20. Sensitivity and Uncertainty Analysis (Enterprise Version Only)

20.1. Sandwich Method for Sensitivity and Uncertainty Calculation

The keff sensitivity analysis utilizes the iterated fission probability method, which can be applied in both traditional power iteration methods and super-history methods. General sensitivity analysis employs collision history methods, GEAR-MC methods, and the new GPT formula method. To perform uncertainty calculations, sensitivity calculations must first be executed. A covariance index file, “covdir”, is required, along with the covariance database pointed to by the covdir directory.

20.1.1. Sensitivity Analysis

An example of the input for keff sensitivity analysis:

Adjoint
method <flag>
KEFF
BlockSize <Number>
Nuclide  <zaid.xxx>   ……
Reaction= <reaction_list_1, reaction _list_2, …>   ……
Constrain <flag_1, flag_2, flag_3,…>
groupoption <Number>
Uncertainty
OUTPUTINTERVAL <Number>
OUTPUTTOTALSEN <flag>

Input Example for General Sensitivity Analysis (Collision Super-History Method, capable of analyzing reaction rate ratios and dynamic parameters on nuclear data):

Adjoint
GENERAL
    BlockSize <Number>
    ResponseNu <zaid.xxx>   <zaid.xxx>
ResponseMT = <reaction_list_1, reaction _list_2>
Ratio = <ratio_list_1, ratio _list_2>
Nuclide  <zaid.xxx>   ……
Reaction= <reaction_list_1, reaction _list_2, …>   ……
groupoption <Number>
Uncertainty
OUTPUTINTERVAL <Number>
Cell = <Number>

Input Example for General Sensitivity Analysis (Super-History GEAR-MC Method, capable of analyzing reaction rate ratios on nuclear data):

Adjoint
GENERAL
    BlockSize <Number>
    GPTMETHOD <Number>
    ResponseNu <zaid.xxx>   <zaid.xxx>
ResponseMT = <reaction_list_1, reaction _list_2>
Ratio = <ratio_list_1, ratio _list_2>
Nuclide  <zaid.xxx>   ……
Reaction= <reaction_list_1, reaction _list_2, …>   ……
Constrain <flag_1, flag_2, flag_3,…>
    groupoption <Number>
Uncertainty
OUTPUTINTERVAL <Number>
Cell = <Number>

Input Example for General Sensitivity Analysis (EGEAR-MC Method, capable of analyzing reaction rate ratios on geometric parameters):

Adjoint
GENERAL
    GEOSENMETHOD <Number>
    BlockSize <Number>
    GPTMETHOD <Number>
    ResponseNu <zaid.xxx>   <zaid.xxx>
ResponseMT = <reaction_list_1, reaction _list_2>
Ratio = <ratio_list_1, ratio _list_2>
OUTPUTINTERVAL <Number>
Surface <Number>
    Cell = <Number>

Input Example for General Sensitivity Analysis (Super-History New GPT Formula Method, capable of analyzing reaction rates on nuclear data):

Adjoint
GENERAL
    BlockSize <Number>
    GPTMETHOD <Number>
    ReaRateType <Number>
Nuclide  <zaid.xxx>   ……
Reaction= <reaction_list_1, reaction _list_2, …>   ……
Constrain <flag_1, flag_2, flag_3,…>
    groupoption <Number>
Uncertainty
OUTPUTINTERVAL <Number>
Cell
    CellTallyID <Number> Cell =<Number>

Input Example for General Sensitivity Analysis (GPT-GE Method, capable of analyzing reaction rates on geometric parameters):

Adjoint
GENERAL
    BlockSize <Number>
    GEOSENMETHOD <Number>
    ReaRateType <Number>
OUTPUTINTERVAL <Number>
Surface <Number>
    Cell = <Number>

Input Example for Random Sampling Method Perturbing a Single Nuclide:

Sampling
SAMPLESIZE=<Number>
Nuclide <zaid.xxx>
Reaction= <reaction_list_1, reaction _list_2>
GROUPOPTION <Number>

