Multidelity approaches for design under uncertainty

Abstract

Uncertainties are present in many engineering applications and it is important to account for their effects during engineering design to achieve robust and reliable systems. One approach is to represent uncertainties as random inputs to the numerical model of the system and investigate the probabilistic behaviour of the model outputs. However, performing optimization in this setting can be computationally expensive, requiring many evaluations of the numerical model to compute the statistics of the system metrics, such as the mean and the variance of the system performance. Fortunately, in many engineering applications, there are one or more lower fidelity models that approximate the original (high-fidelity) numerical model at lower computational costs. This thesis presents rigorous multifidelity approaches to leverage cheap low-fidelity models and other approximations of the expensive high-fidelity model to reduce the computational expense of optimization under uncertainty. Solving an optimization under uncertainty problem can require estimates of the statistics at many different design points, incurring a significant number of expensive high-fidelity model evaluations. The multifidelity estimator is developed based on the control variate method to reduce the computational cost of achieving a specified root mean square error in the statistic estimate by making use of the correlation between the outputs of the expensive high-fidelity model and the outputs of the cheap low-fidelity model. The method optimally relegates some of the computational load to the low-fidelity model based on the relative model evaluation cost and the strength of the correlation. It has demonstrated 85% computational savings in an acoustic horn robust optimization example. When the model is sufficiently smooth in the design space in the sense that a small change in the design variables produces a small change in the model outputs, it has an autocorrelation structure that can be exploited by the control variate method. The information reuse estimator is developed to reduce the computational cost of achieving a specified root mean square error in the statistic estimate by making use of the correlation between the high-fidelity model outputs at one design point and those at a previously visited design point. As the optimization progresses towards the optimum in the design space, the steps taken in the design space often become shorter, increasing the correlation and making the information reuse estimator more efficient. To further reduce the computational cost, the combined estimator is developed to incorporate the features of both the multifidelity estimator and the information reuse estimator. It has demonstrated 90% computational savings in the acoustic horn robust optimization example. The methods developed in this thesis are applied to two practical aerospace applications. In conceptual aircraft design, there are often uncertainties about the future developments of the underlying technologies. The information reuse estimator can be used to efficiently generate a Pareto front to study the trade off between the expected performance and the risk induced by the uncertainties in the different aircraft designs. In a large-scale wing robust optimization problem with uncertainties in material properties and flight conditions, the combined estimator demonstrated a reasonable solution turnaround time of 9.7 days on a 16-processor desktop machine, paving the way to a larger scale wing optimization problem with distributed uncertainties to account for degradation or damage.