Abstract
We introduce the optimizationbased method for the design of thermomechanical metamaterials and, particularly, for the elastostatic cloaking under thermal loads. It consists of solving a largescale, nonlinear constrained optimization problem, where the objective function is the error in the cloaking task accomplishment. The design variables define the required metamaterial distribution. In this way, the cloaking task is accomplished, if not exactly, optimally. Further, the design variables dictate how to fabricate the metamaterial, avoiding the uncertainty of simultaneously mimicking several thermal and mechanical effective properties, as required by transformationbased metamaterial design methods.
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Introduction
Recently, the development of metamaterials has allowed the manipulation of temperature and displacement fields in ways inconceivable in nature, including cloaking^{1,2,3,4,5,6,7}, focusing^{1,4,8,9,10,11,12}, inversion^{4,12,13,14}, shielding^{15,16} and channeling^{17} in heat conduction, and cloaking in elastostatics^{16,18,19} and elastodynamics^{20,21,22}.
Classically, metamaterials have been designed following the transformationoptics (TO) approach, which was originally proposed by Leonhardt^{23} and Pendry^{24} for electromagnetic cloaking. This idea was adapted to design most of the aforementioned metamaterials, either for heat flux manipulation^{1,3,4,5,6,8,9} or mechanical cloaking^{20,21,22}.
Regarding mechanical problems, Shuvalov^{25} demonstrated that the TO approach produces the anisotropic metamaterials known as Willis materials^{26}. In this regard, Milton et al.^{27} demonstrated the need to use, besides the standard fourthorder elasticity tensor, thirdorder tensors for achieving a transformation invariant form of the Navier equations for cloaking purposes, whereas Brun et al.^{28} achieved such a transformation without additional thirdorder tensors but breaking the symmetries of the standard fourthorder elasticity tensor. Additionally, several researchers have made great efforts for the realization of metamaterials issued from TO^{18,19,20,29}. Stenger et al.^{20} designed a circular mechanical cloaking device with radially varying effective elastic properties in an attempt to follow the transformed plate equation derived by Farhat et al.^{30} Kadic et al.^{29} proposed to use pentamodal materials to realize any material from TO. Shortly after, Bückmann et al.^{18} made an elastostatic cloaking using pentamodal metamaterials. As an alternative to TO, Bückmann et al.^{19} proposed the socalled directlattice transformation design approach with application to elastostatic cloaking. Such an approach allows the achievement of realizable metamaterials from lattices. The directlattice transformation dictates the geometry of the latticetype metamaterial, that is, how to fabricate it. On the other hand, the TO defines the inhomogeneous effective material properties required to accomplish a given task, which makes necessary to devise ways to manufacture metamaterials having, or at least mimicking, these particular properties.
Recently, Peralta et al.^{10} proposed a metamaterial design method based on mathematical optimization, avoiding coordinate transformations. It consists of solving an optimization problem to minimize the error in the accomplishment of the task assigned to the metamaterial, taking as design variables the parameters that describe the microstructure. Like direct lattice transformation^{19}, the optimizationbased approach directly prescribes how to fabricate the metamaterial at a point, but it excels the former since not only lattices but any quantitatively characterized material is allowed. It also excels the preceding design methods in terms of versatility since it applies to general heat flux manipulation tasks including concentration^{10,11,12}, shielding^{12} and inversion^{12}. Further, it was easily extended to design very easytofabricate devices made of piecewise homogeneous metamaterials^{11} for heat flux concentration, and even standard isotropic materials for heat flux inversion^{12} and elastostatic cloaking^{16}.
The optimizationbased procedures proposed so far for the design of metamaterials, have been developed in heat conduction^{10,11,12} and the elastostatic problems^{16}. However, an optimizationbased procedure for the design of elastostatic metamaterials under thermal gradients involves the coupled solution of the heat transfer and elasticity problems, namely, a thermoelastic problem. The force equilibrium is then affected by the thermal loads resulting from the temperature variations throughout the material system, which introduces a further complication in the optimization procedure. So, it is worth to mention that the metamaterial design for the accomplishment of a given thermomechanical task will be dictated in terms of a proper spatial arrangement of the thermal conductivity, elastic moduli and thermal expansion. In agreement with the aforementioned aspects, in this work we are interested in the design of metamaterials under coupled thermal and mechanical phenomena, which is unprecedented in the literature. To this end, we will take advantage of the optimizationbased design approach versatility.
Methodology
Let Ω be a body made of an arbitrary material, and u_{0} be the displacement field in Ω under given thermal and mechanical boundary conditions. When the material in the region Ω_{incl} ∈ Ω is replaced by a material with markedly different thermal conductivity, stiffness and/or thermal expansion, the displacement field in Ω is significantly affected. For the purpose of cloaking, let us define a metamaterial occupying the region Ω_{met} around Ω_{incl} such that the resulting displacement field u resembles u_{0} in a given region Ω_{cloak} ∈ Ω.
