Approximate geometric non-linear analysis in topology optimization

Abstract

Density-based topology optimization, when used to design structures that show geometrical non-linear behaviour, currently faces computational effort and stability issues. These issues are caused by the iterative method used in geometric non-linear structural analysis. On top of taking much computational effort to complete, this method encounters instabilities when analyzing low-density elements usually present in the design domain. This study aims to bypass those issues by proposing approximate analysis in the topology optimization routine, which is an analysis based on an approximation of the geometrical non-linear load-deflection curve of a structure, constructed with equilibrium points close to its undeformed configuration. To study the performance and the influence of the parameters that govern approximate analysis, three numerical examples are considered. These indicate that using approximate analysis in topology optimization leads to designs that perform over a finite range of motion, similar to when a non-linear analysis is used. The computational effort needed for approximate analysis is closer to the effort needed for linear analysis than non-linear analysis. A limitation of approximate analysis is that its results are only similar to non-linear analysis as long as the deflections stay in the mildly non-linear domain. When concerning the topology optimization of compliant mechanisms that exhibit mildly geometric non-linear behaviour, we conclude that using approximate analysis is a stable and computationally efficient alternative to non-linear analysis.