Environmental contours as Voronoi cells

Environmental contours are widely used as basis for design of structures exposed to environmental loads. The basic idea the method is decouple description from structural response. This done by establishing an envelope joint extreme values representing critical conditions, such that any structure tolerating loads on this will have a failure probability smaller than prescribed value. Specifically, given n-dimensional random variable $$\mathbf {X}$$ and target $$p_{e}$$ , contour boundary set $$\mathcal {B} \subset \mathbb {R}^{n}$$ with following property: For {F} if {F}$$ does not intersect interior {B}$$ then failure, $$P(\mathbf {X} \in \mathcal {F})$$ bounded above . We work under assumption sets convex, which relevant many real-world applications. In paper, we show may be regarded boundaries Voronoi cells. geometric interpretation leads new theoretical insights suggests simple novel construction algorithm guarantees desired probabilistic properties. illustrated examples in two three dimensions, but results extend arbitrary dimensions. Inspired Voronoi-Delaunay duality numerical discrete scenario, also able derive analytical representation where considered differentiable manifold, criterion its existence established.