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