Input Example for Random Sampling Method Perturbing All Nuclides:

Sampling
SAMPLESIZE = <Number>
All
GROUPOPTION <Number>

Where:

  • Keff is the keyword for the keff sensitivity analysis method input card. By default, this card is disabled.
Method   <flag>

Where:

  • Method is the keyword for the keff sensitivity analysis method input card.
  • flag specifies the keff sensitivity analysis method. flag = 0 (default) corresponds to the traditional power iteration method, and flag = 1 corresponds to the superhistory method, which does not support OpenMP calculations. The general sensitivity analysis does not require enabling this card, and geometric perturbations do not support the superhistory method.
GPTMETHOD   <Number>

Where:

  • GPTMETHOD is the keyword for the general response sensitivity analysis method input card.
  • Number specifies the general sensitivity analysis method. Number = 0 (default) corresponds to the collision superhistory method, flag = 1 corresponds to the superhistory GEAR-MC method, and flag = 2 corresponds to the superhistory new GPT formula method. The general sensitivity analysis does not support OpenMP calculations.
GEOSENMETHOD   <Number>

Where:

  • GEOSENMETHOD is the keyword for the general response sensitivity analysis method input card with respect to geometric parameters.
  • Number specifies the method for general response sensitivity analysis concerning geometric parameters. Number = 0 (default) means that geometric sensitivity analysis is disabled, flag = 1 corresponds to the EGEAR-MC method, and flag = 2 corresponds to the GPT-GE method. The general response sensitivity analysis regarding geometric parameters does not support OpenMP calculations.
Surface   <Number>

Where:

  • Surface is the keyword for the general response sensitivity analysis method input card concerning geometric parameters.
  • Number specifies the specific geometric surface index being analyzed.
ReaRateType   <Number>

Where:

  • ReaRateType is the keyword for the general sensitivity analysis method input card.
  • Number specifies the type of reaction rate in the general response. Number = 0 (default) does not represent any reaction rate, flag = 1 corresponds to fission reaction rate, flag = 2 corresponds to absorption reaction rate, and flag = 3 corresponds to power.
BlockSize   <Number>

Where:

  • BlockSize specifies the keyword for the associated flux (iterated fission probability) or the convergence iterations of the generalized adjoint flux, with Number being the corresponding parameter, generally recommended to be set to 10.
Nuclide
       <zaid.xxx>
       <zaid.xxx>   ……

Where:

  • Nuclide is the keyword for the input card specifying the nuclides for sensitivity analysis.
  • zaid.xxx specifies the ACE cross-section database corresponding to the nuclide, where zaid is the nuclide ID, and the suffix .xxx specifies the type of cross-section database.
ResponseNu
         <zaid.xxx>
         <zaid.xxx>

Where:

  • ResponseNu is the keyword for the input card defining the nuclides involved in the first type of response function for general sensitivity analysis.
  • zaid.xxx specifies the ACE cross-section database corresponding to the nuclide, where zaid is the nuclide ID, and the suffix .xxx specifies the type of cross-section database.
Reaction
        <reaction_list_1, reaction _list_2, …>

  • Reaction specifies the types of reactions for each nuclide in the sensitivity analysis. Each nuclide can correspond to multiple reaction types, separated by commas, e.g., “Reaction= 16, 17, 102, -6, 107”. The correspondence between reaction types and their numbers can be found in the ENDF/B manual, with Table 7-1 providing common reaction type numbers. Note that the nuclides corresponding to the Reaction card must align with those in the Nuclide card. Additionally, the Reaction card cannot be used simultaneously with the Uncertainty card.

    This is because when using the Uncertainty card, the considered reaction types are specified by the reaction types included in the covariance database. When not using the Uncertainty card, the Reaction card must be included.

