불량률 최소화를 통한 강건 최적화의 확률제한조건 처리

이광기(Kwang Ki Lee),박찬경(Chan Kyoung Park),김근연(Geun Yeon Kim),이권희(Kwon Hee Lee),한상욱(Sang Wook Han),한승호(Seung Ho Han) 대한기계학회 2013 大韓機械學會論文集A Vol.37 No.8

강건 최적화 기법은 설계 초기 단계부터 설계변수의 변동이 목적함수에 미치는 효과를 최소화할 수 있는 유일한 방법이다. 강건 최적화의 정식화를 위해서는 분산을 정확히 예측하고 확률제한조건을 정식화하는 것이 가장 중요한 과정이 된다. 분산 및 확률제한조건을 예측하고 정식화하기 위한 방법으로 공정능력지수 및 식스시그마 기법과 같은 여러 가지 방법이 적용되고 있으나, 실제 공정에서 널리 적용되는 불량률을 이용한 확률제한조건 처리 기법에 대한 연구는 아직까지 전무한 상태이다. 본 연구에서는 자동차 로워암의 무게와 최대응력의 평균과 표준편차에 대한 설계영역을 탐색하고, 이후 로워암의 강건 최적화를 수행하였다. 변동을 예측하기 위한 표준편차의 계산은 2 차 테일러 전개를 통해 수치적인 정확도를 기하였다. 강건 최적화는 설계변수의 불연속성을 고려하기 위하여 최적화 과정에서 미분 정보를 적용하지 않은 심플렉스 알고리즘을 적용하였다. A robust optimization is only one of the ways to minimize the effects of variances in design variables on the objective functions at the preliminary design stage. To predict the variances and to formulate the probabilistic constraints are the most important procedures for the robust optimization formulation. Though several methods such as the process capability index and the six sigma technique were proposed for the prediction and formulation of the variances and probabilistic constraints, respectively, there are few attempts using a percent defective which has been widely applied in the quality control of the manufacturing process for probabilistic constraints. In this study, the robust optimization for a lower control arm of automobile vehicle was carried out, in which the design space showing the mean and variance sensitivity of weight and stress was explored before robust optimization for a lower control arm. The 2nd order Taylor expansion for calculating the standard deviation was used to improve the numerical accuracy for predicting the variances. Simplex algorithm which does not use the gradient information in optimization was used to convert constrained optimization into unconstrained one in robust optimization.

비정규 분포에 대한 통계적 모멘트와 확률 제한조건의 민감도 해석

허재성(Jae-Sung Huh),곽병만(Byung-Man Kwak) 대한기계학회 2010 大韓機械學會論文集A Vol.34 No.11

설계단계에서 시스템의 불확실성을 반영하려는 노력이 다양하게 이루어지고 있으며, 강건 최적설계 혹은 신뢰도 기반 최적설계는 이에 대한 대표적인 설계 방법론이다. 이러한 최적화 수식에는 성능함수의 평균, 표준편차와 확률제한조건이 목적함수와 제한조건으로 주로 활용된다. 그러므로, 이러한 통계적 특성치를 효과적으로 계산하는 것은 필수적이며, 더 나아가 최적화 과정에서 비선형 계획법이 일반적으로 활용되므로 민감도가 반드시 필요하다. 본 연구에서는 통계적 모멘트와 확률제한조건에 대해 적분 형태로 정의되는 민감도 수식을 비정규 분포로 확장하고자 한다. 얻어진 민감도 해석 결과는 통계적 모멘트와 손상확률이 설계점에서 계산된 경우, 민감도를 얻기 위해 추가로 성능함수를 계산할 필요가 없음을 보여주므로 효율성 측면에서 우수하다. 그러나, 민감도 수식이 성능함수와 확률밀도함수의 미분과정에서 얻어지는 함수의 곱으로 정의되므로, 동일한 수치적분 방법이 적용되는 경우 민감도 해석 결과는 통계적 모멘트 결과의 정확도에 미치지 못할 가능성이 있다. The efforts of reflecting the system’s uncertainties in design step have been made and robust optimization or reliabilitybased design optimization are examples of the most famous methodologies. The statistical moments of a performance function and the constraints corresponding to probability conditions are involved in the formulation of these methodologies. Therefore, it is essential to effectively and accurately calculate them. The sensitivities of these methodologies have to be determined when nonlinear programming is utilized during the optimization process. The sensitivity of statistical moments and probability constraints is expressed in the integral form and limited to the normal random variable; we aim to expand the sensitivity formulation to nonnormal variables. Additional functional calculation will not be required when statistical moments and failure or satisfaction probabilities are already obtained at a design point. On the other hand, the accuracy of the sensitivity results could be worse than that of the moments because the target function is expressed as a product of the performance function and the explicit functions derived from probability density functions.

