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Advertisement Hide. Convergence and approximation results for non-cooperative Bayesian games: Learning theorems. This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Aliprantis, C.

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Aumann, R. Econometrica 55 , 1—18 Google Scholar. Blume, L. Debreu, G. Fifth Berkeley Symp. II , — Google Scholar. Diestel, J. Survey Providence, R. Dunford, N.

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Part I. New York: Interscience Google Scholar. El-Gamal, M. Theory 4 , — Google Scholar. Feldman, M. Himmelberg, C. Jordan, J. Kalai, E. The Richardson extrapolation can be generalized for a p-th order methods and r -value of grid ratio which does not have to be an integer as:. Thus, the above equation simplifies to:. Otherwise, it will be a third-order estimate. Richardson extrapolation can be applied for the solution at each grid point, or to solution functionals, such as pressure recovery or drag.

This assumes that the solution is globally second-order in addition to locally second-order and that the solution functionals were computed using consistent second-order methods. Other cautions with using Richardson extrapolation non-conservative, amplification of round-off error, etc For our purposes we will assume f is a solution functional i.

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  8. This use will now be examined. We will focus on using f 1 and f 2 to obtain an error estimate. Examining the generalized Richardson extrapolation equation above, the second term on the right-hand side can be considered to be an an error estimator of f 1. The equation can be expressed as:. This quantity should not be used as an error estimator since it does not take into account r or p.

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    This may lead to an underestimation or overestimation of the error. One could make this quantity artificially small by simply using a grid refinement ratio r close to 1. The estimated fractional error E 1 is an ordered error estimator and a good approximation of the discretization error on the fine grid if f 1 and f 2 were obtained with good accuracy i. If such is the case, then another normalizing value should be used in place of f 1.

    If a large number of CFD computations are to be performed i. We will then want to estimate the error on the coarser grid. The Richardson extrapolation can be expressed as:. Richardson extrapolation is based on a Taylor series representation as indicated in Eqn.

    If there are shocks and other discontinuities present, the Richardson extrapolation is invalid in the region of the discontinuity.

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    It is still felt that it applies to solution functionals computed from the entire flow field. Roache suggests a grid convergence index GCI to provide a consistent manner in reporting the results of grid convergence studies and perhaps provide an error band on the grid convergence of the solution. The GCI can be computed using two levels of grid; however, three levels are recommended in order to accurately estimate the order of convergence and to check that the solutions are within the asymptotic range of convergence.

    A consistent numerical analysis will provide a result which approaches the actual result as the grid resolution approaches zero. Thus, the discretized equations will approach the solution of the actual equations. One significant issue in numerical computations is what level of grid resolution is appropriate. This is a function of the flow conditions, type of analysis, geometry, and other variables. One is often left to start with a grid resolution and then conduct a series of grid refinements to assess the effect of grid resolution.

    This is known as a grid refinement study. One must recognize the distinction between a numerical result which approaches an asymptotic numerical value and one which approaches the true solution. It is hoped that as the grid is refined and resolution improves that the computed solution will not change much and approach an asymptotic value i.

    There still may be error between this asymptotic value and the true physical solution to the equations. Roache has provided a methodology for the uniform reporting of grid refinement studies. 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. The GCI is a measure of the percentage the computed value is away from the value of the asymptotic numerical value. It indicates an error band on how far the solution is from the asymptotic value. It indicates how much the solution would change with a further refinement of the grid. A small value of GCI indicates that the computation is within the asymptotic range.

    The refinement may be spatial or in time.

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    The higher factor of safety is recommended for reporting purposes and is quite conservative of the actual errors. When a design or analysis activity will involve many CFD simulations i. DOE study , one may want to use the coarser grid h 2. It is then necessary to quantify the error for the coarser grid. The GCI for the coarser grid is defined as:. It is important that each grid level yield solutions that are in the asymptotic range of convergence for the computed solution. This can be checked by observing two GCI values as computed over three grids,. If a desired accuracy level is known and results from the grid resolution study are available, one can then estimate the grid resolution required to obtain the desired level of accuracy,.

    The grid refinement ratio assumes that the refinement ratio r applies equally in all coordinate directions i,j,k for steady-state solutions and also time t for time-dependent solutions. If this is not the case, then the grid convergence indices can be computed for each direction independently and then added to give the overall grid convergence index,. If one generates a finer or coarser grid and is unsure of the value of grid refinement ratio to use, one can compute an effective grid refinement ratio as:. This effective grid refinement ratio can also be used for unstructured grids.

    The following example demonstrates the application of the above procedures in conducting a grid convergence study. The objective of the CFD analysis was to determine the pressure recovery for an inlet. The flow field is computed on three grids, each with twice the number of grid points in the i and j coordinate directions as the previous grid.

    The number of grid points in the k direction remains the same. Since the flow is axisymmetric in the k direction, we consider the finer grid to be double the next coarser grid. The table below indicates the grid information and the resulting pressure recovery computed from the solutions. Each solution was properly converged with respect to iterations. The column indicated by "spacing" is the spacing normalized by the spacing of the finest grid.

    The figure below shows the plot of pressure recoveries with varying grid spacings.

    Examining Spatial (Grid) Convergence

    As the grid spacing reduces, the pressure recoveries approach an asymptotic zero-grid spacing value. The difference is most likely due to grid stretching, grid quality, non-linearities in the solution, presence of shocks, turbulence modeling, and perhaps other factors. We now can apply Richardson extrapolation using the two finest grids to obtain an estimate of the value of the pressure recovery at zero grid spacing,.

    The grid convergence index for the fine grid solution can now be computed.