本研究為解決非線性可靠度限制式之最佳化問題，提出一個以樣本法與 SQP 為骨架的演算法。此演算法使用平均重要取點法(AAIS)來估計破壞機率與其梯度。平均重要取點法為取樣在函數的極限狀態上，因此可使用較少的樣本數得到穩健且精確的破壞機率估計值。此外，平均重要取點法也可在有限的樣本數下，得到相對精確且變異量小的梯度量值。為防止破壞機率梯度的計算結果遭受機率值低但梯度值高的樣本影響，本研究提出一樣本過濾器，防止少數樣本導致梯度計算結果產生偏差。為確保演算法收斂，本演算法中使用一試驗準則(F-filter)來替代懲罰函數，用以判斷設計點可否被更新，並避免使用懲罰函數所造成的不良條件。
為了提升演算法的效率，將部分已取樣之樣本回收再利用，做為下一次疊代的樣本，因此下一次疊代只需在上一次疊代過程中還未取樣的區域進行取樣即可。文中以一個工程範例以及兩個數學範例說明以本研究所提出演算法之計算結果並與文獻上的方法加以比較，其中一個數學範例為有多個最大可能破壞點，使用現行一階、二階可靠度方法與蒙地卡羅法皆無法有效的解決。本文提出的演算法除了可有效的解決多破壞模式問題，也可提昇處理一般可靠度最佳化問題的效率、穩健度，及收斂性。對於工程上存在的高度非線性可靠度限制式，本研究所提出的取樣 SQP 演算法較文獻上的方法有著更優異的演算性能。

In this work we extend a filter-based sequential quadratic programming (SQP) algorithm to solve reliability-based design optimization (RBDO) problems with highly nonlinear constraints. This filter-based SQP uses the approach of average importance sampling (AAIS) in calculating the values and the gradients of probabilistic constraints. AAIS allocates samples at the limit state boundaries such that relatively few samples are required in calculating constraint probability values to achieve high accuracy and low variance. The accuracy of probabilistic constraint gradients using AAIS is improved by a sample filter to eliminate sample outliers that have low probability of occurrence and high gradient values. To ensure convergence, this algorithm replaces the penalty function by an iteration filter to avoid the ill-conditioning problems of the penalty parameters in the acceptance of a design update. A sample-reuse mechanism is introduced to improve the efficiency of the algorithm by avoiding redundant samples. `Unsampled' region, the region that is not covered by previous samples, is identified by the iteration step lengths, the trust region, and constraint reliability levels. As a result, this filter-based sampling SQP can efficiently handle highly nonlinear probabilistic constraints with multiple most probable points or functions without analytical forms. Several examples are demonstrated and compared with FORM/SORM and Monte Carlo simulation. Results show that by integrating the modified AAIS with the filter-based SQP, overall computation cost can be significantly improved in solving RBDO problems.

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