Identification of microstructural information from macroscopic boundary measurements in steady‐state linear elasticity

We consider the upscaled linear elasticity problem in the context of periodic homogenization. Based on measurements of the deformation of the (macroscopic) boundary of a body for a given forcing, it is the aim to deduce information on the geometry of the microstructure. For a parametrized microstructure, we are able to prove that there exists at least one solution of the associated minimization problem based on the L2$$ {L}&amp;amp;#x0005E;2 $$ ‐difference of the measured deformation and the resulting deformation for a given parameter. To facilitate the use of gradient‐based algorithms, we derive the Gâteaux derivatives using the Lagrangian method of Céa, and we present numerical experiments showcasing the functioning of the method.


INTRODUCTION
For a given periodic microstructure, upscaling of the steady-state linear elasticity problem in three dimensions is a classical result in the context of periodic homogenization. The resulting upscaled system is of the same type, where the homogenized (effective) elasticity tensor depends on the macroscopic variable, and it is computed from solutions of auxiliary problems stated in the periodicity cell.
Here, we consider the associated inverse problem and deduce from measurements on the boundary the interior geometry of the periodicity cell. To achieve this, we combine the methods of periodic homogenization and parameter identification. Therefore, if measured data of the deformation of the exterior boundary of a two-scale composite of two solids under given forcing are available, our results allow to compute parameters characterizing the microscopic structure. Such results are of particular practical relevance if, for example, the microstructure is associated with damage (microscopic holes/domains of weak material of a certain size) and our results allow to compute the extent of the damage (size of the holes/weak domains) from macroscopic boundary measurements.
More concretely, we derive the results on the inverse problem under the assumption that the periodicity cell consists of two perfectly bonded solids, where the one part is completely contained in the cell and its geometry is described by a finite vector of real parameters . The aim of this paper is then to investigate the minimization problem arg min ∈I  ( ) ∶= arg min where the parameter ∈ I , I a compact subset, describes the geometry of the microstructure, u[ ] ∶ Ω → R 3 is the displacement field for a given and u m is the measured displacement. There are several authors studying related parameter identification problems in the context of shape optimization and homogenization. Orlik et al. 1 optimize textile materials via homogenization and beam approximation with a similar approach as in Section 3.1 of this paper, but the result is stated for constant homogenized tensor and under different assumptions on the elasticity tensor. Another paper closely connected to our results is by Allaire et al. 2 where homogenization in connection with shape optimization for linearized elasticity in only two dimensions is studied, where the microstructure consists of a cell with a central rectangular hole (i.e., no material/void space). The aim is to find the optimal shape, that is, the length, width, and rotation of the rectangle. Michailidis 3 considers the linear elasticity equation together with some thermal stress tensor. It is also in the setting of inverse homogenization, but the method of Céa in connection with a smoothed-interface is used instead of a sharp interface as we consider here. It is also worth mentioning Allaire et al., 4 who investigate the damage evolution in linear elasticity via shape optimization, whereby they need to compute the shape derivative. They handle the difficulty that the interface moves instead of the outer boundary, and the full strain and stress tensors are not continuous across the interface, but this work is not set in a multiscale context.
Related research fields in parameter identification in elasticity aim to identify the (microscopic) material parameters and not the shape from measurements on the boundary. [5][6][7] Electrical impedance tomography, 8 for example, where the aim is to find the electrical conductivity and permittivity under special structural assumptions, is another application, where related parameter identification problems arise. Some more general results in shape optimization by homogenization method can be found in the books of, for example, Allaire 9 and Delfour and Zolesio 10 and in the theory of inverse problems of, for example, Isakov 11 and Kirsch. 12 The paper consists of two parts: the study of the direct problem (Section 2) and the study of the inverse problem (Section 3). In Section 2.1, we introduce the homogenized problem and briefly summarize existence and uniqueness of the solutions. While these results are mostly standard, we proof some more detailed properties of the homogenized elasticity tensor required in what follows in Section 2.2. In Section 3.1, we formulate the inverse problem and show the existence of at least one solution of the inverse problem. After computation of the Gâteaux derivative of the homogenized tensor in Section 3.2, we derive the Gâteaux derivative of the functional of the inverse problem in Section 3.3. Focussing on a microstructure consisting of ellipsoids, some numerical experiments showcase the functioning of the method in Section 4. Conclusions are drawn in Section 5.

