Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner

Abstract

This paper applies topology optimisation to the design of structures with periodic microstructural details without length scale separation, i.e. considering the complete macroscopic structure and its response, while resolving all microstructural details, as compared to the often used homogenisation approach. The approach takes boundary conditions into account and ensures connected and macroscopically optimised microstructures regardless of the difference in micro-and macroscopic length scales. This results in microstructures tailored for specific applications rather than specific properties.Dealing with the complete macroscopic structure and its response is computationally challenging as very fine discretisations are needed in order to resolve all microstructural details. Therefore, this article shows the benefits of applying a contrast-independent spectral preconditioner based on the multiscale finite element method (MsFEM) to large structures with fully-resolved microstructural details. * E-mail: joealex@mek.dtu.dk 1 arXiv:1411.3923v1 [cs.CE] 13 Nov 2014The density-based topology optimisation approach combined with a Heaviside projection filter and a stochastic robust formulation is used on various problems, with both periodic and layered microstructures. The presented approach is shown to allow for the topology optimisation of very large problems in Matlab, specifically a problem with 26 million displacement degrees of freedom in 26 hours using a single computational thread.

A practical multiscale approach for optimization of structural damping

“…That said, based on e.g. the recent numerical results presented by Alexandersen and Lazarov (2015), we have reason to believe that a slowly varying microstructure give macroscopic properties close to the ones predicted by homogenization theory.…”

“…An alternative approach is to predefine regions which should have the same material, and then optimize the material structure in each of these regions for a macroscopic response (as done by Andreasen and Sigmund (2012), and with a full solution of the macroscopic problem by Alexandersen and Lazarov (2015)). …”

Length scale and manufacturability in density-based topology optimization

“…Furthermore, the homogenization theory does not account for the boundary conditions and local effects (concentrated loading, loss of local stability). An alternative which provides spatially varying manufacturable microstructures without any post processing is presented in [4] and [3]. The presented methodology resolves all microstructural details and accounts for boundary conditions and localized effects.…”

“…Recently contrast independent preconditioners utilizing ideas both from domain decomposition and multigrid have been presented in [89]. The preconditioners have been applied in the design of robust manufacturable design with full resolution of the microstructural details [4], [3]. The solvers have provided an optimized solution overnight on a single CPU for several million degrees of freedom.…”

“…ρ ∈ D ad (4) where J (ρ, u) is an objective function (weight, material volume, mechanical compliance, displacements or velocities at a subdomain of the design domain), r (ρ, u) = 0 represents the residual of the equations modelling the physical system with solution u ∈ U ad , and {g i (·) ≤ 0, i = 1 . .…”