Topology Optimization of Graded Truss Lattices Based on On-the-Fly Homogenization

Abstract

We introduce a computational framework for the topology optimization of cellular structures with spatially varying architecture, which is applied to functionally graded truss lattices under quasistatic loading. We make use of a first-order homogenization approach, which replaces the discrete truss by an effective continuum description to be treated by finite elements in a macroscale boundary value problem. By defining the local truss architecture through a set of Bravais vectors, we formulate the optimization problem with regards to the spatially varying basis vectors and demonstrate its feasibility and performance through a series of benchmark problems in 2D (though the method is sufficiently general to also apply in 3D, as discussed). Both the displacement field and the topology are continuously varying unknown fields on the macroscale, and a regularization is included for well-posedness. We argue that prior solutions obtained from aligning trusses along the directions of principal stresses are included as a special case. The outlined approach results in heterogeneous truss architectures with a smoothly varying unit cell, enabling easy fabrication with a tunable length scale (the latter avoiding the ill-posedness stemming from classical nonconvex methods without an intrinsic length scale).

X‐Ray Techniques Dedicated to Materials Characterization in Cultural Heritage

“…It is a locus of points in space that spans the entire space by enclosing the planes between neighboring lattice points through perpendicular bisectors. [43] Every Bravais lattice has a reciprocal lattice; [39] the reciprocal lattice is another expression of the crystal lattice in Fourier space. The Brillouin zone is a primitive cell of the reciprocal lattice, where the first Brillouin zone is the same as the Wigner-Seitz cell in the crystal lattice.…”

Cellular topology optimization on differentiable Voronoi diagrams

“…However, this approach requires a very high grid resolution and well‐designed filters to emerge local fine features 36,37 . Homogenization‐based topology optimization plus an additional dehomogenization‐based post‐processing step have been applied to design cellular structures in a multi‐scale way 38‐40 . The resulting optimized designs, however, are considered near‐optimal due to the reconstruction errors from dehomogenization.…”

“…36,37 Homogenization-based topology optimization plus an additional dehomogenization-based post-processing step have been applied to design cellular structures in a multi-scale way. [38][39][40] The resulting optimized designs, however, are considered near-optimal due to the reconstruction errors from dehomogenization. It is noted that a similar framework has also been applied to the optimal design of microreactors with space-filling microchannel flow fields.…”

Cellular Topology Optimization on Differentiable Voronoi Diagrams

“…First-principle methods guided by mathematically rigorous sensitivities and numerically efficient optimization frameworks have remained an unexplored field due to three interleaving challenges in tackling cellular structures' differentiation, generalization, and discretization. topology optimization plus an additional dehomogenization-based post-processing step have been applied to design cellular structures in a multi-scale way [38,39,40]. The resulting optimized designs, however, are considered near-optimal due to the reconstruction errors from dehomogenization.…”