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

**Project No.:** CMMI-0969044

**Duration:** April 1, 2010 – March 31, 2014

**Principal Investigator:** Professor Sharif Rahman

**Graduate Student:** Xuchun Ren

**SUMMARY**

The objective of this research project is to create new theoretical foundations and numerical algorithms for large-scale, reliability-based design optimization (RBDO) of complex engineering systems. The proposed effort is based on: (1) a new extended polynomial dimensional decomposition (X-PDD) method for statistical moment and reliability analyses of a general, high-dimensional, stochastic system (Task 1); (2) integrated X-PDD and score functions for concurrently calculating the design sensitivities of statistical moments and reliability (Task 2); and (3) reliability-based and robust design optimization (RBDO/RDO) algorithms employing X-PDD and score functions (Task 3). The innovative formulation of statistical moment analysis, reliability analysis, and design sensitivities will dramatically accelerate RBDO and RDO processes. Therefore, the research constitutes a new and possibly paradigm-shifting advance towards solving large-scale, complex optimization problems in the presence of uncertainty. If successful, the results of this research are envisioned to be applicable to a broad multidisciplinary design optimization methodology. Potential engineering applications comprise ground vehicle design for improved durability and crashworthiness, fatigue- and fracture-resistant design for civil and aerospace applications, and reliable design of microelectronic packaging under harsh environments. Beyond engineering, potential application areas include energy, finance, management, scheduling, and transportation and logistics, where stochastic optimization plays a vital role. The transfer of knowledge created by this project will take place through organization of engineering design-related symposia, peer-reviewed journal publications, presentations at major conferences and meetings, software development, and student education. Educational goals include graduate student recruitment, implementation of software tools to upgrade existing graduate and undergraduate courses at The University of Iowa, and active participation in Iowa’s outreach programs for middle- and high-school students.

**Task 1: Development of a New Extended Polynomial Dimensional Decomposition (X-PDD) Method for Statistical Moment and Reliability Analyses of a General, High-Dimensional, Stochastic System (Year 1)**

**Orthonormal Polynomial Basis and Fourier-Polynomial Expansions:**A Stieltjes procedure was developed for generating non-classical orthonormal polynomial basis and associated Gauss-quadrature formula that are both consistent with the probability distributions (measures) of the random input. The procedure for generating the recursion coefficients of orthonormal polynomial basis comprises three simple steps: (1) approximate the given measure by a discrete*M*-point measure, where*M*is a positive integer, (2) compute the recursion coefficients of the discrete*M*-point measure, and (3) increase*M*and iterate calculations of the discrete versions of the recursion coefficients until a desired accuracy is achieved. Since no response evaluations are involved, the computational effort associated with the Stieltjes procedure is trivial. Using these orthonormal polynomials, the component functions of X-PDD can be expanded into Fourier series, coined as Fourier-polynomial expansions, leading to an explicit form of X-PDD for representing a high-dimensional response function. Due to the dimensional hierarchy of X-PDD, subsequent reliability analysis of large-scale complex systems can be conducted efficiently.**Dimension-Reduction Integration for Calculating Expansion Coefficients:**The determination of the expansion coefficients of the X-PDD method requires evaluating various*N*-dimensional integrals, where*N*is the dimension of the stochastic or reliability problem. Therefore, calculating coefficients for a large problem dimension, say,*N*> 100, is computationally prohibitive. In this subtask, an innovative dimension-reduction integration technique was developed for estimating the expansion coefficients accurately and efficiently. The technique entails a low-dimensional approximation of the response function, resulting in a finite sum of at most*R*-dimensional integrations, where*R*<*N*. It generates the expansion coefficients approaching their exact values as*R*approaches*N*and is more efficient than performing one*N*-dimensional integration, particularly when*R*<<*N*. Hence, the computational effort in calculating the coefficients is significantly lowered using the dimension-reduction integration. When*R*= 1, 2, or 3, the computational efforts involve one-, at most two-, and at most three-dimensional integrations, respectively. Nonetheless, numerical integration is still required for a general response function. The integration points and associated weights depend on the probability distribution of the random input and are readily available for classical distribution functions, e.g., the Gauss-Hermite quadrature rule when the input follows Gaussian distribution. For an arbitrary probability distribution of the random input, the Stieltjes procedure was employed to generate the appropriate Gauss quadrature formula.

