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Xiaoyang Zheng, Ta-Te Chen, Xiaofeng Guo, [Sadaki Samitsu](https://orcid.org/0000-0002-4139-1656), [Ikumu Watanabe](https://orcid.org/0000-0002-7693-1675)

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[Controllable inverse design of auxetic metamaterials using deep learning](https://mdr.nims.go.jp/datasets/5a9fea9b-7d0b-47d4-a30d-41f91dd5b053)

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Controllable inverse design of auxetic metamaterials using deep learningMaterials & Design 211 (2021) 110178Contents lists available at ScienceDirectMaterials & Designjournal homepage: www.elsevier .com/locate /matdesControllable inverse design of auxetic metamaterials using deep learninghttps://doi.org/10.1016/j.matdes.2021.1101780264-1275/� 2021 The Author(s). Published by Elsevier Ltd.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).⇑ Corresponding author at: Research Center for Structural Materials, NationalInstitute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, Japan.E-mail address: WATANABE.Ikumu@nims.go.jp (I. Watanabe).Xiaoyang Zheng a,b, Ta-Te Chen a,b, Xiaofeng Guo c, Sadaki Samitsu b, Ikumu Watanabe a,b,⇑aGraduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, JapanbResearch Center for Structural Materials, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, Japanc School of Materials Science and Engineering, Southwest University of Science and Technology, Mianyang 621010, Chinah i g h l i g h t s� We propose an inverse designmethod for auxetic metamaterialsusing deep learning.� We designed novel 2D auxeticmetamaterials based on Voronoitessellation for the training dataset.� The trained neural network cangenerate 2D auxetic metamaterialswith user-desired Young’s moduliand Poisson’s ratios.� The proposed method can easily beextended to the inverse design ofother architected materials.g r a p h i c a l a b s t r a c ta r t i c l e i n f oArticle history:Received 12 August 2021Revised 12 October 2021Accepted 15 October 2021Available online 16 October 2021Keywords:Negative Poisson’s ratioMetamaterialGenerative adversarial networkAdditive manufacturingVoronoi tessellationa b s t r a c tAs typical mechanical metamaterials with negative Poisson’s ratios, auxetic metamaterials exhibit coun-terintuitive auxetic behaviors that are highly dependent on their geometric arrangements. The realizationof the geometric arrangement required to achieve a negative Poisson’s ratio relies considerably on theexperience of designers and trial-and-error approaches. This report proposes an inverse design methodfor auxetic metamaterials using deep learning, in which a batch of auxetic metamaterials with a user-defined Poisson’s ratio and Young’s modulus can be generated by a conditional generative adversarialnetwork without prior knowledge. The network was trained based on supervised learning using a largenumber of geometrical patterns generated by Voronoi tessellation. The performance of the network wasdemonstrated by verifying the mechanical properties of the generated patterns using finite elementmethod simulations and uniaxial compression tests. The successful realization of user-desired propertiescan potentially accelerate the inverse design and development of mechanical metamaterials.� 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license(http://creativecommons.org/licenses/by/4.0/).1. IntroductionAuxetic metamaterials, which are mechanical metamaterialswith negative Poisson’s ratios, exhibit counterintuitive deforma-tion behavior [1–7]. Under uniaxial compressive loading, auxeticmetamaterials contract in the orthogonal directions rather thanexpanding; this behavior is in contrast to that of natural and syn-thetic materials, which have positive Poisson’s ratios. Further,under bending loading, an auxetic metamaterial plate deforms intoa convex shape, in contrast to the saddle shape usually seen forcommon materials. Compared with conventional materials, aux-etic metamaterials have higher shear resistance, fracture resis-tance, indentation resistance, impact resistance, and energyabsorption. Such distinctive behaviors make auxetic metamaterialspromising candidates for developing impact absorbers [8,9], strainsensors [10,11], and actuators [12–14], as well as for applicationsin biomedicine [15,16] and electrochemical energy storage andconversion [17,18]. In addition, recent advances in additive manu-http://crossmark.crossref.org/dialog/?doi=10.1016/j.matdes.2021.110178&domain=pdfhttp://creativecommons.org/licenses/by/4.0/https://doi.org/10.1016/j.matdes.2021.110178http://creativecommons.org/licenses/by/4.0/mailto:WATANABE.Ikumu@nims.go.jphttps://doi.org/10.1016/j.matdes.2021.110178http://www.sciencedirect.com/science/journal/02641275http://www.elsevier.com/locate/matdesX. Zheng, Ta-Te Chen, X. Guo et al. Materials & Design 211 (2021) 110178facturing technologies have opened new avenues for the designand fabrication of such complicated structures [19,20].Auxetic behavior can be quantified by the Poisson’s ratiom ¼ �et=el, where et and el are the transverse and longitudinal com-ponents of engineering strain, respectively, for a