 0 as n → ∞, (31) in light of the Rellich compact embedding theorem along with the regularity of the trace operator implies that Similarly, the compact embedding and trace theorems applied to (33) reads as n → ∞. On superimposing (34) and (35), one finds for some θ > 0 independent of n. This implies that v n → 0 strongly in H 1 (B − κ ∪ D κ \ Γ κ ) 3 which is a contradiction. Given (34) and (35), the argument for establishing (29) for the second case of Assumption 4.3 directly follows the proof of Theorem 2.42 in [14]. Lemma 4.8. The scattering operator Λ κ : L 2 (∂B t ) 3 → L 2 (∂B t ) 3 is injective, compact and has a dense range.

Design of imaging functionals
Let us generate a set of sampling points x • ∈ B in the baseline model designating the loci of (monopole and dipole) trial scatterers. Monopole signatures are created via point sources applied along a set of trial directions n, while dipole patterns are constructed by nucleating dislocations L : = x • + RL ⊂ B in the baseline model wherein L is a smooth arbitrary-shaped discontinuity whose orientation is given by the unitary rotation matrix R ∈ U (3). In this setting, the scattering pattern for any admissible density a ∈H 1/2 (L) 3 . Keep in mind that the Green's dyadic G satisfies (10) and its affiliated traction T on L is specified in (9). To construct a sampling-based imaging functional, we deploy Ψ o to explore the range of Λ κ by minimizing the below sequence of cost functions where Λ κ : L 2 (∂B t ) 3 → L 2 (∂B t ) 3 is given by where the middle operator T κ is selfadjoint and positively coercive, i.e., there exists a constant c > 0 independent of Ξ so that Moreover, the range of S * κ coincides with that of Λ 1/2 κ .
Assumptions and lemmas of Section B furnish all the necessary conditions for the fundamental theorems of GLSM [4,44] to apply. These results are required for the differential imaging indicators which for future reference are included in the following.
In addition, if • x • ∈ D κ then there exists a unique solution (u κ , w κ ) to • x • ∈ D o κ then there exists a unique field u o κ satisfying • L ⊂ Γ κ \D κ then there exists a unique v κ ∈H 1/2 (Γ κ \D κ ) 3 such that S * κ v κ = Ψ 1 . In this setting, the affiliated free-field traction t[u f κ ] may be obtained from the second of (5), Now, let us recall from (1) that (a) E i ⊂ B, i ∈ {1, 2, . . .}, is the support of (volumetric) elastic transformation where by Assumption 2.3 supp( i designates the evolution of pore volume which is disjoint from E i since D i ∩ D • i = ∅, and (c)Γ i ∪Γ i represents the support of (geometricΓ i and elasticΓ i ) interfacial evolution. On denoting by D i−1,j , j = 1, 2, . . . , N i−1 (resp. D i,j , j = 1, 2, . . . , N i ) the simply connected components of D i−1 (resp. D i ), one may define the set of stationary inclusionš which remain unchanged between [t i−1 t i ]. By adopting a similar notation, the stationary pores are identified byĎ In this setting,D τ i−1 : = D τ i−1 \Ď τ i−1 , τ = { , o}, signifies the evolved subset of D τ i−1 within the same timeframe. Further, one may introducẽ

Based in this, let us in addition defině
While our objective is to design imaging functionals to reconstruct E i ∪ E o i ∪Γ i ∪Γ i given sequential sensory data at t i−1 and t i , one may observe in what follows that the proposed indicator is capable of recovering then ω > 0 is not a transmission eigenvalue solving Note that special cases such as elastic transformation or growth of intact inclusions are also included in (52) and may be obtained by setting Γ i−1 = ∅ and/orΓ i =Γ i = ∅ in (52). Further, in the case of Γ i−1 ∩Ê i = ∅, pertinent to the transformation of microcracked zones, it is further assumed that ω > 0 does not satisfy ITP f the free-field traction on ∂D o s ( resp. Γ s ) affiliated with the scattering pattern Ψ o (or τ γ ), while (u s , w s ) uniquely solves ITP s at times t s , s ∈ {i − 1, i}. In this setting, (54) • Provided that ω is not a Neumann eigenvalue of (45) per Assumption 4.1, then • Moreover, If L ⊂Γ i−1 then (53) holds.
