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Significance Statement
Bidirectional evolutionary structural optimization BESO using a finite element mesh, first introduced as combination of evolutionary structural optimization and an additive evolutionary structural optimization ESO algorithms determines the optimal topology of a structure according to relative ranking of element sensitive numbers allowing materials to be removed and added simultaneously showing clear advantage over application of evolutionary structural optimization only.
Scientists from The University of Sydney presented a simple alternative to the complicated sensitivity analysis currently involved in dynamic and buckling criterion. Their study is published in journal, Structural and Multidisciplinary Optimization. The aim of the work was first concerned with maximization of fundamental frequency for a given structure followed by maximization of buckling load factor for a given structure.
A sensitivity analysis for both hard-kill and soft-kill bidirectional evolutionary structural optimization methods were outlined in order to know the equivalence between what is termed the dynamic von Mises stress and sensitivity analysis for vibration modes and the similarity between them was shown revealing that both are equivalent in terms of optimization formula. The dynamic von Mises stresses are nominal, not physical, but display the correct sensitivity distribution for the mode being studied.
Sensitivity analysis for buckling topology optimization for soft-kill and hard-kill algorithms was used in order to eliminate artificial buckling modes which may appear in the low density regions of soft-kill algorithms and that equivalence between what is termed the buckling von Mises stress and sensitivity analysis for buckling can be shown which was clearly observable. Likewise, the buckling von Mises stresses are not physical, but display the correct sensitivity distributions for the given buckling mode being studied.
A filter scheme was also used to smoothen element sensitivities across entire design domain including void regions which calculates model sensitivity numbers by taking the average of element sensitivity numbers connected to the node. This was employed in order to guarantee that a solution to the topology optimization problem exist. In cases where objective function and corresponding topology are not convergent, sensitivity numbers are to be averaged with their previous values.
Results shown when considering rectangular plate structure showed that despite difference in absolute magnitudes of dynamic von Mises stress and frequency sensitive number, ranking of elements remained similar. It can be concluded that using dynamic von Mises stress or frequency sensitive number as the sensitivity numbers in ESO/BESO formation will produce comparative results.
Cumulative distribution diagram considering the same rectangular plate structure showed similar trend for relative distribution of sensitivity numbers. Distribution when using dynamic von Mises stress was less which results to steadier convergence.
With plate dimensions of 0.15×0.1m divided equally into 45×30 quadrilateral plane, thickness t=0.01, Young’s modulus E=70GPa, Poisson’s ratio ν=0.3 and density ρ=2700kg/m3. An initial fundamental frequency ωno was set to be 2339Hz in order to maximize fundamental frequency for a prescribed material volume fraction of V=0.5. Results showed that final fundamental frequency using frequency sensitivity number αf and dynamic von Mises stress σvmf was 3368Hz and 3371Hz respectively showing that the optimized topologies are consistent.
The sequence took 46 iterations to converge when using frequency sensitivity analysis but 44 when dynamic von Mises stress was used as sensitivity function. Frequency sensitivity analysis showed slight drop in objective at iteration 29 after which it was admitted back into the optimum path due to the fact that the analysis is based on infinitesimal change in design variables, xi. Hence, it can be said that it’s only accurate for small changes in xi.
Using dynamic von Mises stress as the sensitive function resulted in a final topology consistent in all cases and the optimizer took fewer iterations to converge when using dynamic von Mises stress compared to sensitivity numbers.
When observing buckling criteria, relative distribution of buckling von Mises stress σvmb and buckling sensitivity analysis b had similar trend. Cumulative distributions under buckling objective followed the same trend with dynamic objective.
With dimension 1×1 equally divided by 60 x 60 quadrilateral plane, thickness t=0.01, Young’s modulus E=1, Poisson’s ratio ν=0.3 and density ρ=1 and material buckling fraction V=0.5. The minimum buckling load factor ʎ1 for buckling sensitive number b and buckling von Mises stress σvmb were found to be 0.1253 and 0.1264 respectively. Buckling sensitivity analysis had 59 iterations while buckling von Mises had 42 iterations which show similar trend with the dynamic objective.
Munk et al. (2016) results eradicate numerical issues associated with material models which also allows complex structures to be optimized for dynamic and buckling criteria.
Journal Reference
David J. Munk, Gareth A. Vio, Grant P. Steven. A Simple Alternative Formulation for Structural Optimisation with Dynamic and Buckling Objectives. Structural and Multidisciplinary Optimization, pp 1–18, 2016.
AMME, The University of Sydney, Sydney, Australia
Go To Structural and Multidisciplinary Optimization