deformedmaterialunder a uniaxial load. For isotropic linear elastic materials, m mustsatisfy one of the following conditions [1]: m 2 �1; 0:5½ � for three-dimensional (3D) materials and m 2 �1;1½ � for two-dimensional(2D) materials. Negative Poisson’s ratio materials were foundedin the 1980s [22–25]; since then, considerable efforts have beendevoted designing, modeling, and analyzing auxetic metamaterials[26–29]. Despite such intensive efforts to design new auxeticmetamaterials, it remains challenging to design flexible auxeticmetamaterials with extreme properties, such as materials thatcan maintain negative Poisson’s ratios consistently during largedeformations [21,39,40,32]. For example, some delicately designedstructures have positive Poisson’s ratios under small deformationsand only exhibit negative Poisson’s ratios during further compres-sion [39,40,32]. This is because of the limitations of conventionaldesign methods. Forward design is the mainstream method fordesigning auxetic metamaterials and includes approaches such asbioinspired methods [30,31], mathematical control [32–35], topol-ogy optimization [36–38], and Boolean and lofting operations ofsimple geometries [39–43]. The forward design approach followsa general process: first, a structure is created, and its mechanicalproperties are then investigated by finite element method (FEM)simulations or mechanical testing (Fig. 1). The mechanical proper-ties of the designed materials are known only after time-consuming simulations or experiments. In addition, because theproperties of auxetic metamaterials are determined by the geome-try and assembly of periodic unit cells, the traditional methods arehighly dependent on the prior knowledge of experienced design-ers, resulting in a limited number of design spaces. Further, struc-tural optimization techniques can be employed to achieve extremeproperties such as negative Poisson’s ratios even at finite strain[27,38]; however, nonlinear optimization remains challengingfrom the perspectives of robustness and efficiency.Recent advances in deep learning have facilitated the inversedesign of new materials using various artificial neural networks[44–53]. However, to the best of the authors’ knowledge, deeplearning has not yet been successfully harnessed to create novelauxetic metamaterials and rather has been used only to predictthe mechanical behavior of specific auxetic configurations [48].One important reason for this situation is that the properties ofan auxetic metamaterial are almost completely determined bythe geometry and assembly of periodic unit cells. The realizationof a negative Poisson’s ratio requires a delicate arrangement andFig. 1. Comparison of the conventiona2design of the unit cells; therefore, it is very difficult to build a largedataset consisting of thousands of geometries of auxetic metama-terials and their corresponding properties. For example, successhas been achieved in previous studies only in the property predic-tion and pattern design of simple square-shaped cellular materials[44] or in the realization of isotropic elastic stiffness based on ran-domly generated architectures [47].In this study, we devised a deep learning framework that cansuccessfully generate auxetic metamaterials with assigned Young’smoduli and Poisson’s ratios. First, we created a large dataset com-posed of tens of thousands of geometric patterns and their corre-sponding mechanical properties. The patterns were derived fromVoronoi tessellation, and their properties were calculated using ahomogenization algorithm. The dataset was then used to train aconditional generative adversarial network (CGAN). Unlike a com-mon generative adversarial network (GAN), which consists of agenerator and discriminator, the CGAN uses a linear regressionmodule (solver) to predict the properties of patterns from the gen-erator. The discriminator was trained to push the generator to pro-duce realistic patterns, and the solver was trained to push thegenerator to yield patterns with user-defined properties. Afterthe CGAN had been well trained, it could rapidly generate new pat-terns with user-defined properties and could generate auxeticmetamaterials with constantly negative Poisson’s ratios duringlarge deformations. Finally, the auxetic behaviors of the generatedmetamaterials were verified by FEM simulations and uniaxial com-pression tests.2. Methods2.1. Voronoi tessellation algorithmAs illustrated in Fig. 2, two-dimensional (2D) topology patternswere created using Voronoi tessellation, which is a robust methodthat is capable of creating various porous materials [54–59]. Toensure the periodicity of these 2D patterns, periodic boundary con-ditions were applied in the Voronoi tessellation. Briefly, a seed con-sisting of 64 coordinate points was initially created according toMitchell’s best candidate algorithm [60]. A 2D Voronoi diagramwas created based on the seed. To mimic the nature of actual aux-etic foams, which have both convex and concave cells [61], the 2DVoronoi diagram was modified by merging two adjacent polygons.Finally, a new pattern was formed after smoothing the edges of thepolygons using Chaikin’s algorithm because smooth