论文标题
环境轮廓作为Voronoi细胞
Environmental contours as Voronoi cells
论文作者
论文摘要
环境轮廓被广泛用作设计暴露于环境负荷的结构的基础。该方法的基本思想是将环境描述与结构响应解脱。这是通过建立环境条件的信封来完成的，以便在该信封上容忍负载的任何结构都具有小于规定值的故障概率。具体而言，给定给定$ n $ dimenional随机变量$ \ mathbf {x} $和失败的目标概率$ p_ {e} $，环境轮廓是集合$ \ mathcal {b} \ subset \ subset \ subset \ subset \ mathbb {r}^n} $ sew lafe for sew the the the for not for not for not the for not for not for the the ob for not the tob for not the for not for not for not for not oseb的边界\ Mathbb {r}^{n} $，如果$ \ nathcal {f} $不会与$ \ m nathcal {b} $的内部相交，则失败的可能性，$ p（\ mathbf {x} \ in \ mathcal {f}）$，$ p _} $} $} $}。正如许多现实世界应用所常见的那样，我们在假设故障集为凸的假设下工作。 在本文中，我们表明这种环境轮廓可能被视为伏诺诺细胞的边界。这种几何解释导致了新的理论见解，并提出了一种简单的新型结构算法，可确保所需的概率特性。该方法用两个维度和三个维度的示例进行了说明，但结果扩展到任意维度的环境轮廓。在数值离散方案中受到Voronoi-delaunay二元性的启发，我们还能够得出一个分析表示，其中环境轮廓被视为可区分的歧管，并且建立了其存在的标准。
Environmental contours are widely used as basis for design of structures exposed to environmental loads. The basic idea of the method is to decouple the environmental description from the structural response. This is done by establishing an envelope of environmental conditions, such that any structure tolerating loads on this envelope will have a failure probability smaller than a prescribed value. Specifically, given an $n$-dimensional random variable $\mathbf{X}$ and a target probability of failure $p_{e}$, an environmental contour is the boundary of a set $\mathcal{B} \subset \mathbb{R}^{n}$ with the following property: For any failure set $\mathcal{F} \subset \mathbb{R}^{n}$, if $\mathcal{F}$ does not intersect the interior of $\mathcal{B}$, then the probability of failure, $P(\mathbf{X} \in \mathcal{F})$, is bounded above by $p_{e}$. As is common for many real-world applications, we work under the assumption that failure sets are convex. In this paper, we show that such environmental contours may be regarded as boundaries of Voronoi cells. This geometric interpretation leads to new theoretical insights and suggests a simple novel construction algorithm that guarantees the desired probabilistic properties. The method is illustrated with examples in two and three dimensions, but the results extend to environmental contours in arbitrary dimensions. Inspired by the Voronoi-Delaunay duality in the numerical discrete scenario, we are also able to derive an analytical representation where the environmental contour is considered as a differentiable manifold, and a criterion for its existence is established.
