Chapter 6. Conclusions
6.1 Summary and contributions of this dissertation
This dissertation focuses on developing active learning methods and applications to reliability assessment and design optimization for challenging and complex engineering systems. The four research objectives proposed in Chapter 1 were fulfilled: (1) PAK-Bn method was developed to assess the reliability of complex structural systems using active learning-based Kriging model; (2) The proposed AL- HGP efficiently estimated first-passage probability under stochastic wind excitations, the high-dimensional reliability problem; (3) QS-AGP method was proposed to identify the reliable optimal design of complex structures by constructing quantile surrogates and training the model; and (4) The further developed method, QS2-AGP treated high-dimensional RBDO problems using kernel-based quantile surrogates and sensitivity. The major developments and findings of this study are summarized as follows:
• PAK-Bn method was developed for structural reliability analyses. The main objective of PAK-Bn was to carry out an active learning process, i.e., adaptive selection of simulation points, with low computational costs from a reliability analysis standpoint. The method utilized a new learning criterion designed to identify important points that are located in the vicinity of the limit-state surface and, at the same time, contribute most to the failure probability. The uniformly
distributed samples in n-ball domain could further reduce the number of computational simulations and achieved efficient convergence.
• Several benchmark reliability problems, e.g., a system problem with high non- linearity, small failure probability, and multiple design points, and general engineering problems with moderate dimension, were investigated to demonstrate the accuracy and efficiency of the PAK-Bn method. In each example, the proposed method needed a small number of limit-state function evaluations to achieve accurate and converged estimates. PAK-Bn is expected to effectively deal with such challenging and time-consuming problems in practical engineering with robustness against the types of limit-state surface.
• The first-passage probability problems under stochastic wind loads were solved by the proposed AL-HGP method. AL-HGP utilized the Gaussian-process- based surrogates whose predictive mean and variance were employed to capture the conditional distribution of maximum response given the time-invariant basic random variable while handling heteroscedastic noise. This framework considered both uncertainties arising from the structural systems and the environmental wind loads. The proposed active learning framework further reduced the number of computational simulations by identifying the critical design of experiment (DoE) points that contribute most to the first-passage probability.
• The applications to the eight-story building system and transmission tower structure successfully demonstrated the performance and merits of the proposed AL-HGP method. In each example, the proposed method required fewer dynamic simulations to achieve accurate results, while the "non-adaptive”
surrogate-based estimates could produce inaccurate results with considerable variability. The transmission tower example demonstrated that AL-HGP could deal with high-fidelity computational simulations, e.g., finite element analyses, without losing the benefits and merits of the proposed method. Thus, AL-HGP is expected to effectively deal with such challenging and time-consuming problems in practical engineering.
• A new RBDO method using QS-AGP was proposed. The method utilized a quantile-based formulation to identify the probability-feasible design domain that satisfies the reliability requirements. An adaptive learning procedure was designed to further reduce the number of computational simulations by utilizing the exploration-exploitation trade-off based on quantile surrogates. The proposed optimization scheme relied on design samples and, thus, did not use an optimization algorithm or gradient information on cost and performance functions.
• The accuracy and efficiency of the QS-AGP were successfully tested through several RBDO problems featuring highly nonlinear performance functions, various distribution types, and complexity. In each example, the proposed method needed fewer performance function evaluations to achieve convergence to accurate results. QS-AGP is expected to effectively deal with such challenging and time-consuming RBDO problems in engineering practice.
• The quantile surrogate-based RBDO framework was further developed to handle the high-dimensional RBDO applications. The proposed QS2-AGP aims to find the reliable optimal solution accurately and efficiently by combining the adaptive training process of the quantile surrogates with the design optimization
procedure guided by the parameter sensitivity of quantile surrogates. By avoiding the use of pre-generated design samples and the MC-sample based procedure to fit the quantile surrogate in the previous QS-AGP, QS2-AGP achieved a superior level of efficiency, especially for RBDO problems with a larger number of design parameters. The formulations of the parameter sensitivity of the quantile surrogate also helped further reduce the computational costs.
• The performance and merits of the proposed QS2-AGP method were successfully demonstrated through several numerical examples including high- dimensional RBDO problems up to 15 design parameters and engineering applications. In each example, the proposed QS2-AGP method required fewer performance function evaluations in achieving convergence to accurate results than other RBDO methods. In addition, the proposed QS2-AGP demanded dramatically less computational cost than QS-AGP and treated high- dimensional RBDO problems effectively. Thus, QS2-AGP is expected to effectively deal with a variety of challenging and time-consuming RBDO problems including complex engineering systems and high-dimensional RBDO problems in which the number of the design parameters is considerably large.