surfaces havemore homogeneous stress responses than sharp surfaces [62,63].Note that the relative density of a pattern can be tuned easily byl design method and our method.Fig. 2. Process of structure generation using Voronoi tessellation.X. Zheng, Ta-Te Chen, X. Guo et al. Materials & Design 211 (2021) 110178changing the width of the edges. Here, the width was fixed for sim-plicity. The relative density of the patterns was approximately0.154. By repeating this process, an infinite number of differentpatterns can be created to facilitate big-data-driven materialdesign.2.2. Homogenization algorithmThe elastic moduli (Young’s modulus and Poisson’s ratio) werecalculated according to the theory of homogenization, which hasbeen used extensively to probe the equivalent linear elasticity ofperiodic composites [64–66]. According to the theory of homoge-nization, the effective elasticity tensor eCijkl of a periodic patterncan be computed as:eCijkl ¼ 1Vj jZVCpqrs e0 ijð Þpq � e ijð Þpq� �e0 klð Þrs � e klð Þrs� �dV ð1Þwhere Vj j is the area of a square domain, Cpqrs is the locally varyingstiffness tensor, e0 ijð Þpq represents the prescribed macroscopic strainfields (three strain fields in the case of two dimensions: horizontal,vertical, and shear strains), and e ijð Þpq represents the locally varyingstrain fields and is defined as:e ijð Þpq ¼ epq vij� � ¼ 12vijp;q þ vijq;p� �ð2ÞThe locally varying strain fields are based on the displacement fieldsvij, which can be determined using a prescribed macroscopic strain:ZVCijpqeij vð Þeij vkl� �dV ¼ZVCijpqeij vð Þe0 klð Þpq dV ;8v 2 V ð3Þwhere v denotes the virtual displacement field. The numericalhomogenization procedure is discussed in more detail in the litera-ture [64–66]. After obtaining the effective elasticity tensor eCijkl, theeffective elastic moduli (i.e., Young’s modulus, Poisson’s ratio, shearmodulus, and bulk modulus) can be calculated.In this work, each pattern was first converted into a 256� 256element matrix consisting of 0 and 1, where 0 represents voidregions and 1 represents solid regions. Subsequently, trial strainfields were applied to the element matrix to determine the reac-tion forces and stored elastic energy. Then, a homogenized elastic-ity tensor was obtained after the homogenization calculation.Finally, the effective elastic moduli of the patterns were calculatedaccording to the elasticity tensor. The material model used in thehomogenization was a linear elastic material with a Young’s mod-ulus of 0.6615 MPa and a Poisson’s ratio of 0.49; these values werechosen to fit the equivalent elastic moduli of an incompressibleneo-Hookean solid under a small deformation.2.3. Dataset preparationAfter topology creation and elastic modulus calculation, a largedataset was obtained, in which each datapoint consisted of a pat-3tern and its corresponding labels (Poisson’s ratio and Young’s mod-ulus). The training dataset was composed of 100,000 datapoints.Fig. 3(a) shows the 100,000 randomly created geometric patternsin the property space with the axes representing the Poisson’s ratio(m) and Young’s modulus (E). The randomly created patterns scat-tered in the material property space form a nearly triangular shape,with 2.1 kPa < E < 13.7 kPa and �0:28 < m < 0:38. We refer to thetriangular region as the available E–m space, in which randomlabels were sampled to train the neural network. Fig. 3(b) showsthe distributions of the Poisson’s ratios of these randomly createdpatterns. The distributions indicate that less than 3% of the ran-domly created patterns have negative Poisson’s ratios.2.4. Generative deep learning modelWe devised a CGAN to train the created dataset. A GAN, whichconsists of two models (a generator and discriminator), is a type ofdeep learning network for data generation [67–69]. The generatorand discriminator are trained simultaneously by an adversarialprocess, in which the generator learns to produce data with char-acteristics similar to those of the training data, whereas the dis-criminator learns to distinguish between real data and thegenerated data. A CGAN is a type of GAN in which conditional gen-eration is realized by taking advantage of labels during the trainingprocess. However, with regard to precise data generation, the con-ventional CGAN can hardly provide good guidance for training thegenerator because the discriminator always suffers from overfit-ting [70]. We addressed this problem by employing an indepen-dent module (solver). The solver is a linear regression networkthat acts as a linear elasticity solver to predict the Young’s modulusand Poisson’s ratio of a given pattern (obtained from the dataset orgenerated by the CGAN). Fig. 4(a) illustrates the architecture of theproposed CGAN. A three-player game was conducted in which thegenerator deceived the discriminator in terms of geometry andsimultaneously deceived the solver in terms of elastic moduli. Afterthe CGAN had been well trained using 100,000 datapoints, it couldgenerate a batch of patterns for a given label (Young’s modulus andPoisson’s ratio). More details on the CGAN are discussed in Appen-dix A.2.5. Uniaxial compression testsThe auxetic behaviors of the generated metamaterials were firstinvestigated using a set of 3D-printed samples in uniaxial com-pression tests. Each sample consisted of 3 � 3 unit cells and hadoverall dimensions of 120 mm � 120 mm � 15 mm. The unit cellnumber and size are sufficient to represent a periodic porous mate-rial [47]. To ensure that the specimens underwent large deforma-tion without cracks, they were fabricated using an elasticphotopolymer resin (Elastic 50A resin, Formlabs, USA) by employ-ing a 3D printer (Form 3, Formlabs, USA). A subtle surface finishwas achieved without the use of support structures. The printingparameters were as follows: a layer thickness of 0.05 mm and anFig. 