The shape optimisation of compliant structures to produce a desired snap-through load-placement path

Abstract

The research problem completed in this thesis is to develop an efficient procedure to design the optimal shape of compliant mechanisms for specified load-deflection paths and snap-through behaviour. Here, computationally intensive simulations are required to approximate the entire load path as a function of the shape or spatial variables. Solving this problem efficiently, as will be demonstrated in this thesis, requires unconventional multidisciplinary strategies, as conventional modern techniques are ineffective or impractical. In the case of simulation in the loop, where the numerical simulation is evaluated directly in the optimisation loop, unavoidable numerical discontinuities are present in the objective function. These discontinuities grow in number and size as the complexity and dimensionality of the problem increases. Modern gradient-based optimisers are incapable of bypassing these discontinuities and terminate prematurely, misrepresenting these discontinuities as local minima. Therefore, this research advocates for the use of non-negative gradient projection points, utilised by gradient-only optimisation techniques, to define meaningful shape optimisation solutions. These techniques ignore the discontinuous changes in function value to find non-negative gradient projection points. Simulation in the loop is limited by the sequential nature of iterating from design to design, incurring the time penalty of having to wait for time-consuming simulations. Surrogate-based optimisation parallelises the time-consuming computational simulations, enabling computationally efficient surrogate models to be constructed instead. As the load paths evolve with systematic load application, these models evolve not only spatially but also temporally as a function of a pseudo-time variable. Spatial and temporal variables result in two sources of anisotropy. Firstly, the response anisotropy of the function as the temporal evolution of load-deflection curves is distinct from the spatial evolution as a function of shape variables. Secondly, sampling anisotropy as spatial variables are sampled distinctly from the usually densely sampled temporal variable resulting from iteratively evolving the load-deflection path. Response and sampling anisotropies can result in significant mismatches between the model forms from sampled data and typical isotropic kernels used in surrogate construction. This study develops two solutions that address the response and sampling anisotropies, respectively. The results definitively demonstrate that both sources of anisotropy need to be addressed to construct accurate surrogate models that are meaningful for the shape optimisation problem. First, a novel coordinate transformation scheme is developed to transform the function response to be more isotropic as a data pre-processing step. The key here is to utilise gradient information to estimate an updated isotropic reference frame, which also makes the strategy more tractable for higher dimensions. Secondly, sampling anisotropy is addressed by redistributing the surrogate kernels over the spatio-temporal domain and relying on regression to fit the surrogate response as opposed to limiting the centres to the sampling points. These improved surrogate models require significantly fewer computational resources to complete the optimisation problem as compared to placing the simulations directly in the optimisation loop.