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Boundary and interface conditions in the relaxed micromorphic model: exploring finite-size metastructures for elastic wave control

In this paper, we establish well-posed boundary and interface conditions for the relaxed micromorphic model that are able to unveil the scattering response of fully finite-size metamaterials' samples. The resulting relaxed micromorphic boundary value problem is implemented in finite element simulations describing the scattering of a square metamaterial's sample whose side counts 9 unit cells. The results are validated against a direct finite element simulation encoding all the details of the underlying metamaterial's microstructure. The relaxed micromorphic model can recover the scattering metamaterial's behavior for a wide range of frequencies and for all possible angles of incidence, thus showing that it is suitable to describe dynamic anisotropy. Finally, thanks to the model's computational performances, we can design a metastructure combining metamaterials and classical materials in such a way that it acts as a protection device while providing energy focusing in specific collection points. These results open important perspectives for the short-term design of sustainable structures that can control elastic waves and recover energy.

Common numerical approaches to study the dynamical metamaterials' response include Bloch-Floquet analysis [3,16,48] and direct numerical simulations encoding all the details of the considered metamaterials' microstructures in a Finite Element environment [4,11,26,27,28]. While Bloch-Floquet analysis is limited to the study of elastic wave propagation in unbounded domains, direct numerical simulations can precisely describe the mechanical metamaterials' behavior, also when considering finite-size samples. However, these models may be so computationally expensive that they cannot be conveniently used to investigate metamaterials' response in different configurations combining metamaterials and classical materials. These computational constraints are preventing us from designing large-scale metastructures that can have a true impact in real-world applications.
The awareness of these limitations triggered all the recent advances on dynamical homogenization methods [9,12,13,54,61,62,64]. Such methods share the idea that a periodic infinite-size metamaterial can be replaced by a homogenized continuum, mimicking its response without accounting for all the microstructures' details. This leads to important simplifications of metamaterials' modeling at the macroscopic scale, while still accounting for some influence of the underlying microstructure.
The stream of research on dynamical homogenization methods for metamaterials starts with [61,62,65] and is continued, among others, by [40,44,46,55,56]. While the homogenized elastic parameters obtained through this approach are capable of capturing the behavior of lower (acoustic and first optic) branches of the dispersion curves they cannot recover metamaterials' response at higher frequencies. In the present paper, we focus on the major problem of establishing boundary and interface conditions that are i) intrinsically compatible with the relaxed micromorphic bulk PDEs and ii) able to describe "realistic" metamaterials' dynamic responses when considering specimens of finite size. Thanks to the computational performances of the relaxed micromorphic model, coupled with the introduction of such well-posed boundary and interface conditions, we are able to design and optimize a complex metastructure for protection applications that also allows energy focusing for eventual energy recovery. This metastructure is made of metamaterials and classical (homogeneous) materials bricks that are combined together in such a way that the structure acts as a protection device, while being able to focus energy in specific paths for eventual subsequent re-use.
The paper is now structured as follows. In Section 1 we present an introduction that makes a point about the current state of the art about metamaterials' modelling and about the relaxed micromorphic modelling of metamaterials. Section 2 recalls the bulk equations for both Cauchy and relaxed micromorphic continua and then focuses on the establishment of well-posed boundary and interface duality conditions that can describe "realistic" metamaterials' boundaries, as well as interfaces between different metamaterials and metamaterials/homogeneous materials.
In section 3 we present numerical simulations showing the performances of the relaxed micromorphic model to describe the scattering of finite-size metamaterial's sample. The obtained results clearly indicate the capability of the relaxed micromorphic model of correctly describing metamaterials's dynamic anisotropy.
In section 4 we exploit the computational performances of the relaxed micromorphic model to explore and design a metastructure that acts as a protection tool, while being able to focus energy in specific paths for eventual subsequent reuse.
In section 5, we draw our conclusions and perspective for subsequent works.