Proof. Let x • ∈Ď i−1 (resp. L ⊂Ď i−1 ), then observe that (a) Ψ 0 (resp. Ψ 1 ) of (37) satisfies , and thus, the solutions to ITP s in light of (a) and (45), and (c) in Corollary 5.1, v s = 0 on Γ s \Ď i−1 which with reference to the contact law per the second of (5 (44) and (47) that by definition ITP i−1 = ITP i inĎ i−1 so that which confirms (53). Same argument as above along with the proof of Theorem 4.5 in [44] leads to (53) when , then in light of the above observe that Next, a contradiction argument is adopted to analyze u s inD s , s ∈ {i−1, i}, as the following. Suppose that (54) does not hold i.e., and consider the twelve generic configurations shown in Fig. 2 for microstructural evolution. The argument for similar or compound scenarios may be drawn from the following case studies. Keep in mind that the interior transmission problems of disconnected sets are independent. Based on this,D i−1 , in each case, should be understood as the simply connected domain x • ∈ D i−1,j ⊂D i−1 (or D i−1,j ⊃ L) as defined earlier. Case 1-3 (fracturing of inclusions). With reference to Fig. 2 (a), consider the case where x • (or L) is inD i−1 where evolution occurs either by new internal/boundary fracturesΓ i or by elastically modified interfacesΓ i . Under the premise of (57), let us define w = w i − w i−1 iñ D i−1 \ Γ i . On recalling (24), observe that the Cauchy data of w vanishes on ∂D i−1 \ Γ i which implies by the unique continuation principle that w = 0 inD i−1 . In case 1 -whereD i is endowed with internalΓ i -the contradiction arises from the discontinuity of w i acrossΓ i while w i−1 is continuous. The only exception to the latter, according to the fourth of (24), is when t[w i ] = 0 onΓ i so that w i = 0, which may not be the case per Assumption 4.1. In case 2 -where the contact's elasticity K s with s ∈ {i − 1, i} changes overΓ i i.e., (24), which requires t[w i ] = t[w i−1 ] = 0 overΓ i that contradicts Assumption 4.1. In case 3 -whereΓ i ⊂ ∂D i -the contradiction may be observed from the fifth of (24) where w = 0 reads t[w i ] = 0 onΓ i with similar contradiction to Assumption 4.1.
Case 4 (evolution of elastic contacts). In this case where L ⊂Γ i \D i ⊂ Γ i−1 \Γ i−1 , the fracture stiffness evolves within the matrix, as shown in Fig. 2 (a), such that K i = K i−1 oñ Γ i \D i . Then, (56) is directly concluded from Theorem 4.5 and Theorem 4.7 of [44]. This may also be observed from (46).
Case 5 and 6 (volumetric growth or transformation of intact inclusions). The premise, as depicted in Fig. 2 In this case, the contradiction to (57) may be argued similar to the proof of Theorem 4.2 in [4] which establishes (54).
Case 7-9 (elastic transformation or expansion of fractured inclusions). Let us definew i inD i as the following: Observe in light of (44) and (57) thatw i solves wherein (C,ρ) is given by (51). In this setting, one may show that the Cauchy data To continue, let us consider two configurations: (cases 7, 8) whereD i−1 is not included in any simply connected partẼ i,j ofẼ i , and (case 9) whereD i−1 ⊂Ẽ i,j . Note that in cases 7, 8, ∂D i−1 ∩ ∂D i is of nonzero surface measure, then owing to the equality of Cauchy data associated withw i and w i and the fact that (C i , ρ i ) = (C,ρ) = (C i−1 , ρ i−1 ) on ∂D i−1 ∩ ∂D i one may conclude thatw i = w i onD i−1 \Ẽ i .
