A Newton Solver for Micromorphic Computational Homogenization Enabling Multiscale Buckling Analysis of Pattern-Transforming Metamaterials

Mechanical metamaterials feature engineered microstructures designed to exhibit exotic, and often counter-intuitive, effective behaviour. Such a behaviour is often achieved through instability-induced transformations of the underlying periodic microstructure into one or multiple patterning modes. Due to a strong kinematic coupling of individual repeating microstructural cells, non-local behaviour and size effects emerge, which cannot easily be captured by classical homogenization schemes. In addition, the individual patterning modes can mutually interact in space as well as in time, while at the engineering scale the entire structure can buckle globally. For efficient numerical macroscale predictions, a micromorphic computational homogenization scheme has recently been developed. Although this framework is in principle capable of accounting for spatial and temporal interactions between individual patterning modes, its implementation relied on a gradient-based quasi-Newton solution technique. This solver is suboptimal because (i) it has sub-quadratic convergence, and (ii) the absence of Hessians does not allow for proper bifurcation analyses. Given that mechanical metamaterials often rely on controlled instabilities, these limitations are serious. To address them, a full Newton method is provided in detail in this paper. The construction of the macroscopic tangent operator is not straightforward due to specific model assumptions on the decomposition of the underlying displacement field pertinent to the micromorphic framework, involving orthogonality constraints. Analytical expressions for the first and second variation of the total potential energy are given, and the complete algorithm is listed. The developed methodology is demonstrated with two examples in which a competition between local and global buckling exists and where multiple patterning modes emerge.Comment: 34 pages, 17 figures, 1 table, 1 algorithm, abstract shortened to fulfill 1920 character limi

Microstructural enrichment functions based on stochastic Wang tilings

This paper presents an approach to constructing microstructural enrichment functions to local fields in non-periodic heterogeneous materials with applications in Partition of Unity and Hybrid Finite Element schemes. It is based on a concept of aperiodic tilings by the Wang tiles, designed to produce microstructures morphologically similar to original media and enrichment functions that satisfy the underlying governing equations. An appealing feature of this approach is that the enrichment functions are defined only on a small set of square tiles and extended to larger domains by an inexpensive stochastic tiling algorithm in a non-periodic manner. Feasibility of the proposed methodology is demonstrated on constructions of stress enrichment functions for two-dimensional mono-disperse particulate media.Comment: 27 pages, 12 figures; v2: completely re-written after the first revie

Local tilings informed synthesis of micro-mechanical fields by means of Wang tiles

Grant Agency of the CTU in Prague, the grant No. SGS19/033/OHK1/1T/11 - Design toolchain for modular structures: Accelerated optimization and stochastic analysi

Wang tiles and metal foam micro-structure image synthesis

The authors gratefully acknowledge the financial support from the Grant Agency of the CTU in Prague, the grant No. SGS18/036/OHK1/1T/11 (L. Zr°ubek, M. Doˇsk´aˇr and A. Kuˇcerov´a). Authors would also like to thank for the support from Vicerrector´ıa de Investigaci´on y Extensi´on, project No. 1351022 (M. Meneses-Guzm´an, F. Rodr´ıguez-M´endez and B. Chin´e).Our latest work is focused on image synthesis of metal foam micro-structures using the Wang tiles and Automatic tile design

A Newton solver for micromorphic computational homogenization enabling multiscale buckling analysis of pattern-transforming metamaterials

Mechanical metamaterials feature engineered microstructures designed to exhibit exotic, and often counter-intuitive, effective behaviour such as negative Poisson's ratio or negative compressibility. Such a specific response is often achieved through instability-induced transformations of the underlying periodic microstructure into one or multiple patterning modes. Due to a strong kinematic coupling of individual repeating microstructural cells, non-local behaviour and size effects emerge, which cannot easily be captured by classical homogenization schemes. In addition, the individual patterning modes can mutually interact in space as well as in time, while at the engineering scale the entire structure can buckle globally. For efficient numerical predictions of macroscale engineering applications, a micromorphic computational homogenization scheme has recently been developed (Rokoš et al., J. Mech. Phys. Solids 123, 119–137, 2019). Although this framework is in principle capable of accounting for spatial and temporal interactions between individual patterning modes, its implementation relied on a gradient-based quasi-Newton solution technique. This solver is suboptimal because (i) it has sub-quadratic convergence, and (ii) the absence of Hessians does not allow for proper bifurcation analyses. Given that mechanical metamaterials often rely on controlled instabilities, these limitations are serious. Addressing them will reduce the dependency of the solution on the initial guess by perturbing the system towards the correct deformation when a bifurcation point is encountered. Eventually, this enables more accurate and reliable modelling and design of metamaterials. To achieve this goal, a full Newton method, entailing all derivations and definitions of the tangent operators, is provided in detail in this paper. The construction of the macroscopic tangent operator is not straightforward due to specific model assumptions on the decomposition of the underlying displacement field pertinent to the micromorphic framework, involving orthogonality constraints. Analytical expressions for the first and second variation of the total potential energy are given, and the complete algorithm is listed. The developed methodology is demonstrated with two examples in which a competition between local and global buckling exists and where multiple patterning modes emerge. The numerical results indicate that local to global buckling transition can be predicted within a relative error of 6% in terms of the applied strains. The expected pattern combinations are triggered even for the case of multiple patterns