Department of Mathematics Colloquium


Abstract 

A factor of safety method for quantitative
estimates of gridspacing and timestep 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. Unresolved
issues and challenges are also discussed.