2 Relaxed micromorphic modelling of finite-size metamaterials In this section, we briefly recall the weak and strong form of the governing equations of both classical Cauchy and relaxed micromorphic continua. We then put a particular focus on the establishment of wellposed boundary conditions that are indispensable for the realistic modeling of the free surface of finite-size metamaterial's specimens, as well as of the interfaces between different metamaterials and linear-elastic classical Cauchy materials. The Lagrangian L for the classical Cauchy model is where u is the displacement field, (·) ,t is intended as a derivative with respect to time, ρ is the apparent mass density, and C is the classical 4th order elasticity tensor. The Lagrangian L m for the relaxed micromorphic model is [2] L m (u ,t , ∇u ,t , P ,t , ∇u, P, where u ∈ R 3 is the macroscopic displacement field, P ∈ R 3×3 is the non-symmetric micro-distortion tensor, ρ is the macroscopic apparent density, J micro , J c , T e , T c , are 4th order micro-inertia tensors, and C e , C micro , C c , L are 4th order elasticity tensors. In the numerical applications considered in this work, all the possible characteristic lengths contained in the tensor L will be set equal to zero since their effect can, in a first instance, be neglected when focusing on the dynamic regime. The action functional A of the considered continuum can be defined based on the Lagrangian function as for classical Cauchy and relaxed micromorphic media, respectively. In eq.(3), Ω stands for the volume of the considered continuum in its reference configuration and [0, T ] is a time interval during which the deformation of the continuum is observed. In the case of conservative internal actions, the virtual work of internal actions W int can be defined based on the first variation of the action functional. In formulas, we have: for classical Cauchy and relaxed micromorphic media, respectively. In eq.(4) the variation operator δ indicates that the variation must be taken with respect to the unknown kinematics fields (u for Cauchy media and (u, P ) for the relaxed micromorphic model). Following classical variational calculus, the strong form of the bulk equations of motion for Cauchy and the relaxed micromorphic model can be obtained via a least action principle stating that, in absence of external body loads, the first variation of the action functional must be vanishing. The application of a least-action principle to isotropic Cauchy and relaxed micromorphic models respectively gives [2,43,49,50] ρ u ,tt = Div σ , σ = 2µ sym∇u + λ tr (sym∇u) 1 and ρ u ,tt − Div ( σ ,tt ) = Div ( σ) , where s := C micro sym P , m := L Curl P .

Relaxed micromorphic boundary and interface conditions that are representative of realistic metamaterial's interfaces
The fact of establishing well-posed boundary conditions for the relaxed micromorphic model that are representative of realistic physical situations is of major importance for its sound application in view of the conception of complex metastructures that can control elastic waves and recover energy.
To obtain the conditions that can be established at the boundary ∂Ω of a relaxed micromorphic continuum, the work of external actions W ext m must be introduced as where f ext and Φ ext are the external forces and double forces acting through the boundary ∂Ω of the considered domain and δu and δP are the test functions associated to the micromorphic kinematic fields.
The duality conditions to be imposed at the boundary ∂Ω of the relaxed micromorphic model can be unveiled via the principle of virtual works which states that In this way, the weak conditions that must be verified on the boundary ∂Ω read where t m and τ are the generalized traction and double traction defined as with ν the normal to the considered boundary ∂Ω, and where the cross product × is intended row-wise. It can be shown that, in absence of curvature terms (L = 0), the duality condition condition involving Φ ext and δP must not be assigned on the boundary. There are different possible ways to act on the boundary of the relaxed micromorphic continuum (e.g. assign force or displacement), but in all cases such conditions must verify eq.(10). It is clear that the duality boundary conditions eq.(10) are modified if, instead of having a free metamaterial's boundary, we consider an interface between two metamaterials or between a metamaterial and a Cauchy material. It can be shown that in this case, the interface duality conditions read 1 where t + t − , τ + , and τ − are the tractions and the double tractions at the considered interface, as obtained taking the limit from the "+" and "-" side, respectively. Depending on the fact that the "+" and the "-" regions of the domain are occupied by a relaxed micromorphic continuum or by a classical Cauchy continuum, the traction will be "generalized" or "classical" traction, respectively, while the double traction will be either given by eq.(11) 2 or vanishing. For the sake of completeness, we recall here the definition of the classical Cauchy traction for the isotropic case In the reminder of this paper, we will show how the boundary and/or interface conditions that have to be used in the relaxed micromorphic framework to describe "realistic" metastructures must always be chosen to satisfy the duality conditions (10) and/or (12).
