Quantitative assessment of uncertainties in a structural or mechanical system is one of the major concerns in system design. Reliability analysis is a quantitative method for uncertainties and can be categorized into level II method such as FORM(First Order Reliability Method) or SORM(Second Order Reliability Method), and level III method such as MCS(Monte Carlo Simulation) or moment method.
FORM has been the most popular method because of the easiness of implementation and computational efficiency. However, FORM has a limitation such as the degradation of accuracy induced by the non-linearity of limit state function or the non-normality of random variables. Comparing with FORM, moment method can be applicable to non-linear or non-normal case and give readily available PDF of a limit state function.
Moment method consists of moment calculation and PDF estimation step. In the previous study of moment method, emphasis has been on the improvement of efficiency to calculate moments of limit state function. The Pearson system or the saddle point approximation method is often used for the PDF estimation. However, The PDF estimated from a finite number of moment constraints is not unique and the Pearson system or the saddle point approximation gives only one of the candidates that satisfy the same moment constraints.
Therefore, in this thesis, the concept of maximum entropy, which has been suggested by Shannon in information theory in 1948, is introduced to estimate not the most accurate but the most probable model for reliability analysis and reliability-based design optimization. The maximum entropy principle (MEP) formulated as an optimization problem generates the most probable probability distribution among the many possible that have the same moment conditions. The PDF estimation by the MEP is not restricted by the number of given moments and accommodates higher order moment information and therefore facilitates a higher quality PDF model. The estimated PDF by the MEP using four moments or more is compared with those by the Pearson system. It is observed that the MEP is more flexible and generally applicable than the Pearson system.
A sensitivity analysis of failure probability based on the MEP is performed for RBDO and verified through numerical examples and by a 3-D wing design. It is shown that the MEP based moment method can reduce the number of iterations while maintaining accuracy by increasing the number of moments. The moment method equipped with the MEP is found very flexible and well adopted for reliability analysis and design.
최근 기계 시스템이 갖고 있는 불확성의 정략적 평가 및 이를 이용한 설계기법의 개발이 활발하다. 불확실성을 정량적으로 평가하는 방법으로 일차신뢰도법이 활발히 사용되고 있으나 근사기법으로서의 한계로 인하여 그 적용이 제한적이다. 이러한 일차신뢰도법이 갖고 있는 한계를 극복하고 보다 일반적인 경우에 적용이 가능한 적률기법에 대한 연구가 활발하다. 적률기법은 적률계산 과정과 계산된 적률로부터 확률밀도함수의 추정과정으로 구성되며 기존의 연구에서 보다 효율적인 적률계산 기법에 대한 연구가 주를 이루었고 상대적으로 확률밀도함수 추정 기법에 대한 연구가 미미하였다. 기존의 확률밀도함수 추정기법으로 피어슨 시스템을 이용하는 방법이 활발하였으나 피어슨 시스템은 사용할 수 있는 적률의 개수가 한정적이며 또한 확률밀도함수가 단일모드인 경우에 한정되었다. 따라서 보다 일반적인 확률추정기법의 도입이 필요하다. 이러한 필요에 의하여 본 논문에서는 최대엔트로피법칙을 이용하여 확률밀도함수를 추정하는 기법에 대하여 연구를 수행하였다. 또한 신뢰도기반 최적설계를 위하여 최대엔트로피법칙으로 유도된 손상확률의 민감도 수식화를 제안하였으며 이를 수치예제를 통하여 검증하였다. 또한 수식화된 민감도를 이용하여 3차원 날개 형상의 최적화 등에 이용하였다.
