A symmetry reduction method for continuum structural topology optimization
Section snippets
Introduction and motivation
Continuum structural topology optimization is an increasingly powerful design tool which can be used to optimize the material arrangements in structural systems to achieve a wide variety of performance objectives, including as examples: minimal compliance1, 2, 3, 4; optimal strength5, 6; viscoelastic damping[7]; and compliant mechanisms8, 9, 10. For all of these structural or material topology design optimization applications, if the design domain, the boundary conditions and the loading are
Distribution of materials
The complete undeformed spatial domain of the structure being designed is denoted by ΩS; its designable subset by ΩD; and its non-designable subset, in which the spatial/topological arrangement of materials is taken to be fixed, by ΩN. The arrangement of two pre-selected candidate materials and in ΩD remains to be determined and so this region is called designable. A set of single or multiple loading/boundary conditions to which ΩS will be subjected are specified and a starting design b(0)
Motivation
There are two primary reasons for imposing geometrical symmetry on material layout designs. First, even when the design loads are asymmetrical the designer may still seek a symmetric design so that reversed loading cases can be accommodated equally well. Second, even when both the design loads and boundary conditions applied to ΩS are symmetrical, the resulting material layout field b as obtained through standard optimization techniques, will not necessarily feature the expected geometrical
Motivation
Many structures whose structural stiffnesses and mass distributions are the same and symmetrical in two or more orthogonal directions invariably encounter the phenomenon of repeated, non-simple vibrational eigenvalues. It is recognized that the design sensitivity analysis of repeated eigenvalues is difficult due to the fact that they are generally not continuously differentiable, but are only directionally differentiable[17]. Accordingly, design sensitivities for repeated vibrational
Summary and conclusions
It has been hypothesized that asymmetrical material layout solutions are caused by the non-convexity of highly penalized continuum topology design formulations coupled with limited precision in numerical computations. In this study, a novel symmetry reduction method to control and stabilize non-convex topology design formulations has been investigated and demonstrated on simple, representative examples.
Benefits of the proposed symmetry reduction method are that it produces symmetrical material
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