• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.36 no.8
• /
• pp.873-879
• /
• 2012

Metamodels have been developed with a variety of design optimization techniques in the field of structural engineering over the last decade because they are efficient, show excellent prediction performance, and provide easy interconnections into design frameworks. To construct a metamodel, a sequential procedure involving steps such as the design of experiments, metamodeling techniques, and validation techniques is performed. Because validation techniques can measure the accuracy of the metamodel, the number of presampled points for an accurate kriging metamodel is decided by the validation technique in the sequential kriging metamodel. Because the interpolation model such as the kriging metamodel based on computer experiments passes through responses at presampled points, additional analyses or reconstructions of the metamodels are required to measure the accuracy of the metamodel if existing validation techniques are applied. In this study, we suggest a sensitivity validation that does not require additional analyses or reconstructions of the metamodels. Fourteen two-dimensional mathematical problems and an engineering problem are illustrated to show the feasibility of the suggested method.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.30 no.6 s.249
• /
• pp.691-697
• /
• 2006

Metamodel, model of model, has been widely used to improve an efficiency of optimization process in engineering fields. However, global metamodels of constraints in a constrained optimization problem are required good accuracy around neighborhood of optimum point. To satisfy this requirement, more sampling points must be located around the boundary and inside of feasible region. Therefore, a new sampling strategy that is capable of identifying feasible domain should be applied to select sampling points for metamodels of constraints. In this research, we suggeste sequential feasible domain sampling that can locate sampling points likely within feasible domain by using penalty function method. To validate the excellence of feasible domain sampling, we compare the optimum results from the proposed method with those form conventional global space-filling sampling for a variety of optimization problems. The advantages of the feasible domain sampling are discussed further.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.28 no.11
• /
• pp.1760-1767
• /
• 2004

Sequential sampling approaches of a metamodel that sampling points are updated sequentially become a significant consideration in metamodeling technique. Sequential sampling design is more effective than classical space filling design of all-at-once sampling because sequential sampling design is to add new sampling points by means of distance between sampling points or precdiction error obtained from metamodel. However, though the extremum points can strongly reflect the behaviors of responses, the existing sequential sampling designs are inefficient to approximate extremum points of original model. In this research, new sequential sampling approach using the sensitivity of Kriging model is proposed, so that new approach reflects the behaviors of response sequentially. Various sequential sampling designs are reviewed and the performances of the proposed approach are compared with those of existing sequential sampling approaches by using mean squared error. The accuracy of the proposed approach is investigated against optimization results of test problems so that superiority of the sensitivity approach is verified.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.33 no.12
• /
• pp.1464-1470
• /
• 2009

RBDO approach based on a sampling method with the Kriging metamodel and Constraint Boundary Sampling (CBS), which is sequential sampling method to generate metamodels is proposed. The major advantage of the proposed RBDO approach is that it does not require Most Probable failure Point (MPP) which is essential for First-Order Reliability Method (FORM)-based RBDO approach. The Monte Carlo Sampling (MCS), most well-known method of the sampling methods for the reliability analysis is used to assess the reliability of constraints. In addition, a Cumulative Distribution Function (CDF) of the constraints is approximated using Moving Least Square (MLS) method from empirical distribution function. It is possible to acquire a probability of failure and its analytic sensitivities by using an approximate function of the CDF for the constraints. Moreover, a concept of inactive design is adapted to improve a numerical efficiency of the proposed approach. Computational accuracy and efficiency of the proposed RBDO approach are demonstrated by numerical and engineering problems.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.34 no.5
• /
• pp.541-547
• /
• 2010

The rigorous validation of the accuracy of metamodels is an important topic in research on metamodel techniques. Although a leave-k-out cross-validation technique involves a considerably high computational cost, it cannot be used to measure the fidelity of metamodels. Recently, the mean$_0$ validation technique has been proposed to quantitatively determine the accuracy of metamodels. However, the use of mean$_0$ validation criterion may lead to premature termination of a sampling process even if the kriging model is inaccurate. In this study, we propose a new validation technique based on the mean and variance of the response evaluated when sequential sampling method, such as maximum entropy sampling, is used. The proposed validation technique is more efficient and accurate than the leave-k-out cross-validation technique, because instead of performing numerical integration, the kriging model is explicitly integrated to accurately evaluate the mean and variance of the response evaluated. The error in the proposed validation technique resembles a root mean squared error, thus it can be used to determine a stop criterion for sequential sampling of metamodels.

