**Sponsor: **U.S. National Science Foundation

**Project No.:** CMMI-1933114

**Duration:** September 1, 2019 – August 31, 2022

**Principal Investigator: **Professor Sharif Rahman

**Graduate Students: **Steven Dixler and Dongjin Lee

**SUMMARY**

This research will establish a sound mathematical foundation, create robust numerical algorithms, and build practical computational tools for design optimization of high-dimensional complex systems in the presence of uncertainty. The proposed effort will involve: (1) new multivariate orthonormal basis splines (B-splines) and non-uniform rational B-splines (NURBS), leading to a new spline dimensional decomposition (SDD) method; (2) new formulae and scalable algorithms of the SDD method for calculating the statistical moments, including estimation of design sensitivities of moments from score functions; and (3) new computationally expedient robust design optimization (RDO) algorithms comprising a single or at most a few stochastic simulations. The research is novel, debuting the stochastic version of B-splines and NURBS for the very first time. New formulae and scalable algorithms generated for estimating the statistical moments will account for discontinuous or nonsmooth stochastic responses. The integration with score functions will concurrently determine both the stochastic response characteristics and design sensitivities from the same computational effort, thereby solving RDO problems from a few stochastic simulations. As a consequence, the speed of design process will be substantially enhanced, producing rare or potentially unprecedented solutions to large-scale stochastic design optimization problems.

**Task 1:**

(1) Uncertainty Quantification by Optimal Spline Dimensional Decomposition

**Task 2:**

(1) Robust Design Optimization by Spline Dimensional Decomposition

**Task 3: **

(3) High-Dimensional Stochastic Design Optimization

**Task 1: **

**Uncertainty Quantification by Optimal Spline Dimensional Decomposition:** A novel dimensional decomposition, designated as optimal SDD, is introduced for general high-dimensional uncertainty quantification analysis of complex systems. The method is premised on optimally derived knot vectors of B-splines in some or all coordinate directions, whitening transformation producing measure-consistent orthonormal B-splines equipped with optimal knots, and Fourier-spline expansion of a general high-dimensional output function of interest. Compared with the standard SDD, developed in a prior work, the knot vectors are not mandated to be uniformly distributed or selected instinctively. Two distinct approaches, the one exploiting orthonormal basis and the other utilizing non-orthonormal basis, are advocated for determining optimally the location of knots by minimizing the mean-squared approximation error. In doing so, no expensive-to-evaluate function evaluations, in addition to those required to estimate the expansion coefficients, are incurred, meaning that the optimization can be performed with little extra effort. Indeed, the generation of optimal knot vectors can be viewed as an inexpensive pre-processing step toward creating the optimal SDD. Thereafter, the optimal SDD method is formed by retaining relevant tensor products of univariate splines associated with a chosen degree of interaction from the ANOVA decomposition. Analytical formulae have been proposed to calculate the second-moment properties by the optimal SDD method for a general output random variable in terms of the expansion coefficients involved. Akin to that of the standard SDD method, the computational complexity of the optimal SDD method is polynomial, as opposed to exponential, thus alleviating the curse of dimensionality to an appreciable magnitude.

Numerical results indicate that the optimal SDD method is more precise than the standard SDD method in predicting not only the variance, but also the probability distribution function of an output random variable, both demanding practically the same computational resources. More importantly, a low-order optimal SDD approximation with an adequate mesh size generates a substantially more accurate estimate of the output variance than a high-order approximation from existing polynomial chaos expansion or sparse-grid quadrature. The optimal SDD method proposed is most relevant under locally nonlinear or non-smooth behavior often recognized in engineering applications.

**Task 2: **

**Robust Design Optimization by Spline Dimensional Decomposition:** Two novel, spline-based computational methods were developed for robust design optimization of complex engineering systems. The methods feature SDD of a high-dimensional, discontinuous, or nonsmooth stochastic response for statistical moment analysis, a novel fusion of SDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms, constructing direct and multi-point single-step design processes. In the SDD-based methods, their orthonormal basis functions are derived from compactly supported B-splines, resulting in an excellent approximation power for locally pronounced, highly nonlinear, discontinuous, or nonsmooth stochastic output functions. Hence, a low-variate and/or low-degree SDD approximation with an adequate mesh size can produce remarkably accurate and convergent estimates of response moments. When integrated with score functions, SDD leads to explicit formulae, expressed in terms of the expansion coefficients, for approximating the design sensitivities of moments that are also accurate and theoretically convergent. More importantly, the statistical moments and design sensitivities are determined simultaneously and hence inexpensively from a single stochastic analysis or simulation.