**Task 2: Integrated X-PDD and Score Functions for Concurrently Calculating the Design Sensitivities of Statistical Moments and Reliability (Year 2)**

**Design Sensitivity Analysis of Statistical Moments:**A new computational method, blending PDD and score functions, was developed for calculating the first-order sensitivities of the statistical moments of mechanical system responses with respect to the probability distribution parameters. The methods are based on (1) polynomial dimensional decomposition of a multivariate response function in terms of random variables with increasing dimensions; (2) Fourier-polynomial expansion of the first-order score functions associated with the probability distribution of the random input; and (3) orthogonal structures of the PDD and Fourier expansion and properties of orthogonal polynomials. New closed-form formulae were derived to express the first-order sensitivities of the statistical moments with respect to the design variables. Both the probabilistic response and its design sensitivities can be estimated from a single stochastic analysis without requiring performance function gradients. The effort in obtaining probabilistic sensitivities can be viewed as calculating response at selected deterministic input defined by either integration points or sample points. Therefore, the methods can be easily adapted for solving stochastic design problems involving third-party, commercial finite-element codes.**Global Methods for Robust Design Optimization:**Three new methods were developed for robust design optimization of simple systems subject to uncertainty. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms, encompassing direct, single-step, and sequential design processes. Among the three methods developed, the direct PDD method is the simplest of all, but requires re-calculations of the expansion coefficients at each design iteration, and is, therefore, expensive, depending on the cost of evaluating the objective and constraint functions and the requisite number of design iterations. The single-step PDD method eliminates the need to re-calculate the expansion coefficients from scratch by recycling the old expansion coefficients, consequently holding a potential to significantly reduce the computational effort. However, it depends heavily on the quality of a PDD approximation and the accuracy of the estimated expansion coefficients during design iterations. The sequential PDD method upholds the merits of both the direct and single-step PDD methods by re-calculating the expansion coefficients a few times more than the single-step PDD, incurring a computational complexity that is lower than the direct PDD method.**A Local Method for Robust Design Optimization:**The design methods described in the preceding are founded on PDD approximations of stochastic responses, supplying surrogates of objective and constraint functions for the entire design space. Therefore, these methods are global and may not be cost-effective when the truncation parameters of PDD are required to be exceedingly large to capture high-order responses or high-variate interactions of input variables. Furthermore, a global method using a truncated PDD, obtained by retaining only low-order or low-variate terms, may not even find a true optimal solution. An attractive alternative method, developed in this work and referred to as the multi-point single-step PDD method, involves local implementations of the single-step PDD approximation that are built on a local subregion of the design space. According to this method, the original RDO problem is exchanged with a succession of simpler RDO sub-problems, where the objective and constraint functions in each sub-problem represent their multi-point approximations. The design solution of an individual sub-problem, obtained by the single-step PDD method, becomes the initial design for the next sub-problem, the move limits are updated, and the optimization is repeated iteratively until the optimal solution is attained. Due to its local approach, the multi-point single-step PDD method should solve complex engineering problems using low-order and/or low-variate PDD approximations.

For further information on Task 2, click here to see a conference presentation.

**Task 3: Reliability-Based and Robust Design Optimization (RBDO/RDO) Algorithms Employing X-PDD and Score Functions (Years 3 and 4)**

**Design Sensitivity Analyses of Statistical Moments and Reliability:**Three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random input. The first method represents a novel integration of polynomial dimensional decomposition (PDD) of a multivariate stochastic response function and score functions. Applied to the statistical moments, the method provides mean-square convergent analytical expressions of design sensitivities of the first two moments of a stochastic response. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Numerical examples, including a 100-dimensional mathematical problem, indicate that the new methods developed provide not only theoretically convergent or accurate design sensitivities, but also computationally efficient solutions. A practical example involving robust design optimization of a three-hole bracket illustrates the usefulness of the proposed methods.**Reliability-based Design Optimization:**Two new methods for reliability-based design optimization (RBDO) of complex engineering systems were developed. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, novel integrations of AS-PDD and score functions for calculating the sensitivities of reliability with respect to the design variables, and standard gradient-based optimization algorithms, encompassing a multi-point, single-step design process. The two methods, depending on how the failure probability and its sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the AS-PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For both methods, the failure probability and its design sensitivities are determined concurrently from a single stochastic simulation or analysis. Precisely for this reason, the multi-point, single-step framework of the proposed method affords the ability of solving industrial-scale problems with large design spaces. Numerical results stemming from mathematical functions or elementary engineering problems indicate that the new method provides more accurate or computationally efficient design solutions than existing methods. Furthermore, shape design of a 79-dimensional jet engine bracket was performed, demonstrating the power of the new method to tackle practical RBDO problems.