Periodic and homogenized problem
Let Ω be an open bounded Lipschitz domain in R 3 , Γ D ⊂ Ω closed with |Γ D | > 0, Γ N = Ω∖Γ D , and Y = (0, l 1 ) × (0, l 2 ) × (0, l 3 ) ⊂ R 3 . We consider a bounded sequence {A } of tensors of fourth order in M( , , Ω), which is defined as follows: Definition 1. Let , ∈ R with 0 < < and let  be an open set in R 3 . We denote by M ( , , ) the set of all tensors B = (b i kh ) 1≤i, ,k,h≤3 such that The deformation of the domain Ω under given body load and boundary force g can be described by the displacement field u ∶ Ω → R 3 , which is the solution of the steady-state linear elasticity problem where A describes the properties of the material of the solid and n is the outer unit normal. A classical example for {A } is a sequence of tensors of the form with A Y-periodic, that is, the material properties only depend on the microstructure. The linearized strain tensor e(u ) is given by the symmetric gradient of the displacement field, that is, Due to the assumptions on A , there exists a unique solution u for every .
where C is a constant only depending on Ω.
Proof. The proof can be found in Theorem 10.6., 13 whereby we use additionally Korn's inequality for the estimate.
We introduce the periodic unfolding operator from chapter 1.1. 14

Definition 2. For a Lebesegue measurable function on
, is defined as follows: for a.e. (x, ) ∈ Ω × Y , Y is the unique linear combination of the unit vectors e ∈ R 3 with integer coordinates ∈ Z, that is, Under an additional assumption on A , we can pass to the limit → 0 to find the homogenized solution. We denote by H 1 per,0 (Y ) the space of all functions u ∈ H 1 (Y ), which are Y -periodic and have mean value zero, that is, Then, B ∈ M( , , Ω × Y ) and there exists u ∈ H 1 and (u,û) is the unique solution of Proof. The assumption (2) implies that B ∈ M( , , Ω×Y ). Due to Theorem 1, the solutions u are uniformly bounded in H 1 (Ω) and so we can apply standard periodic unfolding results (see chapter 1.4 14 ) to get convergences (3)- (5).
Choosing appropriate test functions in in the weak formulation of (1) and passing to the limit, we receive (6). We can ) 1∕2 . Note that due to the Y -periodicity and Korn's inequality for functions with zero trace on part of the boundary and for periodic functions with zero mean value, there holds for every ) .
Using this inequality and the fact that B ∈ M( , , Ω × Y ), we can apply the Lax-Milgram theorem to problem (6) to show that there exists a unique solution.
We can split the homogenized problem into a macroscopic problem and a cell problem.

Theorem 3. The homogenized problem (6) is of the form
for a.e. x ∈ Ω and w kl , for all v ∈ [H 1 per,0 (Y )] 3 . If B is Y -periodic, we can write the cell problem in the strong form for a.e. x ∈ Ω.
Proof. This result follows by standard arguments. It can be shown as in the proof of Proposition 3.7 14 for the diffusion problem.
The variational formulation of the homogenized problem (7) is given by

Properties of the homogenized tensor and homogenized problem
We want to show that the homogenized tensor A hom (see Equation (8)) is in the set M(̃,̃, Ω) for some constants̃,̃> 0 under an additional assumption, whereby we refer to the book of Jikov, Kozlov and Oleinik 15 in the general setting of G-convergence. For the proof, we need the boundedness of the cell solutions.