Between the two methods developed, the direct SDD method is most straightforward, but it demands re-calculations of the SDD expansion coefficients at each design iteration. Therefore, it easily becomes computationally expensive. However, due to simplicity in the optimization algorithm, it was employed to evaluate the performance of SDD approximations in solving the elementary mathematical RDO problems. In contrast, the multi-point single-step SDD method adjusts local enforcement of SDD approximation, where the original RDO problem is converted into a series of local RDO problems defined on subregions of the entire design space. As a result, the method permits employing a low-degree SDD approximation to obtain a reliable RDO solution in the case of a large design space and locally highly nonlinear stochastic responses. Also, the latter method avoids the necessity of recomputing the expansion coefficients by reprocessing the old expansion coefficients, thus dramatically reducing the computational cost. Therefore, the multi-point single-step SDD method is capable of solving practical engineering problems, as successfully demonstrated by shape design optimization of an industrial-scale robotic gripper jaw.

**Task 3: **

**High-dimensional Stochastic Design Optimization under Dependent Random Variables by a Dimensionally Decomposed Generalized Polynomial Chaos Expansion**: Two innovative reconfigurations of generalized polynomial chaos expansion (GPCE), leading to the single-step dimensionally decomposed-GPCE (DD-GPCE) and the multi-point-single-step DD-GPCE (MPSS-DD-GPCE) method, were invented for high-dimensional stochastic design optimization of complex mechanical systems in the presence of input random variables with arbitrary, dependent probability distributions. The methods feature the DD-GPCE approximation for statistical moment and reliability analyses of a high-dimensional stochastic response; a novel synthesis between DD-GPCE 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 single-step DD-GPCE and MPSS-DD-GPCE methods. In these new design methods, the multivariate orthonormal basis functions are built consistent with the desired degree of interaction between input variables and the polynomial order, thus helping to alleviate the curse of dimensionality to a substantial magnitude. In addition, when integrated with score functions, the DD-GPCE approximation leads to analytical formulae for calculating the design sensitivities. More significantly, the statistical moments, failure probability, and their respective design sensitivities are determined concurrently from a single stochastic analysis or simulation.

Of the two design methods developed, the single-step DD-GPCE method, formulated globally on the entire design space, is highly efficient due to recycling of the expansion coefficients. However, it may not remain accurate or effective when confronted with overly large design spaces and/or high-dimensional stochastic responses. In contrast, the MPSS-DD-GPCE method ushers in a local enforcement of DD-GPCE approximations, where the original RDO or RBDO problem is converted into a series of concomitant local problems defined on subregions of the entire design space. As a result, the method allows employing a low-degree DD-GPCE approximation to obtain a reliable design solution even when a design space is large. Also, the latter method avoids the necessity of recomputing the expansion coefficients by reprocessing the old expansion coefficients whenever possible, thus dramatically reducing the computational cost. Therefore, the MPSS-DD-GPCE method is capable of solving practical engineering problems, as demonstrated by shape design optimization of an industrial-scale bogie side frame with 41 random variables.

**Supervision of Graduate Students: **Two graduate students, Messrs. Steven Dixler and Dongjin Lee, were involved in this project. Both students completed their respective studies, receiving M.S. (August 2022) and Ph.D. (December 2021) degrees in Mechanical Engineering. The PI was the advisor of both graduate students.

**Development of a New Course: **The PI developed a new intermediate-level course titled *Uncertainty Quantification and Design Optimization*, which was offered in Fall 2022 semester for the first time. In this course, the PI exposes undergraduate and graduate students to modern methods of probabilistic analysis and design optimization in the presence of uncertainty. The course materials focuses on two principal exemplars of design optimization: (1) robust design optimization (RDO) by propagating input uncertainties to system behavior, resulting in a reduced sensitivity of optimal design; and (2) reliability-based design optimization (RBDO) by fulfilling a prescribed reliability level, leading to optimal design with an acceptably low risk of failure.

**Insertion of Research Results into Undergraduate Graduate Courses: **The topics of RDO and RBDO were discussed in an intermediate-level course *Computer-Aided Engineering* (Spring 2021) and graduate-level course, *Probabilistic Mechanics and Reliability* (Fall 2021). In addition, students, in conjunction with the latter course, performed an independent study by introducing the new SDD-based methods developed in this project for solving RBDO problems.

**Invited and Plenary Lectures: **The PI’s group earned invitation to present lectures on uncertainty quantification and design optimization methods at national and international conferences as follows:

(1) 2020 EMI/PMC, New York, NY, May 2020 (two papers).