**Supervision of Graduate Student: **A Ph.D. student, Mr. Xuchun Ren, is (was) involved in this project. Mr. Ren is making satisfactory progress towards completion of his Ph.D. program. The PI is (was) the sole advisor of this graduate student.

**Insertion of Research Results into Undergraduate and Graduate Courses: **The newly developed decomposition methods from Tasks 1 were already integrated into the syllabus of a graduate course, 58:257 *Probabilistic Mechanics and Reliability*, which was last offered in the Fall 2010 semester. The PI will integrate new stochastic methods developed in Task 2 for the Fall 2012 offering of the above-mentioned course. The PI strongly believes that the insertion of new research results will motivate learning by students in the areas of computational dynamics, stochastic methods, and high-performance computing.

**Undergraduate Curriculum Enhancement and Capstone Design: **From 2010-present, the PI served as a faculty advisor to at least one capstone design team in conjunction with the Program for Enhanced Design Experience - a flagship capstone design program at The University of Iowa that the PI had earlier expanded in an NSF CAREER project (Award No. 9733058). The design projects, mostly sponsored by John Deere Dubuque Works (Dubuque, IA), involved dynamic analysis and design optimization of earth-moving and other construction vehicles. In 2010, the PI advised two student groups on dynamic finite-element analysis of vehicle components and system design for mini-Baja competition sponsored by the Society of Automotive Engineers.

**Development of Symposia: **Since the beginning of the project (2010), the PI developed several technical sessions on stochastic methods, design optimization, and computational mechanics in conjunction with various national and international conferences. He also chaired two technical sessions on probabilistic methods at *Engineering Mechanics Institute Annual Conferences* in 2010 (Los Angeles, CA) and 2012 (Boston, MA).

**Invited Seminars and Lectures: **Since the beginning of this project (2010), the PI and his student presented 15 technical papers or posters on the newly developed stochastic method and design optimization algorithms at (1) 3rd Workshop on Sparse Grids and Applications, Stuttgart, Germany, 2014; (2) ASCE Engineering Mechanics Institute Annual Conference, Hamilton, ON, 2014; (3) 4th International Symposium on Inverse Problems, Design and Optimization, Albi, France, 2013; (4) 11th International Conference on Structural Safety & Reliability, New York, NY, June 2013; (5) 14^{th} AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Indianapolis, IN, 2012; (6) NSF Engineering Research and Innovation Conference, Boston, MA, July 2012; (7) 11^{th} ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, South Bend, IN, June 2012; (8) ASCE Engineering Mechanics Institute Annual Conference, Boston, MA, June 2011; (9) NSF Engineering Research and Innovation Conference, Atlanta, GA, January 2011; (10) 13^{th} AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Fort Worth, TX, September 2010; and (11) ASCE Engineering Mechanics Institute Annual Conference, Los Angeles, CA, August 2010.

**Journal Articles:**

- Ren, X., Rahman, S., and Yadav, V., “Reliability-based Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition,” submitted to Structural and Multidisciplinary Optimization, 2014.
- Ren, X. and Rahman, S., “Stochastic Design Optimization Involving Structural and Distributional Parameters,” submitted to ASME Journal of Mechanical Design, 2014.
- Rahman, S., “A Generalized ANOVA Dimensional Decomposition for Dependent Probability Measures,” submitted to SIAM Journal of Uncertainty Quantification, 2013.
- Rahman, S., and Ren, X., “Novel Computational Methods for High-Dimensional Stochastic Sensitivity Analysis,” International Journal for Numerical Methods in Engineering, Vol. 98, pp. 881–916, 2014. (Sensitivity-IJNME.pdf)
- Rahman, S., “Approximation Errors in Truncated Dimensional Decompositions,” accepted in Mathematics of Computation, to appear in 2014.
- Rahman, S., “Uncertainty Quantification by Alternative Decompositions of Multivariate Functions,” SIAM Journal on Scientific Computing, Vol. 35, pp. A3024-A3051, 2013.
- Ren, X. and Rahman, S., “Robust Design Optimization by Polynomial Dimensional Decomposition,” Structural and Multidisciplinary Optimization, Vol. 48, No. 1, pp. 127-148, 2013. (rdo-smo.pdf)
- Rahman, S. and Xu, H., “Comments on High-dimensional Model Representation for Structural Reliability Analysis,” International Journal for Numerical Methods in Biomedical Engineering, Vol. 27, pp. 1652-1659, 2011.
- Rahman, S., “Global Sensitivity Analysis by Polynomial Dimensional Decomposition,” Reliability Engineering & System Safety, Vol. 96, No. 7, pp. 825-837, 2011.
- Rahman, S., “Statistical Moments of Polynomial Dimensional Decomposition,” Journal of Engineering Mechanics, Vol. 136, No. 7, pp. 923-927, 2010.

