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.

Projective dynamics was introduced a few years ago as a fast method to yield an approximate yet stable solution to the dynamics of nodal systems subject to stiff internal forces. Previous attempts to include contact forces in that framework considered adding a quadratic penalty energy to the global system, which however broke the simple - constant matrix - structure of the global linear equation, while failing to treat contact in an implicit manner. In this paper we propose a simple yet effective method to integrate in a unified and semi-implicit way contact as well as dry frictional forces into the nested architecture of Projective dynamics. Assuming that contacts apply to nodes only, the key is to split the global matrix into a diagonal and a positive matrix, and use this splitting in the local step so as to make a good prediction of frictional contact forces at next iteration. Each frictional contact force is refined independently in the local step, while the original efficient structure of the global step is left unchanged. We apply our algorithm to cloth simulation and show that contact and dry friction can be captured at a reasonable precision within a few iterations only, hence one order of magnitude faster compared to global implicit contact solvers of the literature.