Lemma 1.
The solution w kh of the cell problem (9) is bounded as follows: We additionally need some auxiliary lemmas.
Proof. Let v ∈ [H 1 per,0 (Y )] 3 be extended Y -periodically, denoted byṽ. We only have to prove thatṽ ∈ [H 1 loc (R 3 )] 3 . Therefore, let K be a compact subset of R 3 . We define  ⊂ Z 3 and Then, using the transformation formula, and analogously, Since K was arbitrary, we get the desired result. 13 there holds for all With the results from above, we get for all where n is the normal of Y , which proves that v is Y -periodic. As in Lemma 2, we get that v can be extended Y -periodically to an element of The sum disappears since v is periodic and either is continuous on Y + or already zero.
Let A ∈ M( , , ) and m ∈ R 3×3 be a symmetric matrix. We introduce the Voigt notation to rewrite the tensor of fourth order as a 6 × 6 matrix and the symmetric matrix as a vector of R 6 :

Lemma 4. Let A ∈ M( , , ). Then, the inverse of A V exists and is symmetric. Furthermore, there holds
Proof. Let A ∈ M( , , ). Let V ∈ R 6 . Then, the associated symmetric matrix is Since A is elliptic for symmetric matrices, we get where (·, ·) denotes the standard scalar product. This shows that A V is positive definite and, due to assumption, symmetric. Thus, the inverse of A V exists and is symmetric. To prove the inequality, we follow the proof of Propostion 8.3, 13 where the same result is shown for tensors of second order. Let w ∈ R 3×3 be a symmetric matrix. Then, there holds for m = A −1 w defined as in (11) the equation Am = w and Since A is a linear operator, we can estimate the operator norm Thus, Together with estimate (12), we get So now we can prove the following result.

Theorem 4. If B is additionally Y -periodic in the second argument, there holds
Proof. We prove the theorem for a.e.x ∈ Ω. Since B ∈ M( , , Ω × Y ) and Lemma 1 holds, we get that A hom i kl ∈ L ∞ (Ω). Using w kl as a test function in (9) for the cell solution w i (and the other way round) and the symmetry of B, we can easily compute that A hom is symmetric. To prove the coercivity of A hom with the coercivity constant , we extend B Y -periodically in the second argument. So the tensor weakly in L 2 (Ω) for all i, = 1, 2, 3. In the next step, we want to prove for all ∈ C ∞ 0 (Ω) We notice that Using the symmetry of B(x, ·), we can apply and thus, since we can approximate v by C ∞ (Ω) functions. Using this result and the strong and weak convergences from above, The last equation holds since The coercivity of B together with the weak lower semicontinuity of the L 2 -norm yields for → 0 and for all ∈ ) .
If we apply Lemma 4 to w = B e(v ), whereby (B ) −1 w is defined as in (11), we get for all Passing to the limit yields because the same convergence holds as in the proof of the coercivity due to the symmetry of B and A hom (x). We needed the symmetry of the matrix only to get the weak convergence of e(v ) to m. Since ≥ 0 was arbitrary and we can apply the Cauchy-Schwarz inequality, we get Thus, (10). Furthermore, for a constant C independent of the structure of the cell Y .
Proof. Although we have already proven in Theorem 2 the existence and uniqueness, we can use the last result to show this directly by applying the Lax-Milgram theorem. Using the properties of , Korn's inequality and the trace operator, we receive the inequality (13).

