Machine-Learning-Powered, Rapid, Accurate, and Multi-Target Mechanical Metamaterials Inverse Design.

Mechanical and phononic metamaterials exhibiting negative elastic moduli, gapped vibrational spectra, or topologically protected modes enable precise control of structural and acoustic functionalities. While much progress has been made in their experimental and theoretical characterization, the inverse design of mechanical metamaterials with arbitrarily programmable spectral properties and mode localization remains an unsolved problem. Here, we present a flexible computational inverse-design framework that allows the efficient tuning of one or more gaps at nearly arbitrary positions in the spectrum of discrete phononic metamaterial structures. The underlying algorithm optimizes the linear response of elastic networks directly, is applicable to ordered and disordered structures, scales efficiently in 2D and 3D, and can be combined with a wide range of numerical optimization schemes. We illustrate the broad practical potential of this approach by designing mechanical bandgap switches that open and close pre-programmed spectral gaps in response to an externally applied stimulus such as shear or compression. We further show that the designed structures can host topologically protected edge modes, and validate the numerical predictions through explicit 3D finite element simulations of continuum elastica with experimentally relevant material parameters. Generally, this network-based inverse design paradigm offers a direct pathway towards manufacturing phononic metamaterials, DNA origami structures and topolectric circuits that can realize a wide range of static and dynamic target functionalities.

Controlling stress and deformation induced by thermo-mechanical stimulation in high-precision mechanical systems can be achieved by mechanical metamaterials (MM) exhibiting negative thermal expansion (NTE) and negative Poisson's ratio (NPR). However, the inverse design of MM exhibiting a wide range of arbitrary target NTEs and NPRs is a challenging task due to the low design flexibility of analytical methods and parametric studies based on numerical simulation. In this study, we propose Bézier curve-based programmable chiral mechanical metamaterials (BPCMs) and a deep autoencoder-based inverse design model (DAIM) for the inverse design of BPCMs. Through iterative transfer learning with data augmentation, DAIM can generate BPCMs with a curved rib shape inaccessible with the Bézier curve, which improves the inverse design performance of the DAIM in the data sparse domain. This approach decreases the mean absolute error of NTE and NPR between the inverse design target and the numerical simulation results of inverse designed BPCMs on the data-sparse domain by 79.25% and 83.33% on average, respectively. A 3D-printed BPCM is validated experimentally and exhibits good coincidence with the target NTE and NPR. Our proposed BPCM and the corresponding inverse design framework enable the inverse design of BPCMs with NTE in the range of -1100 to 0 ppm K-1 and NPR in the range of -0.6 to -0.1. Furthermore, programmable thermal deformation modes with a fixed Poisson's ratio are realized by combining various inverse designed BPCM unit cells. BPCMs and the DAIM for their inverse design are expected to improve the mechanical robustness of high-precision mechanical systems through tunable modulation of thermo-mechanical stimulation.

Mechanical metamaterials - engineered materials whose behavior is defined by their internal structure rather than their base material - have received great at- tention for the unusual properties they can exhibit. However, while the forward problem of predicting the mechanical behavior of a given metamaterial can be reliably done through simulation or experiments, the inverse problem - designing metamaterials with specific properties - remains difficult, especially for heteroge- neous metamaterials composed of multiple unit cells. This dissertation explores the use of generative machine learning to address this inverse design problem. The first part of the work investigates how inverse and generative models can be conditioned to produce unit cells with prescribed mechanical properties and how they can be used to generate multiple valid solutions for the same target. This is examined using a curved-beam metamaterial, which allows a compact geometric representation without complex design restrictions. The effect these restrictions have on the generative models is investigated in the second part. For this, a number of generative models—Variational Autoencoders, Generative Adversarial Networks, and Diffusion Models—are evaluated on a kirigami metamaterial where dependencies between the cuts cause intricate design constraints. The final part covers the challenges that arise when generative models are applied to heteroge- neous metamaterials, in which multiple unit cells are combined to create spatially varying mechanical properties. The results show that the applicability of generative models for the design of metamaterials strongly depends on the choice of geometric parameterization. Generative models make assumptions about their input data that generally hold for images, but can be invalid for metamaterials when complex dependencies be- tween the input parameters exist. B-splines can provide a representation without those dependencies and are therefore well suited to be paired up with generative models. Furthermore, it is also important how these geometrical parameters are sampled to create the training data for the generative models, as this sampling also affects the distribution of the properties in the training data. Overall, while generative machine learning models prove effective for the inverse design of mechanical metamaterials their successful application requires careful parameterization and selection of the training data.

