Metamodeling for High Dimensional Simulation-Based Design Problems

Abstract

Computational tools such as finite element analysis and simulation are widely used in engineering. But they are mostly used for design analysis and validation. If these tools can be integrated for design optimization, it will undoubtedly enhance a manufacturer's competitiveness. Such integration, however, faces three main challenges: 1) high computational expense of simulation, 2) the simulation process being a black-box function, and 3) design problems being high dimensional. In the past two decades, metamodeling has been intensively developed to deal with expensive black-box functions, and has achieved success for low dimensional design problems. But when high dimensionality is also present in design, which is often found in practice, there lacks of a practical method to deal with the so-called Highdimensional, Expensive, and Black-box (HEB) problems. This paper proposes the first metamodel of its kind to tackle the HEB problem. This work integrates Radial Basis Function (RBF) with High Dimensional Model Representation (HDMR) into a new model, RBF-HDMR.The developed RBF-HDMR model offers an explicit function expression, and can reveal the 1) contribution of each design variable, 2) inherent linearity/nonlinearity with respect to input variables, and 3) correlation relationships among input variables. An accompanying algorithm to construct the RBF-HDMR has also been developed. The model and the algorithm fundamentally 1 Corresponding author, Tel: 778 782 8495 Fax: 778 782 7514 Email: gary_wang@sfu.ca 2 change the exponentially growing computation cost to be polynomial. Testing and comparison confirm the efficiency and capability of RBF-HDMR for HEB problems.

High dimensional Kriging metamodelling utilising gradient information

“…ANOVA-HDMR is constructed by eval- to be known about the function to be modelled while considering the sample budget. In this work, the Cut-HDMR variant is chosen to model high dimensional problems due to its arithmetic simplicity while providing accurate high dimensional metamodels with least sample budget [20,25,27]. A Cut-HDMR variant using radial basis functions was recently introduced in [20] along with an adaptive sampling and model construction algorithms with promising results.…”

“…Subsequently, the Kriging models with gradient data are compared with Kriging models without gradient data (termed as Ordinary Kriging based HDMR) and also with Radial basis function based models without gradient data (termed as RBF-HDMR) from [27]. The principal motivation for the introduction of Gradient Enhanced Kriging based HDMR (GEK-HDMR) is to investigate how much reduction in number of training sample points can be achieved while modelling high dimensional problems by incorporating the cheaply available gradient data.…”

“…The gradient values of the benchmark functions with respect to the design variables are analytically calculated. • Function1 [27] …”

“…Moreover, the advantage of incorporating additional gradient data is significantly visible from the comparison with the RBF-HDMR in [27] (see "Function1" and "Function2" results in Tables 1, 3 …”

Optimization of the Complex-RFM optimization algorithm

“…Many different SMs exist and several attempts have been made to compare their performances for both analytical functions and engineering problems [10], [11], [22], [24]. However, it is difficult to draw any general conclusion about which SM performs best [25].…”