Factors of Safety for Richardson

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. DOI: 10.1115/1.4001771

[1]  Hugh W. Coleman,et al.  VERIFICATION AND VALIDATION OF CFD SIMULATIONS , 1999 .

[2]  Frederick Stern,et al.  Computational Towing Tank Procedures for Single Run Curves of Resistance and Propulsion , 2008 .

[3]  Martin Hoekstra,et al.  Discretization Uncertainty Estimation based on a Least Squares version of the Grid Convergence Index , 2006 .

[4]  J. Cadafalch,et al.  Verification of Finite Volume Computations on Steady-State Fluid Flow and Heat Transfer , 2002 .

[5]  I. Demirdzic,et al.  Fluid flow and heat transfer test problems for non‐orthogonal grids: Bench‐mark solutions , 1992 .

[6]  Joel H. Ferziger,et al.  Computation of turbulent flow over a backward-facing step - Zonal approach , 1988 .

[7]  C. Bruneau,et al.  The 2D lid-driven cavity problem revisited , 2006 .

[8]  O. Botella,et al.  BENCHMARK SPECTRAL RESULTS ON THE LID-DRIVEN CAVITY FLOW , 1998 .

[9]  Roger W. Logan,et al.  Comparing 10 Methods for Solution Verification, and Linking to Model Validation , 2005, J. Aerosp. Comput. Inf. Commun..

[10]  S. Thangam,et al.  Turbulent Flow Past a Backward-Facing Step: A Critical Evaluation of Two-Equation Models , 1992 .

[11]  Assensi Oliva,et al.  Numerical experiments in turbulent natural and mixed convection in internal flows , 1995 .

[12]  William L. Oberkampf,et al.  AIAA COMMITTEE ON STANDARDS FOR COMPUTATIONAL FLUID DYNAMICS: STATUS AND PLANS , 2004 .

[13]  Lmt Bart Somers.  The simulation of flat flames with detailed and reduced chemical models , 1994 .

[14]  Patrick J. Roache,et al.  Verification and Validation in Computational Science and Engineering , 1998 .

[15]  T. Xing,et al.  Factors of Safety for Richardson Extrapolation , 2010 .

[16]  N. Sinha,et al.  Error Quantification for Computational Aerodynamics Using an Error Transport Equation , 2007 .

[17]  Hugh W. Coleman,et al.  Comprehensive Approach to Verification and Validation of CFD Simulations—Part 1: Methodology and Procedures , 2001 .

[18]  Frederick Stern,et al.  FACTORS OF SAFETY FOR RICHARDSON EXTRAPOLATION FOR INDUSTRIAL APPLICATIONS , 2008 .

[19]  Ismail Celik,et al.  Numerical Experiments on Application of Richardson Extrapolation With Nonuniform Grids , 1997 .

[20]  Assensi Oliva,et al.  A parallel algorithm for the detailed numerical simulation of reactive flows , 2000 .

[21]  Peter A. Cavallo,et al.  Viscous Error Transport Equation for Error Quantification of Turbulent Flows , 2008 .