3. Dataset for neural network training. (a) Young’s moduli and Poisson’s ratios of 100,000 randomly created patterns from Voronoi tessellation. (b) Distributions ofPoisson’s ratios for randomly created patterns and for CGAN-outputted patterns with an input condition of m ¼ �0:28.Fig. 4. Inverse design using CGAN. (a) CGAN architecture. (b) CGAN performance for different training datapoints. (c) Comparison between user-input and CGAN-output realelastic moduli.X. Zheng, Ta-Te Chen, X. Guo et al. Materials & Design 211 (2021) 110178operation temperature of 33 �C. All the specimens were fully curedat 60 �C for 20 min after washing with isopropanol. A typical 3D-printed sample is shown in Fig. 5(a).The mechanical properties of the 3D-printed specimens wereinvestigated by performing static compression tests using a motor-ized test stand (EMX-500 N, IMADA, Japan). A constant displace-ment rate of 10 mm/min was set during the tests, in which thesamples were uniaxially compressed between two plates. Thedeformation process was captured using a high-speed camera,and the stress–strain curves were plotted using the recordedload–displacement data. The Hencky (logarithmic) strain compo-nents of the longitudinal (compression) and transverse directionswere calculated usingFig. 5. Example of an evaluated object: (a) The 3D printed-sample and (b) a finiteelement meshing of an RVE.4�i ¼ ln 1þ �uiLi� �¼ ln 1þ ei½ � for i 2 l; tf g; ð4Þwhere �ui is the average boundary displacement between the topand bottom or the left and right of the red-marked interior unit cell,and Li is the initial length of the interior unit cell. In this study,Ll ¼ Lt = 40 mm. The average boundary displacements of the interiorunit cell were measured by post-processing the recorded movies.Using the Hencky strain, Poisson’s ratio at finite strain was definedas mlt ¼ ��t=�l.The Young’s modulus El for the longitudinal (compression)direction was calculated by linear fitting of the initial linear por-tions of the stress–strain curves. Least-squares approach was usedto find the suitable value of El, in which the optimization problemwas defined asminXNk¼1F kð ÞA� e kð ÞEl !224 35; ð5Þwhere N is number of data, F is the applied load, and A is the initialcontact area between the sample and the measuring instrument. Inthis study, the initial contact area was calculated as A ¼ 1;800 mm2(120 mm � 15 mm).In this study, compression tests were carried out on the struc-ture in two directions, namely, x and y, as shown in Fig. 5(b); thatis, mxy; myx; Ex, and Ey were evaluated.X. Zheng, Ta-Te Chen, X. Guo et al. Materials & Design 211 (2021) 1101782.6. Finite element method simulationsA deformation problem with a periodic microstructure, i.e., arepresentative volume element (RVE), was solved using a nonlin-ear FEM to simulate the deformation state at finite strain[31,71,72]. The displacement field w in an RVE is divided into theuniform part �u and the periodic part ~u : w ¼ �uþ ~u. The uniformdisplacement �u is described by the macroscopic displacement gra-dient �H as �u ¼ �HY , where Y is the coordination system in the RVE.The boundary value problem of an RVE is formulated as a self-equilibrium problem for a periodic displacement field ~u:ZXYP : rY ~gdXY ¼ 0 8~g 2 Wperiodic; ð6Þwhere P is first Piola–Kirchhoff stress, ~g is the variation of the peri-odic displacement ~u;XY is the volume of the overall RVE, andWperiodic is the real solution space of the periodic function.Based on the periodicity of the displacement field, the differ-ence in displacements at two points A and B, which satisfy the peri-odicity on the corresponding surfaces of the RVE, is derived aswA �wB ¼ �H YA � YBð Þ. The above node-based boundary condi-tions are set on the finite element model of an RVE, and the dis-placement field w is then controlled by the macroscopicdisplacement gradient �H. Following the definition of an RVE, amacroscopic variable can be calculated as the volume average ofthe corresponding microscopic variable. Thus, the macroscopicstress �P can be evaluated as�P :¼ 1XYZXYPdXY : ð7ÞBased on the above equations, the mechanical properties of the pro-posed patterns were further validated using an FEM simulationplatform (COMSOL Multiphysics Ver. 5.4, COMSOL, Sweden). A 2Dplane strain model was utilized under periodic boundary condi-tions. To mimic the properties of the 3D-printed material, the mate-rial model in the simulations was defined as an incompressible neo-Hookean model with a Young’s modulus of 0.6615 MPa that was