The latter according to Assumption 5.1 implies that w i =w i = 0 inẼ i which by unique continuation reads w i = 0 inD i . This requires u i = −Ψ o inD i−1 which is a contradiction since u i is smooth by definition while Ψ o features a singularity at x • . In case 9, one may directly deduce fromẼ i ⊃Ẽ i,j =D i and Assumption 5.1 that w i = 0 inD i which leads to the same contradiction.
Case 10 (elastic transformation of microcracked damage zones). With reference to Fig. 2 (c), consider the case where L coincides with the binder's fractures at t i−1 (L ⊂ Γ i−1 ) within a neighborhood that undergoes elastic evolution at t i such that L ⊂ Γ i−1 ∩Ê i . Contrary to (56), let wherein both free fields u f i−1 and u i satisfy ∇·C : ∇( · )+ρ ω 2 ( · ) = 0 inÊ i . Then observe that the latter also governs u Based on which, one may defineŵ i = u i +Ψ 1 inÊ i and similar to Cases 7-9 conclude that (w , which by Assumption 5.1 readsŵ i = 0 requiring that u i = −Ψ 1 inÊ i which is a contradiction since u i is smooth by definition while Ψ 1 has a discontinuity across L. Case 11-12 (expansion of pores). Consider the case shown in Fig. 2 where the evolution of cavities occurs. In contrast to (55), let where (G i−1 , G i )(Ψ o ; γ) are the constructed minimizers of (J γ i−1 , J γ i ) in (38) according to (42). Then, in light of factorization (12) and (13), Theorem 5.2 reads Differential evolution indicators. Let us introduce the imaging functionals D i : Then, it follows that In other words, D i (resp.D i ) assumes near-zero values except at the loci ofD where the indicator increases and remains finite as γ → 0. By building on [4,Theorem 4.4] and [44,Theorem 4.8], it is quite straightforward to formulate pertinent results for noisy data which for brevity are not included in this paper.

Synthetic experiments
The evolution indicators of (60) are put to test in this section by a set of numerical experiments. The primary focus is on a randomly heterogeneous and discontinuous background with evolving microstructure due to elastic transformation and/or fracturing. The special cases of monolithic solids endowed with crack or pore networks are reported in [44]. In this section, the synthetic scattered fields v obs i , i = {•, 1, 2, 3}, are simulated via the boundary element method [12], see [43] for more on the computational platform.
With reference to Fig. 3, the testing configuration at t • i.e., the background domain entails a composite slab of dimensions 2.5 × 2.5 × 0.01 comprised of an elastic binder endowed with ellipsoidal inclusions of arbitrary distribution and size. The in-plane diameters of scatterers ranges from one to five shear wavelengths λ s = 0.04, while their pairwise distances are greater than 2λ s . The normalized shear modulus, mass density, and Poisson's ratio of the matrix are taken as µ m = 1, ρ m = 1 and ν m = 0.25, while that of the scatterers are µ s = 2, ρ s = ρ m and Given that λ s is four times greater than the specimen thickness, the leading contributions to scattered fields are the in-plane components which are then used for data inversion.
The obtained scattered signatures are used to compute the synthetic wavefront densities G i , i = {•, 1, 2, 3}, as approximate minimizers of the cost functionals in (38). The latter follows the common three steps required for constructing any sampling-based indicator, namely: (1) forming the discrete scattering operators Λ i at every t i , (2) assembling the trial signatures Ψ o of (37) as the right-hand side of the scattering equation, and (3) solving the latter by minimizing the discretized cost function (38) by invoking the Morozov discrepancy principle. A detailed account of this process is provided in [44]. Given Λ i and G i , one may then evaluate the imaging functionals D i from (60) which is expected to achieve its highest values at the loci of new and evolved scatterers in each sensing sequence. Fig. 4 illustrates the successive evolution indicators D j , j = {1, 2, 3}, over the sampling area. For each time window t j−1 −t j , the "true" support of elastic variations is provided in the top row. Keep in mind that in any sequence, D j is by design insensitive to the scatterers at t j−1 provided that they remain unchanged by t j . The differential maps within t • − t 2 feature relatively sharp localizations with minimal artifacts which, given the shear wavelength, may be attributed to the  rather sparse distribution of scatterers in this timeframe. Some artifacts emerge in the D 3 map, however, since the elastic variations between t 2 − t 3 occur within a densely packed network of fractures and inclusions which are assumed to be unknown.
