Environmental contours as Voronoi cells

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.