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

- Rahman, S., “Adaptive-Sparse Polynomial Dimensional Decomposition,” Proceedings of 3rd Workshop on Sparse Grids and Applications, Stuttgart, Germany, September 2014.
- Ren, X., Yadav, V., and Rahman, S., “Reliability-based Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition,” Proceedings of EMI2014 Engineering Mechanics Institute Conference, Hamilton, ON, August 2014.
- Ren, X. and Rahman, S., “Novel Computational Methods for High-Dimensional Stochastic Sensitivity Analysis,” Proceedings of EMI2014 Engineering Mechanics Institute Conference, Hamilton, ON, August 2014.
- Ren, X. and Rahman, S., “Stochastic Design Sensitivity Analysis and Optimization of Complex Systems,” Proceedings of 11th International Conference on Structural Safety & Reliability, New York, NY, June 2013.
- Rahman, S. and Ren, X., “Stochastic Design Optimization of Complex Systems,” 4th International Symposium on Inverse Problems, Design and Optimization, Albi, France, June 2013.
- Ren, X. and Rahman, S., “Robust Design Optimization by Polynomial Dimensional Decomposition,” Proceedings of 14th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Indianapolis, IN, September 2012.
- Rahman, S., “Approximation Error in Truncated Dimensional Decomposition for Uncertainty Quantification,” Proceedings of 14th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Indianapolis, IN, September 2012.
- Ren, X. and Rahman, S., “Reliability-based Design Optimization by Polynomial Dimensional Decomposition,” Proceedings of 11th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, South Bend, IN, June 2012.
- Rahman, S., “A Generalized ANOVA Dimensional Decomposition for Arbitrary Probability Measures,” Proceedings of 11th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, South Bend, IN, June 2012.
- Rahman, S., “Reliability-Based Design Optimization of Large-Scale Complex Systems,” Proceedings of 2012 NSF Engineering Research and Innovation Conference, Boston, MA, July 2012.
- Rahman, S., “Global Sensitivity Analysis by Polynomial Dimensional Decomposition,” Proceedings of 2011 Engineering Mechanics Institute Conference, Boston, MA, June 2-4, 2011.
- Rahman, S., “Reliability-Based Design Optimization of Large-Scale Complex Systems,” Proceedings of 2011 NSF Engineering Research and Innovation Conference, Atlanta, GA, January 2011.
- Rahman, S., “An Extended Polynomial Dimensional Decomposition Method for Arbitrary Probability Distributions,” Proceedings of 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Fort Worth, TX, September 13-15, 2010.
- Rahman, S. and Wei, D., “Reliability-based Design Optimization by a Univariate Decomposition Method,” Proceedings of 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Fort Worth, TX, September 13-15, 2010.
- Rahman, S., “Extended Polynomial Dimensional Decomposition for Arbitrary Probability Measures,” Proceedings of EMI 2010 Engineering Mechanics Conference, Los Angeles, CA, August 8-11, 2010.

**Poster Presentations:**

The computer programs and related files created during this research are organised according to the following journal papers published. Click on the title of each paper to open a list of computer programs created for solving RDO/RBDO problems published in the paper.

- Rahman, S., and Ren, X., “Novel Computational Methods for High-Dimensional Stochastic Sensitivity Analysis,” International Journal for Numerical Methods in Engineering, Vol. 98, pp. 881–916, 2014.
- Ren, X. and Rahman, S., “Robust Design Optimization by Polynomial Dimensional Decomposition,” Structural and Multidisciplinary Optimization, Vol. 48, No. 1, pp. 127-148, 2013.