Assuming linear thermoelastic behavior and small strains, the displacement field in Ω is governed by the equilibrium equation
where ρ is the density, b is the body force, and σ is the Cauchy stress tensor defined by
being ε = [grad u + (grad u)^{T}]/2 the infinitesimal strain tensor, C the elastic moduli tensor, α the thermal expansion tensor, T the temperature and T_{ref} the temperature for zero thermal strain. The dependence of u on T obliges to couple the solution of the equilibrium Eq. (1) to that of the heat conduction equation
where k is the thermal conductivity tensor and Q is the internal heat source.
Here, this coupled thermomechanical problem is solved using the finite element method (FEM), as detailed by Fachinotti et al.^{31} With Ω divided in finite elements, let the microstructure at each finite element Ω^{(e)} in the nonhomogeneous metamaterial region Ω_{met} be quantitatively characterized by the vector p^{(e)} of microparameters. Consequently, any effective material property at Ω^{(e)} is a function of p^{(e)}, as it is currently the case of the thermal conductivity, the thermal expansion and the elastic moduli. To couple the thermal fields to the mechanical response, introduces a further complication in the optimization based design of metamaterials for cloaking purposes. Although the objective function is defined only in terms of the displacement field, the cloaking task fulfillment will be affected by the dependence of both displacement and temperature on the microparameters distribution. The current approach can be seen as a generalization of that proposed by Sigmund and Torquato^{32} for the design of metamaterials with extreme thermal expansion features, where only the mechanical properties were considered, since the temperature distribution was prescribed to be homogeneous irrespectively of the inhomogeneity in thermal conductivity. Note that the problem addressed by Sigmund and Torquato does not involve either the coupling between the force equilibrium and heat conduction equations, or the temperature dependence on the microparameters. On the other hand, such aspects are properly addressed in the fully coupled thermomechanical problem proposed in this communication.
Now, with the effective material properties throughout the metamaterial region Ω_{met} characterized by the vector P = [p^{(1)} p^{(2)} …] containing the characteristic microparameters p^{(e)} of all the elements Ω^{(e)} ∈ Ω_{met}, the displacement in the whole body Ω depends on P, i.e., u = u(x, P) for all x ∈ Ω; see the work of Fachinotti et al.^{31} for details on this dependence when the equilibrium and heat conduction equations are solved using FEM.
Then, the problem of metamaterial design for elastostatic cloaking consists of finding P such that u(x,P) = u_{0}(x) for all x ∈ Ω_{cloak}. Numerically approximated, it can be stated as: to find P such that
at a series of predefined checkpoints \({\bar{{\bf{x}}}}^{(i)}\in {{\rm{\Omega }}}_{{\rm{cloak}}}\).
To guarantee the realizability of the microstructure, the search of P must be constrained to a feasible design set \({\mathscr{D}}\). In general, this precludes the exact accomplishment of the cloaking task (3) leading to a nearelastostatic cloaking. The nearcloaking problem in elastostatic has also recently been adressed by Craster et al.^{33} through a different formulation, which is based on the regularized change of variable originally developed by Kohn et al.^{34} for electromagnetic cloaking. In the frame of the optimization based procedure proposed in this communication, let us obtain a design for which the root mean square error (RMSE) in the accomplishment of the task reaches a minimum by solving the nonlinear constrained optimization problem
where P are the design variables. As an application example, let Ω be a square plate made of nylon (See Table 1 for physical properties) under plane strain conditions, free of internal heat sources and body forces, subject to thermal and mechanical boundary conditions as shown in Fig. 1. Assuming a fixed Cartesian reference frame xyz, let z be the zerostrain direction. After making a circular hole Ω_{incl} at the center of the plate, let us design a metamaterial device occupying the region Ω_{met} (around the hole) such that the displacement field in Ω_{cloak} (outside the device) be affected to a minimum extent.