(2) 14th WCCM-ECCOMAS 2020, Paris, France, July 2020 (one paper).

(3) 16^{th} U.S. National Congress on Computational Mechanics, Chicago, IL, 2021 (two papers).

(4) 1^{st} International Conference on Mechanical System Dynamics, Nanjing, China, August 2022 (plenary paper).

(5) 2022 SIAM Conference on Uncertainty Quantification, Atlanta, GA, April 2022 (one paper).

**Journal Articles**

- Rahman, S., “Global Sensitivity Analysis by a Generalized Polynomial Dimensional Decomposition,” submitted to SIAM/ASA Journal on Uncertainty Quantification, 2022.
- Lee, D., Jahanbin, R., and Rahman, S., “High-Dimensional Stochastic Design Optimization under Dependent Random Variables by a Dimensionally Decomposed Generalized Polynomial Chaos Expansion,” submitted to International Journal for Uncertainty Quantification, 2022.
- Lee, D., Jahanbin, R., and Rahman, S., “Robust Design Optimization by Spline Dimensional Decomposition,” Probabilistic Engineering Mechanics, Vol. 8, Article 103218, 2022, pp. 1-27.
- Rahman, S. and Jahanbin, R., “A Spline Dimensional Decomposition for Uncertainty Quantification in High Dimensions,” SIAM/ASA Journal on Uncertainty Quantification, Vol. 10, pp. 404-438, 2022.
- Lee, D. and Rahman, S., “Reliability-based Design Optimization under Dependent Random Variables by a Generalized Polynomial Chaos Expansion,” Structural and Multidisciplinary Optimization, Vol. 65, Article 21, 2022, pp. 1-29.
- Dixler, S., Jahanbin, R., and Rahman, S., “Uncertainty Quantification by Optimal Spline Dimensional Decomposition,” International Journal for Numerical Methods in Engineering, 2021, pp. 1-37.
- Rahman, S. and Jahanbin, R., “Orthogonal Spline Expansions for Uncertainty Quantification in Linear Dynamical Systems,” Journal of Sound and Vibration, Vol. 512, 2021, Article 116366.
- Lee, D. and Rahman, S., “Robust Design Optimization under Dependent Random Variables by a Generalized Polynomial Chaos Expansion,” Structural and Multidisciplinary Optimization, Vol. 63, 2021, pp. 2425-2457.
- Rahman, S., “A Spline Chaos Expansion,” SIAM/ASA Journal on Uncertainty Quantification, Vol. 8, 2020, pp. 27-57.
- Lee, D. and Rahman, S., “Practical Uncertainty Quantification Analysis Involving Statistically Dependent Random Variables,” Applied Mathematical Modelling, Vol. 84, 2020, pp. 324-356.

**Papers in Conferences, Symposiums, and Congresses**

- Rahman, S., “Uncertainty Quantification in Linear Dynamical Systems by Orthogonal Spline Expansion,” Proceedings of First International Conference on Mechanical System Dynamics, Nanjing, China, August 2022.
- Rahman, S. and Jahanbin, R., “Stochastic Isogeometric Analysis on Arbitrary Multipatch Domains by Spline Dimensional Decomposition,” Proceedings of 2022 SIAM Conference on Uncertainty Quantification, Atlanta, GA, April 2022.
- Dixler, S. and Rahman, S., “An Adaptive-Sparse Spline Dimensional Decomposition Method for High-Dimensional Uncertainty Quantification,” Proceedings of 16
^{th}U.S. National Congress on Computational Mechanics, Chicago, IL, July 2021. - Lee, D. and Rahman, S., “Reliability-based Shape Design Optimization under Dependent Random Variables by a Generalized Polynomial Chaos Expansion,” Proceedings of 16
^{th}U.S. National Congress on Computational Mechanics, Chicago, IL, July 2021. - Rahman, S., “A Spline Chaos Expansion,” Proceedings of 14th World Congress on Computational Mechanics, Paris, France, July 19-24, 2020.
- Dixler, S., Jahanbin, R., and Rahman, S., “A Spline Dimensional Decomposition with Optimized Knot Sequences,” Proceedings of 2020 Engineering Mechanics Institute and Probabilistic Methods Conference, New York, NY, May 26-29, 2020.
- Lee, D. and Rahman, S., “Robust Design Optimization with a Generalized Polynomial Chaos Expansion under Dependent Input Variables,” Proceedings of 2020 Engineering Mechanics Institute and Probabilistic Methods Conference, New York, NY, May 26-29, 2020.

**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.