Uncertainties related to external loads and material properties are not uncommon and should be taken into account for design of reliable structures. Deterministic approaches and reliability-based approaches are usually adopted to handle such uncertainties. The former is very simple and based on empirical equations, but liable to over-design. The latter has been developed during the last three decades, and has become one of the most rational method for considering uncertainties in design problems. The reliability-based approach is particularly well established to evaluate the failure probability of structures.
Probabilistic optimization generally involves the failure probability evaluation for a limit state function, which is used to determine the failure or success of a system. The failure probability can be approximated by either a reliability index approach(abbreviated as RIA) or a fixed norm formulation(abbreviated as FNF). Suboptimizations are required for the approximation of the failure probability of each constraint. Sub-problems are defined as finding the most probable failure point where the failure probability is the largest in random variable space in RIA, or searching for the minimum performance target point, which has a prescribed failure probability in FNF. As such, probabilistic optimization has an inner loop for probability calculation and an outer loop for design optimization.
In probabilistic optimization problems, the numerical cost of evaluating probabilistic constraints is quite large compared with that of deterministic problems and hence efficient evaluation is required for structural design problems that have large state equations. In this paper, we adopt the envelope function to reduce the number of probabilistic constraints and a deterministic constraint to limit the maximum constiuent function of the envelope function. The limit value is included in the process of design optimization and automatically decided like design variables at the end of design optimization. It is recommended that a user-defined parameter of the envelope function is chosen as 0.1 or 0.2.
Several numerical examples are tested adopting the reliability index approach or the fixed norm formulation with or without the envelope function. Also. the proposed algorithm is tested through the examples. The results show that the proposed method requires less number of the function evaluations. Efficiency improvement is remarkable for large structural problems which require a lot of number of finite element analyses.
신뢰성 있는 구조를 설계할 때 하중 또는 물성치 등에 존재하는 불확실성을 고려하여야 한다. 확정론적 최적설계에서는 제한조건을 보수적으로 설정하거나 안전 계수의 개념을 이용하여 그러한 변동을 간접적으로 고려하여 왔다. 신뢰도 기반 최적설계는 시스템의 손상 여부를 한계 상태식의 만족 확률로 나타내어 최적설계를 진행하게 된다. 이 때, 확률 제한조건의 평가시 요구되는 수치적 비용이 확정론적 최적설계보다 훨씬 커지기 때문에 본 논문에서는 덮개 함수를 이용하여 확률 제한조건의 개수를 감소시키고, 덮개 함수의 구성함수 중 최대값을 한정하는 제한조건을 도입하는 수식화를 제안하여 신뢰도 기반 최적설계의 효율을 개선하였다. 시스템의 파손 여부를 판정하기 위해 유한 요소해석이 필요한 자동차 단품의 신뢰도 기반 최적설계 예제와 몇몇 수치 예제 문제에 제안한 방법을 적용하여 그 효율성과 정확성을 검증하였다. 본 논문에서 제안한 방법을 통해 기존의 방법보다 더 작은 함수 계산만으로도 적절한 최적해를 얻을 수 있었다.
