Large-scale buckling-constrained topology optimization based on assembly-free finite element analysis

Abstract

In this article, we propose a fast method to solve large-scale three-dimensional topology optimization problems subject to buckling constraints. Buckling analysis entails the solution of a generalized eigenvalue problem. For problems with large degrees of freedom, the current numerical methods tend to be memory-hungry, leading to high computational costs. First, a low-memory assembly-free linear buckling analysis method is proposed. Specifically, this method is based on the voxelization model, an assembly-free version of the deflated conjugate gradient is used to accelerate the iteration solution of linear systems of equations, where neither the stiffness matrix nor the deflation matrix is assembled, and the parallelization of matrix-vector multiplication is achieved by the congruency of voxels. Due to the particularity of the stress stiffness matrix in the buckling analysis, the inverse iteration is used to solve the general eigenvalue problem, which can reduce the operations of stress stiffness matrix considerably. Based on the efficient buckling analysis method, we extend the level-set method for buckling constraints in a semi-analytical manner. Several numerical experiments demonstrate that the proposed method can solve large-scale three-dimensional buckling analysis and topology optimization against buckling constraints effectively.

Topological gradient in structural optimization under stress and buckling constraints

“…Considering buckling phenomena in the context of topological optimization brings two kinds of difficulties: one related to the computational cost of the buckling analysis (Bian & Fang (2017)) and the other one to the poor convergence of some optimization processes (Bruyneel et al (2008); Gao et al (2020)).…”

“…Based on a shift-invert technique and a matrix factorization, Dunning et al (2016) improved the rate of convergence of the Block Jacobi Conjugate Gradient method, but their aproach is highly memory consuming. The authors of Bian & Fang (2017) proposed an assembly-free iterative solver to efficiently estimate the critical buckling load.…”

“…In the context of structural optimization under buckling constraints, it is usual to compute only the critical buckling load as it is considered enough to assess the structural resistance to buckling (Lindgaard & Dahl (2013);Luo & Tong (2015); Bian & Fang (2017)). However, as highlighted by Bruyneel et al (2008), as the topology of the structure changes during optimization, the buckling mode associated to the critical buckling load may also change, leading to a slow convergence or to divergence of the optimization process.…”

A general global-local modelling framework for the deterministic optimisation of composite structures

“…The same authors, in Ijsselmuiden et al (2010), use the approach of Setoodeh et al (2009) for VAT buckling load maximisation. Interesting is the work of Bian and Fang (2017), who propose an assembly-free method for TO. In Thomsen et al (2018), authors derive the buckling gradient for TO problems in the framework of the Solid Isotropic Material Penalisation (SIMP) approach.…”

Towards solving large-scale topology optimization problems with buckling constraints at the cost of linear analyses

“…Distribution of coefficients c jr , as defined by(14) for the two optimized designs ofFigure 1(b,c) (left and right, respectively). Values c jr > 0.95 are shown in red and values c jr ∈ [0.85, 0.95] are shown in blue, while c jr < 0.85 are not represented.…”