Parametrization of the microstructure
As metamaterial we choose a laminate of aluminum and polyethylene (Materials A and B in Table 1, respectively), just as depicted in Fig. 1. Although this laminate can be constructed with any pair of materials, it has already been proven that the background material should have intermediate mechanical properties regarding the laminate constituents, in order to guarantee the achievement of an optimal design for cloaking purposes^{16}. The microstructure at any finite element Ω^{(e)} ∈ Ω_{met} is completely characterized by p^{(e)} = [t^{(e)} θ^{(e)}], being t^{(e)} the relative thickness of aluminum and θ^{(e)}π the orientation of the laminate; note that the relative thickness of polyethylene is 1 − t^{(e)}. Although all these materials are isotropic, the aluminumpolyethylene laminate is markedly anisotropic. At each finite element, let us adopt a local Cartesian frame λτz, with τ normal to the laminate, λ and τ lying in the xy plane, and z being the zerostrain direction. The relevant tensorial components of the effective properties referred to the λτz frame are listed in Table 2. Analytical expressions for a bilayered laminated under plane stress conditions were obtained by Vasiliev and Morozov^{35} using micromechanical analysis. Following these authors, we derive the current analytical expressions for plane strain conditions. When referred to the local frame λτz, all the physical properties of a bilayered laminate depend on the relative thicknesses of the layers, i.e. t_{A} and t_{B} = 1 − t_{A}. The dependence on the orientation θ^{(e)} is introduced when these properties are referred to the Cartesian frame xyz. The displacement field u is determined by solving the equilibrium equation after the heat conduction equation (that gives the temperature field T). Both equations are solved using FEM. Given the symmetry of the domain and boundary conditions, only the quarter of Ω with x ≥ 0 and y ≥ 0 is modeled using an uniform mesh of 200 × 200 = 40000 bilinear finite elements.
Results
Considering the nylon plate without the hole, the FEM solution for the displacement and temperature fields, that is u_{0} and T_{0}, are that given in Fig. 2a–c; note that \({\rm{\max }}\Vert {{\bf{u}}}_{0}\Vert =11.63\,{\rm{mm}}\). For the plate with the hole, we use the mesh of 35053 square bilinear finite elements obtained by simply discarding the elements whose centers lie in Ω_{incl}. Without a cloaking device, the displacement and temperature fields in the holed plate are that shown in Fig. 2d–f, and the error in the accomplishment of the cloaking task is \({{\rm{RMSE}}}_{0}=6.18\,{\rm{mm}}=0.53\,{\rm{\max }}\Vert {{\bf{u}}}_{0}\Vert \).
For the cloaking problem, the 15086 elements of the holed mesh whose centers lei in Ω_{met} are assumed to have variable microstructure (to be determined), whereas the remaining elements (those lying outside Ω_{met}) are still made of nylon. The microparameters t^{(e)} and θ^{(e)} of the former make the design variable vector P, whereas the centers of the latter are the checkpoints for the cloaking task accomplishment. Note that P has dimension dimP = 2 × 15086 = 30172. Further, each admissible P_{i} must lie in the interval [0, 1], which is a bound constraint. Like in standard material distribution problems^{36}, the checkerboardtype instabilities are prevented using a linear density filter^{37} identically as done by Peralta et al.^{10} for the design of purely thermal metamaterials.
Finally, the nonlinear largescale constrained optimization problem given by equation (4) is solved using the method of moving asymptotes (MMA)^{38}. The socomputed variable microstructure is shown in Fig. 3, where the relative thickness of aluminum and the orientation are represented by a colormap and oriented segments, respectively. The displacement and temperature fields computed by including a cloaking device with such a microstructure, are shown in Fig. 2g–i. Quantitatively, using this optimal device, the cloaking task is achieved up to an error RMSE = 0.119 mm = 0.0192 RMSE_{0}.
Extension to multiple thermal and mechanical boundary conditions
In order to prove the versatility of the methodology proposed in this communication, let us examine the case of cloaking under multiple thermal and mechanical boundary conditions. For this purpose, let us consider a straightforward redefinition of the objective function to be minimized. Let \({{\bf{u}}}_{0}({\bf{x}},{ {\mathcal B} }^{(\alpha )})\) be the displacement under the set of thermal and mechanical boundary conditions \({ {\mathcal B} }^{(\alpha )}\) in the domain Ω without the inclusion, and \({\bf{u}}({\bf{x}},{\bf{P}},{ {\mathcal B} }^{(\alpha )})\) be the displacement under that set of boundary conditions in the presence of the inclusion Ω_{incl} together with the cloaking metamaterial occupying Ω_{met}, where the effective thermal and mechanical properties throughout the metamaterial are defined by P.
In order to account for multiple boundary conditions \({ {\mathcal B} }^{\mathrm{(1)}},{ {\mathcal B} }^{\mathrm{(2)}},\ldots ,{ {\mathcal B} }^{({N}_{{\rm{set}}})}\), the minimization problem must be defined in terms of the weighted sum of the RMSEs in the cloaking task accomplishment under each set of boundary conditions \({ {\mathcal B} }^{(\alpha )}\):
where ω_{α} is the weight assigned to the task accomplishment under the boundary conditions \({ {\mathcal B} }^{(\alpha )}\).