fit-ted from the compression tests. All the model geometries weremeshed using approximately 2:5� 105 second-order triangularsolid elements. An example of meshing is shown in Fig. 5(b). A con-tact condition based on an augmented Lagrangian method was setin the finite element model. For the large deformation, a parametricsweep of the longitudinal displacement was used with a stop con-dition of el ¼ 0:2. The Poisson’s ratios and Young’s moduli in thesimulation results were calculated using the same method thatwas employed to obtain the experimental results.3. Results and discussion3.1. Conditional generative adversarial networkWe tested the performance of our CGAN model during eachepoch by generating 1024 patterns with labels that were randomlysampled from the available E–m space. The performance was eval-uated in terms of the mean squared error (MSE) of the sum of E andm:MSE ¼ 1NXNi¼1E ið Þ � bE ið Þ� �2þ m ið Þ � m̂ ið Þ� �2�  ; ð8Þwhere N ¼ 1024 is the number of labels sampled from the availableE–m space, E is the normalized input Young’s modulus, bE is the nor-malized output Young’s modulus, m is the normalized input Pois-son’s ratio, and m̂ is the normalized output Poisson’s ratio. Tofacilitate deep learning, both the Young’s modulus and Poisson’s5ratio were normalized to the range from 0 to 1 based on the maxi-mum and minimum values of the E–m space shown in Fig. 3(a). Asmaller MSE indicates better performance. Fig. 4(b) shows thechange in the MSE during the training epoch. Two convergencestages are observed in the MSE curve. The first one is before epoch5, where the solver learns very rapidly to generate patterns similarto the real patterns. The next stage is 5 < epoch< 50, where the sol-ver learns relatively slowly to generate patterns with the corre-sponding input elastic moduli. Initially, the MSE decreasesrapidly; then, it decreases gradually, finally reaching a minimumof approximately 0.014 after epoch 60. The low MSE indicates thatour CGAN is capable of generating patterns with a user-definedYoung’s modulus and Poisson’s ratio. In addition, the line plots forthe loss (see Appendix A for the definition) of the generator and dis-criminator are shown in Fig. 4(b). The plots represent typical loss–epoch graphs of a stable GAN training process: the losses of the gen-erator and discriminator begin erratically and gradually converge toa stable equilibrium after epoch 20. This finding further demon-strates the stability of our CGAN.3.2. Inverse design of auxetic metamaterialsAfter the CGAN had been well trained using appropriate param-eters, it could invert the design of auxetic metamaterials: a label(Young’s modulus and Poisson’s ratio) was input, and the CGANgenerated a batch of geometrical patterns with the correspondingYoung’s modulus and Poisson’s ratio. The performance of thetrained CGAN was evaluated by comparing each input value andits output values (i.e., the Young’s modulus and Poisson’s ratio ofthe generated pattern). Fig. 4(c) compares 1024 samples with inputlabels sampled from the available E–m space. The coordinates ofeach point correspond to the input Young’s modulus or Poisson’sratio (X coordinate) and the output Young’s modulus or Poisson’sratio (Y coordinate). A position closer to the bisection line(Y ¼ X) represents better performance of the CGAN. The narrowbandwidth of the scatter distributions indicates the good perfor-mance of the trained CGAN, demonstrating that the CGAN can gen-erate a batch of geometries with user-desired Young’s moduli andPoisson’s ratios. These results also show that the CGAN can effec-tively perform extrapolation from the training data to provide acontrollable inverse design, in contrast to the random generationobtained by Voronoi tessellation.We further demonstrated that the trained CGAN facilitates theinverse design of auxetic metamaterials with very low negativePoisson’s ratios. Considering that the lower boundary of the Pois-son’s ratio in the available E–m space had m ¼ �0:28 and E ¼ 3kPa, we input the label with these values and the CGAN generateda batch of auxetic metamaterials (Fig. 6). In this figure, the inputand output values are also compared below each pattern and showgood agreement. Fig. 3(b) compares the distribution of Poisson’sratios between the CGAN-generated patterns and randomly cre-ated patterns. The comparison shows that it is easy to generateauxetic metamaterials with very low negative Poisson’s ratiosusing the CGAN, in contrast to the random generation method.Althugh among the structures randomly generated with the Voro-noi tessellation algorithm in Section 2.3, only 3% had a negativePoisson’s ratio, the proposed approach can almost certainly gener-ate the intended disordered structures having negative Poisson’sratios. More importantly, this inverse design method does notrequire delicate arrangement of the shapes, distributions, and com-binations of geometrical elements. This method is independent ofprior knowledge about the design of auxetic metamaterials. Morepatterns generated by the CGAN with different elastic moduli areshown in Figs. A3 and A4 in Appendix A.Fig. 6. Auxetic metamaterials generated using the CGAN with input