ResponseMT
          <reaction_list_1, reaction _list_2>
  • ResponseMT specifies the reaction types involved in the first type of response function for general sensitivity analysis for each nuclide. Each nuclide can correspond to multiple reaction types, separated by commas, e.g., “Reaction= 16, 17, 102, -6, 107”. The correspondence between reaction types and their numbers can be found in the ENDF/B manual, with Table 15-1 providing common reaction type numbers. Note that the nuclides corresponding to the ResponseMT card must align with those in the RespnseNu card.
表20.2 Correspondence between Reaction Types and Numbers (Partial ENDF Reaction Types Listed)
MT Number Reaction Type Remarks
1 Total Cross Section For continuous energy ACE cross-sections, when the cross-section temperature does not match the fuel element temperature, the Doppler broadening is applied to the elastic scattering cross-sections and the total cross-section. The adjusted cross-section is what is being reported here.
-2 Absorption Does not include fission
2 Elastic Scattering  
4 Inelastic Scattering  
18 Total Fission  
16 (n,2n) Applicable only to continuous energy ACE cross-sections
17 (n,3n)
102 (n,γ)
103 (n,p)
107 (n,α)
452 Average Number of Fission Neutrons |
455 Average Number of Prompt Fission Neutrons |
456 Average Number of Delayed Fission Neutrons |
-1018 Total Fission Neutron Spectrum |
-1455 Prompt Fission Neutron Spectrum |
-1456 Delayed Fission Neutron Spectrum | 
Ratio
     <ratio_list_1, ratio _list_2>
  • Ratio specifies the composition of the first type of response function in the general sensitivity analysis. This card needs to be used in conjunction with RespnseNu and ResponseMT . For example:
ResponseNu
  92235.60c
  92238.60c
ResponseMT =
  18,
  18
Ratio =
  2 1

In this case, ResponseNu defines two nuclides, 92235 and 92238, and ResponseMT defines their reaction type as MT=18, which is total fission. According to the order of appearance of the two nuclides, the total fission index for 92235 is 1, and for 92238 is 2. Thus, Ratio=2 1 means that index 2 is divided by index 1, forming a first-type response function, i.e., the fission reaction rate of U-238 divided by the fission reaction rate of U-235. In the current version, Ratio can only define one response.

GENERAL

Where:

  • GENERAL is the keyword for the general sensitivity analysis method input card. By default, this card is disabled.
Constrain   <flag_1, flag_2, flag_3,…>

Where:

  • Constrain is the keyword for the input card to compute the bound fission neutron spectrum.
  • flag specifies whether to compute the bound fission neutron spectrum. flag = 0 (default) means not to compute the bound fission neutron spectrum, and flag = 1 means to compute it. It is recommended to compute the bound fission neutron spectrum for the analyzed nuclides to obtain accurate results during uncertainty calculations.
  • General sensitivity analysis does not require enabling this card.
Cell = Cell_vector

Where:

  • Cell is the keyword for the cell input card. This card is used to define the geometric scope of the generalized response functions for the reaction rate ratios and the adjoint flux-weighted reaction rate ratios. For details, refer to the Cell card for the tally.
CellTallyID = IDNumber

Where:

  • CellTallyID is used to define the geometric region number for which the sensitivity coefficients of the reaction rate types of the generalized response function need to be output. IDNumber is the number of the cell tally for reference in the output. This card needs to be used in conjunction with the Cell card to specify which cell’s sensitivity coefficients are being statistically evaluated.
GroupOption   <flag>
  • GroupOption specifies the number of energy groups for the output of sensitivity coefficients. flag = 0 means user-defined energy framework, with specific energy points specified through the Energybin card; flag = 1 (default) calculates energy-integrated sensitivity coefficients; flag = 252 uses the program’s built-in 252 energy grid; flag=44 uses the program’s built-in 44-group energy grid; flag=56 uses the program’s built-in 56-group energy grid.
Energybin   <energy_1 energy_2 energy_3 …>
  • Energybin specifies the energy intervals for the output of sensitivity coefficients, with parameters as energy boundary points (MeV). For example, “Energybin 6.25E-7 20 ” indicates the tally range from 0.625 eV to 20 MeV, and from 20 MeV to positive infinity. The Energybin card can only be used when the GroupOption card’s flag=0 .

