Factors of Safety for Richardson Extrapolation

2010 ◽  
Vol 132 (6) ◽  
Author(s):  
Tao Xing ◽  
Frederick Stern
Keyword(s):  
Correction Factor ◽  
Confidence Level ◽  
Factor Of Safety ◽  
Richardson Extrapolation ◽  
Time Step ◽  
Order Of Accuracy ◽  
P Values ◽  
Grid Convergence ◽  
Small Uncertainty ◽  
Grid Convergence Index

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.

Perspective: A Method for Uniform Reporting of Grid Refinement Studies

1994 ◽  
Vol 116 (3) ◽  
pp. 405-413 ◽  
Author(s):  
P. J. Roache
Keyword(s):  
Richardson Extrapolation ◽  
Coarse Grid ◽  
Error Estimator ◽  
Grid Refinement ◽  
Order Method ◽  
A Posteriori ◽  
Asymptotic Approach ◽  
Grid Convergence ◽  
Grid Convergence Index ◽  
Do So

This paper proposes the use of a Grid Convergence Index (GCI) for the uniform reporting of grid refinement studies in Computational Fluid Dynamics. The method provides an objective asymptotic approach to quantification of uncertainty of grid convergence. The basic idea is to approximately relate the results from any grid refinement test to the expected results from a grid doubling using a second-order method. The GCI is based upon a grid refinement error estimator derived from the theory of generalized Richardson Extrapolation. It is recommended for use whether or not Richardson Extrapolation is actually used to improve the accuracy, and in some cases even if the conditions for the theory do not strictly hold. A different form of the GCI applies to reporting coarse grid solutions when the GCI is evaluated from a “nearby” problem. The simple formulas may be applied a posteriori by editors and reviewers, even if authors are reluctant to do so.

Richardson Extrapolation-Based Discretization Uncertainty Estimation for Computational Fluid Dynamics

2014 ◽  
Vol 136 (12) ◽  
Author(s):  
Tyrone S. Phillips ◽  
Christopher J. Roy
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Convergence Rates ◽  
Richardson Extrapolation ◽  
Local Solution ◽  
Navier Stokes ◽  
Grid Convergence ◽  
Computational Fluid Dynamics Cfd ◽  
Grid Convergence Index ◽  
2D And 3D

This study investigates the accuracy of various Richardson extrapolation-based discretization error and uncertainty estimators for problems in computational fluid dynamics (CFD). Richardson extrapolation uses two solutions on systematically refined grids to estimate the exact solution to the partial differential equations (PDEs) and is accurate only in the asymptotic range (i.e., when the grids are sufficiently fine). The uncertainty estimators investigated are variations of the grid convergence index and include a globally averaged observed order of accuracy, the factor of safety method, the correction factor method, and least-squares methods. Several 2D and 3D applications to the Euler, Navier–Stokes, and Reynolds-Averaged Navier–Stokes (RANS) with exact solutions and a 2D turbulent flat plate with a numerical benchmark are used to evaluate the uncertainty estimators. Local solution quantities (e.g., density, velocity, and pressure) have much slower grid convergence on coarser meshes than global quantities, resulting in nonasymptotic solutions and inaccurate Richardson extrapolation error estimates; however, an uncertainty estimate may still be required. The uncertainty estimators are applied to local solution quantities to evaluate accuracy for all possible types of convergence rates. Extensions were added where necessary for treatment of cases where the local convergence rate is oscillatory or divergent. The conservativeness and effectivity of the discretization uncertainty estimators are used to assess the relative merits of the different approaches.