labels of E ¼ 3 kPa and m ¼ �0:28.X. Zheng, Ta-Te Chen, X. Guo et al. Materials & Design 211 (2021) 1101783.3. Investigation of auxetic behaviorTo probe the auxetic behavior of the generated metamaterialsunder large deformation, we conducted a systematic analysis byperforming uniaxial compression tests and FEM simulations.Fig. 7(a) shows a sequence of progressively deformed shapes ofthe generated auxetic metamaterials under four different levelsof compressive engineering strain. The experimental and simula-tion results show a consistent deformation tendency wherein themetamaterial gradually contracts when compressed uniaxiallyalong with the shrinkage of its interior holes. The overall shrinkagephenomenon proves that the metamaterial is an auxetic metama-terial with a negative Poisson’s ratio. The progressively deformedshapes of other patterns with Poisson’s ratios ranging from �0:2to 0.3 are shown in Fig. A3; the patterns with a positive Poisson’sratio expand laterally when compressed uniaxially.Fig. 7(b) presents a quantitative evaluation of Poisson’s ratiosunder different compression strains. The transverse strains weredetermined from the average transverse strains of the interior unitcell to reduce the influence of the boundary conditions. The figureshows that the calculated Poisson’s ratios monotonically decreasewith increasing strain (e 6 0:2). A more detailed Poisson’s ratio–strain curve obtained from the simulation results was also com-Fig. 7. Auxetic behavior of structures. (a) Progressively deformed configurations of FEMstrain curves and (c) stress–strain curves from FEM simulations and uniaxial compressi6pared with the curve obtained from the experimental results. Ini-tially, the Poisson’s ratio decreases; then, it gradually increasesduring compression because the empty space is insufficient, whichcauses the ligaments to bend when they are in contact with eachother. Overall, the experimental and simulation results demon-strate that the negative Poisson’s ratio can maintain a wide rangeof compressive strain (e ¼ 0:2). Notably, the training dataset wascalculated by the homogenization algorithm at small strain. In thiscase, a negative Poisson’s ratio can be attained in the small strainregion, as expected; however, it does not necessarily maintain thisratio during a large deformation. It typically changes during defor-mation, as shown in Fig. 7(b). The proposed approach can generatea suitable structure if the training dataset is produced in consider-ation of finite strain and the objective is set as maintaining a neg-ative Poisson’s ration during a large deformation.We also analyzed the stresses of the auxetic metamaterials dur-ing deformation. As shown in Fig. 7(c), the stress–strain curvesexhibit good linearity. The lack of transformation from the linearelastic region to the plateau region indicates that the auxeticbehavior of the designed metamaterials is a result not of buckling,but rather of ligament bending. It is noteworthy that the designedauxetic metamaterials are considerably different from typicalbuckling-induced auxetic metamaterials, whose stress–strainmodel and 3D-printed sample under uniaxial compressive load. (b) Poisson’s ratio–on tests.X. Zheng, Ta-Te Chen, X. Guo et al. Materials & Design 211 (2021) 110178curves have an additional plateau region between the linear elasticand densification regions [31,41–43].4. ConclusionsWe developed a controllable inverse design method for auxeticmetamaterials using deep learning. The proposed deep learningmodel, CGAN, was trained using 100,000 randomly created pat-terns generated by Voronoi tessellation. The trained CGAN canfacilitate the mass generation of 2D auxetic metamaterials withuser-desired elastic moduli. The proposed method can be easilyextended to the inverse design of 3D auxetic metamaterials incombination with Voronoi tessellation. In addition, this studyopens new avenues to harness deep learning in the realization ofuser-desired properties for applications in which specific materialproperties are required (e.g., actuator fabrication, sensor manufac-turing, and catalysis). Finally, although this work was focused onlyon 2D auxetic metamaterials with random structures, it is highlyrecommended that inverse design of 3D auxetic metamaterialsbe developed in future work, paving the way for further applica-tions in which complicated 3D geometries are preferred.FundingThis work was supported by JSPS KAKENHI [Grant Nos.21H02006].Declaration of Competing InterestThe authors declare that they have no known competing finan-cial interests or personal relationships that could have appearedto influence the work reported in this paper.AcknowledgmentsThe authors would like to acknowledge the assistance providedby Dr. Sachiko Taniguchi in the experiments.Appendix A. Details of conditional generative adversarialnetworkThe CGAN used in this study was composed of three neural net-work structures: a generator, discriminator, and solver. The gener-ator was trained to produce patterns of auxetic metamaterialsfrom latent variables (multivariate normal distribution) and user-defined labels (Young’s modulus and Poisson’s ratio) and simulta-neously aimed