Purpose Mechanical meta-materials have aroused great interest during the past few decades. Unique or even unprecedented mechanical behaviors and functionalities have been created through the rational design of the micro-structure of mechanical meta-materials. Moreover, the development of additive manufacturing (AM) techniques further promotes their applications in various fields like aerospace, automobile, tissue engineering, etc. In recent years, mechanical meta-materials have also been applied to the design of aeroengine structures due to their superior performance and lightweight features. This paper aims to give a comprehensive review of the design of mechanical meta-materials and show their recent applications to aeroengine structures. Design/methodology/approach This paper presents a state-of-the-art review on the design of mechanical meta-materials and their applications to the rotating and stationary components in aeroengine structures. Findings Mechanical meta-materials are categorized into two kinds, including homogeneously architected meta-materials that have one unique unit cell periodically arranged to fill the space and heterogeneously architected meta-materials that may integrate different unit cell designs or introduce gradient changes to the whole structure. A promising design strategy, i.e. “the right micro-structure deployed to the right position,” is proposed for heterogeneous meta-materials, and it is concluded that the design methods for mechanical meta-materials are moving from simple cell design to the direction of customized and optimized design enabled by artificial intelligence. Besides, the integration of metal AM and mechanical meta-materials paved a novel path for the design of lightweight high-performance engine structures. Originality/value The present paper systematically reviews the design of mechanical meta-materials, and for the first time summarizes the application of mechanical meta-materials to aeroengine structures.

A shift is occurring from avoiding structural geometric nonlinearities to deliberately exploiting them for new functionalities. Forward design–predicting nonlinear responses from given parameters–is well established, but the inverse problem–identifying parameters that yield a desired nonlinear behaviour–remains a major challenge due to the complexity of the response. Developing an efficient inverse design framework is therefore essential to enable broader application of nonlinear mechanical metamaterials. To address this, we introduce an enhanced conditional generative adversarial network (c-GAN) framework that synergizes three key innovations: (i) surrogate-based physics-informed supervision via a pre-trained simulator network, (ii) residual generator architecture for stable gradient flow, and (iii) structured noise encoding for enhanced diversity and controllability. This c-GAN-Physics-Stochastic architecture generates precise geometric parameters in a single forward pass, bypassing traditional iterative optimization bottlenecks. We demonstrate the framework’s effectiveness through two case studies: mechanical metamaterials designed for recoverable energy dissipation via sequential snap-through instabilities, and auxetic metamaterials maintaining a constant Poisson’s ratio under large deformations. Experimental results validate the physical realisability of the generated designs. Inference with the conditional GAN provides millisecond-level design prediction, while a hybrid c-GAN + genetic algorithm (GA) refinement strategy achieves exceptional agreement with high-fidelity evaluations (up to R 2 > 0.99 ) at increased computational cost. This work demonstrates the potential of data-driven methods, particularly c-GANs, in addressing the inverse design challenge for nonlinear mechanical metamaterials for novel functionalities. By delivering an accessible toolbox, we aim to promote its widespread use and practical implementation by engineers, paving the way for innovative applications.

Complex uncertainty-oriented robust topology optimization for multiple mechanical metamaterials based on double-layer mesh

A deep learning approach for inverse design of gradient mechanical metamaterials

A defining feature of mechanical metamaterials is that their properties are determined by the organization of internal structure instead of the raw fabrication materials. This shift of attention to engineering internal degrees of freedom has coaxed relatively simple materials into exhibiting a wide range of remarkable mechanical properties. For practical applications to be realized, however, this nascent understanding of metamaterial design must be translated into a capacity for engineering large-scale structures with prescribed mechanical functionality. Thus, the challenge is to systematically map desired functionality of large-scale structures backward into a design scheme while using finite parameter domains. Such "inverse design" is often complicated by the deep coupling between large-scale structure and local mechanical function, which limits the available design space. Here, we introduce a design strategy for constructing 1D, 2D, and 3D mechanical metamaterials inspired by modular origami and kirigami. Our approach is to assemble a number of modules into a voxelized large-scale structure, where the module's design has a greater number of mechanical design parameters than the number of constraints imposed by bulk assembly. This inequality allows each voxel in the bulk structure to be uniquely assigned mechanical properties independent from its ability to connect and deform with its neighbors. In studying specific examples of large-scale metamaterial structures we show that a decoupling of global structure from local mechanical function allows for a variety of mechanically and topologically complex designs.

TPMS-based strut-shell interpenetrating lattice metamaterial with wide-range customizable mechanical properties and superior energy absorption

Advances in 3D/4D printing of mechanical metamaterials: From manufacturing to applications

Mechanical metamaterials are hierarchical structures that can be designed to possess exotic properties such as auxeticity, tunable stiffness, and bistability by optimizing the geometry of their microscale unit cells. This work develops a computational framework in which an arbitrary set of homogenized stress–strain goal points is defined, and topology optimization is employed to synthesize a unit cell to achieve the desired response. The framework extends the current state‐of‐the‐art by incorporating highly nonlinear phenomena such as internal contact, snap‐through buckling, and bistability in a single approach to capture a wide range of mechanical responses without the need for generating datasets, tuning parameters, or possessing prior knowledge of the optimal solution. Unit cells are generated for three challenging nonlinear responses and their behavior is validated via mechanical testing. The framework will enable the generation of highly nonlinear unit cells for metamaterials in applications such as morphing structures, soft robotics, and energy absorbing materials.

