Lagrangian approach and shape gradient for inverse problem of breaking line identification in solid: contact with adhesion

A class of inverse identification problems constrained by variational inequalities is studied with respect to its shape differentiability. The specific problem appearing in failure analysis describes elastic bodies with a breaking line subject to simplified adhesive contact conditions between its faces. Based on the Lagrange multiplier approach and smooth Lavrentiev penalization, a semi-analytic formula for the shape gradient of the Lagrangian linearized on the solution is proved, which contains both primal and adjoint states. It is used for the descent direction in a gradient algorithm for identification of an optimal shape of the breaking line from boundary measurements. The theoretical result is supported by numerical simulation tests of destructive testing in 2D configuration with and without contact.

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Introduction
We prove rigorously the shape gradient for a class of inverse identification problems, which are constrained by penalty equations approximating variational inequalities (VIs). The specific problem describes identification of a breaking line in a body, where the fracture (crack) separating the single solid into two pieces allows adhesive contact.
The main drawback of classical hypotheses of brittle fracture mechanics according to Griffith [18] concerns infinite stresses in the vicinity of a crack which are physically inconsistent. Within a quasi-brittle fracture approach, there are several concepts that provide finite stresses by assuming a plastic zone around the crack tip, or taking into account inelastic phenomena at the crack faces being in contact. In [39] there is given a historical overview on modeling of non-ideal contact beginning from Coulomb's work on friction [10] up to the work of Johnson-Kendall-Roberts on adhesion of elastic bodies [26], validated with experiments. A comprehensive review of modern theories and experimental studies for adhesive joints and their failure can be found in [45]. Based on observations of hydraulic fracturing [46], Barenblatt underlied a cohesive crack model, which laid the foundation of the nonlinear fracture mechanics today. In opposite to the classical stress-free crack, in the work [3] he introduced two crucial hypotheses that crack faces close smoothly, and the normal stress is a function of the crack opening. Two representative functions f(δ) of the material dependence between the interaction force f and the crack opening δ according to Barenblatt's model and the bi-linear traction-separation law adopted in hydraulic fracturing are sketched in figure 1. A simplified model can be derived by applying the method of asymptotic expansion with respect to the thickness of interface adhesive layer, see e.g. [40], which asymptotic limit results in the spring model with linear f = αδ. The corresponding potential of the surface energy at the interface is quadratic with respect to opening. In this work we examine the case in relation to identifiability of the adhesive crack as a part of breaking line in a body.
Let Ω t denote a set of geometric variables depending on a time-parameter t, which is determined by a manifold (the breaking line) Σ t with a normal vector ν t . Motivated by the applications in fracture, we consider a functional of the total energy E(u; Ω t ) given over a Hilbert space V(Ω t ) as the sum of bulk and surface energies. The contact condition for the normal opening ν t · [[u]] ⩾ 0 across Σ t (see [27]) determines the feasible set K(Ω t ) ⊂ V(Ω t ), which topology implies a convex cone. For differentiable functions E, minimization of E(u; Ω t ) over u ∈ K(Ω t ) yields the first order optimality condition The VI (1.1) constitutes the forward problem. The variational formulation (1.1) was employed earlier [28,30] for the description of nonpenetrating cracks (implying that Σ t has singularity at the crack tip), and supported by appropriate numerical methods [23,29]. The surface energies were specified taking into account for adhesion [15] and cohesion [32,41], where the latter results in non-smooth and non-convex functionals E. For non-differentiable energies, see respective hemi-VI approaches in [19,42]. We cite [43] for the concept of the conical differential of a solution to the Signorini VI. In [22] sensitivity estimates in shape optimization problems were investigated for a class of semilinear elliptic VIs based on material derivatives. In [24] shape sensitivity analysis for an inverse obstacle problem and its regularization via penalization was performed by use of geometric properties of active and biactive sets. The inverse problem consists in identification of the manifold Σ t from a measurement z given at an observation boundary Γ O t . Using