Uncertainty Verification process for assessing simulation numerical

# Uncertainty verification process for assessing

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Uncertainty: Verification : process for assessing simulation numerical uncertainties U SN and, when conditions permit, estimating the sign and magnitude Delta δ * SN of the simulation numerical error itself and the uncertainties in that error estimate U SN I: Iterative, G : Grid, T: Time step, P: Input parameters Validation : process for assessing simulation modeling uncertainty U SM by using benchmark experimental data and, when conditions permit, estimating the sign and magnitude of the modeling error δ itself. D : EFD Data; U V : Validation Uncertainty Validation achieved SN SM S T S 2 2 2 SN SM S U U U J j j I P T G I SN 1 2 2 2 2 2 P T G I SN U U U U U ) ( SN SM D S D E 2 2 2 SN D V U U U V U E
45 Results (UA, Verification) Convergence studies : Convergence studies require a minimum of m=3 solutions to evaluate convergence with respective to input parameters. Consider the solutions corresponding to fine , medium ,and coarse meshes (i). Monotonic convergence: 0<R k <1 (ii). Oscillatory Convergence: R k <0; | R k |<1 (iii). Monotonic divergence: R k >1 (iv). Oscillatory divergence: R k <0; | R k |>1 Grid refinement ratio : uniform ratio of grid spacing between meshes. Monotonic Convergence Monotonic Divergence Oscillatory Convergence 1 k S 2 k S 3 k S 21 2 1 k k k S S 32 3 2 k k k S S 21 32 k k k R 1 2 3 1 2 m m k k k k k k k x x x x x x r
46 Results (Verification, RE) Generalized Richardson Extrapolation (RE): For monotonic convergence , generalized RE is used to estimate the error δ * k and order of accuracy p k due to the selection of the kth input parameter. The error is expanded in a power series expansion with integer powers of x k as a finite sum. The accuracy of the estimates depends on how many terms are retained in the expansion, the magnitude (importance) of the higher-order terms, and the validity of the assumptions made in RE theory
47 Results (Verification, RE) Power series expansion Finite sum for the k th parameter and m th solution order of accuracy for the i th term Three equations with three unknowns ε SN is the error in the estimate S C is the numerical benchmark ) ( i k p i k p n i k k g x i k m m 1 * 1 21 1 1 * * k k p k k RE k r k k k k r p ln ln 21 32 J k j j jm k C I k k m m k m m S S S , 1 * * * ˆ J k j j jm i k p n i k C k g x S S i k m m , 1 * ) ( 1 ) ( ˆ J k j j j k p k C k g x S S k , 1 * 1 ) 1 ( ) 1 ( 1 1 ˆ J k j j j k p k k C k g x r S S k , 1 * 3 ) 1 ( 2 ) 1 ( 1 3 ˆ J k j j j k p k k C k g x r S S k , 1 * 2 ) 1 ( ) 1 ( 1 2 ˆ SN SN SN * * SN C S S
48 Results (UA, Verification, cont’d) Monotonic Convergence : Generalized Richardson Extrapolation Oscillatory Convergence : Uncertainties can be estimated, but without signs and magnitudes of the errors.

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