A proper general decomposition approach for an efficient structural reliability analysis

所有工程学科的目标都是坚固的产品和结构。因此，对这类产品和结构的可靠性评估在其整个生命周期--开发、建造、运营和退役--中具有重要意义。可靠性分析的目标是计算失效概率。特别是在结构可靠性分析的情况下，由于结构失效非常罕见，因此预计失效概率非常小。在文献中，可以找到用于估计故障概率的各种方法。这些计算通常涉及复杂的非线性有限元模拟，这导致了计算成本高昂的分析。最近的方法旨在减少所需的确定性计算的次数，以减少高计算量。然而，这种方法仍然需要对基础有限元模型进行几千次计算。该方案的思想是利用模型降阶技术来提高可靠性分析的计算效率。为了减少复杂数值模拟的计算量，模型降阶是一个流行的概念。这一概念最近被用于各种应用中。该提案的目的是开发一种模型阶次方法，该方法可用于可靠性分析的基本计算密集型有限元模拟。将可靠性方法与模型降阶相结合，可以有效地计算失效概率。为此，发展了以随机参数为附加坐标的非线性结构问题的广义分解方法。在非线性的情况下，位移和内部变量由包含可靠性分析所产生的随机参数的分离形式来近似。这种方法导致了整个参数空间的复杂有限元模拟的分离形式的求解。简化后的模型自适应地耦合到可靠性分析中。因此，可靠性问题被分解为几个具有较高失效概率的子问题。故障概率被估计为条件故障概率的乘积。基于为每个子问题生成的PGD模拟来计算条件概率。借助于简化模型的解，每个极限状态函数可以以分离的形式导出。该函数可以用很低的计算工作量来评估。因此，可以以非常有效的方式估计每个条件概率。该方法可普遍应用于各种结构可靠度问题。通过研究徐变对预应力钢筋混凝土结构可靠度的影响，验证了该方法的有效性。

All engineering disciplines aim at robust products and structures. Therefore, the assessment of the reliability of such products and structures is important over their whole life time: development, construction, operation and decommission. The goal of reliability analysis is the computation of the probability of failure. Especially in case of structural reliability analysis, very small failure probabilities are expected, because structural failure is very rare. In the literature, a variety of approaches for estimating the probability of failure can be found. These calculations generally involve a complex nonlinear finite element simulation, which leads to a computationally expensive analysis. Recent methods aim to decrease the number of required deterministic calculations, to reduce the high computational effort. Nevertheless, such approaches still require a few thousand calculations of the underlying finite element model. The idea of the proposal is to use model reduction techniques to improve the computational efficiency of the reliability analysis. Model reduction is a popular concept to decrease the computational effort of complex numerical simulations. The concept is recently used in a variety of applications. The aim of the proposal is to develop a model order approach which can be used for the underlying computationally intensive finite element simulations of a reliability analysis. The combined reliability method with model reduction leads to an efficient computation of the probability of failure. For this purpose, the proper general decomposition (PGD) method is developed for nonlinear structural problems with random parameters as additional coordinates. The displacements as well as the internal variables, in case of nonlinearity, are approximated by a separated form including the random parameters resulting from the reliability analysis. The approach leads to solution of a complex finite element simulation in a separated form for the whole parameter space. The reduced model is adaptively coupled within the reliability analysis. Therefore, the reliability problem is decomposed in several sub-problems with higher failure probabilities. The probability of failure is estimated as product of the conditional failure probabilities. The conditional probabilities are computed based on PGD simulations generated for each sub-problem. By means of the solution of the reduced model, each limit state function can be derived in a separated form. This function can be evaluated with low computational effort. Consequently, each conditional probability can be estimated in a very efficient way. The developed method can be generally applied to all kinds of structural reliability problems. In the proposal, the efficiency of the method is validated by investigating the influence of creep on the reliability of a prestressed reinforced concrete structure.