Sensitivity Analysis and Uncertainty Quantification in OFFBEAT

OFFBEAT provides built-in capabilities for performing Sensitivity Analysis (SA) and Uncertainty Quantification (UQ) by allowing users to perturb input parameters systematically. This is achieved through a dictionary located in system/userParameters, which enables modifying material properties and behavioral model parameters.

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Input Parameter Perturbation

The userParameters dictionary contains scale factors F and offset values delta that are used to perturb the nominal value computed by specific material properties and behavioral models. The perturbed value \(v'\) is computed from the nominal value \(v\) according to the following formula:

\[ \begin{aligned} v' = v \cdot F + delta \end{aligned} \]

The perturbation factors are defined in the system/userParameters dictionary according to the following structure:

• Material Properties: The perturbation factors defined in the materials dictionary refer to the different thermo-mechanical properties of OFFBEAT materials (e.g., density, emissivity, Young's modulus, etc.). If a specific perturbation is to be applied to a specific material, for example, UO2, it is possible to include the specific perturbation factors in the subdictionary materials/UO2.

• Behavioral Models: The perturbation factors defined in the behavioralModels dictionary refer to the different behavioral models (e.g., relocation, densification, swelling, etc.). It is possible to define correlation-specific perturbation by using a sub-dictionary. For example, if only the UO2FRAPCON model for swelling is to be perturbed, its perturbation factor can be included in the subdictionary behavioralModels/swelling/UO2FRAPCON.

  Note

  A comprehensive list of all the available perturbation parameters is printed in the OFFBEAT's log at the beginning of the simulation.

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Usage Example system/userParameters

/*--------------------------------*- C++ -*----------------------------------*\
  =========                 |
  \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
   \\    /   O peration     | Website:  https://openfoam.org
    \\  /    A nd           | Version:  9
     \\/     M anipulation  |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version         2;
    format          ascii;
    class           dictionary;
    location        "system";
    object          userParameters;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
materials
{
    // Perturbations defined here will impact the models for each property
    // across all the materials defined in the simulations
    F_rho           [ 0 0 0 0 0 ] 1;
    F_Cp            [ 0 0 0 0 0 ] 1;
    F_k             [ 0 0 0 0 0 ] 1;
    F_emissivity    [ 0 0 0 0 0 ] 1;
    F_E             [ 0 0 0 0 0 ] 1;
    F_nu            [ 0 0 0 0 0 ] 1;
    F_alphaT        [ 0 0 0 0 0 ] 1;

    // UO2-specific perturbations
    UO2
    {
        F_rho           [ 0 0 0 0 0 0 0 ] 1.05704;
        F_Cp            [ 0 0 0 0 0 0 0 ] 1.04758;
        F_k             [ 0 0 0 0 0 0 0 ] 0.973783;
        F_emissivity    [ 0 0 0 0 0 0 0 ] 0.904494;
        F_E             [ 0 0 0 0 0 0 0 ] 1.02771;
        F_nu            [ 0 0 0 0 0 0 0 ] 0.951355;
        F_alphaT        [ 0 0 0 0 0 0 0 ] 1.00606;
    }

    // zircaloy-specific perturbations
    zircaloy
    {
        F_rho           [ 0 0 0 0 0 0 0 ] 1.03358;
        F_Cp            [ 0 0 0 0 0 0 0 ] 1.08002;
        F_k             [ 0 0 0 0 0 0 0 ] 0.960583;
        F_emissivity    [ 0 0 0 0 0 0 0 ] 0.974784;
        F_E             [ 0 0 0 0 0 0 0 ] 1.05035;
        F_nu            [ 0 0 0 0 0 0 0 ] 0.952649;
        F_alphaT        [ 0 0 0 0 0 0 0 ] 0.905154;
    }
}

behavioralModels
{
    densification
    {
        F_epsilonDensification [ 0 0 0 0 0 0 0 ] 1.08;
    }
    relocation
    {
        F_epsilonRelocation [ 0 0 0 0 0 0 0 ] 0.98;
    }
    swelling
    {
        // It is possible to define correlation-specific perturbation
        UO2FRAPCON
        {
            F_epsilonSwelling [ 0 0 0 0 0 0 0 ] 1.25;
        }
    }
}