INVERSE PROBLEM
For the inverse problem, we consider a reference cell Y = (0, l 1 ) × (0, l 2 ) × (0, l 3 ) consisting of two parts, one of which is a Lipschitz domain Y 0 completely contained in Y , whose geometry can be described by a (finite) vector of real parameters . A concrete example for such a geometry is an ellipsoid of material embedded in a matrix of other material.
The following results generally hold for parametrized microstructures, but we will consider an ellipsoidal microstructure as illustrated in Figures 1 and 2 for explicit examples and computations: I for some small , where 1 , 2 , and 3 are the lengths of the axis. Furthermore, Y 0 [ ] is centered in the middle of the cuboid Y with axis in direction of the standard unit vectors, Since the structure in the cube Y depends on , we sometimes write Y [ ] instead of Y to emphasize this property. As mentioned in Section 1, we consider a perfectly bonded composite of two materials. Therefore, we define the elasticity tensor A [ ] as follows: and some fourth-order tensors A 0 , A 1 ∈ M( , , Ω) such that for a.e. (x, ) ∈ Ω × Y . In this case, ] 3 due to Theorem 6.2 in Chapter I. 16 In the previous section, we were in the setting that we know the microstructure, that is, the value of . So if , g are given, we can easily compute the solution u[ ]. From now on, we only know , g and some measured data u m . With this information, we want to find the structure of the reference cell. We define the input-output operator: where u[ ] is the solution of the homogenized problem (10) for given .
The operator  is linear and continuous due to (13) and the properties of the trace operator. We consider the following inverse problem.
Definition 4 (Inverse problem). Let 0 < < min{l 1 , l 2 , l 3 }. Find ∈ I such that for given measured data u m ∈ [L 2 ( Ω)] 3 , when forces ( , g) are applied, is the solution of the minimization problem arg min ∈I  ( ) ∶= arg min Of course, other functionals than  could be used.

Existence result
In this section, we want to show that there exists at least one solution of the inverse problem (14). Theorem 6. The operator  ,g is continuous.
Proof. Let n ,̂∈ I with n →̂for n → ∞ and u[ k ], u[̂] the corresponding weak solutions of the homogenized problem (10). Then, for all ∈ H 1 (Ω) → R is the bilinear form of the left-hand side of (10) and F ∶ H 1 Γ D (Ω) → R is the -independent functional of the right-hand side of (10), that is, The third argument of a only emphasizes that the bilinear form is considered for some given . Taking the difference of both equations yields Choosing the test function = u[ n ] − u[̂] and using the coercivity of A hom , we estimate Due to Korn's inequality for functions with zero value on part of the boundary, there holds for some constants c > 0 independent of . Using B[ ] ∈ M( , , Ω × Y ) and Lemma 1, for a.e. x ∈ Ω and for some constant C > 0 independent of and x. Thus, ||A hom i kl [ ]|| L ∞ (Ω) ≤ C for all ∈ I and i, , k, l ∈ {1, 2, 3} and for a.e. x ∈ Ω and C > 0 independent of x. By Theorem 7, pointwise for a.e. x ∈ Ω. So by the dominated convergence theorem, the right-hand side of (15) converges to 0 as n → ∞, which proves that  ,g is continuous. Proof. Let x ∈ Ω and n ∈ I with n →̂. Clearly,̂∈ I and We consider the terms  1 n and  2 n separately. We estimate where we have applied Lemma 1. Since B i hr [ n ] ∈ L ∞ (Ω × Y ) and Y is bounded, we obtain that B i hr [ n ](x, ·) ∈ L 2 (Y ) for a.e. x ∈ Ω and where we have used that A 0 , A 1 ∈ L ∞ (Ω). The second term,  2 n (x), converges to zero if we show that for n → ∞, because we already know the strong convergence in L 2 (Y ) from above. Due to Lemma 1, the solutions w kl [ n ] of the cell problem are uniformly bounded in [L ∞ (Ω, H 1 per,0 (Y ))] 3 . Thus, there exists a subsequence (again denoted by n ) and a functionw ∈ [H 1 per,0 (Y )] 3 such that We equate the cell problems (9)  With this theorem, we can easily show that there exists at least one solution of the inverse problem (14).

