High-dimensional numerical optimization of fiber reinforced polymers with variational autoencoders and Bayesian optimization

Abstract

Designing engineering structures relies on accurate numerical simulations to predict the behaviour of a structure before its realization. In the design process, many variables influence the final structure. There is an incentive for optimizing the design based on the desire for cheaper structures and less material usage. Although a wide variety of optimization techniques exist, in practice many structures are not fully optimized and could fulfill their purpose with less material usage. This partly results from optimization techniques generally requiring many simulations for problems with a high number of variables. In creating increasingly complex models, simulating its performance can take up to days or weeks to compute, incentivizing the development of optimization methods that use as little simulations as possible. The goal of this thesis is to provide a method of optimization which significantly reduces the required number of simulations to be performed, aiming to overcome the issue of required computational effort. The optimization problem used in this thesis is the geometric optimization of a fiber reinforced polymer (FRP) microstructure. Each evaluation requires a Finite Element analysis, and many parameters are required to describe the geometry. The goal is to find the maximum stress at perfect plasticity during uniaxial tension, interpreted as a measure of strength. While generally only fibers with a circular cross-section are used, here a morphing parameter is introduced that changes this shape between being circular and square. Furthermore, fibers are allowed to overlap, effectively creating a single fiber with a complex shape. This geometry optimization provides a challenging problem for which different optimization techniques can be compared. The approach taken in attempting to minimize the number of function evaluations is a combination of several machine learning methods. Based on a limited number of initial samples, Bayesian optimization (BO) is applied. In BO a prediction model in the form of a Bayesian Neural Network (BNN) is created and used to inform further sampling. This prediction model provides a mean and a standard deviation in its prediction, both of which are used to find the best point to sample next. The number of initial samples required to create an accurate prediction model increases exponentially with the number of parameters. It is therefore opted to first use a variational autoencoder (VAE) to reduce the number of parameters in which BO is performed, by encoding the parameters in a lower dimensional representation. The variational autoencoder does this without requiring any function evaluation. Results show that when encoding the original parameters using the VAE, the encoded representation is unable to recreate all possible configurations of the original parameters. Furthermore, by transforming to a lower dimensional representation, within this representation the complexity of a function increases compared to the original function, generally leading to many local optima. This complexity proves difficult for the BNN to accurately predict based on a limited number of samples. As a result, BO does optimize the result, but does not reliably find the global optimum of the reduced parameter space. No clear conclusion can be made on the overall performance of the method compared to alternative optimization methods. Recommendations are made for what additional studies could be performed. Further recommendations are given for solving issues limiting the current performance as well as possible additions to the framework. The fiber geometry optimization serves as a good case to compare different optimization methods. It is shown that the geometry has a significant influence on the mechanical performance of a FRP microstructure, and optimization is therefore beneficial. The load case considered is however too specific for the optimized result to have direct practical use. Still, it does demonstrate that the results of an optimization study can be generalized. As the optimization framework is data driven, the optimization objective could easily be extended to study more realistic problems.