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

**Contract No.:** CMMI-0653279

**Duration:** August 15, 2007 – July 31, 2011

**Principal Investigator:** Professor Sharif Rahman

**Graduate Students:** V. Yadav

**Partner Organizations: **Rockwell Collins, Cedar Rapids, IA

**SUMMARY**

The overall objective of this research project was to conduct fundamental research on computational methods for solving random eigenvalue problems in modeling and simulation of stochastic dynamic systems. The proposed effort involved: (1) development of a new decomposition for lower-dimensional approximations of general complex-valued eigensolutions of random eigenvalue problems (Task 1); (2) development of a novel polynomial dimensional decomposition method for probabilistic characteristics of eigensolutions (Task 2); and (3) error analyses of polynomial dimensional decomposition and polynomial chaos expansion and global sensitivity analysis (Task 3). The new decomposition methods and computational tools developed in this project will aid in accurate and efficient probabilistic characterization of dynamic system responses, such as natural frequencies and modes shapes. The statistics and rare-event probabilities (e.g., failure probability) predicted by these methods may potentially lead to new or improved designs in the presence of uncertainties due to insufficient information, limited understanding of underlying phenomena, and inherent randomness. Therefore, the research results will be of significant benefit to several commercial and industrial applications, such as civil, automotive, and aerospace infrastructure. Potential engineering applications include analysis and design of civil structures; noise-vibration-harshness, and crashworthiness of ground vehicle systems; and fatigue durability of aerospace structures. For the microelectronics industry, applications in reliability of microelectronics and interconnects, reliability of micro-electro-mechanical systems for sensors and actuators are relevant. In summary, any application discipline that can be simulated or analyzed using computer software/tools should benefit from this research.

**Task 1: New Decomposition Method for Lower-Dimensional Approximations of General Complex-Valued Eigensolutions of Random Eigenvalue Problems**

**Multivariate Function Decomposition:**A new polynomial decomposition was developed for a general multivariate function representing complex-valued eigensolutions. The decomposition involves a hierarchical expansion of a multivariate function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier-polynomial expansion of component functions, and an innovative dimension-reduction integration for calculating the coefficients of the expansion. The new decomposition does not require sample points as in the previous version, yet it generates a convergent sequence of lower-variate estimates of an eigensolution. The method is non-intrusive in the sense that the expansion coefficients are obtained by calculating responses at selected deterministic input defined by the integration points. Therefore, the method can be easily adapted to solving large-scale eigenvalue problems requiring external commercial codes.**Statistical Moments and Probability Densities of Complex-Valued Eigensolutions:**A new dimensional decomposition method was developed for calculating the probabilistic characteristics of complex-valued eigenvalues and eigenvectors of linear, stochastic, dynamic systems. The method is based on: (1) a finite, hierarchical decomposition allowing lower-dimensional approximations of eigensolutions; (2) Lagrange interpolation of lower-dimensional component functions; and (3) Monte Carlo simulation. The effort in finding the probabilistic characteristics of eigensolutions can be viewed as performing eigenvalue analyses at selected deterministic input defined by sample points. Compared with the commonly used perturbation method, neither the assumption of small input variability nor the calculation of the derivatives of eigensolutions is required by the method developed. Hence, the method can be easily adapted for solving random eigenvalue problems involving third-party, commercial, finite-element codes. The proposed method was applied to a nonproportionally damped spring-mass-damper system, a non-proportionally damped cantilever beam, and an industrial-scale disc brake system with an unsymmetric stiffness matrix. Results indicate that the decomposition method provides excellent estimates of the moments and/or probability densities of eigenvalues and eigenvectors for various cases, including large statistical variations of input. The computational efforts required by the univariate and bivariate decomposition methods are linear and quadratic, respectively, with regard to the number of random variables involved. Therefore, the method developed is also computationally efficient when compared with the direct Monte Carlo simulation.