To recover the evolution support over an extended timeframe as the microstructure becomes progressively complex, one may superimpose the consecutive reconstructions (of Fig. 4) as in

Conclusion
This study furnishes the theoretical foundation for differential evolution indicators for ultrasonic imaging of elastic variations within a heterogeneous and discontinuous background of random structure. In this vein, the wellposedness conditions for the forward and inverse scattering problems are established; in light of which, the pairwise relation between scattering solutions -associated with distinct datasets, is determined. For this purpose, twelve scenarios for microstructural transitions are investigated including (a) fracturing of inclusions, (b) evolution of discontinuity surfaces within each material component or at bimaterial interfaces, (c) elastic transformation and/or expansion of fractured inclusions, (d) conversion of microcracked damage zones, and (e) expansion of pores. In all cases, it is shown that certain measures of the synthetic incident fields constructed based on the scattering solutions are, in the limit, equivalent at the loci of unknown scatterers which remain both geometrically and mechanically invariant between a given pair of time steps. This allows for exclusive reconstruction of evolved features without the knowledge (or need for recovery) of stationary components in the background. This is particularly useful in uncertain environments as showcased by the synthetic experiments -provided that the illuminating wavelength is sufficiently smaller than the relevant microstructural length scales e.g., pairwise distance between the scatterers. Relaxing such constraints may be possible through time-domain inversion as a potential direction in future studies.

Acknowledgements
The corresponding author kindly acknowledges the support provided by the National Science Foundation (Grant No. 1944812). This work utilized resources from the University of Colorado Boulder Research Computing Group, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado State University.

A Wellposedness of the direct scattering problem
Observe that ∀v ∈ H 1 (B − κ ∪ D κ \Γ κ ) 3 , the variational form of (5) reads The sesquilinear form on the left-hand side of (61) may be decomposed as In light of the Korn inequality [43], observe that A(v κ , v ) is coercive. Moreover, the antilinear form B(v κ , v ) is compact by the application of Cauchy-Schwarz inequality to |B(v κ , v )|, the compact embedding of H 1 (B − κ ∪D κ \Γ κ ) 3 into L 2 (B − κ ∪D κ \Γ κ ) 3 , and the compactness of the trace operator v κ → v κ as a map from H 1 (B − κ ∪D κ \Γ κ ) 3 to L 2 (Γ κ ) 3 owing to the compact embedding ofH 1/2 (Γ κ ) 3 into L 2 (Γ κ ) 3 . As a result, (5) is of Fredholm type, and thus, is wellposed as soon as the uniqueness of a solution is guaranteed. Let (u f | D κ \Γκ , ∇u f | D κ \Γκ , t[u f ]| Γκ , t[u f ]| ∂D o κ ) = 0, then on setting v = v κ , observe from (61) that ( D κ \Γκ ∇v κ : C κ : ∇v κ dV ξ + Γκ v κ · K κ v κ dS ξ ) = 0, implying that v κ = 0 on Γ κ , and v κ = 0 in D κ \Γ κ owing to Assumption 2.2 and the first of (5). Note that the jump in v κ vanishes not only on the intersection Γ κ ∩ ∂D κ , but also on the perfectly continuous interface ∂D κ \Γ κ according to the fourth of (5). Thus, the Holmgren's theorem implies that the scattered field v κ vanishes in an open neighborhood of ∂D κ which by virtue of the unique continuation theorem leads to v κ (ξ) = 0 in ξ ∈ B \D o κ . This completes the proof for the uniqueness of a scattering solution in B − κ ∪ D κ \ Γ κ , and thus, substantiates the wellposedness of the forward problem.