Bayesian buckling load optimisation for structures with geometric uncertainties

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.

• The paper presents a framework combining Bayesian optimization and robust methods to account for geometric uncertainties in structural buckling load optimization.
• Numerical results show improved computational efficiency using Quasi-Monte Carlo methods and highlight the framework's ability to balance structural performance and variability.
• The framework is applicable to various structural types and could integrate with machine learning for future advancements in adaptive structure optimization.

An Analysis of "Bayesian Buckling Load Optimisation for Structures with Geometric Uncertainties"

The paper "Bayesian Buckling Load Optimisation for Structures with Geometric Uncertainties" by Liu, Xiao, and Cirak presents an innovative framework for optimising the buckling loads of structures with inherent geometric imperfections. Such structures are susceptible to sudden buckling failures due to non-linear geometric properties and imperfections introduced during manufacturing or assembly. This research addresses how these uncertainties can be robustly accounted for in structural optimization, significantly advancing the stability analysis domain.

Methodological Framework

The proposed framework incorporates robust optimisation, uncertainty quantification, and advanced computational techniques to enhance the reliability of buckling load estimates. Key methodological elements include:

1. Geometric Nonlinearity and Random Imperfections: The geometric imperfection parameters are modelled as random variables, encapsulating real-world deviations from ideal structural configurations. The study particularly models these imperfections through a multivariate normal distribution reflecting realistic manufacturing deviations.
2. Monte Carlo Sampling: It utilizes Monte Carlo sampling to evaluate different outcomes by generating a range of possible imperfections. This process is computationally intensive as it involves multiple nonlinear finite element (FE) computations per sample.
3. Extended System Method: To efficiently compute buckling loads, the research adopts the extended system method. This approach streamlines calculations by avoiding traditional, computationally expensive path-following analysis.
4. Quasi-Monte Carlo Sampling: Sobol's sequence is employed for quasi-Monte Carlo sampling, ensuring a more uniform distribution of sample points, which enhances convergence rates and reduces computational burden compared to traditional random sampling techniques.
5. Bayesian Optimisation: The optimisation considers both the mean and standard deviation of buckling loads, thus supporting robust designs that exhibit less sensitivity to geometric uncertainties. The Bayesian optimisation technique ensures an efficient search for the global optimum without derivative calculations, which are infeasible for complex objective functions like those derived from nonlinear FE analysis.

Numerical Results and Implications

The paper reports significant improvements in computational efficiency, notably when using Sobol sampling, which decreased the number of required FE analyses without sacrificing accuracy. Remarkably, an empirical test involving a von Mises truss showed only minimal error with a reduced sample size, validating the high fidelity of the proposed model.

The paper's examples, ranging from simple to complex truss structures, illustrate the breadth of the framework's applicability. These cases highlight how trade-offs between structural mean performance and its variability (due to geometric imperfection) can be systematically analysed and optimized. The study found that optimising for both metrics shifted the distribution of buckling loads towards configurations that balance performance stability and uncertainty sensitivity.

Future Directions

The investigation into Bayesian optimisation for structural design optimisation opens several avenues for future research. It suggests the possibility of extending this framework beyond truss structures to other forms such as beams, shells, and even complex composite materials. Moreover, this methodology could integrate with machine learning models to predict nonlinear structural behaviours more commonly, perhaps paving the way for real-time optimisation in adaptive structures.

In conclusion, the research offers considerable advancements in buckling load optimisation, combining probabilistic methods with state-of-the-art computational techniques to address the challenges posed by geometric uncertainties in structural engineering. Such work reinforces the need for robust optimisation solutions, bridging theoretical rigor with practical engineering applications.