Dissertation
Stochastic optimization for design under uncertainty with dependent random variables
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Autumn 2021
DOI: 10.17077/etd.006340
Abstract
The overarching goal of this study is to provide a rigorous theoretical foundation accompanied by robust numerical algorithms for design optimization of complex systems subject to random input following arbitrary, dependent probability distributions. To achieve the goal, four research tasks were established: (1) construction of a practical version of a generalized polynomial chaos expansion (GPCE) for uncertainty quantification (UQ) analysis; (2) development of novel robust design optimization (RDO) methods by integrating GPCE and design sensitivity analysis of statistical moments; (3) development of novel reliability-based design optimization (RBDO) methods by integrating GPCE and design sensitivity analysis of reliability; and (4) development of novel stochastic design optimization methods featuring RDO and RBDO for high-dimensional complex systems.
The four major achievements of this study are as follows. First, a practical refinement of GPCE was developed for UQ analysis of stochastic mechanical systems in the presence of arbitrary, dependent probability measures of input variables. Unlike the Rodrigues-type formula, which exists for select probability measures, a three-step computational algorithm is put forward to generate a sequence of any approximate measure-consistent multivariate orthonormal polynomials. For UQ analysis under dependent random variables, two regression methods, comprising existing standard least squares (SLS) and newly developed partitioned diffeomorphic modulation under observable response preserving homotopy (D-MORPH), are proposed to estimate the coefficients of GPCE for the very first time. The partitioned D-MORPH regression is applicable to either an underdetermined or an overdetermined system, thus substantially enhancing the ability of the original D-MORPH regression. Second, new computational methods were invented for RDO of complex engineering systems subject to input random variables with arbitrary, dependent probability distributions. The methods are built on the GPCE for determining the second-moment statistics of a general output function of dependent input random variables, an innovative coupling between the GPCE and score functions for calculating the statistical moment sensitivities with respect to the design variables, and a standard gradient-based optimization algorithm, establishing direct GPCE, single-step GPCE, and multi-point single-step GPCE design processes. New analytical formulae were developed for design sensitivity analysis that is synchronously performed with statistical moment analysis. Third, a new computational method was created for RBDO of complex mechanical systems subject to input random variables following arbitrary, dependent probability distributions. It involves a GPCE for reliability analysis subject to dependent input random variables, a novel synthesis between the GPCE and score functions for estimating the sensitivities of the failure probability with respect to design variables, and a standard gradient-based optimization algorithm, resulting in a multi-point single-step design process. The method, designated as the multi-point single-step GPCE method or simply the MPSS-GPCE method, yields analytical formulae for computing the failure probability and its design sensitivities concurrently from a single stochastic simulation or analysis. Finally, new computational methods for high-dimensional design optimization in the presence of input random variables with arbitrary, dependent probability distributions were generated. The methods feature a dimensionally decomposed GPCE (DD-GPCE) for statistical moment and reliability analyses associated with a high-dimensional stochastic response; a novel synthesis between the DD-GPCE approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability; and a standard gradient-based optimization algorithm, constructing the single-step DD-GPCE and multi-point single-step DD-GPCE (MPSS-DD-GPCE) methods. In these new design methods, the multivariate orthonormal basis functions are assembled consistent with the chosen degree of interaction between input variables and the polynomial order, thus helping to deflate the curse of dimensionality to the extent possible. In addition, when coupled with score functions, the statistical moments, failure probability, and their design sensitivities are determined concurrently from a single stochastic analysis or simulation.
The innovative formulations or methods for RDO and RBDO and associated numerical tools developed in this study are applicable to a broad multidisciplinary design optimization methodology. Numerous mathematical and solid-mechanics problems were tested to establish high accuracy and computational efficiency of the proposed methods. In addition, these methods were demonstrated to solve industrial-scale, complex design optimization problems from several engineering disciplines.
Details
- Title: Subtitle
- Stochastic optimization for design under uncertainty with dependent random variables
- Creators
- Dongjin Lee
- Contributors
- Sharif Rahman (Advisor)Jia Lu (Committee Member)Hiroyuki Sugiyama (Committee Member)Shaoping Xiao (Committee Member)Osnat Stramer (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mechanical Engineering
- Date degree season
- Autumn 2021
- DOI
- 10.17077/etd.006340
- Publisher
- University of Iowa
- Number of pages
- xvii, 341 pages
- Copyright
- Copyright 2021 Dongjin Lee
- Language
- English
- Description illustrations
- illustrations (some color)
- Description bibliographic
- Includes bibliographical references (pages 323-341).
- Public Abstract (ETD)
The design of real-life engineered systems and products involves uncertainties in applied forces, material properties, and manufacturing processes. These uncertainties directly impact the performance of a product, resulting in a significant loss of revenue or even catastrophic failure. In this Ph.D. study, design optimization of complex systems in the presence of uncertainty or random variables following arbitrary, dependent probability distributions was conducted. The research involves fundamental developments of novel computational methods to solve two major archetypes of design under uncertainty: (1) robust design optimization, which improves product quality by reducing the sensitivity of an optimal design; and (2) reliability-based design optimization, which focuses on attaining an optimal design by warranting a sufficiently low risk of failure. Depending on the objective set forth by a designer, uncertainty is effectively controlled by these design optimization methods. The innovative formulations or algorithms of statistical moment and reliability analyses, design sensitivity analysis, and optimization algorithms–essential components of the computational design methods developed–have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new design methods have the power to tackle industrial-scale design optimization. Potential engineering applications include any transportation system design for improved durability and crashworthiness; fatigue- and facture-resistant design for civil, biotech, and aerospace applications; and reliable design of microelectronic packaging in harsh environments, to name a few.
- Academic Unit
- Mechanical Engineering
- Record Identifier
- 9984210943602771
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