Sponsor: U.S. National Science Foundation
Project No.: CMMI-2317172
Duration: September 2023 – August 2026
Principal Investigator: Professor Sharif Rahman
Graduate Students: Md. Rashel Talukdar
SUMMARY
The chief goal of this research is to conduct foundational research on inventing efficient computational algorithms and creating practical computational tools for robust and reliability-based design optimization (RDO and RBDO) of high-dimensional complex systems subject to random input following an arbitrary dependent probability distribution.
The technical activities to meet the project goal, as outlined in the original proposal, comprise the following three tasks:
- Task 1: Novel mathematical developments of a generalized analysis-of-variance expansion, leading to a generalized spline dimensional decomposition (GSDD) for tackling dependent random variables head on;
- Task 2: New scalable algorithms of the GSDD method for calculating relevant probabilistic response characteristics and design sensitivities of a high-dimensional, complex mechanical system; and
- Task 3: Innovative GSDD-driven optimization algorithms for efficiently solving high-dimensional RDO and RBDO problems, including stochastic shape and topology design.
Task 1:
(1) Higher-Order Moment Analysis of Spline Chaos Expansion:
Task 2:
Task 3:
Task 1:
Higher-Order Moment Analysis of Spline Chaos Expansion: A UQ analysis entailing high-order moments calculated from SCE approximations of a real-valued, general output function of input random variable was conducted. The approximation quality of SCE was assessed in terms of the modulus of smoothness of the function. When the largest element of the mesh from SCE approaches zero, the modulus of smoothness vanishes, resulting in the higher-order moment convergence of SCE to the correct limit. Therefore, the moment of SCE of an arbitrary order converges to the exact moment for any degree of splines as the element size decreases. Moreover, the weaker modes of convergence, such as those in probability and in distribution, transpire naturally.
Numerical computations of moments by SCE and polynomial chaos expansion (PCE), conducted for a collection of simple yet relevant examples, indicate the following:
When the output function is smooth or nonsmooth and the random domain is bounded or unbounded, SCE and PCE both provide convergent estimates of the variance. However, their relative convergent rates may differ, depending on the smoothness of the function.
When the function is globally smooth, PCE is likely to provide more efficient estimates of variance than SCE for the same computational effort. However, for a nonsmooth function, the trend reverses, and the convergence properties of PCE in estimating variance may degrade appreciably.
Higher-order moments, such as skewness and kurtosis, calculated using SCE converge for all examples considered in this study. In contrast, moments of PCE of orders larger than two may or may not converge, depending on the regularity of the output function or the probability measure of input random variables.
Task 2:
Task 3:
Supervision of Graduate Students: A new Ph.D. student, Mr. Md. Rashel Talukdar, is going to join the PI’s group soon in August 2024. The student, to be appointed as a half-time graduate research assistant, will be supported by this NSF project. The PI is the advisor of the graduate student.
Insertion of Research Results into Undergraduate and Graduate Courses: In 2022, the PI developed a new intermediate-level course, titled Uncertainty Quantification and Design Optimization, exposing undergraduate and graduate students to modern methods of probabilistic analysis and design optimization in the presence of uncertainty. During the second offering in Fall 2024, the PI will discuss novel methods of UQ and robust/reliability-based design optimization developed from this and past NSF projects.
Invited Seminars and Lectures: The PI’s group earned invitations to present two lectures on uncertainty quantification and design optimization at Engineering Mechanics Institute Conference and Probabilistic Mechanics & Reliability Conference (EMI/PMC 2024), Chicago, IL, May 28-31, 2024. The PI presented both papers.
Journal Articles
Rahman, S., “Higher-Order Moments of Spline Chaos Expansion,” accepted in Probabilistic Engineering Mechanics, 2024 (DOI: https://doi.org/10.1016/j.probengmech.2024.103666).
Lee, D. and Rahman, S., “High-Dimensional Stochastic Design Optimization under Dependent Random Variables by a Dimensionally Decomposed Generalized Polynomial Chaos Expansion,” International Journal for Uncertainty Quantification, Volume 13, No. 4, pp. 23-59, 2023.
Papers in Conferences, Symposiums, and Congresses
Rahman, S., “A Spline Chaos Expansion for Uncertainty Quantification in Linear Dynamical Systems,” Proceedings of 2024 Engineering Mechanics Institute Conference and Probabilistic Mechanics & Reliability Conference (EMI/PMC 2024), Chicago, IL, May 28-31, 2024.
Lee, D. and Rahman, S., “A Generalized Polynomial Chaos Expansion for High-Dimensional Design Optimization under Dependent Random Variables,” Proceedings of 2024 Engineering Mechanics Institute Conference and Probabilistic Mechanics & Reliability Conference (EMI/PMC 2024), Chicago, IL, May 28-31, 2024.
Others
A number of computer codes were developed for performing various numerical simulations required in this project. For further information, please contact the PI (S. Rahman) at sharif-rahman@uiowa.edu.