**Task 2: New Polynomial Dimensional Decomposition for Probabilistic Characteristics of Eigensolutions**

**Extended Polynomial Dimensional Decomposition:**An extended polynomial dimensional decomposition method was developed for solving stochastic problems subject to independent random input following an arbitrary probability distribution. The method involves Fourier-polynomial expansions of component functions by orthogonal polynomial bases, the Stieltjes procedure for generating the recursion coefficients of orthogonal polynomials and the Gauss quadrature rule for a specifed probability measure, and dimension-reduction integration for calculating the expansion coefficients. The extension, which subsumes non-classical orthogonal polynomials bases, generates a convergent sequence of lower-variate estimates of the probabilistic characteristics of a stochastic response. Numerical results indicate that the extended decomposition method provides accurate, convergent, and computationally efficient estimates of the tail probability of random mathematical functions or reliability of mechanical systems. The convergence of the extended method accelerates significantly when employing measure-consistent orthogonal polynomials.**Statistical Moments of Polynomial Dimensional Decomposition:**The polynomial dimensional decomposition involving Fourier-polynomial expansions of lower-dimensional component functions was studied. By exploiting the orthogonal structure of the decomposition and the properties of orthogonal polynomials, explicit formulae for calculating the response moments in terms of the expansion coefficients were derived. The results of an industrial-scale stochastic problem indicate that the formulae provide accurate, convergent estimates of the first two moments examined at modest computational effort.**Probability Distributions of Natural frequencies of Uncertain Dynamic Systems:**A polynomial dimensional decomposition method was developed for calculating the probability distributions of random eigenvalues commonly encountered in dynamic systems. The method involves a hierarchical decomposition of a multivariate function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier-polynomial expansion of component functions, and an innovative dimension-reduction integration for calculating the expansion coefficients. Unlike the previous development, the new decomposition does not require sample points, yet it generates a convergent sequence of lower-variate estimates of the probability distributions of eigensolutions. Numerical results, including frequency distributions of a piezoelectric transducer, indicate that the decomposition method developed provides accurate, convergent, and computationally efficient estimates of the tail probabilistic characteristics of eigenvalues.

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

**Task 3: New Sensitivity Formulations for Probabilistic Measures of Random ****Eigensolutions**

**Orthogonal Polynomial Expansions for Solving Random Eigenvalue Problems:**Two stochastic expansion methods stemming from polynomial dimensional decomposition (PDD) and polynomial chaos expansion (PCE) were investigated for solving random eigenvalue problems commonly encountered in dynamics of mechanical systems. Although the infinite series from PCE and PDD are equivalent, their truncations endow contrasting dimensional structures, creating significant differences between the resulting PDD and PCE approximations in terms of accuracy, efficiency, and convergence properties. When the cooperative effects of input variables on an eigenvalue attenuates rapidly or vanish altogether, the PDD approximation commits smaller error than the PCE approximation for identical expansion orders. Numerical analyses of mathematical functions or simple dynamic systems reveal markedly higher convergence rates of the PDD approximation than the PCE approximation. From the comparison of computational efforts, required to estimate with the same precision the frequency distributions of dynamic systems, including a piezoelectric transducer, the PDD approximation is significantly more efficient than the PCE approximation.**Global Sensitivity Analysis by Polynomial Dimensional Decomposition:**A polynomial dimensional decomposition (PDD) method was developed for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol's method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent.**Sensitivity of Moments and Probability Distributions of Eigensolutions:**Underway.

**Supervision of Graduate Student: **A Ph.D. student, Mr. Vaibhav Yadav, was involved in this project. Mr. Yadav is making satisfactory progress towards completion of his Ph.D. program and is expected to graduate in 2012. 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 and 2 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 3 for the Fall 2012 offering of the above-mentioned course. In addition, the brake-squeal eigenvalue analysis, conducted in Task 1, was incorporated into the syllabus of an undergraduate, senior-level course, 58:110 *Computer-Aided Engineering*, which is offered every Spring semester. 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 2007-2011, 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 2008 and 2010, the PI advised two student groups on dynamic finite-element analysis of vehicle components and systems for mini-Baja competition sponsored by the Society of Automotive Engineers.

**Development of Symposia: **Since the beginning of the project (2007), the PI developed several minisymposia or technical sessions on stochastic methods and computational mechanics in conjunction with various national and international conferences. The PI, in cooperation with faculty at University of Massachusetts at Amherst and Johns Hopkins University, has developed a minisymposium on stochastic aspects of modeling and simulation at the *10 ^{th} U.S. National Congress on Computational Mechanics*, Columbus, OH, July 2009. He also developed or 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 (2007), the PI gave several invited seminars and lectures on the newly developed dimensional decomposition method at the following institutions: (1) The University of Iowa (Applied and Computational Mathematics Program), Iowa City, IA (2008); (2) Norwegian University of Science and Technology, Trondheim, Norway (2007); (3) Statoil, Trondheim, Norway (2007); and (3) University of Illinois, Chicago, IL (2007). In addition, the PI and/or his student presented technical papers on the same topic at (1) 11^{th} U.S. National Congress on Computational Mechanics, Minneapolis, MN (2011); (2) ASCE Engineering Mechanics Institute Annual Conference, Boston, MA (2011); (3) 13^{th} AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Fort Worth, TX (2010); (4) ASCE Engineering Mechanics Institute Annual Conference, Los Angeles, CA (2010); (5) 50^{th} AIAA Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, May 2009; and (6) 49^{th} AIAA Structures, Structural Dynamics, and Materials Conference, Chicago (2008).