to deceive the discriminator and solver. The dis-criminator was trained to distinguish between the patternsproduced by the generator and those from the real dataset. The sol-ver was trained to predict the Young’s modulus and Poisson’s ratioof a given pattern. The CGAN was optimized by a minimax gameusing the following equations:Table A.1Network architecture of generator.Description Kernel sConcatenate (Z, L) -Fully connected + Batch normalization + Reshape -2D transposed convolution + Batch normalization + Leaky ReLU 4 � 42D transposed convolution + Batch normalization + Leaky ReLU 4 � 42D transposed convolution + Batch normalization + Leaky ReLU 4 � 42D transposed convolution + Batch normalization + Leaky ReLU 4 � 42D transposed convolution + Batch normalization + Leaky ReLU 4 � 42D transposed convolution 4 � 4Tanh -7ĥD ¼ argminhDLD tD;D X; hDð Þð Þ þ LD uD;D G Z; L; hGð Þ; hDð Þð Þf g ðA:1ÞĥG ¼ argmaxhGLD uD;D G Z; L; hGð Þ; hDð Þð Þ � aLS L; S G Z; L; hGð Þ; hSð Þð Þf gðA:2ÞĥS ¼ argmaxhSLS L; S X; hSð Þð Þf g ðA:3Þwhere hD; hG, and hS are the sets of parameters of the discriminator,generator, and solver, respectively. D;G, and S denote the discrimi-nator, generator, and solver, respectively. X 2 Rn�p is the trainingdataset (vectors of auxetic metamaterials), L 2 Rn�l are the labelsof the dataset X (i.e., Young’s moduli and Poisson’s ratios), andZ 2 Rn�l are the latent variables from a multivariate normal distri-bution during each iteration. LD is a loss function (binary cross-entropy function) for the discriminator. tD and uD are target labelsand are generally set to 1 and 0, respectively. However, we appliedthe label smoothing technique for the target labels [73]: tD wasreplaced by a random number between 0.7 and 1.2, and uD wasreplaced by a random number between 0 and 0.3. The moderatingweights, a, determined the extent to which the generator focusedon the training of the input labels and was set to be 0.1 in our study.Deep learning calculations were performed using TensorFlow[74]. An Adam optimizer with a learning rate of 0.0001 and b1 of0.5 was used to train the model. The batch size for the trainingwas set to 32. The detailed network structures used in this studyare listed in Tables A.1,A.2,A.3. In short, the layers used includeda 2D convolutional layer, a 2D transposed convolutional layer, 2Dmax pooling, a fully connected layer, batch normalization, anddropout, and the activation functions used included Leaky ReLUand tanh. Note that circular padding was used in the 2D convolu-tional layer and 2D transposed convolutional layer to maintainand identify the periodicity of the patterns. Examples of down-and up-samplings with circular and zero paddings are shown inFig. A1. This figure demonstrates that compared to the commonlyused zero padding, circular padding can more effectively help theoutput tensor retain its periodicity.Because the solver is independent of the generator and discrim-inator, we first trained the solver with supervised learning. Theprocess took approximately 23 h to train 200 epochs with100,000 datapoints on a single NVIDIA RTX A6000 graphic card.The solver is a type of linear regression model that can be usedto predict the Young’s modulus and Poisson’s ratio (elastic moduli)of a given pattern. Fig. A2 shows the MSE between the predictedand reference elastic moduli for different numbers of datasets. Asmaller MSE value represents a better performance of the trainedsolver. Each dataset was split as follows: 80% was used as the train-ing set and 20% was used as the testing set. To prevent the solverfrom overfitting, early stopping was performed. As shown inFig. A2(a), the MSE reaches a very low value of 0.003 after 20epochs when using 100,000 datapoints, and it becomes difficultize Resampling Input shape Output shape- 128 + 2 130- 130 4 � 4 � 512Up 4 � 4 � 512 8 � 8 � 256Up 8 � 8 � 256 16 � 16 � 128Up 16 � 16 � 128 32� 32� 64Up 32 � 32 � 64 64� 64� 32Up 64 � 64 � 32 128� 128� 16Up 128 � 128 �16 256� 256� 1- 256 � 256 � 1 256 � 256 � 1Table A.2Network architecture of discriminator.Description Kernel size Resampling Input shape Output shape2D convolution + Leaky ReLU + Dropout 4� 4 Down 256� 256� 1 128� 128� 162D convolution + Leaky ReLU + Dropout 4� 4 Down 128� 128� 16 64� 64� 322D convolution + Leaky ReLU + Dropout 4� 4 Down 64� 64� 32 32� 32� 642D convolution + Leaky ReLU + Dropout 4� 4 Down 32� 32� 64 16� 16� 1282D convolution + Leaky ReLU + Dropout 4� 4 Down 16� 16� 128 8� 8� 2562D convolution + Leaky ReLU + Dropout 4� 4 Down 8� 8� 256 4� 4� 512Flatten - - 4� 4� 512 8192Fully connected - - 8192 1Tanh - - 256� 256� 1 256� 256� 1Table A.3Network architecture of solver.Description Kernel size/pool size Resampling Input shape Output shapeUnit 1 2D convolution 3 � 3 - 256 � 256 � 1 256 � 256 � 162D convolution 3 � 3 - 256 � 256 � 16 256 � 256 � 162D max pooling 2 � 2 Down 256 � 256 � 1 128 � 128 � 16Unit 2 2D convolution 3 � 3 - 128 � 128 � 16 128 � 128 � 322D convolution 3 � 3 - 128 � 128 � 32 128 � 128 � 322D max pooling 2 � 2 Down 128 � 128 � 32 64 � 64 � 32Unit 3 2D convolution 3 � 3 - 64 � 64 � 32 64 � 64 � 642D convolution 3 � 3 - 64 � 64 � 64 64 � 64 � 642D max pooling 