Application of grid convergence index in FE computation

2013 ◽  
Vol 61 (1) ◽  
pp. 123-128 ◽  
Author(s):  
L. Kwaśniewski
Keyword(s):  
Fluid Dynamics ◽  
Finite Element ◽  
Order Of Convergence ◽  
Richardson Extrapolation ◽  
Simple Problem ◽  
Plastic Response ◽  
Grid Convergence ◽  
The Difference ◽  
Grid Convergence Index ◽  
Selection Of

Abstract This paper presents an application of the grid convergence index (GCI) concept based on the Richardson extrapolation to a selected simple problem of a cantilever beam loaded with vertical forces at the tip end. The GCI method, popular in computational fluid dynamics, has been recently recommended for finite element (FE) applications in solid and structural mechanics. Based on the results obtained usually for three meshes, the GCI method enables one to determine, in an objective manner, the order of convergence to estimate the asymptotic solution and the bounds for discretization error. The example shows that the characteristics of the convergence depend on the selection of the quantity of interest, which can be local or a global functional such as the deflection considered here. The results differ for different FE formulations, and the difference is bigger when the nonlinearities (e.g., due to plastic response) are taken into account

Numerical Experiments on Application of Richardson Extrapolation With Nonuniform Grids

1997 ◽  
Vol 119 (3) ◽  
pp. 584-590 ◽  
Author(s):  
Ismail Celik ◽  
Ozgur Karatekin
Keyword(s):  
Viscous Sublayer ◽  
Standard Uncertainty ◽  
Richardson Extrapolation ◽  
Grid Refinement ◽  
Uncertainty Measure ◽  
Nonuniform Grids ◽  
Wall Functions ◽  
Convergence Error ◽  
Grid Convergence ◽  
Grid Convergence Index

Some unresolved problems related to Richardson extrapolation (RE) are elucidated via examples, and possible remedies are suggested. The method is applied to the case of turbulent flow past a backward facing step using nonuniform grid distributions. It is demonstrated that RE can be used to obtain grid independent solutions using the same grid refinement factors in both coordinate directions. The use of generalized wall functions together with the standard k-ε model seems to work well even if the grid refinement extends into the viscous sublayer. In addition, the grid convergence index and other standard uncertainty measures are compared, and a new uncertainty measure is suggested which seems to be a better indicator for the grid convergence error.

Enhanced Explicit Scheme to Solve Transient Heat Conduction Problem

2007 ◽  
Vol 2 (1) ◽  
Author(s):  
Ganesh Hegde ◽  
Madhu Gattumane
Keyword(s):  
Heat Conduction ◽  
Correction Factor ◽  
Truncation Error ◽  
Transient Heat Conduction ◽  
Explicit Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Time Step ◽  
One Dimensional ◽  
Partial Derivatives

Improvement in accuracy without sacrificing stability and convergence of the solution to unsteady diffusion heat transfer problems by computational method of enhanced explicit scheme (EES), has been achieved and demonstrated, through transient one dimensional and two dimensional heat conduction. The truncation error induced in the explicit scheme using finite difference technique is eliminated by optimization of partial derivatives in the Taylor series expansion, by application of interface theory developed by the authors. This theory, in its simple terms gives the optimum values to the decision vectors in a redundant linear equation. The time derivatives and the spatial partial derivatives in the transient heat conduction, take the values depending on the time step chosen and grid size assumed. The time correction factor and the space correction factor defined by step sizes govern the accuracy, stability and convergence of EES. The comparison of the results of EES with analytical results, show decreased error as compared to the result of explicit scheme. The paper has an objective of reducing error in the explicit scheme by elimination of truncation error introduced by neglecting the higher order terms in the expansion of the governing function. As the pilot examples of the exercise, the implementation is aimed at solving one-dimensional and two-dimensional problems of transient heat conduction and compared with the results cited in the referred literature.

scholarly journals Grid convergence study of a cyclone separator using different mesh structures

2017 ◽  
Vol 23 (3) ◽  
pp. 311-320 ◽  
Author(s):  
R.A.F. Oliveira ◽  
G.H. Justi ◽  
G.C. Lopes
Keyword(s):  
Pressure Drop ◽  
Collection Efficiency ◽  
Fluid Dynamic ◽  
Two Phase ◽  
Mesh Structure ◽  
Pressure Drops ◽  
Regular Elements ◽  
Grid Convergence ◽  
Grid Convergence Index ◽  
Mesh Structures

