Multiscale topology optimization with embedded TPMS architected materials

Topology optimization has emerged as a critical computational tool for designing lightweight, robust, resilient and efficient structures. Recent advances in additive manufacturing technologies enable the production of complex objects across multiple scales, fostering the development of novel architectures endowed with diverse topologies and material classes, tailored to specific performance requirements. In this work we explore the use of Triply Periodic Minimal Surface (TPMS) architected materials, which mimic natural and biological systems to achieve exceptional mechanical efficiency and scalability. To this aim, we present a multiscale, multi-material topology optimization framework that leverages a gradient-based scheme to minimize compliance under multiple volume constraints. TPMS microstructures are generated via the Fourier Series Function (FSF) method, seamlessly integrated into the optimization process through homogenization theory. The Solid Isotropic Material with Penalization (SIMP) model is coupled with Discrete Material Optimization (DMO) interpolation, introducing a continuation parameter that transitions smoothly from a convex problem to a non-convex problem. To handle volume constraints effectively, the ZPR-BFGS design variable update scheme is adapted to the continuum setting, allowing constraints to be updated independently, sequentially, or in parallel. This framework enables flexible volume constraints, which can govern either all or a subset of materials at both global and local scales. Additionally, we introduce a voxel-based post-processing strategy to ensure scalable designs, smooth material transitions, and tunable scale separation. Key insights are illustrated through meaningful numerical examples, demonstrating the effectiveness of the proposed framework. The methodology highlights the versatility of TPMS-based architectures in achieving optimal material distribution with arbitrary geometric complexity.