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A Comparative Study on Metamodel Based Stochastic Analysis of Composite Structures
Published in Sudip Dey, Tanmoy Mukhopadhyay, Sondipon Adhikari, Uncertainty Quantification in Laminated Composites, 2018
Sudip Dey, Tanmoy Mukhopadhyay, Sondipon Adhikari
The Least Interpolating Polynomials use polynomial basis functions and also interpolate responses. They choose a polynomial basis function of “minimal degree” as described by Fang and Horstemeyer (2006) and hence are called “least interpolating polynomials”. This type of metamodel deserves more study. In addition, Pérez et al. (2002) transformed the matrix of second-order terms of a quadratic polynomial model into the canonical form to reduce the number of terms. Messac and his team developed an extended RBF model (Mullur and Messac 2005) by adding extra terms to a regular RBF model to increase its flexibility, based on which an optimal model could be searched for. Turner and Crawford proposed a NURBS-based metamodel, which was applied only to low dimensional problems (Turner and Crawford 2005). If gradient information can be reliably and inexpensively obtained, it can be utilized in metamodeling (Morris et al. 1993). High dimensional model representation is found to be successfully applied in the problems related to optimization and system identification (Mukhopadhyay et al. 2015c, 2016d). A multipoint approximation (MPA) strategy has also received some attention (Wang et al. 1996, Rasmussen 1998, Shin and Grandhi 2001). MPA uses blending functions to combine multiple local approximations, and usually gradient information is used in metamodeling. Metamodels can also be constructed when design variables are modeled as fuzzy numbers (Madu 1995, Kleijnen 2004). Each metamodel type has its associated fitting method. For example, polynomial functions are usually fitted with the (weighted) least square method; the kriging method is fitted with the search for the Best Linear Unbiased Predictor (BLUP). Simpson et al. (1997) illustrated a detailed review on the equations and fitting methods for common metamodel types. In general computer experiments have very small random error which might be caused by the pseudorandom number generation or rounding (Kleijnen 2004). Giunta et al. (1994) found that numerical noises in computing the aerodynamic drag of High Speed Civil Transport (HSCT) caused many spurious local minima of the objective function. The problem was due to the discontinuous variations in calculating the drag by using the panel flow solver method. Madsen et al. (2000) stated that noises could come from the complex numerical modelling techniques. In case of physical or noisy computer experiments, it is found that Kriging and RBF are more sensitive to numerical noise than polynomial models (Jin et al. 2003). However, Kriging, RBF, and ANN could be modified to handle noises, assuming the signal to noise ratio is acceptable (Van Beers and Kleijnen 2004).
Shape optimization for blended-wing–body underwater glider using an advanced multi-surrogate-based high-dimensional model representation method
Published in Engineering Optimization, 2020
Ning Zhang, Peng Wang, Huachao Dong, Tianbo Li
Metamodelling construction is the most important phase in surrogate-based optimization methods. Well-known metamodelling techniques include the polynomial response surface method (PRS) (Box and Draper 1987), radial basis function (RBF) (Fang and Horstemeyer 2006), neural network (Papadrakakis, Lagaros, and Tsompanakis 1998), support vector regression (SVR) (Smola and Schölkopf 2004) and kriging (Cressie 1988). However, the samples increase exponentially and the established high-dimensional surrogate models are not accurate with finite samples when the dimension of the underlying problem grows. Various approaches based on surrogate model techniques have been developed to tackle these challenges (Regis and Shoemaker 2013; Regis 2014; Bouhlel et al.2019). High-dimensional model representation (HDMR) is one of the most promising techniques among these approaches. It has been proposed to solve high-dimensional expensive black-box (HEB) problems (Rabitz et al.1999; Rabitz and Aliş 1999), as introduced by Sobol (1993). There are two main types of HDMR: analysis of variance–HDMR (ANOVA-HDMR) and cut-HDMR (Rabitz et al.1999; Li, Rosenthal, and Rabitz 2001). The ANOVA-HDMR is commonly used for sensitivity analysis and identifying important variables and correlations. Differently from ANOVA-HDMR, the cut-HDMR is relatively more efficient and integral computation is completely unnecessary. Accordingly, the cut-HDMR has been widely studied to generate highly accurate approximation for HEB problems (Shan and Wang 2010, 2011; Cai et al.2016; Chen et al.2019).