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Research Papers: Techniques and Procedures

# Factors of Safety for Richardson Extrapolation

[+] Author and Article Information
Tao Xing

Department of Mechanical Engineering, College of Engineering, Architecture and Physical Sciences, Tuskegee University, Tuskegee, AL 36088taox@tuskegee.edu

Frederick Stern

IIHR-Hydroscience and Engineering, C. Maxwell Stanley Hydraulics Laboratory, University of Iowa, Iowa City, IA 52242-1585frederick-stern@uiowa.edu

J. Fluids Eng 132(6), 061403 (Jun 23, 2010) (13 pages) doi:10.1115/1.4001771 History: Received February 11, 2009; Revised April 19, 2010; Published June 23, 2010; Online June 23, 2010

## 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.

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## Figures

Figure 1

Comparison of error estimates for the one-dimensional wave equation

Figure 2

Effectivity indices for different error estimates

Figure 3

Factor of safety for different verification methods with pth=2 and r=2 for the CF method

Figure 4

Actual factor of safety for sample 3, sample 3 averaged using ΔP=0.01, and FSA¯±tSFSA¯ for samples 9–25: (a) GCI method, (b) GCI1 method, (c) GCI2 method, (d) CF method, and (e) FS method

Figure 5

Standard deviation for the mean SX¯ and coefficient of variation SX¯(%X¯) for samples 9–25: (a) SX¯ and (b) SX¯(%X¯)

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