**Journal articles:**

- Rahman, S., “Approximation Errors in Truncated Dimensional Decompositions,” submitted, 2011.
- Yadav, V. and Rahman, S. and “Multiplicative Polynomial Dimensional Decomposition for High-Dimensional Stochastic Computing,” submitted, 2011.
- Rahman, S. and Yadav, V., “Orthogonal Polynomial Expansions for Solving Random Eigenvalue Problems,” International Journal for Uncertainty Quantification, Vol. 1, No. 2, pp. 163-187, 2011.
- Rahman, S., “Global Sensitivity Analysis by Polynomial Dimensional Decomposition,” Reliability Engineering & System Safety, Vol. 96, No. 7, pp. 825-837, 2011.
- Rahman, S., “Dimensional Decomposition Methods Revisited,” Probabilistic Engineering Mechanics, 2009.
- Rahman, S., “Statistical Moments of Polynomial Dimensional Decomposition,” Journal of Engineering Mechanics, Vol. 136, No. 7, pp. 923-927, 2010.
- Rahman, S., “Extended Polynomial Dimensional Decomposition for Arbitrary Probability Distributions,” Journal of Engineering Mechanics, Vol. 135, No. 12, pp. 1439-1451, 2009.
- Rahman, S., “Stochastic Sensitivity Analysis by Dimensional Decomposition and Score Functions,” Probabilistic Engineering Mechanics, Vol. 24, pp. 278-287, 2009.
- Rahman, S., “Probability Distributions of Natural Frequencies of Uncertain Dynamic Systems,” AIAA Journal, Vol. 47, No. 6, pp. 1579-1589, 2009.
- Rahman, S., “A Polynomial Dimensional Decomposition for Stochastic Computing,” International Journal for Numerical Methods in Engineering, Vol. 76, pp. 2091-2116, 2008.
- Rahman, S., “Stochastic Dynamic Systems with Complex-Valued Eigensolutions,” International Journal for Numerical Methods in Engineering, Vol. 71, pp. 963-986, 2007.

**Papers in conferences, symposiums, and congresses:**

- Yadav, V. and Rahman, S., “Multiplicative Polynomial Dimensional Decomposition for High-Dimensional Stochastic Computing,” Proceedings of 11th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, Notre Dame, IN, June 2012.
- Rahman, S., “Approximation Errors in Truncated Dimensional Decompositions,” Proceedings of 11th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, Notre Dame, IN, June 2012.
- Yadav, V. and Rahman, S., “A Comparison of Two Stochastic Expansion Methods for Solving Random Eigenvalue Problems,” Proceedings of 11th U.S. National Congress on Computational Mechanics, Minneapolis, MN, July 2011.
- Yadav, V. and Rahman, S., “Multiplicative Polynomial Dimensional Decomposition for Stochastic Computing,” Proceedings of 2011 Engineering Mechanics Conference, Boston, MA, June 2-4, 2011.
- Rahman, S., “Global Sensitivity Analysis by Polynomial Dimensional Decomposition,” Proceedings of 2011 Engineering Mechanics Conference, Boston, MA, June 2-4, 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., “Error Analysis of Dimensional Decomposition Methods,” Proceedings of 2010 Engineering Mechanics Conference, Los Angeles, CA, August 8-11, 2010.
- Rahman, S., “A Polynomial Dimensional Decomposition Method for Arbitrary Probability Measures,” Proceedings of 2010 Engineering Mechanics Conference, Los Angeles, CA, August 8-11, 2010.
- Rahman, S., “Stochastic Computing by a New Polynomial Dimensional Decomposition Method,” Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, May 2009.
- Rahman, S., “A Coupled Dimensional Decomposition and Score Function Method for Probabilistic Sensitivity Analysis,” Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, May 2009.
- Rahman, S., “Probabilistic Analysis of Dynamic Systems with Complex-valued Eigensolutions,” Proceedings of 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Schaumburg, IL, April 2008.