Performance-based inverse design of nonlinear mechanical metamaterials via enhanced conditional generative adversarial networks

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 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.

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.

Multi-target inverse design, which involves designing multiple targets with different optimization objectives, becomes a key focus in mechanical metamaterials (MMs) design. Specifically, many practical applications impose varying requirements for different sections. For instance, the heel of a sole demands to provide support, while the arch should be comfortable, adequately supportive, and lightweight. However, existing approaches, such as topology optimization, typically focus on optimizing MMs for specific objectives, e.g., high strength. Worse, these approaches are often inaccurate and time-consuming, even just for a single target. In this work, based on graded triply periodic minimal surface (TPMS) architectures, a machine-learning-powered approach is proposed for rapid, accurate, and multi-target MM inverse design by employing a six-parallel pipeline network architecture and utilizing deep networks to map structural parameters to mechanical curves. The most suitable results are selected based on the target curves and other performance requirements, which can be derived from the structural parameters. The approach achieves a normalized root-mean-square error (NRMSE) of 2.49% on the test dataset and outputs corresponding design parameters within seconds, simultaneously meeting multiple targets. Finally, such an approach is demonstrated in designing soles suitable for various gait scenarios and foot deformity treatments.

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.

Numerical simulations have revolutionized material design. However, although simulations excel at mapping an input material to its output property, their direct application to inverse design has traditionally been limited by their high computing cost and lack of differentiability. Here, taking the example of the inverse design of a porous matrix featuring targeted sorption isotherm, we introduce a computational inverse design framework that addresses these challenges, by programming differentiable simulation on TensorFlow platform that leverages automated end-to-end differentiation. Thanks to its differentiability, the simulation is used to directly train a deep generative model, which outputs an optimal porous matrix based on an arbitrary input sorption isotherm curve. Importantly, this inverse design pipeline leverages the power of tensor processing units (TPU)—an emerging family of dedicated chips, which, although they are specialized in deep learning, are flexible enough for intensive scientific simulations. This approach holds promise to accelerate inverse materials design.

Development and validation of a hybrid aerodynamic design method for curved diffusers using genetic algorithm and ball-spine inverse design method

The conventional approach to nanophotonic metasurface design and optimization for a targeted electromagnetic response involves exploring large geometry and material spaces. This is a highly iterative process based on trial and error, which is computationally costly and time consuming. Moreover, the non-uniqueness of structural designs and high non-linearity between electromagnetic response and design makes this problem challenging. To model this unintuitive relationship between electromagnetic response and metasurface structural design as a probability distribution in the design space, we introduce a framework for inverse design of nanophotonic metasurfaces based on cyclical deep learning (DL). The proposed framework performs inverse design and optimization mechanism for the generation of meta-atoms and meta-molecules as metasurface units based on DL models and genetic algorithm. The framework includes consecutive DL models that emulate both numerical electromagnetic simulation and iterative processes of optimization, and generate optimized structural designs while simultaneously performing forward and inverse design tasks. A selection and evaluation of generated structural designs is performed by the genetic algorithm to construct a desired optical response and design space that mimics real world responses. Importantly, our cyclical generation framework also explores the space of new metasurface topologies. As an example application of the utility of our proposed architecture, we demonstrate the inverse design of gap-plasmon based half-wave plate metasurface for user-defined optical response. Our proposed technique can be easily generalized for designing nanophtonic metasurfaces for a wide range of targeted optical response.

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.

Over the past two decades, colloidal particles with a variety of shapes, sizes, and compositions have been synthesized and characterized successfully. One of the most important applications for colloidal building blocks is to engineer functional structures as mechanical, electrical, and optical metamaterials. However, complex interaction dynamics between the building blocks as well as sophisticated structure-property relationships make it challenging to design structures with predictable target properties. In this paper, we implement an inverse material design framework using Genetic Algorithm (GA)-based techniques to streamline the design of colloidal structures based on target properties. We investigate spherical particles as well as colloidal molecules of different sizes and shapes and evaluate a Geometric Landscape Accessibility parameter that identifies the size of feasible domains within the geometric phase space of each structure. Considering target photonic properties, our GA-assisted framework is further utilized to identify sets of building blocks and structures that lead to various target values for the size of the photonic band gaps. The proposed framework in this study will provide new insight for predictive computational material design approaches and help establish more efficient ways of understanding structure-property relations in sub-micrometer-scale materials.

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.

A nonlinear metamaterial plate for suppressing vibration and sound radiation

Most of the inverse design methods of turbomachinery experience the shortcoming where the target aerodynamic parameters need to be manually specified depending on the designers’ experience and insight, making the design result aleatory and even deviated from the real optimal solution. To tackle this problem, an experience-independent inverse design optimization method is proposed and applied to the redesign of a compressor cascade airfoil in this study. The experience-independent inverse design optimization method can automatically obtain the target pressure distribution along the cascade airfoil through the genetic algorithm, rather than through the manual specification approach. The shape of cascade airfoil is then solved by the adjoint method. The effectiveness of the experience-independent inverse design optimization method is demonstrated by two inverse design cases of the compressor cascade airfoil, i.e. the inverse design of only the suction surface and the inverse design of both the suction and pressure surfaces. The results show that the proposed inverse design method is capable of significantly improving the aerodynamic performance of the compressor cascade. At the examined flow condition, a thin airfoil profile is beneficial to flow accelerations near the leading edge and flow separation avoidance near the trailing edge. The proposed inverse design method is quite generic and can be extended to the three-dimensional inverse design of advanced compressor blades.

Polarization-dependent boundary modes in nonlinear mechanical metamaterials

An inverse design framework incorporating a physics-based surrogate model and global optimization is proposed to assist in the design of low warpage ultra-thin packages by adjusting the metal densities over substrate subsections and layers. The surrogate model is derived from two finite element analysis (FEA) models. The first one describes the relationship between the metal density in the substrate layer to the coefficient of thermal expansion (CTE) while the second one describes the relationship between in-plane CTE variation of the substrate to the warpage profile. Results from these two FEA models are used to train separate artificial neural networks (ANN). When these ANNs are run sequentially, the surrogate model can accurately determine the warpage profile for any set of metal densities. Three global optimization algorithms, Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and Cross Entropy (CE) were then evaluated using this surrogate model. Three case studies consisting of different warpage profiles (original and 20% reduced warpage) and constraints to the optimization search space (±20% or ±50% change to metal density) were then evaluated using these algorithms. For all three cases, the three algorithms converged to similar solutions, indicating that indeed the global minimum has been attained and determined. However, GA took a significantly longer time to converge than PSO and CE. Based on these results, PSO and CE are recommended to be suitable algorithms to carry out inverse design for this type of problem.