Metamodel-based Shape Optimization of Connecting Rod Considering Fatigue Life

To optimize a connecting rod satisfying fatigue life, metamodel-based design optimization is proposed. To approximately predict both volume and fatigue life of connecting rod, kriging metamodel is constructed based on maximin eigenvalue sampling. Fatigue analysis is accomplished for the calculation of fatigue life. The results of metamodel-based design optimization are compared with those of classical optimization. The advantages of metamodel-based optimization are discussed. Introduction In engineering optimization, direct coupling between optimization algorithm and simulation model may be inefficient since iterative computer simulations during optimization process usually require enormous computational costs. In this case, an efficient solution is to use approximate model for responses that expresses relationship between design variables and responses with a moderate number of computer simulations. This approximate model is usually referred to as metamodel. A variety of metamodels such as response surface model [1], radial basis function, and kriging model [2] have been developed. Optimization technique incorporating these metamodels is called as metamodel-based design optimization. Metamodel-based design optimization consists of four steps: to determine sampling points, to build the metamodel, to assess accuracy of the metamodel, and then to use the validated metamodel for design optimization. In this paper, metamodeling technique is applied to shape optimization of connecting rod considering fatigue life. For fatigue analysis, commercial finite element software ANSYS is used. Optimization problem is defined as volume minimization of connecting rod while satisfying life cycles. Results of metamodel-based optimization are compared with those of typical optimization. Benefits of metamodel-based optimization are also discussed. Metamodeling Sampling technique. A proper choice of sampling strategy is very important in metamodeling since the accuracy of metamodel sensitively depends on selection of sample points. Usually, sampling points used to build metamodel should be uniformly distributed over entire design space without replication. It can lead to the good performance of metamodel at predicted points. A variety of sampling techniques based on this concept, the so-called space-filling sampling, have been developed: Latin hypercube sampling, maximin distance sampling, maximum entropy sampling, and maximin eigenvalue sampling [3]. In this paper, maximin eigenvalue sampling is employed. Maximin eigenvalue sampling is the optimal sampling based on eigenvalue problem as follows: 0 u R = − ) ( λ (1) where R is ) ( n n× correlation matrix that presents spatial correlation among n pre-sampled points. The entities of correlation matrix are defined by correlation function, i.e., Key Engineering Materials Vols. 306-308 (2006) pp 211-216 online at http://www.scientific.net © (2006) Trans Tech Publications, Switzerland Online available since 2006/Mar/15 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.33-14/04/08,13:32:50)       ∑ − − = = d n k p k j k i k ij x x R 1 ) ( exp θ (2) where k i x denotes the k-th component of sample points d n i R ∈ x . Superscript p represents the type of correlation function: 1 = p and 2 = p denote exponential and Gaussian correlation, respectively. Correlation parameter k θ is pre-determined as 0.01 before the optimization to obtain sample points. Since correlation matrix is positive definite, all eigenvalues of Eq. (1) are positive. Maximin eigenvalue sampling maximizes minimum eigenvalue min λ as follows: min maximizeλ (3) In maximin eigenvalue sampling, maximization of the minimum eigenvalue can be interpreted as the selection of sample set to acquire the maximum amount of information. That is, when sample points spread evenly over design domain, much information for design domain can be obtained from the sample points. Fig. 1 shows results of maximin eigenvalue sampling with different sample size for 2-dimensional problem. It is interesting that sample points are geometrically symmetric with respect to dotted line or point-symmetric with respect to the center of design domain. Note that design domain is normalized to ] 1 , 0 [ . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 1 Illustrations of maximin eigenvalue sampling Metamodel. Various metamodels exist as mentioned in the previous section, but here we review two metamodels, response surface model and kriging model, that are widely used in the application of design optimization. Response surface model has much success in fitting the data from physical experiment because it is based on regression technique to minimize the effect of random error. Second-order form of response surface model can be expressed as follows: ∑ ∑∑ + + ∑ + = = = < = d n