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# Stochastic Challenge Problems

To promote further the adoption of stochastic methods and tools in the engineering simulation area, the NAFEMS Stochastic Working Group (SWG) regularly sets out challenge problems for the community to solve. The SWG review the responses to the challenge problems to determine what approaches are being used by industry. The challenge problems also help to inform the group on what guidance material is needed.

The challenge problems and the summarised responses can be found below.

## Current Challenge Problem:

### Uncertain Knowledge

This new challenge problem, published in the April 2022 issue of the NAFEMS Benchmark Magazine is focused on the consideration of epistemic uncertainties. An irreducible uncertainty (aleatory uncertainty) is an inherent variation associated with the physical system being modeled which is characterized by probability distributions. Reducible uncertainty (epistemic uncertainty) is the lack of certitude in a measured or calculated value that can be reduced by gathering more data, observations, or information. Reducible uncertainty is typically addressed by attempting to include conservative assumptions in the formulation of the uncertainties. Reducible uncertainty is also called ‘lack of knowledge uncertainty’.

There are a variety of methods available to solve this problem. For example, double-loop Monte-Carlo simulations, where the inner loop samples from the aleatory uncertainty and the outer loop samples from the epistemic uncertainties. The given challenge problem does not, however, prescribe a specific solution method. The aim is rather to see what methods are applied in the community to address mixed uncertainty problems. The focus of this challenge problem is to estimate the uncertainty in the probabilistic output quantities and separate them according to aleatory and epistemic sources.

To respond to the Current Challenge Problem and to correspond with the NAFEMS Stochastics Working Group

Email: swg@nafems.org

T​he deadline for responses is Friday 25th November 2022

(in line with the deadline for Abstracts for NAFEMS' 2023 World Congress)

#### Challenge Problem:What are the Chances of the Ship Snapping?

This challenge problem represents the case of a damaged oil tanker. An explosion has ruptured the deck structure with the consequence that the resulting section modulus at this location is significantly decreased. Of course there is residual
strength; however, to keep the bending stress within acceptable limits, the bending moment generated by sea waves must be carefully monitored whilst sailing in open water to reach the repair yard. Is the predicted sea state allowable?

#### Challenge Problem:Uncertainty Quantification in a Tensile Material Test Specimen

The challenge is to determine the probability that the tensile stress will exceed the material yield stress during a tensile test. This problem has a closed form solution but the challenge is to apply engineering stochastics simulation methods to reproduce the theortical solution.

#### Challenge Problem:Uncertainty Quantification and Value of Information Challenge Problem: Electrical Problem

It is difficult to execute model validation exercises when uncertainties due to variation in parameters and operational conditions have to be accounted. A challenge problem is devised to develop statements of measures of confidence that the provided model can meet the functional requirements. Information provided from various sources for a specified set of conditions (intended application) will be used to assess the reliability of the device based on the functional requirements. The purpose of the challenge problem is to showcase and share different approaches to UQ methods among SWG members. Methods are not judged and this is not a competition to find the true solution.

A typical electronic component may be represented by an equivalent R-L-C (resistive, inductive and capacitive) series circuit. The electric transients that occur within the system are of interest. This particular challenge problem was chosen because the fundamental equations based on Kirchhoff’s current and voltage laws are well known and R-L-C parameters are usually readily available to electrically characterize components or systems. Further, variation and uncertainties from sources is presented in different forms and such responses can be easily generated using readily available tools.