Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy

Li Zheng, Siddhant Kumar, Dennis M. Kochmann
2021 Computer Methods in Applied Mechanics and Engineering  
We present a two-scale topology optimization framework for the design of macroscopic bodies with an optimized elastic response, which is achieved by means of a spatially-variant cellular architecture on the microscale. The chosen spinodoid topology for the cellular network on the microscale (which is inspired by natural microstructures forming during spinodal decomposition) admits a seamless spatial grading as well as tunable elastic anisotropy, and it is parametrized by a small set of design
more » ... all set of design parameters associated with the underlying Gaussian random field. The macroscale boundary value problem is discretized by finite elements, which in addition to the displacement field continuously interpolate the microscale design parameters. By assuming a separation of scales, the local constitutive behavior on the macroscale is identified as the homogenized elastic response of the microstructure based on the local design parameters. As a departure from classical FE 2 -type approaches, we replace the costly microscale homogenization by a data-driven surrogate model, using deep neural networks, which accurately and efficiently maps design parameters onto the effective elasticity tensor. The model is trained on homogenized stiffness data obtained from numerical homogenization by finite elements. As an added benefit, the machine learning setup admits automatic differentiation, so that sensitivities (required for the optimization problem) can be computed exactly and without the need for numerical derivatives -a strategy that holds promise far beyond the elastic stiffness. Therefore, this framework presents a new opportunity for multiscale topology optimization based on data-driven surrogate models. on the achievable effective stiffness of cellular media [6, 8] . However, both types of architectures suffer from stress localization at the junctions of beams or plates, which results in early failure and poor recoverability [9] [10] [11] . Smooth architectures like those based on triply-periodic minimal surfaces (TPMS) [12] [13] [14] address the aforementioned issue, but they do not overcome a common limitation in the fabrication of architectures based on periodic unit cells: classically, all truss, plate, and TPMS designs produce periodic structures with high sensitivity of mechanical properties to symmetry-breaking imperfections and defects and limited opportunity for introducing smoothly spatially variant structures. In light of these challenges, spinodal metamaterials have emerged recently as a new class of non-periodic architected material [15] [16] [17] [18] [19] [20] . Their design is inspired by topologies observed during spinodal decomposition [21, 22] , which occurs, e.g., during rapid diffusion-driven phase separation in nanoporous alloys [23, 24] , polymer blends [25] , and micro-emulsions [26] . The computational design of spinodal microstructures relies on simulating the phase separation process by phase field methods whose kinetics are modeled by Cahn-Hilliard-type evolution equations [19, [27] [28] [29] [30] [31] [32] or, as a shortcut, by Gaussian random fields [20] . As a two-phase mixture spontaneously decomposes into two spatially-separated stable phases, the process is artificially arrested and solid-or shell-based topologies are extracted by, respectively, removing one of the two phases or by retaining the interfaces and eliminating both phases. The resulting topologies are composed of smooth, bi-continuous, and non-self-intersecting surfaces were shown to have intriguing properties. Portela et al. [16] demonstrated that spinodal topologies exhibit better (and close to optimal) stiffness scaling with respect to relative density, and they were shown to exhibit a significantly improved mechanical resilience over comparable truss and plate metamaterials, as seen in the recovery after repeated nonlinear compression. Unlike beam and plate architectures, spinodal metamaterials (like TPMS) avoid stress concentrations at beam or plate junctions, thus reducing stress concentrations, which considerably contributes to the observed high mechanical resilience. Compared to TPMS, the non-periodicity of spinodal architectures renders them insensitive to symmetry-breaking defects and fabrication-induced imperfections [17] . Guell Izard et al. [18] further demonstrated ultra-high energy absorption characteristics of spinodal architectures. Spinodal metamaterials can also self-assemble across several length scales -from centimeters to nanometers [16] , which is a promising avenue to overcome the scalability challenge of additive manufacturing. (Note that in this study we do not take into account the manufacturability of structures as, e.g., [33] ). We recently introduced a computational shortcut to generate spinodal-like topologies, referred to as spinodoid topologies [15] . This approach replaces computationally expensive phase field simulations for topology generation and provides a simple parametrization based on anisotropic Gaussian random fields (GRFs) [34, 35] ; unlike the isotropic formulation of Soyarslan et al. [20] , spinodoids allow for an efficient exploration of a wide design space of anisotropic mechanical properties. When designing functionally-graded metamaterials, the GRF-based approach admits seamless, spatially-variant topologies, which in contrast to periodic unit-cell-based designs bypasses discontinuities and tessellation-related limitations. Spinodoid metamaterials bear potential for applications ranging from energy absorption and impact protection to heat exchange and to synthetic bone. An ongoing challenge in the design of patient-specific bone implants is to match the anisotropic topological and mechanical properties of bone -which can be highly heterogeneous across patients as well as within the same bone. Functionally-graded spinodoid metamaterials were shown to be promising candidates for inverse-designed synthetic bones [15] for improved biomechanical compatibility and reduced bone atrophy. Yet, aside from optimizing the properties of specific spinodoid topologies, they have not been used in any two-scale design challenge, such as, e.g., identifying the optimal macroscale shape of a bone implant while optimizing the local, spatially-varying spinodoid microstructure. To this end, we here address the systematic design of spatially-variant, functionally-graded bodies with a spinodoid microstructure through a data-driven topology optimization approach. Topology optimization is a well-established technique (see [36] and [37] for detailed reviews). The classical Solid Isotropic Material Penalization (SIMP) method and its extensions [38-42] define a continuous volume fraction field ρ : Ω → [0, 1] over a body Ω ⊂ R d in d dimensions, such that the local linear elastic modulus tensor is approximated as where C 0 and C S are the stiffness tensors of void and solid regions, respectively, and p ≥ 3 is a penalization exponent to promote (approximately) purely void or solid states. Most SIMP-based methods optimize the material
doi:10.1016/j.cma.2021.113894 fatcat:7airfcahpnhmpbknyrt4mrxk74