강건 최적설계에서 통계적 모멘트와 확률 제한조건에 대한 효율적인 민감도 해석

허재성(Jae-Sung Huh),곽병만(Byung-Man Kwak) 대한기계학회 2008 大韓機械學會論文集A Vol.32 No.1

The efforts of reflecting the system’s uncertainties in design step have been made and robust optimization or reliability-based design optimization are examples of the most famous methodologies. In their formulation, the mean and standard deviation of a performance function and constraints expressed by probability conditions are involved. Therefore, it is essential to effectively and accurately calculate them and, in addition, the sensitivity results are required to obtain when the nonlinear programming is utilized during optimization process. We aim to obtain the new and efficient sensitivity formulation, which is based on integral form, on statistical moments such as the mean and standard deviation, and probability constraints. It does not require the additional functional calculation when statistical moments and failure or satisfaction probabilities are already obtained at a design point. Moreover, some numerical examples have been calculated and compared with the exact solution or the results of Monte Carlo Simulation method. The results seem to be very satisfactory.

불량률 최소화를 통한 강건 최적설계의 확률제한조건 처리에 관한 연구

한승호(Seung Ho Han),이광기(Kwang Ki Lee) 대한기계학회 2011 대한기계학회 춘추학술대회 Vol.2011 No.10

In a design process, a robust optimization is only a way to minimize effects of variances in design variables on the design objectives. It should be carried out before beginning with manufacturing process by taking into account of the variances. Therefore, to predict the variances and to solve the formulation of constraints are the most important procedures for the robust optimization. Though several methods such as the process capability index and the six sigma technique were proposed for the prediction and solution of the variances and constraints, respectively, there are few attempts using a percent defective which has been widely applied in the quality control of the manufacturing process. In this study, the robust optimization for a case study of the control arm in automobile vehicle was carried out, in which the design space showing the mean and variance sensitivity of weight and stress was explored. Simplex algorithm was used to find the robustness of the optimal solution by the minimization of the percent defective and the formulation of constrained optimization converting to unconstrained one.

불량률 최소화를 통한 강건 최적화의 확률제한조건 처리

이광기(Kwang Ki Lee),박찬경(Chan Kyoung Park),안대균,한승호(Seung Ho Han) 대한기계학회 2012 대한기계학회 춘추학술대회 Vol.2012 No.11

In a design process, a robust optimization is only a way to minimize effects of variances in design variables on the design objectives. It should be carried out before beginning with manufacturing process by taking into account of the variances. Therefore, to predict the variances and to solve the formulation of constraints are the most important procedures for the robust optimization. Though several methods such as the process capability index and the six sigma technique were proposed for the prediction and solution of the variances and constraints, respectively, there are few attempts using a percent defective which has been widely applied in the quality control of the manufacturing process. In this study, the robust optimization for a case study of the control arm in automobile vehicle was carried out, in which the design space showing the mean and variance sensitivity of weight and stress was explored. Simplex algorithm was used to find the robustness of the optimal solution by the minimization of the percent defective and the formulation of constrained optimization converting to unconstrained one.

덮개 함수를 이용한 강건 최적설계의 제한 조건 단일화

이정준,정도현,이병채,Lee, Jeong-Jun,Jeong, Do-Hyeon,Lee, Byeong-Chae 대한기계학회 2002 大韓機械學會論文集A Vol.26 No.8

Design variables and design parameters are rarely deterministic in practice. Robust optimal design takes into consideration of the uncertainties in the design variables and parameters. Robust optimization methodology with probability constraints requires a lot of system analyses fer calculating failure probability of each constraint. By introducing an envelope function to reduce the number of constraints, efficiency of robust optimization techniques can be considerably improved. Through four illustrative examples, it is shown that the number of system analyses is greatly decreased while little differences in the optimum results are observed.

부 최적화 문제의 근사적인 계산을 통한 신뢰도 최적설계 방범의 효율개선

정도현,이병채,Jeong, Do-Hyeon,Lee, Byeong-Chae 대한기계학회 2001 大韓機械學會論文集A Vol.25 No.10

Alternative computational scheme is presented fur reliability based optimal design using a modified advanced first order second moment (AFOSH) method. Both design variables and design parameters are considered as random variables about their nominal values. Each probability constraint is transformed into a sub -optimization problem and then is resolved with the modified Hasofer- Lind-Rackwitz-Fiessler (HL-RF) method for computational efficiency and convergence. A method of design sensitivity analysis for probability constraint is presented and tested through simple examples. The suggested method is examined by solving several examples and the results are compared with those of other methods.