The rational design of mechanical metamaterials has the potential to grant them with properties unattainable by bulk materials. This paper describes twisting mechanical metamaterials (TMMs), a new class of programmable matter whose rationally designed architecture endows them with compression-twist coupling, a property absent in conventional materials. Upon compression, the cellular architecture of TMMs efficiently transforms longitudinal strain into unidirectional or bidirectional twisting—free from mechanical frustrations and independent of the flexible material chosen to fabricate the TMM. TMMs can be rapidly designed using an automated generative algorithm, which enables the tailoring of their twisting properties (up to 3.24° twist per % strain) and Poisson’s ratio (−0.6 to +1.22) using only two design parameters. The scalable architecture of TMMs does not suffer from size effects which impair their twist per unit strain. Bidirectional TMMs autonomously switch their direction of rotation during uniform displacement-controlled compressive loading, conferring them with non-interchangeable compression-twist coupling. The capability of TMMs to controllably convert translation to rotation blurs the boundary between materials and machines, expanding the range of mechanical properties currently accessible to 3D printed materials and paving the way toward the creation of future programmable matter.

Lattice structures with architecture based on triply periodic minimal surfaces (TPMS) have recently attracted much attention due to their mathematically controlled topologies, which give rise to predefined and fascinating mechanical properties. Such materials with topology-dependent properties are generally known as “metamaterials.” The research on the TPMS-based metamaterials was restricted to theoretical knowledge until the recent development of additive manufacturing technology (AM). Herein, the design, properties, challenges, and applications, of TPMS-based mechanical metamaterials are reviewed. This review provides the researcher with a collective resource of numerical and experimental data for the AM of TPMS-based structures.

Mechanical metamaterials are emerging as an important tool in many engineering applications, where microarchitectured lattices have advantages of high strength-to-weight ratios, energy absorption capabilities and versatility in designed mechanical behaviours. Whilst a variety of microarchitectured lattices have been previously investigated for their mechanical properties, they have been mostly constrained to strut and plate-based designs and subjected to additional post-processing to modify their functionality. In this work, we demonstrate a novel approach utilising the two-photon polymerisation (2PP) method to generate complex, defect-free microarchitectured lattices. Unlike traditional methods, our technique allows for the direct fabrication of shell-based triply periodic minimal surfaces (TPMS) without requiring any additional post-processing. We utilise gyroid and diamond TPMS unit cells to perform uniaxial micro-compression testing to characterise their mechanical properties. In doing so, we gain insights into the mechanical behaviours of microlattices, with the finding that diamond cells possess superior stiffness and energy absorption potential compared to gyroid cells. To unravel the size-dependency property alterations at this scale, we further analyse a variety of bending beam samples and observed a strain rate-dependent mechanical behaviour attributable to the material's viscoelastic nature. We employ finite element analysis to enhance our understanding of the deformation mechanisms inherent in TPMS topologies, finding good agreement with measured mechanical properties. This work establishes a comprehensive framework that spans from design and geometry characterisation to robust microstructure testing using indentation, combined with comprehensive numerical simulation.

We present a data-efficient neural-network model for predicting linear-elastic properties of spinodoid metamaterials from their mesoscale structure. Our machine-learning model requires 75 data points for training, greatly improving data efficiency over previous models that required thousands of training samples. We achieve this by leveraging concepts from geometric learning. Specifically, we exploit physical properties, such as positive semi-definiteness of the elasticity tensor, as well as structural invariances and equivariances of the problem, for example with respect to coordinate axes permutations. The neural network model is designed to exactly fulfill these constraints; it does not have to learn them from data. The resulting model enables data- and compute-efficient inverse design of spinodoid metamaterials. In inverse design, the goal is to find a material mesostructure that leads to desired mechanical properties on the macroscale. Exactly fulfilling physical and structural constraints, the present neural network model remains differentiable. This allows using fast gradient-based optimizers for inverse design. We demonstrate this by inversely designing spinodoid metamaterials that achieve desired linear elastic target properties in three dimensions. Inverse design is treated as a constrained optimization problem over the parameters describing the metamaterial. The results confirm that the present approach requires significantly less training data than previous machine-learning approaches and allows incorporating multiple objectives in the inverse design process. Since the structure of the design space is independent of the target material properties, we hope that such data-efficient models will be useful also for inverse design of spinodoids beyond linear elasticity.