the optimization formalism [25], we minimize the least-squares objective where u t solves (1.1), and a parameter ρ > 0 implies the perimeter regularization. The VI (1.1) is involved in the optimization as an equilibrium constraint. We cite the optimization-based inverse problems in acoustic scattering [1,4], electrical tomography [8,21], fluid mechanics [33], and free boundary problems [20]. The classical theory of inverse problems and its applications in mathematical physics can be found in [35]. For relevant tasks, see optimal control of partial differential equations (PDEs) [7], shape control of VIs [2], and optimal object location [36]. The shape optimization approach was applied for the inverse problem of identification of interfaces [14], geometric objects [31], inhomogeneities [6], and breaking lines [16].
Our main goal consists in deriving optimality conditions for the equilibrium constrained minimization (1.2) with respect to variations of the shape Σ t . This implies a property of directional differentiability. To construct a differentiable approximation of the VI (1.1), for a regularization parameter ε > 0 and a smooth penalty β ε based on the Lavrentiev regularization (see [38]): we introduce the penalized equilibrium equation: find u ε t ∈ V(Ω t ) such that is conjectured to be not shape-differentiable. For this reason we compute a directional derivative with respect to specific shape perturbations, which give descent directions for the shape optimization problem. It coincides with the usual shape derivative in case of linear state equations.
The principal difficulty concerns non-linearity of the operator ∂ ε u E due to the presence of penalty term in (1.3), even for linear ∂ u E. Let the second variation 3) exist, be surjective with respect to u ε t , and the Lagrange identity hold at the solution: where V(Ω t ) ⋆ denotes the dual space. Then the associated adjoint operator see [5,34,37]. We set a corresponding Lagrangian linearized at the solution u ε t as Korn-Poincaré inequality (2.4). Let an observation boundary We assume that these geometric properties are preserved for all t ∈ (t 0 , t 1 ) under shape perturbations specified below.
We define a parameter-dependent set of geometric objects which includes the Dirichlet, Neumann, observation boundaries, and the breaking line (manifold in 3D), respectively. An example geometry of Ω t is sketched in figure 2 in 2D.
For fixed t, we consider a linear elastic body occupying the broken domain where the latter inequality guarantees non-penetration, see [27]. Using (2.1) we prescribe adhesion at Σ t with the help of quadratic surface energy (see [15,42]): The symmetric tensors of linearized strain ϵ = (ϵ ij ) d i,j =1 and Cauchy stress σ = (σ ij ) d i,j =1 are given by the symmetric gradient and Hooke's law: where the gradient ∇u = (∂u i /∂x j ) d i,j =1 , the transposition ⊤ swaps columns for rows. A fourth order tensor of elastic coefficients C(x) ∈ W 1,∞ (Ω) d×d×d×d is symmetric: C ijkl = C jikl = C klij for i, j, k, l = 1, . . . , d, and positive definite. The scalar product of tensors in (2.3) satisfies the Korn-Poincaré inequality: there exists K KP > 0 such that over the Sobolev space accounting for the Dirichlet boundary condition: The feasible set corresponding to the constraint in (2.1) due to contact reads and its topology implies a convex and closed cone.
Theorem 1 (Solvability of the adhesive contact problem). There exists a unique solution u t ∈ K(Ω t ) to the constrained minimization problem: where the total energy E is composed according to (2.2) and (2.6) as the sum The solution satisfies a first-order optimality condition in the form of VI: The H 2 -smooth solution satisfies the boundary value problem: Proof. Indeed, applying standard variational arguments to the quadratic functional in (2.9) implies the VI (2.10). Its operator ∂ u E builds a continuous bilinear form in V(Ω t ) 2 , which is coercive by the virtue of Korn-Poincaré inequality (2.4). Then a unique minimizer is argued by the Lions-Stampacchia theorem.
From lemma 1 we deduce solvability for the ε-penalized problem.
Theorem 2 (Solvability of the Lavrentiev penalization). There exists a unique solution u ε t ∈ V(Ω t ) to the penalty problem: where the identity transformation id(s) = s. The H 2 -smooth solution satisfies the following boundary value problem: Using the Cauchy-Schwarz, Korn-Poincaré (2.4), and the trace inequality thus, it is coercive. The penalty β ε is uniformly continuous preserving L 2 -convergence, then . Therefore, applying a Galerkin approximation and the Brouwer fixed point theorem (see [13]) justifies a solution to the variational problem (2.15). The uniqueness due to the strict monotony, and the boundary value formulation (2.16a)-(2.16f ) can be derived in a standard way.
Next we consider the inverse identification problem.