As an applied example, let us consider the design of a single device for the fulfillment of the cloaking task under two different sets of thermal and mechanical boundary conditions, just as depicted in the cases 1 and 2 of Fig. 4, respectively. The domain dimensions remain the same as depicted in Fig. 1. The optimizationbased procedure proposed in this study can also be conducted including the dependence of the mechanical properties with temperature. In this extended version of the thermomechanical cloaking problem, the metamaterial will be designed including the dependence of Young’s modulus on temperature. The Young’s modulus temperaturedependent functions of the background material (nylon) and the constituent materials of the laminate (aluminum and polyethylene) have been obtained from a nonlinear regression of the data reported in the references of Table 3. The remaining materials properties are kept constant, with the values reported in Table 1. However, this methodology allows the possibility of introducing the temperature dependence of any property without further complications. It should be noted that even if such properties are constant for each constituent material of the laminate, the single dependency of the Young’s modulus with temperature introduces a change in all the effective mechanical properties listed in Table 2. Without a cloaking device, the error in the accomplishment of the extended cloaking task is RMSE_{0} = 11.325 mm, with \({{\rm{RMSE}}}_{0}^{(1)}=19.23\,{\rm{mm}}=0.75\,{\rm{\max }}\Vert {{\bf{u}}}_{0}^{(1)}\Vert \) and \({{\rm{RMSE}}}_{0}^{(2)}=3.42\,{\rm{mm}}=0.29\,{\rm{\max }}\Vert {{\bf{u}}}_{0}^{(2)}\Vert \), for the boundary conditions of cases 1 and 2, respectively. The displacement and temperature fields in the nylon full plate and the nylon holed plate under the boundary conditions of Case 1, are shown in Fig. 5(a–f), respectively. The variable microstructure of the device designed to fulfill the global cloaking task is shown in Fig. 6.
The displacement and temperature fields computed by including a cloaking device with such a microstructure, are shown in Fig. 5(g–i). Using this optimal device, the cloaking task is achieved up to an error RMSE = 1.535 mm = 0.136 RMSE_{0}, with \({{\rm{RMSE}}}^{(1)}=2.52\,{\rm{mm}}=0.131\,{{\rm{RMSE}}}_{0}^{(1)}\) and \({{\rm{RMSE}}}^{(2)}=0.55\,{\rm{mm}}=0.161\,{{\rm{RMSE}}}_{0}^{(2)}\). Such results demonstrate the possibility of designing a metamaterial to accomplish a thermomechanical cloaking task under multiple boundary conditions, by means of the optimizationbased procedure proposed in this work. Also, it is worth noting that a device designed for multiple boundary conditions performs the cloaking task for each individual case poorly when compared with the device specifically designed for one of them, as done in the previous section.
In conclusion, we introduced the design of thermomechanical metamaterials, which is an unprecedented topic in the literature, using the optimizationbased approach. With the task of elastostatic cloaking under thermal loads as an example, any task measured by an objective function depending on temperature and displacements (extreme thermal expansion, for instance) can be accounted for. This method allows the fulfillment of such a task, if not exactly, optimally, using realizable metamaterials. Actually, it dictates how to fabricate the metamaterial. To avoid the need of simultaneously mimicking several thermal and mechanical properties, which is highly uncertain, is a crucial advantage of this novel approach with respect to the TO procedures (in case they are to be used for thermomechanical metamaterials). Although the realizability of the sodetermined microstructure is assured, we will evaluate strategies for the design of easiertofabricate thermomechanical devices, following our developments for thermal^{11,12} and mechanical metadevices^{16}, where piecewise homogeneous either isotropic^{12,16} or anisotropic^{11} materials were used. Thus, in future works, we plan to introduce such constraints in order to facilitate the fabrication of thermomechanical metadevices, taking into account the capabilities of polymer and metal 3D printers.
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Acknowledgements
We thank the following institutions for supporting this work: The Argentine Agency for Scientific and Technological Promotion (ANPCyT), through the project “Computational Design of Metamaterials” (PICT20162673). The National Littoral University (UNL) at Santa Fe, Argentina, through the project “Metamaterials: Computational Design, Thermal, Mechanical and Acoustic Applications, and Prototyping” (CAI+D 2016 087LI).
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V.D.F. developed the optimizationbased methodology for general mechanical problems, and J.C.A.H. extended it to thermomechanical coupled problems. J.C.A.H. and I.P. are developing applications to the design of thermomechanical metamaterials. V.D.F. is the leading researcher, J.C.A.H. and I.P. are postdoctoral and doctoral fellows respectively, both under the tutorship of V.D.F.
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Álvarez Hostos, J.C., Fachinotti, V.D. & Peralta, I. Metamaterial for elastostatic cloaking under thermal gradients. Sci Rep 9, 3614 (2019). https://doi.org/10.1038/s41598019405176
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DOI: https://doi.org/10.1038/s41598019405176
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