poster - International Conference of Agricultural Engineering

is computationally more efficient. But, the RSM require many approximation when problems
are strongly nonlinear and sometimes shows large errors in the calculation of the sensitivity
of the reliability index. In order to overcome these problems, Kang et al.(2010) proposed an
efficient RSM applying a moving least squares (MLS) approximation.
In this study, for reliable risk assessment of groundwater pollution, stochastic modelling was
performed for the flow of Benzene considering dispersion, advection and retardation using a
moving least squares response surface method (MLS-RSM). Two parameters are
considered as input random variables and transformed into standard normal distribution
using Hermite polynomials. The accuracy of analysis results are evaluated in comparison
with MCS results.
2. Theory and Background
2.1. Hermite polynomials
Hermite Polynomials utilizes a series of orthogonal polynomials to facilitate stochastic
analysis. These polynomials are used as orthogonal basis to decompose a random process.
One-dimensional Hermite polynomials are given by:
H ( ) 1
H
0
H ( )
1
2
H ( ) 1
2
3
H ( ) 3
3
k 1
( ) H
k
( ) kH
k 1
( )
(1)
where is a standard Gaussian random variable (mean=0 and variance=1). Hermite
polynomials over [- , ] satisfy the following orthogonal relation:
1
e
2
1
u
2
2
H
m
( u)
H
n
( u)
du
mn
n!
(2)
where
mn
is the Kronecker delta function. Any random variable can be approximated by
Hermite polynomials expansion as follows (Phoon, 2003):
Y a H ( i i
)
(3)
i0
where a
i
is the deterministic Hermite polynomials expansion coefficient which depends on
the distribution of Y . H ( ) is multi-dimensional Hermite polynomials of degree i .
i
2.2. MLS-RSM
The RSM is a collection of mathematical and statistical techniques for the approximation and
optimization of stochastic models that was developed by Box and Draper(1987). The RSM is