Environmental contours based on inverse SORM

Abstract

It is well known that the full long-term response analysis is recognized as the most accurate and reliable method for evaluation of the extreme response in the design of ships and marine structures. However, such a method is time consuming for large and complex systems. To improve efficiency, the environmental contour method (ECM) is frequently used to approximate the long-term extreme response. The ECM is based on an environmental contour, which is traditionally obtained by the inverse first order reliability method (IFORM) with the assumption that the failure surface in the U space is approximated as a tangent plane at the design point. However, such an approximation underestimates the true failure probability if the failure surface in the U space is a concave set, and then the corresponding environmental contour would not be conservative for possible designs. In this work, a more conservative ISORM (inverse second order reliability method) contour is proposed. In this method, a specific secondorder surface is applied to approximate the failure surface at the design point, and then the failure domain in the U space would always be overestimated since the corresponding safe domain is approximated and underestimated as a sphere, regardless of the shapes of the failure surfaces. Therefore, the generated environmental contour can be always conservative for design purposes. The differences of the environmental contours generated by different methods, i.e., the traditional IFORM and the proposed ISORM, are illustrated by relevant examples, such as the wave statistics, wind wave statistics, and first-year ice ridge statistics.

Environmental contours for circular‐linear variables based on the direct sampling method

“…This is also discussed in Vanem, where different contouring methods were compared assuming some simple structural problems. Possibly, contouring methods based on the second‐order reliability method (SORM) could then be useful …”

“…In particular, the direct sampling contours will by definition always enclose a convex set, whereas the IFORM contours will tend to follow the scatter of the data more closely. Other approached for environmental contours exist, see, eg, previous studies . The most common applications of environmental contours are restricted to two‐dimensional problems and with only linear variables, although generalizations to higher dimensions are, in principle, straightforward .…”

Design for Extremes: A Contour Method for Defining Requirements Based on Multivariate Extremes

“…Based on a given value for α the contour construction technique described in Section 2.2 can be applied. Other contour construction methods are suitable too and based on the specific design project, one might apply another technique (see, for example, Winterstein et al, 1993;Huseby et al, 2013;Jonathan et al, 2014;Chai and Leira, 2018). Constructing a contour based on a highest density region, however, has a potential advantage for the design process: for a given probability content this region occupies the smallest possible volume in the sample space.…”

“…Later, Winterstein et al (1993) proposed a new way of constructing environmental contours, the socalled inverse first-order reliability method (IFORM), which was developed within the framework of structural reliability theory [see, for example, Madsen et al (2006)]. To this day, numerous methods for the construction of an environmental contour have been proposed (see, for example, Huseby et al, 2013;Jonathan et al, 2014;Haselsteiner, Ohlendorf, Wosniok and Thoben, 2017;Chai and Leira, 2018;Manuel et al, 2018;Vanem, 2018). The process of developing an environmental contour typically involves the estimation of the joint probability density function of the variables of interest and contour construction in the narrower sense.…”

Comparing different contour methods with response-based methods for extreme ship response analysis

“…Other approached for environmental contours exist, see e.g. [10,11,12,13,14,15,16,17] or the recent review in [18]. In this paper various approaches to environmental contours will be used, and the results from an approximate long-term analysis using such contours will be compared to an approximate long-term analysis based on a statistical response emulator.…”