• Journal of Mechanical Science and Technology
• /
• v.20 no.8
• /
• pp.1169-1182
• /
• 2006

Robust design technology has been applied to versatile engineering problems to ensure consistency in product performance. Since 1980s, the concept of robust design has been introduced to numerical optimization field, which is called the robust optimization. The robustness in the robust optimization is determined by a measure of insensitiveness with respect to the variation of a response. However, there are significant difficulties associated with the calculation of variations represented as its mean and variance. To overcome the current limitation, this research presents an implementation of the approximate statistical moment method based on kriging metamodel. Two sampling methods are simultaneously utilized to obtain the sequential surrogate model of a response. The statistics such as mean and variance are obtained based on the reliable kriging model and the second-order statistical approximation method. Then, the simulated annealing algorithm of global optimization methods is adopted to find the global robust optimum. The mathematical problem and the two-bar design problem are investigated to show the validity of the proposed method.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.29 no.9 s.240
• /
• pp.1217-1224
• /
• 2005

To improve the accuracy of a metamodel, additional sample points can be selected by using a specified criterion, which is often called sequential sampling approach. Sequential sampling approach requires small computational cost compared to one-stage optimal sampling. It is also capable of monitoring the process of metamodeling by means of identifying an important design region for approximation and further refining the fidelity in the region. However, the existing critertia such as mean squared error, entropy and maximin distance essentially depend on the distance between previous selected sample points. Therefore, although sufficient sample points are selected, these sequential sampling strategies cannot guarantee the accuracy of metamodel in the nearby optimum points. This is because criteria of the existing sequential sampling approaches are inefficient to approximate extremum and inflection points of original model. In this research, new sequential sampling approach using the sensitivity of metamodel is proposed to reflect the response. Various functions that can represent a variety of features of engineering problems are used to validate the sensitivity approach. In addition to both root mean squared error and maximum error, the error of metamodel at optimum points is tested to access the superiority of the proposed approach. That is, optimum solutions to minimization of metamodel obtained from the proposed approach are compared with those of true functions. For comparison, both mean squared error approach and maximin distance approach are also examined.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.31 no.1 s.256
• /
• pp.55-61
• /
• 2007

Recently simulation model becomes an essential tool for analysis and design of a system but it is often expensive and time consuming as it becomes complicate to achieve reliable results. Therefore, high-fidelity simulation model needs to be replaced by an approximate model, the so-called metamodel. Metamodeling techniques include 3 components of sampling, metamodel and validation. Cross-validation approach has been proposed to provide sequnatially new sample point based on cross-validation error but it is very expensive because cross-validation must be evaluated at each stage. To enhance the cross-validation of metamodel, sequential sampling method using candidate points and representative cross-validation is proposed in this paper. The candidate and representative cross-validation approach of sequential sampling is illustrated for two-dimensional domain. To verify the performance of the suggested sampling technique, we compare the accuracy of the metamodels for various mathematical functions with that obtained by conventional sequential sampling strategies such as maximum distance, mean squared error, and maximum entropy sequential samplings. Through this research we team that the proposed approach is computationally inexpensive and provides good prediction performance.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.35 no.9
• /
• pp.1007-1013
• /
• 2011

Because of their high flexibility and durability, multilayer bellows are manufactured for use in commercial vehicles, while single-layer bellows are manufactured for use in passenger vehicles. A study based on the finite element method (FEM) and shape optimization for the single-layer bellows has been actively performed; however, until now, a study based on the FEM has rarely been performed for the multilayer bellows with gaps between the layers. This paper presents a finite-element modeling scheme for the multilayer bellows to improve simulation reliability during the evaluation of stress and flexibility. For performing shape optimization for the multilayer bellows, DOE (design of experiment) and the Kriging metamodel followed by the D-optimal method are used.