Comparison of Response Surface Construction Methods for Derivative Estimation Using Moving Least Squares, Kriging and Radial Basis Functions

* NASA Langley Research Center, Hampton, VA 23681, U.S.A. Abstract Response construction methods using Moving Least Squares (MLS), Kriging and Radial Basis Functions (RBF) are compared with the Global Least Squares (GLS) method in three numerical examples for derivative generation capability. Also, a new Interpolating Moving Least Squares (IMLS) method adopted from the meshless method is presented. It is found that the response surface construction methods using the Kriging and RBF interpolation yields more accurate results compared with MLS and GLS methods. Several computational aspects of the response surface construction methods also discussed. I. Introduction: Structural reliability engineering analysis involves determination of the probability of structural failure taking into account the uncertainty in the geometric parameters, material properties and loading conditions (1). The uncertain quantities are treated as random variables, and often a large number of simulations (structural analyses) with different sets of random variables are necessary to estimate the reliability of a structure. Hence, the computational effort required to perform a structural reliability analysis can be very high. In order to minimize the computational time, response surface functions are often used as simple and inexpensive replacements for computationally expensive structural analyses in reliability methods. Most of the response surface construction methods use Global Least Squares (GLS) methods. The GLS methods use a single quadratic or cubic polynomial representing the entire parametric space of the random variables. In general, a single polynomial to represent the entire parametric space introduces large errors in the response estimation or limits the size of the parametric space. In order to overcome such difficulties, local (piecewise) polynomial functions with higher order derivative continuity were used in response surface approximation (2-4). In arriving at an interpolated value at some point in the parametric space, the local methods more heavily weight data samples that are "nearby" rather than giving all data samples equal weight. It is found that higher order derivative continuity methods require fewer sampling points for the same accuracy (2) compared to GLS methods. A new local Moving Least Squares (MLS) response surface construction method was developed in reference 2. The MLS method was compared with other local methods such as Kriging (3) and found to be more accurate and computationally effective for the examples considered. Another class of local response surface construction using Radial Basis Functions (RBF) was proposed in reference 4. Along with the classical RBF, polynomial augmented RBF and compactly supported radial basis functions (5-7) were used for response surface construction in reference 4. The methods based on MLS, Kriging, and RBF were compared in two numerical examples in references 2 and 4 and found to predict the response almost with the same accuracy. Even though the MLS, Kriging, and RBF methods produce higher accuracy in response prediction compared to the GLS methods, the computational efficiency or the ability of these local methods to reproduce derivatives is not investigated in the literature. The ability to estimate the derivatives accurately is very critical for the response construction methods in optimization and reliability studies. Hence, it is necessary to examine the local methods for derivative generation capabilities. The purpose this paper is to study and compare the computational efficiency and derivative generation capability of the local MLS, Kriging and RBF methods by applying these methods to several numerical examples. Also, a new Interpolating Moving Least Squares (IMLS) method adopted from the meshless method is presented.