Theorem 8. There exists at least one solution of the minimization problem (14).
Proof. The operator ( ) =  • ,g ( ) is continuous, since the trace operator  and  ,g are continuous (see Theorem 6). The set I is compact, so we can apply the extreme value theorem to guarantee that there exists at least onê∈ I , which minimizes (14).
We even get the compactness of the solution space. Proof. Consider a sequence of solutions (u n ) n ⊂ L ,g . Then, there exists n ∈ I such that u n = u[ n ]. Since I is a compact set, there exists a subsequence of ( n ) n (again denoted by n) such that n converges to somê∈ I . Since  ,g is continuous (see Theorem 6), we receive the convergence of u n to u [̂] in H 1 Γ D

(Ω).
In the rest of the section, we want to derive the Gâteaux derivative of  to facilitate the use of gradient-based optimization algorithms. As we will see later, we need the Gâteaux derivative of A hom for this, which we will compute in the following section.

Gâteaux derivative of A hom
To compute the Gâteaux derivative of A hom , we apply the concept of shape derivatives. More precisely, we use the Lagrangian method of Céa following the idea of Allaire et al. 4 For this, we define for all x ∈ Ω a Lagrangian function x i kl which coincides with The main advantage is that the computation of the shape derivative of x i kl is much easier than of x i kl since we can apply standard shape derivative results. We cannot apply these directly to x i kl because the solutions of the cell problems also depend on Y 0 . For readability, we omit the index in the divergence div (·) and in the symmetric gradient e (·) because all the computations in this section are for some fixed x ∈ Ω. Since some spatial derivatives of w kl may be discontinuous at the interface Σ Y , we write the cell problem as a transmission problem: For a.e. x ∈ Ω find (w x (e(w x,1 kl ) − e kl ))n 1 + A 0 x (e(w x,0 kl ) − e kl ))n 0 = 0 on Σ Y (17) and where and n = n 0 = −n 1 is the outward unit normal vector of the interface Σ Y with direction from Y 0 to Y 1 . It can be easily shown that the transmission problem is equivalent to (16). Clearly, the restriction of the solution w kl (x, ·) of (16) to Y 0 resp. Y 1 solves the transmission problem, that is, w kl (x, ·) = w x,1 kl in Y 1 and w kl (x, ·) = w x,0 kl in Y 0 . Now, we define the general Lagrangian, where q 1 , q 0 play the role of Lagrange multipliers, 3 . In the next two lemmas, we compute some conditions for optimal points. Lemma 6. The solution (u 1 , u 0 ) of the transmission problem satisfies the optimality condition for all ∈ [H 1 per,0 (Y )] 3 . Therefore, the solution w x kl ∶= w kl (x, ·) of (16) fulfills the condition Proof. Let ∈ [H 1 per,0 (Y )] 3 . We compute the directional derivatives by using integration by parts (19) and the second statement of the lemma follow directly.
We define the adjoint transmission problem: x (e(p 1 ) + e i ))n 1 + A 0 x (e(p 0 ) + e i ))n 0 = 0 on Σ Y (20) for = 0, 1, which is equivalent to The equivalence can be proven as before.