3 Finite element implementation of a relaxed micromorphic boundary value problem for finite-size metamaterials All the finite element analyses presented in this paper have been performed in the time harmonic domain (plane incident wave) and the following interface conditions on the displacement and on the traction have been enforced at the interface ∂Ω (see Fig. 1): 2 where u − , t − , u + , and t + are the displacements and the tractions on the interface ∂Ω, computed as the limit from the "-" and the "+" side, respectively. It is clear that these strong-form boundary conditions, also enforce the validity of their weak counterparts (12) 1 , so that the considered differential problem is well-posed. We stress once again that, since the region Ω − is occupied by a Cauchy continuum and the region Ω + is occupied by a relaxed micromorphic continuum, the tractions t − and t + are respectively given by Taking advantage of the superposition principle and assuming that an incident wave is travelling in the domain Ω − , it is possible to decompose the external displacement u − and traction t − into an incident and scattered component, while the internal displacement u + only consists of a scattered component. We can thus rewrite the interface conditions (14) as In this way it is possible to assign the "incident wave load" at the interface by setting the jump between the internal and the external scattered field to be equal to the incident wave contribution. This allows us to write Therefore, the final unknowns determined by the solution of the finite-element analysis are the internal (in Ω + ) and external (in Ω − ) scattered displacement fields.

Relaxed micromorphic model validation on the anisotropic scattering of a finite-size metamaterial's sample
In this section, we consider a square metamaterial's specimen whose side is constituted by 9 unit cells of the type depicted in Fig. 2 and whose geometric and elastic properties are given in Table 1.  This metamaterial's sample is embedded in a Cauchy continuum of the same type (elastic properties given in Table 1). This means that the considered metastructure can be realized by drilling holes of the type presented in Fig. 2 inside a homogeneous aluminum plate which is much larger than the internal metamaterial's sample (Fig. 3(center )). The external Cauchy continuum is modelled as an infinite medium by the mean of using a perfectly matched layer (PML). All the analyses are carried out under the planestrain hypothesis. Since there is no discontinuity of the material properties at the boundary between the external and internal material, no specific interface conditions must be imposed when implementing the detailed micro-structured finite-element simulation. The scattered profile will be produced by the fact that the incident wave encounters the traction-free surface of the holes. To validate the relaxed micromorphic model for the description of the scattering of this finite-size metamaterial's sample, we also implement the corresponding finite-element simulation in which the internal domain is modelled by means of the eq.(2) and the interface conditions (17) (Fig. 3(right)). The planestrain hypothesis is accounted for by setting u 3 = 0 and P 13 = P 23 = P 31 = P 32 = P 33 = 0. The relaxed micromorphic parameters for the metamaterial of Fig. 2 are defined in eq.(18), they have been calibrated in [2] on an infinite metamaterial, and their values are summarized in Table 2.
In eq.(18) we report the elasticity tensors expressed in Voigt notation, where only the parameters involved under the plane-strain hypothesis are presented.  The calibration procedure presented in [2] is performed by calibrating the relaxed micromorphic parameters on the two principal directions (θ = 0°and θ = 45°). However, due to the fact that the microand macro-tetragonal symmetry is taken into account through eq.(2) of the relaxed micromorphic elasticity tensors, the metamaterial's scattering response is recovered for all intermediate angles of incidence.
To clearly underline that the relaxed micromorphic model is able to unveil the complete anisotropic behaviour of the considered metamaterial, we show in this section the scattering solution obtained for two intermediate angles, namely θ = 15°and θ = 30°. However, for the interested reader, we also report in the Appendix A some particularly interesting simulation solutions obtained for θ = 0°.
Since the incident wave comes from the external Cauchy medium, we are able to set the incident wave to be either a pressure or a shear incident wave. In formulas we have u I (x 1 , x 2 , t) = a P/S,I ψ P/S,I e i k P /S,I 1 x1+k P /S,I 2 x2−ω t , with where P/S is related to the pressure or shear wave, I means incident, a P/S is the amplitude, ψ P/S,I is the normalized eigenvector associated with the pressure or shear wave, k P/S,I 1 and k P/S,I 2 are the components of the wave vector, and ω is the frequency. The amplitude, frequency and second component of the wave vector are all assigned, while the first component of the wave vector is computed accordingly to eq.(19) 2 which gives the classical dispersion relations for Cauchy continua.
Finally, in order to show the performances of the relaxed micromorphic model for the whole considered frequency range (up to the upper band-gap limit), we also implement a reference finite-element simulation in which the internal region is occupied by a classical Cauchy continuum whose elastic parameters are obtained as the long-wave limit of the relaxed micromorphic model and are given in Table 2. This configuration is schematically represented in Fig.3(left) and the associated finite-element simulation will show that the correct framework to deal with the medium-high frequency homogenized metamaterial's response is given by the relaxed micromorphic model. Clearly, in this case, the implemented boundary value problem is given by eq.(5) and eq.(13). Figures 4-7 show the metamaterial's scattering response, as obtained with the equivalent Cauchy (left), the micro-structured (center ) and the relaxed micromorphic simulations (right), for the two considered angles of incidence and for two frequency laying in the metamaterial's band-gap region (lower band-gap and higher band-gap regions). By direct inspection of Fig. 4 and Fig. 5, it is possible to notice that the relaxed micromorphic model correctly describes the metamaterial's scattering response for all considered frequencies, angles of incidence, and for both pressure and shear incidence waves. On the other hand, it can be directly inferred from these figures that a classical Cauchy continuum model is not able to describe the metamaterial's scattering patterns at the considered band-gap frequency.