Inverse problem and shape gradient for the linearized Lagrangian
For a given observation z ∈ H 1 (∂Ω) d , we consider the least-squares objective in (1.2): where u ε t solves the penalty equation (2.15). From the fundamental theorem of calculus, the following representation holds for continuous β ′ ε : We introduce a quadratic Lagrangian linearized according to (3.2) around the solution u ε t to penalty equation (2.15) as follows (see [32]): due to (2.15) coincides with the objective (3.1): (3.5)

Theorem 3 (Solvability of the saddle-point problem). There exists the unique saddle-point
is a solution to the adjoint equation: for all v ∈ V(Ω t ). The H 2 -smooth solution satisfies the boundary value problem: Proof. The Lagrangian functionL ε from (3.3) is quadratic and convex in u, and linear in v. Therefore, the first order optimality condition for the former inequality in (3.4) is expressed using (3.2) by the primal variational equation (2.15), and by the adjoint variational equation (3.6) for the latter inequality in (3.4). The unique solution u ε t to (2.15) was proven in theorem 2. For fixed u ε t , the bilinear form A ε (u ε t ) in the left-hand side of (3.6) is bounded and coercive by virtue of the Korn-Poincaré inequality (2.4) and β ′ ε ⩾ 0 in lemma 1, hence (3.6) has a unique solution by the Lax-Milgram theorem. The boundary value formulation (3.7a)-(3.7g) follows straightforwardly.
Proof. The perturbed LagrangianL ε (s) in (3.16) is quadratic and convex in u, and linear in v.
Due to the asymptotic representation in (T2) and the mean value theorem, the equation (3.25) can be expressed using the operator A ε from (3.6) as with weight α v s ∈ (0, s) and bounded bilinear residual R v : V(Ω t ) 2 → R. The operator A ε (u ε t ) is coercive and weakly continuous. By the Brouwer fixed point theorem, for ε small enough the variational equation (3.26) has a unique solutionũ ε t+s ∈ V(Ω t ). The other optimality condition ∂ uL ε (s, u ε t ,ũ ε t+s ,ṽ ε t+s ; Ω t ) = 0 has the form , which admits the asymptotic decomposition as with weight α u s ∈ (0, s) and bounded bilinear residual R u : V(Ω t ) 2 → R. Its unique solutioñ v ε t+s ∈ V(Ω t ) is guaranteed at least for small s. This finishes the proof.
(T4) Strongly convergent subsequence for k → ∞ exists such that Proof. The proof is split into the three steps: uniform estimate, weak convergence, and strong convergence.
We consider a trial breaking-line Σ t ∈ S from the feasible set Let V t,h be the FE-space of piecewise-linear functions such that We solve the ε-penalized forward problem (2.15): find u ε t,h ∈ V t,h such that for all u h ∈ V t,h . By this, we disretize the penalty function β ε in (2.12) as The discrete adjoint equation (3.6) consists in finding v ε t,h ∈ V t,h such that Assuming the observation boundary Γ O = Γ N , we synthesize the measurement z h from (4.4) and consider the inverse problem of shape identification: find Σ t such that where u ε t,h solves (4.7). Zero minimum in (4.8) would attained at Σ t = Σ and u ε t,h = z h without the contact and regularization. In this case, different meshes generated for z h and u ε t,h avoid the inverse crime.  is plotted, which attains as minimum 28, 73%. We note that the computation is presented for the small penalty parameter ε = 10 −10 , while lager values may cause some increase of the ratio curves after reaching the minimum. From the simulation it can be observed in figure 5(a) that the left part of Σ without contact is recovered well, whereas the right part of interface being in contact (see figure 4(a)) is not approached during the iteration.
To remedy the hidden part, we apply the boundary force g 1 ≡ 0, g 2 (x 1 , 0.5) = (1 − 1.25x 1 )µ L , g 2 (x 1 , 0) = −g 2 (x 1 , 0.5), (4.13) which is more stretching than g in (4.3). Now the whole Σ is open as can be seen in figure 6. The numerical result of the identification algorithm is depicted in figure 7. Here plot (a) presents the selected iterations of Σ (n) , plot (b) shows the objective ratio attaining as minimum 0, 02%, and plot (c) demonstrates the shape misfit ratio R(n) from (4.12), which decays to 0, 66%. Now the whole Σ seen in figure 7 is recovered very accurate by algorithm 1.

Conclusion
The paper is a part of research on directional differentiability of shape control problems subjected to VIs and its applications to inverse problems in nonlinear fracture mechanics. In the previous work [32] we developed the general theory of shape differentiability for noncovex problems, and we applied it to the contact problem for a cohesive energy, which is non-convex one. The new result is obtained for the surface energy, which is now convex one, but it was not considered before in the context of inverse identification problems. From the point of view of the theory of inverse and ill-posed problems, we have investigated how the key property of convexity affects identifiability of a shape being under unilateral contact conditions. On the basis of this contribution we conclude that the identification result is influenced not at first by convexity, rather contact conditions in the complementarity form or its penalty approximation.
From our numerical simulation tests we make a conclusion that the suggested algorithm of breaking-line identification is physically consistent with the setup of destructive physical analysis (DPA), where a defect is being opened. The DPA is widely used experimental technique to detect the failure of a specimen.

Data availability statement
No new data were created or analysed in this study.