In a cyclone design, pressure drop and collection efficiency are two important performance parameters to estimate its implementation viability. The optimum design provides higher efficiencies and lower pressure drops. In this paper, a grid independence study was performed to determine the most appropriate mesh to simulate the two-phase flow in a Stairmand cyclone. Computational fluid dynamic (CFD) tools were used to simulate the flow in an Eulerian-Lagrangian approach. Two different mesh structure, one with wall-refinement and the other with regular elements, and several mesh sizes were tested. The grid convergence index (GCI) method was applied to evaluate the result independence. The CFD model results were compared with empirical correlations from bibliography, showing good agreement. The wall-refined mesh with 287 thousand elements obtained errors of 9.8% for collection efficiency and 14.2% for pressure drop, while the same mesh, with regular elements, obtained errors of 8.7% for collection efficiency and 0.01% for pressure drop.

scholarly journals Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

2011 ◽  
Vol 1 (3) ◽  
pp. 215-234 ◽  
Author(s):  
Lizhen Chen ◽  
Jie Shen ◽  
Chuanju Xu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Splitting Schemes ◽  
Time Step ◽  
Order Of Accuracy ◽  
Navier Stokes Equations ◽  
Pressure Stabilization ◽  
Legendre Spectral Method ◽  
Direction Splitting ◽  
The Stability

AbstractWe propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.

scholarly journals Assessment of Susceptibility to Liquefaction of Saturated Road Embankment Subjected to Dynamic Loads

2014 ◽  
Vol 36 (1) ◽  
pp. 15-22 ◽  
Author(s):  
Anna Borowiec ◽  
Krzysztof Maciejewski
Keyword(s):  
Factor Of Safety ◽  
Stress Component ◽  
Critical Time ◽  
Water Reservoir ◽  
Constitutive Relationship ◽  
Plastic Behavior ◽  
Shear Stresses ◽  
Time Step ◽  
The Road ◽  
Liquefaction Susceptibility

Abstract Liquefaction has always been intensely studied in parts of the world where earthquakes occur. However, the seismic activity is not the only possible cause of this phenomenon. It may in fact be triggered by some human activities, such as constructing and mining or by rail and road transport. In the paper a road embankment built across a shallow water reservoir is analyzed in terms of susceptibility to liquefaction. Two types of dynamic loadings are considered: first corresponding to an operation of a vibratory roller and second to an earthquake. In order to evaluate a susceptibility of soil to liquefaction, a factor of safety against triggering of liquefaction is used (FSTriggering). It is defined as a ratio of vertical effective stresses to the shear stresses both varying with time. For the structure considered both stresses are obtained using finite element method program, here Plaxis 2D. The plastic behavior of the cohesionless soils is modeled by means of Hardening Soil (HS) constitutive relationship, implemented in Plaxis software. As the stress tensor varies with time during dynamic excitation, the FSTriggering has to be calculated for some particular moment of time when liquefaction is most likely to occur. For the purposes of this paper it is named a critical time and established for reference point at which the pore pressures were traced in time. As a result a factor of safety distribution throughout embankment is generated. For the modeled structure, cyclic point loads (i.e., vibrating roller) present higher risk than earthquake of magnitude 5.4. Explanation why considered structure is less susceptible to earthquake than typical dam could lay in stabilizing and damping influence of water, acting here on both sides of the slope. Analogical procedure is applied to assess liquefaction susceptibility of the road embankment considered but under earthquake excitation. Only the higher water table is considered as it is the most unfavorable. Additionally the modified factor of safety is introduced, where the dynamic shear stress component is obtained at a time step when its magnitude is the highest - not necessarily at the same time step when the pore pressure reaches its peak (i.e., critical time). This procedure provides a greater margin of safety as the computed factors of safety are smaller. Method introduced in the paper presents a clear and easy way to locate liquefied zones and estimate liquefaction susceptibility of the subsoil - not only in the road embankment.

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