Delft University of Technology Bayesian Machine Learning in metamaterial design Fragile becomes supercompressible Bessa, Miguel A.; Głowacki, Piotr; Houlder, Michael DOI 10.1002/adma.201904845 Publication date 2019 Document Version Final published version Published in Advanced Materials Citation (APA) Bessa, M. A., Głowacki, P., & Houlder, M. (2019). Bayesian Machine Learning in metamaterial design: Fragile becomes supercompressible. Advanced Materials, 31(48), [1904845]. https://doi.org/10.1002/adma.201904845 Important note To cite this publication, please use the final published version (if applicable). Please check the document version above. Copyright Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim. This work is downloaded from Delft University of Technology. For technical reasons the number of authors shown on this cover page is limited to a maximum of 10. COMMUNICATION www.advmat.de Bayesian Machine Learning in Metamaterial Design: Fragile Becomes Supercompressible Miguel A. Bessa,* Piotr Glowacki, and Michael Houlder data-driven approach to explore new Designing future-proof materials goes beyond a quest for the best. The next design possibilities, while reducing experi- generation of materials needs to be adaptive, multipurpose, and tunable. This mentation to a minimum (validation). is not possible by following the traditional experimentally guided trial-and- Without loss of generality, we illustrate error process, as this limits the search for untapped regions of the solution this paradigm shift by focusing on a new concept for a low density mechanical met- space. Here, a computational data-driven approach is followed for exploring amaterial (Figure 1). The aim is to addi- a new metamaterial concept and adapting it to different target proper- tively manufacture a building block that ties, choice of base materials, length scales, and manufacturing processes. achieves recoverable supercompressibility Guided by Bayesian machine learning, two designs are fabricated at different while maintaining high strength and stiff- length scales that transform brittle polymers into lightweight, recoverable, ness. This concept results from a combina- tion of a deployable mast[9,10] developed for and supercompressible metamaterials. The macroscale design is tuned for highly deformable space structures, and a maximum compressibility, achieving strains beyond 94% and recoverable thin-walled conical frustum[11] common in strengths around 0.1 kPa, while the microscale design reaches recoverable impact absorption applications. strengths beyond 100 kPa and strains around 80%. The data-driven code is Remarkably, even without inverting available to facilitate future design and analysis of metamaterials and struc- the design process, additively manufac- tured mechanical metamaterials with low tures (https://github.com/mabessa/F3DAS). 3 densities (ρ ≪ 1 g cm− ) have achieved compressive strengths above 100 MPa while exhibiting brittle fracture using Structure-dominated materials (metamaterials) are pushing the carbon lattices,[12] on the order of 1 MPa with partially recover- envelope of known electromagnetic,[1] classic,[2] and quantum[3] able compressive strains in excess of 50% using microlattices of mechanical properties by exploring new geometries. Additive alumina[13] or graphene aerogels,[14] and on the order of 0.5 kPa manufacturing has been a major driving force in this explora- and exceeding 80% strains for another graphene aerogel.[15] tion since virtually any topology can be obtained to probe the These investigations demonstrate that tuning the geometry of vast design space created by geometric changes in the mate- the metamaterial to explore symmetry-breaking instabilities rial structure. This has led to discoveries of materials with new (buckling) is the key to achieve large effective compressibility properties and functionality, for example exhibiting negative while causing small deformation of a high strength base mate- linear compressibility,[4] tunable negative stiffness,[5] controllable rial. Yet, the process for designing these metamaterials and reconfigurability via programmable surfaces[6] or via origami,[7] assessing their mechanical limits is cumbersome and time-con- and symmetry breaking elastic surface patterns.[8] However, suming due to the absence of general design principles arising metamaterial design has relied on extensive experimentation from highly nonlinear, unstable, and imperfection sensitive and a trial-and-error approach where analytical or computa- responses that depend on many geometric parameters. More tional models only provide a posteriori explanations. Here, we importantly, even when successful designs are found,[12–15] argue in favor of inverting the process by using a computational tuning them for new applications requiring different properties and functionality is not trivial. In this context, machine learning can provide significant advantages to the design process. Prof. M. A. Bessa, P. Glowacki, M. Houlder Machine learning and deep learning are already able to sur- Department of Materials Science and Engineering pass the limits of a human mind in specific tasks. Algorithms Delft University of Technology are capable of winning against human champions in increasingly 2628 CD Delft, The Netherlands difficult games,[16–18] they recognize faces with near human-level E-mail: [email protected] accuracy,[19] and predict ratings of unseen movies based on past The ORCID identification number(s) for the author(s) of this article [20] can be found under https://doi.org/10.1002/adma.201904845. user selection. These achievements are permeating to different scientific disciplines such as materials science[21,22] and chemi- © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, stry,[23] and recently they are reaching the design of electromag- Weinheim. This is an open access article under the terms of the Creative netic metamaterials via nonprobabilistic machine[24] and deep Commons Attribution License, which permits use, distribution and repro- [25–28] [29] duction in any medium, provided the original work is properly cited. learning methods, and even generative deep learning. However, metamaterials usually derive their unprecedented prop- DOI: 10.1002/adma.201904845 erties from exploring imperfection-sensitive behavior because Adv. Mater. 2019, 31, 1904845 1904845 (1 of 6) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advmat.de exploration and exploitation. The Supporting Information includes two sections (“Why Bayesian machine learning?” and “Bayesian machine learning vs Bayesian optimization”) with addi- tional references and considerations for the interested reader. Our data-driven exploration follows a recent framework created to design materials and structures.[32,33] This modular framework integrates: 1) design of experiments to sample the input variables, 2) efficient predictive analyses to generate the output database, 3) machine learning to establish input–output relationships from the database, and 4) optimization to deter- mine optimum designs from the machine learning model. Different methods can be chosen for each module depending on the problem’s dimensionality, size of database needed for the learning process, and whether the phenomena of interest is probabilistic or deterministic. Our mechanical metamaterial undergoes probabilistic responses due to buckling/postbuck- ling mechanisms, and involves a moderate number of design parameters that is expected to lead to large training datasets. Recent Bayesian machine learning methods called sparse Gaussian processes[30] have been developed to fulfill these requirements while being sufficiently scalable. We considered several algorithms, as detailed in the Supporting Information (“Sparse Gaussian processes: scalability & accuracy”), and con- cluded that the sparse Gaussian process regression (SGPR) algorithm[34] was the most adequate for the regression tasks, while the scalable variational Gaussian process classification (SVGP)[35] was the most adequate for classification. Data-driven design of the metamaterial concept is then summarized in Figure 2. First, the building block (Figure 1) is parameterized according to the design variables. The geometry is defined by the top and bottom base diameters, D1 and D2, height P, and four parameters that define the cross-section of the ver- tical elements (longerons): the cross-sectional area A, moments of inertia Ix and Iy, and torsional constant Jτ. In addition, since the metamaterial is targeted to be reversibly deformable, any chosen base material is defined by its elastic constants: the Young’s modulus E and the shear modulus G. Given that this is a nonlinear elastic mechanics problem, we know a priori that the geometric variables can be scaled by one of the dimensions, chosen here as D1, and that we can consider the ratio of elastic Figure 1. Supercompressible metamaterial building block with general- constants G/E. Finally, since no contact between the deformed ized cross-section for the longerons (x is the radial direction; y is the longerons