Random Matrix Theory for Stochastic Structural Dynamics

Adhikari, S.
6th International Congress on Industrial and Applied Mathematics (ICIAM 2007), Zurich, Switzerland, July 2007.

Uncertainties need to be taken into account for credible results from high fidelity, physics and engineering-based numerical simulations. As a result, the quantification and propagation of uncertainties play a crucial role in today's predictive engineering science. Uncertainties can be broadly divided into parametric uncertainty (inherent variability/data uncertainty) and nonparametric uncertainty (lack of knowledge/model uncertainty). Within the past three decades the deterministic finite element method has been generalized to consider parametric uncertainties in a systematic manner by means of the Stochastic Finite Element Method (SFEM). In the context structural dynamics, using SFEM one usually obtains the random mass, stiffness and damping matrices. More recently a systematic approach to model nonparametric uncertainty is proposed. Using this method one can obtain the probability density function of the system matrices. Therefore, both parametric and nonparametric uncertainty modelling lead to matrix variate distributions describing the system matrices of a linear dynamic system. In this paper it is first proved that both parametric and non-parametric uncertainty can be modelled using a unified matrix variate distribution (similar to a noncentral Wishart distribution). Parametric and nonparametric uncertainty models can be identified as special cases of this general distribution. Once the random system matrices are `known' via this distribution, using the random matrix theory exact analytical methods have been proposed for the solution of a set of random algebraic equations (K y = f) and the random eigenvalue problem (K fj = lj M fj).
BiBTeX Entry
@INPROCEEDINGS{cp41,
    AUTHOR={S. Adhikari},
    TITLE={Random matrix theory for stochastic structural dynamics},
    BOOKTITLE={6th International Congress on Industrial and Applied Mathematics (ICIAM 2007)},
    YEAR={2007},
    Address={Zurich, Switzerland},
    Month={July},
    Note={}
}

by Sondipon Adhikari