2 � 2 Down 64 � 64 � 64 32 � 32 � 64Unit 4 2D convolution 3 � 3 - 32 � 32 � 64 32 � 32 � 1282D convolution 3 � 3 - 32 � 32 � 128 32 � 32 � 1282D max pooling 2 � 2 Down 32 � 32 � 128 16 � 16 � 128Unit 5 2D convolution 3 � 3 - 16 � 16 � 128 16 � 16 � 2562D convolution 3 � 3 - 16 � 16 � 256 16 � 16 � 2562D max pooling 2 � 2 Down 16 � 16 � 256 8 � 8 � 256Unit 6 2D convolution 3 � 3 - 8 � 8 � 256 8 � 8 � 3842D convolution 3 � 3 - 8 � 8 � 384 8 � 8 � 3842D max pooling 2 � 2 Down 8 � 8 � 384 4 � 4 � 384Unit 7 2D convolution 3 � 3 - 4 � 4 � 384 4 � 4 � 5122D convolution 3 � 3 - 4 � 4 � 512 4 � 4 � 5122D max pooling 2 � 2 Down 4 � 4 � 512 2 � 2 � 512Unit 8 2D convolution 3 � 3 - 2 � 2 � 512 2 � 2 � 5122D convolution 3 � 3 - 2 � 2 � 512 2 � 2 � 5122D max pooling 2 � 2 Down 2 � 2 � 512 1 � 1 � 512Flatten + Fully connected - - 1 � 1 � 512 256Fully connected 256 128Fully connected - - 128 2Fig. A1. Examples of the use of circular padding and zero padding. Compared to the patterns generated using zero padding, those produced using circular padding remainmore periodic.X. Zheng, Ta-Te Chen, X. Guo et al. Materials & Design 211 (2021) 1101788Fig. A2. (a) Performance of the solver evaluated using the MSE between the predicted and reference elastic moduli for different numbers of data points. (b) Comparisonbetween the solver-predicted and reference elastic moduli. The latter are the elastic moduli of tested patterns calculated by the homogenization algorithm.X. Zheng, Ta-Te Chen, X. Guo et al. Materials & Design 211 (2021) 110178to decrease the MSE further even when using a larger dataset (e.g.,200,000 datapoints). To be conservative, we chose 100,000 data-points for the training process. Fig. A2(b) presents the predictedYoung’s moduli and Poisson’s ratios of the tested patterns, in whichthe values predicted by the solver are close to the reference values,thus demonstrating the good performance of the solver. The linearrelationship of the solver-predicted results is better than that ofthe CGAN-generated results. After the solver had been well trained,the checkpoints of the solver were saved and the solver was usedto train the generator and discriminator, in which the saved check-points were used to calculate the loss of the patterns produced bythe generator. The process spent approximately 35 h to train 200epochs on a single NVIDIA RTX A6000 graphic card.Appendix B. Patterns generated by CGANThe well-trained CGAN is capable of generating numerous pat-terns with a user-defined Young’s modulus and Poisson’s ratio.Fig. A3. Patterns generated by the CGAN with di9Fig. A3 shows some patterns produced by the CGAN with differentYoung’s moduli and Poisson’s ratios. The results show that the val-ues of the input elastic moduli are very close to those of the outputelastic moduli, which further demonstrates the good performanceof the CGAN. Note that the differences between the elastic modulicalculated by the homogenization algorithm and the FEM simula-tion are caused by the algorithms themselves. The close agreementamong the results of the CGAN predictions, FEM simulations, andexperiments confirms that the proposed CGAN-based techniqueis a robust method for the inverse design of auxetic metamaterials.We also investigated the deformation behaviors of differentCGAN-generated patterns under uniaxial compression (Fig. A4).The good curve fitting of the stress–strain curves and Poisson’sratio curves along the x- and y-axes demonstrates the isotropicproperties of this type of pattern. Furthermore, some patterns thatinitially have positive Poisson’s ratios exhibit auxetic behavior dur-ing further compression, which is caused by the shrinking of inte-rior concave pores.fferent Young’s moduli and Poisson’s ratios.Fig. A4. Deformation behavior of patterns with different elastic moduli. (a) Progressively deformed configurations of FEM model under uniaxial compressive loading. (b)Poisson’s ratio–strain curves and (c) stress–strain curves from FEM simulations.X. Zheng, Ta-Te Chen, X. Guo et al. Materials & Design 211 (2021) 110178References[1] G.N. Greaves, A. Greer, R.S. Lakes, T. Rouxel, Poisson’s ratio and modernmaterials, Nat. Mater. 10 (11) (2011) 823–837.[2] R.S. Lakes, Negative-Poisson’s-ratio materials: Auxetic solids, Annu. Rev.Mater. Res. 47 (2017) 63–81.[3] X. Yu, J. Zhou, H. Liang, Z. Jiang, L. Wu, Mechanical metamaterials associatedwith stiffness, rigidity and compressibility: A brief review, Prog. Mater Sci. 94(2018) 114–173.[4] Z. Wang, C. Luan, G. Liao, J. 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Controllable inverse design of auxetic metamaterials using deep learning 1 Introduction 2 Methods 2.1 Voronoi tessellation algorithm 2.2 Homogenization algorithm 2.3 Dataset preparation 2.4 Generative deep learning model 2.5 Uniaxial compression tests 2.6 Finite element method simulations 3 Results and discussion 3.1 Conditional generative adversarial network 3.2 Inverse design of auxetic metamaterials 3.3 Investigation of auxetic behavior 4 Conclusions Funding Declaration of Competing Interest Acknowledgments Appendix A Details of conditional generative adversarial network Appendix B Patterns generated by CGAN References