Computational Design of Flexible Planar Microstructures

Abstract

Mechanical metamaterials enable customizing the elastic properties of physical objects by altering their fine-scale structure. A broad gamut of effective material properties can be produced even from a single fabrication material by optimizing the geometry of a periodic microstructure tiling. Past work has extensively studied the capabilities of microstructures in the small-displacement regime, where periodic homogenization of linear elasticity yields computationally efficient optimal design algorithms. However, many applications involve flexible structures undergoing large deformations for which the accuracy of linear elasticity rapidly deteriorates due to geometric nonlinearities. Design of microstructures at finite strains involves a massive increase in computation and is much less explored; no computational tool yet exists to design metamaterials emulating target hyperelastic laws over finite regions of strain space. We make an initial step in this direction, developing algorithms to accelerate homogenization and metamaterial design for nonlinear elasticity and building a complete framework for the optimal design of planar metamaterials. Our nonlinear homogenization method works by efficiently constructing an accurate interpolant of a microstructure's deformation over a finite space of macroscopic strains likely to be endured by the metamaterial. From this interpolant, the homogenized energy density, stress, and tangent elasticity tensor describing the microstructure's effective properties can be inexpensively computed at any strain. Our design tool then fits the effective material properties to a target constitutive law over a region of strain space using a parametric shape optimization approach, producing a directly manufacturable geometry. We systematically test our framework by designing a catalog of materials fitting isotropic Hooke's laws as closely as possible. We demonstrate significantly improved accuracy over traditional linear metamaterial design techniques by fabricating and testing physical prototypes.

Computational Homogenization for Inverse Design of Surface-based Inflatables

“…Our coarsened design optimization relies on an understanding of the metric change induced by inflating a pattern as well as the bending stiffness around the inflated state. Periodic homogenization [Nakshatrala et al 2013;Schumacher et al 2018;Sperl et al 2020;Zhang et al 2023a] is a rigorous approach for defining a periodic metamaterial's properties by studying the behavior of an infinite tiling of its pattern unit cell 𝑌 . The conceptually infinite simulation is made tractable by an assumption that the tiling's translational symmetry is preserved by the deformation 1 .…”

“…We compute homogenized stretching and bending stiffnesses of an inflated tiling around its equilibrium state using analytical formulas obtained via sensitivity analysis, the general idea of which has been introduced in past works [Chan-Lock et al 2022;Zhang et al 2023a]. However, our method differs from the previous ones on calculating bending stiffnesses [Schumacher et al 2018;Sperl et al 2020], which fit constitutive models to a large number of sampled deformations at finite strains.…”

“…Homogenization. Many works in graphics and computational fabrication have employed homogenization to accelerate elasticity simulations [Chan-Lock et al 2022;Chen et al 2015;Kharevych et al 2009;Sperl et al 2020] and rigid-body simulation [Tang et al 2023], and to design volumetric [Chen et al 2018;Panetta et al 2017Panetta et al , 2015Schumacher et al 2015], sheet Martínez et al 2019;Schumacher et al 2018;Tozoni et al 2020;Zhang et al 2023a] and ribbon [Rodriguez et al 2022] metamaterials. To our knowledge, we are the first to develop a periodic homogenization technique for analyzing the contraction and the bending and stretching stiffnesses of inflated pattern tilings.…”