Bayesian buckling load optimisation for structures with geometric uncertainties

TL;DR

This work addresses robust optimisation of buckling loads in geometrically nonlinear structures under random geometric imperfections by modeling imperfections as mode amplitudes $eta_i hicksim ext{N}(0,\sigma_{eta_i}^2)$ along buckling modes $oldsymbol{\phi}_i$ and computing the resulting buckling-load distribution via Monte Carlo sampling. It introduces an efficient workflow that combines the extended system method for direct stability-point computation, Sobol quasi-Monte Carlo sampling to generate imperfection samples, and Gaussian-process-based Bayesian optimisation to select cross-sectional areas $oldsymbol{a}$ using the objective $g(oldsymbol{a})=oldsymbol{ extstylerac{ar{oldsymbol{\lambda}}_c(oldsymbol{a})}{ar{oldsymbol{\lambda}}_c^*}}$ and standard-deviation term, i.e. $g(oldsymbol{a})= abla$. Buckling loads are computed with a direct stability-point search, while the GP surrogate and EI acquisition guide the design search, achieving accurate mean and std estimates with far fewer FE evaluations thanks to Sobol sampling. Demonstrations on a two-ring and a five-ring star dome and a truss column show improved mean buckling loads and reduced variability for appropriate trade-offs between mean and std, with the approach scalable to modest-dimensional problems. The framework holds potential for extension to shells, beams, and topology optimization, and for integration with commercial FE packages.

Abstract

Optimised lightweight structures, such as shallow domes and slender towers, are prone to sudden buckling failure because geometric uncertainties/imperfections can lead to a drastic reduction in their buckling loads. We introduce a framework for the robust optimisation of buckling loads, considering geometric nonlinearities and random geometric imperfections. The mean and standard deviation of buckling loads are estimated by Monte Carlo sampling of random imperfections and performing a nonlinear finite element computation for each sample. The extended system method is employed to compute the buckling load directly, avoiding costly path-following procedures. Furthermore, the quasi-Monte Carlo sampling using the Sobol sequence is implemented to generate more uniformly distributed samples, which significantly reduces the number of finite element computations. The objective function consisting of the weighted sum of the mean and standard deviation of the buckling load is optimised using Bayesian optimisation. The accuracy and efficiency of the proposed framework are demonstrated through robust sizing optimisation of several geometrically nonlinear truss examples.

Figures (19)

• Figure 1: Optimisation of a pin-jointed star dome truss. The aim is to maximise the mean and to minimise the standard deviation (std) of the buckling load, and the design variables are the cross-sectional areas of struts. The optimisation with respect to only the mean buckling load in figure (c) yields the mean buckling load $\overline{\lambda}_c/\overline{\lambda}_c^* = 1.0$ and standard deviation $\sigma_{c}/\sigma_{c}^* = 1.0$. The optimisation with respect to the mean and standard deviation of the buckling load in figure (d) yields the mean buckling load $\overline{\lambda}_c/\overline{\lambda}_c^* = 0.646$ and standard deviation $\sigma_{c}/\sigma_{c}^* = 0.491$. The two parameters $\lambda_c^*$ and $\sigma_{c}^*$ are normalising constants. For further details see Section \ref{['sec:examples']}.
• Figure 2: Computation of stability points of a Von Mises truss using the extended system method, with and without path-following. The yellow curves represent the analytical solution of the load-displacement relationship. Figure (b) shows the combination of the nonlinear path-following and extended system method converging to the first stability point. Figures (c) and (d) are the convergence of the extended system method to the first and second stability points, respectively.
• Figure 3: A $2$D Sobol sequence with consecutive sets. Figures (a) and (b) show two consecutive sets with $2^7=128$ points in each set. Figure (c) shows the combination of the two sets with $256$ points. Note that each set is fairly uniformly distributed.
• Figure 4: Illustration of determining the buckling loads of a von Mises truss with random geometric imperfections. Positive and negative amplification factors are separated and sorted. The repeated computations start with the as-designed truss using the combination of the path-following and the extended system method and then iterate through every geometric imperfection with the extended system method only.
• Figure 5: Comparison of the quasi-random Sobol sampling and pseudorandom sampling of the geometric imperfection parameter $\beta$ and the corresponding histograms of the buckling load $\lambda_c$. All histograms use 1024 samples. The solid lines are kernel density estimates.