Two‐scale topology optimization in computational material design: An integrated approach

Abstract

SummaryIn this work, a new strategy for solving multiscale topology optimization problems is presented. An alternate direction algorithm and a precomputed offline microstructure database (Computational Vademecum) are used to efficiently solve the problem. In addition, the influence of considering manufacturable constraints is examined. Then, the strategy is extended to solve the coupled problem of designing both the macroscopic and microscopic topologies. Full details of the algorithms and numerical examples to validate the methodology are provided.

Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework

“…where U χ (x) is the nominal elastic energy density 21 . The result in equation ( 86) is close to the one obtained for the sensitivity in the (regularized/smoothed) method SIMP [8], except for the material exchange term ∆χ(x) (see equation (19)). Now the original compliance functional J (h) , in equation ( 81)-(a) can be extended to account for the restriction in equation ( 81)-(b).…”

“…where sgn(∆χ(x)) = −1 ∀x ∈ Ω + and sgn(∆χ(x)) = 1 ∀x ∈ Ω − (see equation (19)). The corresponding topological derivative, at pointx ∈ Ω, can be now computed from equations (22) and (30) as…”

“…c) The Laplacian smoothing,ξ (i) t n ←ξ (i) τ,t n , is neither affected by the initialization operations 19 .…”

“…t n (x) and λ (i) t n , their difference, ψ (i) t n ≡ ξ (i) t n (x) − λ (i) t n , and the corresponding topology,χ (i) t n = H β (ψ (i) t n ), is not affected by that shift. Also the normalization in equation (75) does not affect neither the sign of ψ (i)t n nor the topology.c) The Laplacian smoothing,ξ (i) t n ←ξ (i) τ,t n , is neither affected by the initialization operations19 .d) In consequence, if the discrimination function ψ (i) t n , in equation (76)-(a), is applied in terms of the shifted and normalized entities,ξ (i) τ,t n (x) andλ (i) t n at Ω + (see equation (76)), the topology, at convergence, is not affected. However, the robustness of the iterative process is remarkably increased 17.…”

Linear and nonlinear topology optimization design with projection‐based ground structure method (P‐GSM)

“…Guo and Zhang et al proposed a homogenization framework with the use of asymptotic analysis to achieve the fast design of devices filled with quasiperiodic microstructure, where a mapping function is implemented to transform an infill graded microstructure to a spatially periodic configuration. Some other advanced multiscale design methods for simultaneous achieving macro‐ and microscale topology optimization or nonlinear metamaterial design can be found in References .…”