Novel Computational Methods for Solving Random Eigenvalue Problems

Sponsor:                           U.S. National Science Foundation
Award No.:                        CMMI-1130147
Duration:                           September 1, 2011 – August 31, 2015
Principal Investigator:    Professor Sharif Rahman
Graduate Student:           V. Yadav, X. Ren

The major goal of this research project is to create theoretical foundations and numerical algorithms of innovative computational methods for solving a general random eigenvalue problem in modeling and simulation of high-dimensional stochastic dynamic systems.  The proposed effort involves: (1) development of a new polynomial dimensional decomposition (PDD) method for predicting the statistical moments and probability distributions of eigensolutions of stochastic dynamic systems, including innovative dimension-reduction and sampling techniques for estimating the expansion coefficients (Task 1); (2) quantification and comparison of approximation errors from the referential and ANOVA dimensional decomposition (RDD and ADD) methods and of approximation errors from the PDD and polynomial chaos expansion (PCE) methods (Task 2); and (3) development of new multiplicative PDD methods, hybridization, and respective error analyses (Task 3).  The new methods and computational tools to be 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 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.  It is anticipated that the implementation of the new stochastic methods developed herein will lead to next-generation scientific simulation codes, facilitating nuanced mathematical analysis of increasingly complex dynamical systems.