Structural optimization using Kriging approximation

Abstract

An optimization method using Kriging approximation is applied to a structural optimization problem. The method involves two main processes. The first is a space estimation process that uses the Kriging method, and the second is an optimization process. The use of the Kriging method makes it easier to perform the approximation optimization. As an example of the estimation performed as part of structural optimization, a response surface for layout optimization of beam reinforcement is estimated. To evaluate the applicability of the Kriging method, Kriging estimation is compared with neural network approximation.

As a numerical example, the optimization of a stiffened cylinder for an eigenfrequency problem is illustrated. The obtained results clearly show the applicability of the method.

Introduction

Lower computational cost, generality, robustness, and accuracy are all required for structural optimization processes. Given these concerns, it would seem that large-scale problems, eigenfrequency problems, impact problems, and nonlinear problems with noisy functions or the like would be expensive and difficult to optimize practically. While optimization using experimental data or the results of CAE analyses with manual operation is required in industry or education, structural optimization using experimental results or discrete sample data that includes some errors is also difficult to solve.

For such problems, approximation optimization methods for structural optimization have recently been gaining attention. The response surface method (RSM), which is an approximation method, is a very effective approach. Barthelemy et al. and Haftka et al. reported on their survey of optimization using RSM [1], [2]. Box et al. discussed the contribution of polynomial approximation to optimum design using experimental data [3]. Kashiwamura et al. applied RSM to the optimization of an automotive seat frame [4]. Sobieski et al. applied response surface estimation to collaborative optimization for large-scale multidisciplinary design [5]. Hosder et al. reported on the application of polynomial response surface approximation to the multidisciplinary optimization of high-speed civil transport [6].

An optimization method using a neural network (NN) is another means of solving these difficult optimization problems. Hagiwara et al. applied holographic NN approximation to crash optimization [7], [8]. Shi et al. applied NN approximation to the crash-worthiness of a vehicle frame with the most probable optimal design method [9]. Nikolaidis et al. reported on NN and response surface polynomials for vehicle joints design [10]. Berke et al. also reported on the application of NN to structural mechanics [11]. Papilia et al. compared radial-basis NN with the polynomial-based technique [12].

Such approximation optimization methods can be readily applied to practical industrial applications, although there are some problems with the approximation process. RSM, which is based on experimental programming, normally requires the assumption of the order of the approximated base function because the approximation process is performed using the least square method for the coefficients of the function. Therefore, the designer must evaluate the schematic shape of the objective function over an entire solution space. This will sometimes be difficult because it requires an understanding of the qualitative tendency of the entire design space.

For a noisy known objective function, for example, it is acceptable to provide a subjectively assumed base function. For general structural optimizations, however, which use a deterministic and accurate evaluation for an unknown objective function, the observed value at a sample point should be more dependable for the approximation. This is because it would be difficult to determine the order of the base function to minimize the approximation error without any knowledge of the solution space. Another problem, pointed by Shi et al. is the difficulty of applying RSM based on experimental programming to a design problem having many design variables [9].

A NN approximation generally minimizes the sum of the approximation errors at the sample points, so that the accuracy of the approximated value at a sample point is relatively high. Carpenter et al. reported that a neural net approximation offers more flexibility to allow fitting than RSM [13]. A NN, however, presents some practical difficulties. One is the computational cost incurred for learning, while the other is, for example, the need for the operator to be skilled or experienced in using NN [13].

Against this background, there has been a demand for the development of an approximation optimization method that is more accurate, flexible, and easy to use. One good approximation method is Kriging approximation, which is a spatial estimation method. Sacks et al. applied Kriging approximation to the design for a computational experiment on circuit simulation [14], and Simpson et al. reported the optimization of an aerospike nozzle problem using Kriging approximation [15]. Giunta et al. compared Kriging models and polynomial regression models [16], while Jin et al. compared some metamodeling techniques for mathematical test problems and one engineering problem [17]. In this paper, to evaluate the applicability of the Kriging estimation method to engineering optimization, optimization using Kriging estimation and a gradient method is applied to structural optimization problems. Some examples of estimation and structural optimizations, as well as a comparison between the Kriging and NN results, are attempted using the proposed method, and the applicability of the method to a structural optimization problem is investigated.