Lemma 7.
The solution (p 0 , p 1 ) of the adjoint transmission problem satisfies the optimality condition 3 . In particular, the function −w x i ∶= −w i (x, ·), where w i is the solution of (16) for k = i, l = , is a solution of (21), and thus fulfills the condition Proof. The proof is similar to the proof of the last lemma.
In order to proceed, we introduce the shape derivative. The following definitions and propositions and further details can be found in Michailidis. 3
The following two propositions give the shape derivative for functionals, where the integrand does not depend on the domain.
The proposition still holds if Ω 0 is regular enough to apply the transformation formula and Gauß's theorem.
where H = div n is the mean curvature of Ω 0 .
The weak solution w kl of the cell problem (16) is not shape differentiable. However, the next lemma shows that the restricted functions w x,0 kl and w x,1 kl are shape differentiable. Lemma 8. The solutions w x,1 kl of (17) and w x,0 kl of (18) are shape differentiable for a.e. x ∈ Ω and ∈ [W 1,∞ 0 (Y )] 3 .
Proof. The lemma can be shown as in the proof of Theorem 5.3.2. 17 The main idea is to consider the cell problem (16) on the transformed domain Y ∶= (Id + )(Y ) for some ∈ [W 1,∞ 0 (Y )] 3 . By the change of variable theorem, the weak formulation can be rewritten as an integral over the reference cell Y . Since the integrand thus obtained is of class C 1 with respect to and v ∈ [H 1 per,0 (Y )] 3 , we can apply the implicit function theorem to get the desired result. In the next lemma, we prove that the Lagrangian x i kl is equal to the functional x i kl (Y 0 ) in the optimal point (w x,0 kl , w x,1 kl , −w x,0 i , −w x,1 i , Y 0 ), whereby we write w x,h kl , h = 0, 1, instead of w x kl to emphasize which problem w x kl solves and that only the values in Y h are relevant for calculation of x i kl . With this result, we can compute the shape derivative of x i kl instead of x i kl (Y 0 ), which is much easier. Lemma 9. The shape derivative of the objective function x i kl (Y 0 ) exists and is given by Proof. The following identity holds using the solution properties of w x,1 kl and w x,0 kl . Differentiating this identity with respect to the shape yields In the special case where q 0 = −w x,0 i and q 1 = −w x,1 i the last two terms disappear, which proves the lemma. Since v 0 , v 1 , q 0 , q 0 do not depend on the structure of Y 0 , we can apply the standard results of Propositions 1 and 2 to compute the shape derivative of the Lagrangian x i kl : where H is the mean curvature. The terms involving H vanish on Σ Y , since w x,1 kl = w x,0 kl and w x,1 i = w x,0 i on Σ Y . The same argument and the fact that A 1 x (e(w x,1 kl ) − e kl )n = A 0 x (e(w x,0 kl ) − e kl )n on Σ Y leads to where we denote by A x (e(w x kl ) − e kl )n and A x (e(w x i ) − e i )n the continuous quantities through the interface. This general formula can be simplified under additional assumptions on the materials. In what follows, we consider isotropic materials, that is, the material tensors are of the form where = 0, 1, ∈ L ∞ (Ω) and ∈ L ∞ (Ω) are the Lamé parameters depending on the macrovariable and I 2 and I 4 are the identity tensors of second and fourth orders. At each point of Σ Y , we define the unit normal vector n and both unit tangential vectors as a collection t such that (t, n) is a local orthonormal basis. A 3 × 3 matrix in local basis can be written as follows: where the square brackets denote the jump on the interface, that is, Proof. The statements follow by direct calculation using ,k,l≤3 and the similarity invariance of the trace.
With this lemma, we can compute Thus, tr(e(w x kl ) − e kl ) tt tr(e(w x i ) − e i ) tt which is an expression of only continuous functions at the interface. Proof. Since q = 0 on Σ Y , there holds ∇qt = 0 for all tangential vectors t. Thus, if (t 1 , t 2 , n) is an orthonormal basis, (A x (e(w x kl ) − e kl )) nn e(q) nn = n T (A x (e(w x kl ) − e kl ))(nn T + t 1 t T 1 + t 2 t T 2 )(∇q) T n = (A x (e(w x kl ) − e kl ))n · (∇q) T n, where we have used the fact that nn T + t 1 t T 1 + t 2 t T 2 = (t 1 , t 2 , n)(t 1 , t 2 , n) T = I 3 . The rest of the proof follows by rewritting the right-hand side of the equation.
We apply this lemma to our problem Summing up all the results, we obtain the shape derivative of x i kl . Theorem 9. The shape derivative of the Lagrangian tr(e(w x kl ) − e kl ) tt tr(e(w x i ) − e i ) tt (Ω) such that Then, due to the uniqueness of the solutions, We derive the first variation of  , Using the fact that u is Gâteaux differentiable, we can determine the Gâteaux derivative of the objective function

