Bayesian optimisation technique with a Gaussian process as the surrogate model for structural problems with random imperfections

Optimisation of linear structures is a well-developed and highly relevant research field. In particular, for linear problems with differentiable objective functions, gradient-based methods are well suited. In engineering practice, structural nonlinear behaviour and imperfections different from ideal design often need to be considered. Optimised lightweight structures, such as long-span bridges and stadium roofs, are more sensitive to sudden failure through buckling. Imperfections or perturbations in geometry and loading can drastically reduce the buckling load of a structure compared to that of a perfect structure. It is, however, computationally expensive to analyse structural problems with nonlinearities and imperfections and to compute the derivatives of the respective objective functions, rendering gradient-based methods unsuitable. In the present work, we use a Bayesian optimisation technique with a Gaussian process as the surrogate model for the described structural problems with random imperfections. Bayesian optimisation is exceptionally well suited for expensive black-box objective functions with no analytical expressions. To rapidly determine the random buckling loads, we consider an extended system of equilibrium equations for directly computing the buckling loads, i.e. bifurcation and snap-through points on the load-displacement path. This approach is exceedingly efficient for repeatedly computing buckling loads for different imperfections. We use the complex-step derivative approximation to determine the required directional derivatives of matrices. A low discrepancy sequence, Sobol sequence, is adopted for sampling from imperfections