Factors of Safety for Richardson Extrapolation

Abstract

A factor of safety method for quantitative estimates of grid-spacing and time-step uncertainties for solution verification is developed. It removes the two deficiencies of the grid convergence index and correction factor methods, namely, unreasonably small uncertainty when the estimated order of accuracy using the Richardson extrapolation method is greater than the theoretical order of accuracy and lack of statistical evidence that the interval of uncertainty at the 95% confidence level bounds the comparison error. Different error estimates are evaluated using the effectivity index. The uncertainty estimate builds on the correction factor method, but with significant improvements. The ratio of the estimated order of accuracy and theoretical order of accuracy P instead of the correction factor is used as the distance metric to the asymptotic range. The best error estimate is used to construct the uncertainty estimate. The assumption that the factor of safety is symmetric with respect to the asymptotic range was removed through the use of three instead of two factor of safety coefficients. The factor of safety method is validated using statistical analysis of 25 samples with different sizes based on 17 studies covering fluids, thermal, and structure disciplines. Only the factor of safety method, compared with the grid convergence index and correction factor methods, provides a reliability larger than 95% and a lower confidence limit greater than or equal to 1.2 at the 95% confidence level for the true mean of the parent population of the actual factor of safety. This conclusion is true for different studies, variables, ranges of P values, and single P values where multiple actual factors of safety are available. The number of samples is large and the range of P values is wide such that the factor of safety method is also valid for other applications including results not in the asymptotic range, which is typical in industrial and fluid engineering applications. An example for ship hydrodynamics is provided.

A comparative study on the design of an environmentally friendly RoPax ferry using CFD

“…It is expected that the numerical uncertainties for the other cases are of the same order. Xing and Stern (2010) state that the Richardson extrapolation (RE) method (1911) is the basis for existing quantitative numerical error/uncertainty estimates for time-step convergence and grid-spacing.…”

A comparison of adjoint and data-centric verification techniques.

“…The Grid Convergence Index (GCI) [18] is perhaps the first attempt to infer these asymptotic parameters from data along with an estimate of the uncertainty in the estimates. A variety of similar techniques have subsequently been developed (see e.g., [20,16,15] and the references therein): each using different assumptions and/or user experience to determine the appropriate safety or correction factors to improve the reliability of the uncertainty estimates without gross overestimation.…”

Uncertainty quantification for a sailing yacht hull, using multi-fidelity kriging

“…Initial error analysis for marine applications focused on grid convergence and time discretisation errors [34,40,33]. Recently, as a result of these efforts (i.e.…”