SIMULATION RESULTS
We present a typical result of numerical experiments showcasing the functioning of the method. The aim is to derive the length of the axes of the ellipsoids making up the microstructure from measurements of the deformation on the boundary of a beam of 60 mm × 30 mm × 30 mm volume. We assume that the beam is made up of concrete and ellipsoidal polyvinyl chloride (PVC) aggregates arranged periodically on a microscopic scale. Due to the different scales, we nondimensionalize the cell problem, that is, we consider the (nondimensional) reference cell of side lengths 2 × 1 × 1 with the PVC ellipsoid centered in the middle of the cuboid with axis lengths ( 1 , 2 , 3 ) ∈ [0.12, 1.88] × [0.12, 0.88] 2 and the rest of the cell filled up with concrete. In our model, we assume fixed constraints on one of the small lateral faces of the beam, no volume forces, and some boundary load on part of the surface. Although we know that the PVC is of ellipsoidal structure, we want to find the exact dimension, that is, the (vector-valued) parameter . Therefore, we formulate the parameter identification problem arg min ∈I  ( ) ∶= arg min where I = [0.12, 1.88]×[0.12, 0.88] 2 , u m is the deformation of the beam computed for the target value target = (1.5, 0.6, 0.4) and u[ ] is the deformation for given . The scaling with the constant 1 | Ω| 2 has no impact on the derived analytical results from the last sections apart from a scaling factor.
Our implementation is based on MATLAB® (version R2020a) and COMSOL LiveLink TM for MATLAB®. The main computation is done with the finite element simulation software COMSOL Multiphysics®, 18 that is, we solve numerically the cell problem (9) and the homogenized problem (10), whereby quadratic serendipity finite elements are used. We take these results to compute the homogenized tensor (8), the target functional (29), and its Gâteaux derivative (28). All these values are needed to apply the gradient method fmincon in MATLAB ® , which solves the minimization problem (29).  We start the iteration with the initial guess = (0.12, 0.12, 0.12), which is a boundary value of I . In Figure 3, the values of 1 , 2 , and 3 in every iteration step are plotted, whereby the constant function shows the value of . The algorithm terminates after 84 steps when the relative changes in all elements of is less than the step tolerance of 10 −6 . We obtain = (1.492, 0.602, 0.400). The corresponding functional values  in every iteration step can be seen in Figure 4.
Summing up, the simulation results show that the method works. A proper stability and sensitivity analysis, which is beyond the scope of this work, would be required to quantify this properly.

CONCLUSION AND OUTLOOK
We considered the homogenized problem of linear elasticity, in which the microstructure is accounted for by the effective elasticity tensor, the elements of which are based on solutions of elliptic cell problems in the representative cell. We proved that there exists at least one solution of the corresponding inverse problem identifying the parametrised microstructure from macroscopic boundary measurements. With formula (28) for the Gâteaux derivative of  , wherefore we have to compute the shape derivative of the homogenized tensor and solve several weak partial differential equation problems, we were able to apply generally known numerical gradient-based algorithms to get a solution of the minimization problem when measured data are given. Numerical experiments for an ellipsoidal microstructure illustrated that the length of the axes of the ellipsoids could be recovered from boundary measurements. Although we have only considered isotropic materials with ellipsoidal microstructure at the end, the results can also be applied to the anisotropic materials (using (22) instead of Theorem 9) or to more general microscopic geometries as long as there holds an equation of the form (23) for appropriate . In this context, it may as well be of interest to consider more advanced microscopic models, for example, linear elasticity with slip-displacement conditions. 19 Since we only considered the case in which the microstructure is independent of the macroscopic variable, that is, the effective elasticity tensor is constant, the microstructure needs to be the same everywhere in the domain. For future work, it may be of interest to consider as a function of x, that is, the microscopic ellipsoids are of different sizes depending on the macroscopic variable x. While this is a classic generalization of the forward problem, the inverse problem then requires finding a vector-valued function rather than a (constant) vector, which make the optimization problem infinite-dimensional.

ACKNOWLEDGEMENT
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CONFLICT OF INTEREST
This work does not have any conflicts of interest.