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

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.

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.

scholarly journals The Performance of a Gradient-Based Method to Estimate the Discretization Error in Computational Fluid Dynamics

Computation ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 10
Author(s):  
Adhika Satyadharma ◽  
Harinaldi
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Discretization Error ◽  
Computational Time ◽  
Simulation Data ◽  
Manufactured Solutions ◽  
Grid Convergence ◽  
Gradient Based ◽  
New Perspective ◽  
Grid Convergence Index

Although the grid convergence index is a widely used for the estimation of discretization error in computational fluid dynamics, it still has some problems. These problems are mainly rooted in the usage of the order of a convergence variable within the model which is a fundamental variable that the model is built upon. To improve the model, a new perspective must be taken. By analyzing the behavior of the gradient within simulation data, a gradient-based model was created. The performance of this model is tested on its accuracy, precision, and how it will affect a computational time of a simulation. The testing is conducted on a dataset of 36 simulated variables, simulated using the method of manufactured solutions, with an average of 26.5 meshes/case. The result shows the new gradient based method is more accurate and more precise then the grid convergence index(GCI). This allows for the usage of a coarser mesh for its analysis, thus it has the potential to reduce the overall computational by at least by 25% and also makes the discretization error analysis more available for general usage.

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.

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.

COMBINATORIAL POLYNOMIALLY COMPUTABLE CHARACTERISTICS OF SUBSTITUTIONS AND THEIR PROPERTIES

2020 ◽  
Vol 7 (2) ◽  
pp. 34-41
Author(s):  
VLADIMIR NIKONOV ◽  
◽  
ANTON ZOBOV ◽  
Keyword(s):  
Mathematical Expectation ◽  
Point Of View ◽  
Small Range ◽  
Computational Point ◽  
Wide Range ◽  
Bijective Function ◽  
The Difference ◽  
Element Base ◽  
Selection Of

The construction and selection of a suitable bijective function, that is, substitution, is now becoming an important applied task, particularly for building block encryption systems. Many articles have suggested using different approaches to determining the quality of substitution, but most of them are highly computationally complex. The solution of this problem will significantly expand the range of methods for constructing and analyzing scheme in information protection systems. The purpose of research is to find easily measurable characteristics of substitutions, allowing to evaluate their quality, and also measures of the proximity of a particular substitutions to a random one, or its distance from it. For this purpose, several characteristics were proposed in this work: difference and polynomial, and their mathematical expectation was found, as well as variance for the difference characteristic. This allows us to make a conclusion about its quality by comparing the result of calculating the characteristic for a particular substitution with the calculated mathematical expectation. From a computational point of view, the thesises of the article are of exceptional interest due to the simplicity of the algorithm for quantifying the quality of bijective function substitutions. By its nature, the operation of calculating the difference characteristic carries out a simple summation of integer terms in a fixed and small range. Such an operation, both in the modern and in the prospective element base, is embedded in the logic of a wide range of functional elements, especially when implementing computational actions in the optical range, or on other carriers related to the field of nanotechnology.

scholarly journals A New Mixed Functional-probabilistic Approach for Finite Element Accuracy

2020 ◽  
Vol 20 (4) ◽  
pp. 799-813
Author(s):  
Joël Chaskalovic ◽  
Franck Assous
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Asymptotic Relation ◽  
Probabilistic Approach ◽  
Random Variable ◽  
High Order ◽  
Relative Accuracy ◽  
The Difference ◽  
New Perspective

AbstractThe aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble–Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements {P_{k}} and {P_{m}} ({k<m}). Then we analyze the asymptotic relation between these two probabilistic laws when the difference {m-k} goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.

scholarly journals Age-dependency in mortality of Finnish family caregivers: a nationwide register-based study

2020 ◽  
Vol 30 (Supplement_5) ◽  
Author(s):  
T M Mikkola ◽  
H Kautiainen ◽  
M Mänty ◽  
M B von Bonsdorff ◽  
T Kröger ◽  
...  
Keyword(s):  
General Population ◽  
Family Caregivers ◽  
Family Caregiver ◽  
Causes Of Death ◽  
Lower Mortality ◽  
All Cause Mortality ◽  
The Difference ◽  
Caregiver Role ◽  
Selection Of

Abstract Purpose Mortality appears to be lower in family caregivers than in the general population. However, there is lack of knowledge whether the difference in mortality between family caregivers and the general population is dependent on age. The purpose of this study was to analyze all-cause mortality in relation to age in family caregivers and to study their cause-specific mortality using data from multiple Finnish national registers. Methods The data included all individuals, who received family caregiver's allowance in Finland in 2012 (n = 42 256, mean age 67 years, 71% women) and a control population matched for age, sex, and municipality of residence (n = 83 618). Information on dates and causes of death between 2012 and 2017 were obtained from the Finnish Causes of Death Register. Flexible parametric survival modeling and competing risk regression adjusted for socioeconomic status were used. Results The total follow-up time was 717 877 person-years. Family caregivers had lower all-cause mortality than the controls over the follow-up (8.1% vs. 11.6%) both among women (hazard ratio [HR]: 0.64, 95% CI: 0.61-0.68) and men (HR: 0.73, 95% CI: 0.70-0.77). Younger adult caregivers had equal or only slightly lower mortality than their controls, but after age 60, the difference increased markedly resulting in over 10% lower mortality in favor of the caregivers in the oldest age groups. Caregivers had lower mortality for all the causes of death studied, namely cardiovascular, cancer, neurological, external, respiratory, gastrointestinal and dementia than the controls. Of these, the lowest was the risk for dementia (subhazard ratio=0.29, 95%CI: 0.25-0.34). Conclusions Older family caregivers have lower mortality than the age-matched controls from the general population while younger caregivers have similar mortality to their peers. This age-dependent advantage in mortality is likely to reflect selection of healthier individuals into the family caregiver role. Key messages The difference in mortality between family caregivers and the age-matched general population varies considerably with age. Advantage in mortality observed in family caregiver studies is likely to reflect the selection of healthier individuals into the caregiver role, which underestimates the adverse effects of caregiving.

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