Similar results can be retrieved in Fig. 6 and Fig. 7 that are relative to a higher frequency value taken in the upper band-gap region.

Shield device metastructure allowing energy focusing
Once the relaxed micromorphic boundary value problem has been validated to describe the scattering of a finite-size metamaterial, we exploit the computational performances of the relaxed micromorphic model to explore a complex metastructure that can act as a protection device, while focusing energy for its eventual subsequent reuse. We present here the metastructure in Fig. 8, whose intent is to channel and focus the energy of the incident wave along specific paths with the aim of harvesting such energy, while, in addition, also acts as a shield.
The continuity conditions enforced at the interfaces between the external Cauchy medium and the equivalent relaxed micromorphic medium are the same reported in eqs. (17), where ∂Ω is composed of all the boundary delimiting the domains made-up of relaxed micromorphic material.    Table 1. The metamaterial's unit cell in the left panel is given in Fig. 2 and the relaxed micromorphic elastic coefficients used to model the metamaterial in the right panel are given in Table 2.
This metastructure is subjected to a downward time harmonic pressure wave (1.2E6 [Hz]) which propagates parallel to the vertical axis. The blue rectangles in Fig. 8 represent dumpers, which are designed to reduce the reflected wave in the higher part of the domain. In Fig. 9 is reported the comparison between the norm of the dimensionless displacement field (scaled with respect to the amplitude of the incident wave) for the micro-structured material (left) and the equivalent relaxed micromorphic material (right). As a measure of how much energy is focused in the four channels, we evaluate the ratio between the average energy density stored in the channels and the average energy density stored in the outer Cauchy material. For a quantitative comparison, we take the weighted sum of the green and yellow domain for the micro-structured simulation (Fig. 10(left)), while only the green portion for the equivalent relaxed micromorphic simulation (Fig. 10(right)). This has been done since in the micro-structured material simulation we do not have a neat interface between the domain in which we focus the incident wave and the first layer of unit cells, while we have one in the equivalent relaxed micromorphic material one. This makes the scattered wave focus on the boundary of the cross shaped holes in the yellow region for the micro-structured material simulation, while the energy is focused on the boundary of the green interface when considering the relaxed micromorphic simulation. After these considerations, we calculate that the ratio between the average energy density stored in the channels and the average energy density stored in the remaining Cauchy material is approximately 5 and the results given by the two simulations are in quantitative agreement. In formulas, we can write where A 1 is the area of the "yellow+green" region, A 2 is the area of the "green" region, W 1 is the weighted energy contained in A 1 , and W 2 is the energy contained in A 2 . Moreover, A ext and W ext are the area of the outer Cauchy material and the energy stored, respectively, for both the models. This means that the average concentration of energy is 5 times higher in the channels than outside. Energy converters can be locally used to recover and reuse the energy initially carried out by the incident wave, and an optimization of the proposed network can be explored to further increase the efficiency of the structure.

Conclusions
In this paper, we have shown that, once well-posed boundary and interface conditions are established, the relaxed micromorphic model can be efficiently used to model the scattering patterns of finite-size metamaterials. While the relaxed micromorphic model was previously validated on metamaterials' slabs that are finite in one direction, but unbounded in the other [2,49,50], it is the first time that the model is validated on fully finite-size specimens. This validation holds for a wide range of frequencies and for all possible directions of propagation of the incident wave, thus showing that our model is able to fully unveil the metamaterials' anisotropic responses in the dynamic regime. Moreover, exploiting the computational performances of the model, we are able to design a metastructure (combining metamaterials and homogeneous Cauchy materials) that acts as a protection device, while focusing energy in specific paths for eventual subsequent conversion and reuse. In future works, we will build on these results to further optimize metastructures of the type presented here in view of their application for acoustic and vibration control. The present paper thus opens new perspectives for the shot-term design of realistic sustainable metastructures that can control elastic waves while recovering energy.