A Non-parametric Approach for Uncertainty Quantification in Elastodynamics

Adhikari, S.,
47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Newport, Rhode Island, USA, May 2006.

Uncertainties are unavoidable in the description of real-life engineering systems. The quantification of uncertainties plays a crucial role in establishing the credibility of a numerical model. Uncertainties can be broadly divided into two categories. The first type is due to the inherent variability in the system parameters, for example, different cars manufactured from a single production line are not exactly the same. This type of uncertainty is often referred to as aleatoric uncertainty. If enough samples are present, it is possible to characterize the variability using well established statistical methods and consequently the probably density functions (pdf) of the parameters can be obtained. The second type of uncertainty is due to the lack of knowledge regarding a system, often referred to as epistemic uncertainty. This kind of uncertainty generally arise in the modelling of complex systems, for example, in the modeling of cabin noise in helicopters. Due to its very nature, it is comparatively difficult to quantify or model this type of uncertainties.
Broadly speaking, there are two approaches to quantify uncertainties in a model. The first is the parametric approach and the second is the non-parametric approach. In the parametric approach the uncertainties associates with the system parameters, such as Young's modulus, mass density, Poisson's ratio, damping coefficient and geometric parameters are quantified using statistical methods. This type of approach is suitable to quantify aleatoric uncertainties. Epistemic uncertainty on the other hand do not explicitly depend on the systems parameters. For example, there can be unquantified errors associated with the equation of motion (linear on non-linear), in the damping model (viscous or non-viscous), in the model of structural joints, and also in the numerical methods (e.g, discretisation of displacement fields, truncation and roundoff errors, tolerances in the optimization and iterative algorithms, step-sizes in the time-integration methods). It is evident that the parametric approach is not suitable to quantify this type of uncertainties and a non-parametric approach is needed for this purpose.
In this paper a general uncertainty quantification tool for structural dynamic systems is developed. The aim is to quantify non-parametric uncertainty arising in complex systems. A new approach based on the random matrix theory (RMT) and the maximum entropy method (MEM) is proposed. It is assumed that only the mean of the system matrices are known. The derived probability density function of the random system matrices are completely characterized by the dimension of the matrices and their mean values. The main outcome of this study is that if only the mean value of a system matrix is known then the matrix follows a Wishart distribution (with proper parameters). It will be shown in the paper that the discovery of the Wishart distribution in this context has many interesting consequences. The applications of the derived matrix variate distribution are illustrated by using examples involving complex systems.


BiBTeX Entry
@INPROCEEDINGS{cp19,
    AUTHOR={S. Adhikari},
    TITLE={A non-parametric approach for uncertainty quantification in elastodynamics},
    BOOKTITLE={47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics \& Materials Conference},
    YEAR={2006},
    Address={Newport, Rhode Island, USA},
    Month={May},
    Note={}
}

by Sondipon Adhikari