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Minimal Convex Environmental Contours

Åsmund Hausken Sande, Johan S. Wind

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.

Minimal Convex Environmental Contours

Abstract

We develop a numerical method for the computation of a minimal convex and compact set, , in the sense of mean width. This minimisation is constrained by the requirement that for all unit vectors given some Lipschitz function . 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.
Paper Structure (18 sections, 22 theorems, 113 equations, 7 figures)

This paper contains 18 sections, 22 theorems, 113 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main results in two dimensions
  3. basic intro to needed theory
  4. Standard Notation
  5. Convexity
  6. Numerical Definitions
  7. Minimal Valid Contours
  8. The Continuous Problem
  9. The Discrete Problem
  10. Convergence and Near-Optimality
  11. Numerical examples
  12. Connection with Hausdorff Topology
  13. Basic Concepts and Definitions
  14. Existence of Solutions to the Continuous Problem
  15. Convergence in Hausdorff Metric
  16. ...and 3 more sections

Key Result

Proposition 3.3

Let $\mathcal{B} \subset \mathbb{R}^N$ be a non-empty, convex, and compact set. Denote the maximum radius $R \coloneqq \sup_{p \in \mathcal{B}} \left\|p\right\|_2$. Then $B(\mathcal{B},u) = \max_{p \in \mathcal{B}} \langle p, u\rangle$ is $R$-Lipschitz as a function of $u \in S^{N-1}$. Note also $|B

Figures (7)

  • Figure 1: We discretise the true outreach requirement (black circle) into only nine directions (blue). This highlights the difference between $\mathcal{B}^*$ from \ref{['prop:Bstar']} (green) and $\mathcal{B}'$ from \ref{['cor:Bprime']} (orange). $\mathcal{B}^*$ always lies inside of $\mathcal{B}'$.
  • Figure 2: Left: Periodic constraint function $C$ constructed such that \ref{['cpprob']} has several shapes with minimal perimeter. Right: The outreach requirement boils down to requiring the shape to touch the four sides of a square (blue). A specific numerical implementation of our method \ref{['lpprob_2d1']} output the green shape. We highlight two other optimal shapes (orange).
  • Figure 3: The naive method fails to output a convex shape satisfying the requirements, when the outreach function $C$ contains noise. We highlight a violated outreach constraint in blue. The plots are the same, with the right one being zoomed in.
  • Figure 4: Shape found by \ref{['lpprob']} for the 3d example described in \ref{['exam:3d']}.
  • Figure 5: Left: A 3-Lipschitz continuous constraint function $C$. Right: The empirical convergence rate of our method matches the rate $O(1/m)$ given by our theory (\ref{['thm:DPepsinCPeps_discrete']}).
  • ...and 2 more figures

Theorems & Definitions (58)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • Definition 3.6
  • Definition 3.7
  • Remark 3.8
  • Definition 4.1
  • ...and 48 more