기계 구조의 강건 설계를 위한 최적화 기법의 개발

정도현,이병채,Jeong, Do-Hyeon,Lee, Byeong-Chae 대한기계학회 2000 大韓機械學會論文集A Vol.24 No.1

In order to reduce the variation effects of uncertainties in the engineering environments, new robust optimization method, which considers the uncertainties in design process, is proposed. Both design variables and system parameters are considered as random variables about their nominal values. To ensure the robustness of performance function, a new objective is set to minimize the variance of that function. Constraint variations are handled by introducing probability constraints. Probability constraints are solved by the advanced first order second moment (AFOSM) method based on the reliability theory. The proposed robust optimization method has an advantage that the second derivatives of the constraints are not required. The suggested method is examined by solving three examples and the results are compared with those for deterministic case and those available in literature.

〈학술논문〉 : 함수근사모멘트방법의 민감도 해석에 적용 가능성에 대한 연구

허재성(Jae-Sung Huh),곽병만(Byung-Man Kwak) 대한기계학회 2010 대한기계학회 춘추학술대회 Vol.2010 No.11

설계단계에서 시스템의 불확실성을 반영하려는 노력이 다양하게 이루어지고 있으며, 강건최적설계 혹은 신뢰도 기반 최적설계는 이에 대한 대표적인 설계 방법론이다. 실제 문제에 이러한 방법론을 적용하기 위해서는 성능함수의 통계적 모멘트와 손상확률에 대한 정확하고 효율적인 추정방법이 필요하고, 더불어 최적화를 위한 방향탐색과정에서 요구되는 민감도 해석의 정확성 및 효율성이 확보되어야 한다. 본 연구에서는 함수근사모멘트 방법을 기존에 유도된 적분 형태의 민감도 해석 식에 적용하여 그 민감도 해석 결과의 정확성을 확인하고자 한다. 민감도 해석 결과를 타 방법의 결과와 비교하여 함수근사모멘트 방법의 타당성을 입증하고자 한다. 활용된 적분 형태의 민감도 해석은 손상확률 혹은 통계적 모멘트가 계산된 경우 추가적인 함수 계산 없이 민감도를 얻을 수 있는 효율적인 방법이다. The efforts of reflecting the system’s uncertainties in design step have been made and robust optimization or reliability-based design optimization are examples of the most famous methodologies. For applying them into industrial problems, an accurate and efficient method of estimating statistical moments or failure probability is required, and further the results of sensitivity analysis, which is needed for searching direction during optimization process, should also be accurate and efficient. This research aims to apply the function approximation moment method into the sensitivity analysis formulation which is expressed as integral form and then to show the accuracy of the sensitivity results. Their results are compared of those of other moment methods and then show the feasibility of the function approximation moment method. The sensitivity analysis with integral form is the efficient formulation of obtaining sensitivity information without any additional function calculation if the failure probability or statistical moments are calculated.

함수근사모멘트방법의 신뢰도 기반 최적설계에 적용 타당성에 대한 연구

허재성(Jae-Sung Huh),곽병만(Byung-Man Kwak) 대한기계학회 2011 大韓機械學會論文集A Vol.35 No.2

설계단계에서 시스템의 불확실성을 반영하려는 노력이 다양하게 이루어지고 있으며, 강건 최적설계 혹은 신뢰도 기반 최적설계는 이에 대한 대표적인 설계 방법론이다. 실제 문제에 이러한 방법론을 적용하기 위해서는 성능함수의 통계적 모멘트와 손상확률에 대한 정확하고 효율적인 추정 방법이 필요하고, 더불어 최적화를 위한 방향탐색과정에서 요구되는 민감도 해석의 정확성 및 효율성이 확보되어야 한다. 본 연구에서는 함수근사모멘트 방법을 기존에 유도된 적분 형태의 민감도 해석 식에 적용하여 그 민감도 해석 결과의 정확성을 확인하고, 이를 대표적인 신뢰도 기반 최적설계 문제에 적용하고자 한다. 민감도 해석 결과 및 신뢰도 기반 최적설계 결과를 타 방법의 결과와 비교하여 함수근사모멘트 방법의 타당성을 입증하고자 한다. 활용된 적분 형태의 민감도 해석은 손상확률 혹은 통계적 모멘트가 계산된 경우 추가적인 함수 계산 없이 민감도를 얻을 수 있는 효율적인 방법이다. Robust optimization or reliability-based design optimization are some of the methodologies that are employed to take into account the uncertainties of a system at the design stage. For applying such methodologies to solve industrial problems, accurate and efficient methods for estimating statistical moments and failure probability are required, and further, the results of sensitivity analysis, which is needed for searching direction during the optimization process, should also be accurate. The aim of this study is to employ the function approximation moment method into the sensitivity analysis formulation, which is expressed as an integral form, to verify the accuracy of the sensitivity results, and to solve a typical problem of reliability-based design optimization. These results are compared with those of other moment methods, and the feasibility of the function approximation moment method is verified. The sensitivity analysis formula with integral form is the efficient formulation for evaluating sensitivity because any additional function calculation is not needed provided the failure probability or statistical moments are calculated.