Minimal Convex Environmental Contours

TL;DR

This paper introduces a numerical method to compute minimal convex environmental contours based on mean width minimization, aiding reliability analysis in marine structure design with efficient linear programming solutions.

Contribution

It presents a novel linear programming approach for constructing minimal convex environmental contours constrained by a Lipschitz function, with proven convergence properties.

Findings

Method effectively computes minimal convex contours.

Convergence analysis confirms reliability of the approach.

Numerical examples demonstrate practical applicability.

Abstract

We develop a numerical method for the computation of a minimal convex and compact set, $\mathcal{B}\subset\mathbb{R}^N$, in the sense of mean width. This minimisation is constrained by the requirement that $\max_{b\in\mathcal{B}}\langle b , u\rangle\geq C(u)$ for all unit vectors $u\in S^{N-1}$ given some Lipschitz function $C$. This problem arises in the construction of environmental contours under the assumption of convex failure sets. Environmental contours offer descriptions of extreme environmental conditions commonly applied for reliability analysis in the early design phase of marine structures. Usually, they are applied in order to reduce the number of computationally expensive response analyses needed for reliability estimation. We solve this problem by reformulating it as a linear programming problem. Rigorous convergence analysis is performed, both in terms of convergence of mean widths and in the sense of the Hausdorff metric. Additionally, numerical examples are provided to illustrate the presented methods.