The output format of the sensitivity calculation is .Adjoint files.

------------------ Nuclide = 92235, Reaction Type = 4 ------------------
Group       Energy Bin         Ave            RE
  1         1.0000E-11     0.0000E+00     0.0000E+00
  2         3.0000E-09     0.0000E+00     0.0000E+00
  3         7.5000E-09     0.0000E+00     0.0000E+00
  ...
================================ sum of energy of first response  ================================
------------------ Nuclide = 92235, Reaction Type = 4 ------------------
Group       Energy Bin         Ave            RE
  0         8.1900E+00     3.5373E-03     6.7842E-01

In the previous section, the first part represents the sensitivity coefficients for each energy interval, consistent with the energy intervals specified by the GroupOption card; the latter part is the energy-integrated sensitivity coefficients, which are the cumulative sensitivity coefficients across all energy intervals, and this value is independent of the chosen energy intervals. The standard deviations corresponding to the sensitivity coefficients are all relative errors.

20.1.2. Uncertainty Analysis

Uncertainty
  • Uncertainty is the keyword for the input card for uncertainty analysis; adding this option indicates that uncertainty analysis will be performed. During the uncertainty calculations, sensitivity coefficient calculations are performed first. The nuclides involved in the sensitivity analysis are specified through the Nuclide card, while the reaction types for each involved nuclide will be determined by reading the covariance index file “covdir”, rather than through the Reaction card. When the Uncerainty card is enabled, the flag in the GroupOption card will be set internally to 44, meaning that the uncertainty will be calculated using the 44-group covariance database.

20.1.3. Output Options

OUTPUTINTERVAL   <Number>

Where:

  • OUTPUTINTERVAL specifying how many generations interval to output the result.
  • Number specifies the number of generations after which results are output. By default, sensitivity coefficients and uncertainty calculation results are only output once the calculation simulation has finished.
OutputTotalSen   <flag>

Where:

  • OutputTotalSen is the keyword for specifying whether to output the energy-integrated sensitivity coefficients.
  • Flag specifies whether to output the energy-integrated sensitivity coefficients. flag = 0 (default value) means do not calculate energy-integrated sensitivity coefficients, while flag = 1 means to calculate them. The flag=1 option can only be used when the flag in the GroupOption is not equal to 1.

20.2. Uncertainty Calculation Using Random Sampling Method

Uncertainty analysis using the random sampling method requires not only the covariance database index file “covdir” and the covariance database it points to, but also the disturbance database index file “samdir” and the disturbance database it points to.

Nuclide
       <zaid.xxx>
       <zaid.xxx> ……

Where:

  • Nuclide is the keyword for the input card specifying the nuclides for sensitivity analysis.
  • zaid.xxx specifies the ACE cross-section database corresponding to the nuclide, where zaid is the nuclide ID, and the suffix .xxx specifies the type of cross-section database.
Reaction
        <reaction_list_1, reaction _list_2>
  • Reaction specifies the reaction types for the nuclides that need to be disturbed. Each nuclide can only analyze one pair of reaction types per calculation, separated by commas. The correspondence between reaction types and their identifiers can be found in the ENDF/B manual, where Table 15-1 provides some common reaction type identifiers. It is important to note that the nuclides corresponding to Reaction must be consistent with the Nuclide option card. Furthermore, if the reaction pair for the analyzed nuclide does not exist in the disturbance factor database, the program will not disturb it internally. Therefore, it is advisable to check whether the corresponding reaction pair exists in the disturbance factor database before using this option card.
SampleSize  <Number>

Where:

  • SampleSize is the keyword for specifying the disturbance sample size, with Number as the corresponding parameter. The maximum supported Number is currently 300.
All

Where:

  • All is the keyword for disturbing all nuclides and all reaction types (depending on the covariance database). Currently, the uncertainties in the covariance database include uncertainties in scattering cross-sections, absorption cross-sections, fission cross-sections, average number of fission neutrons, and fission neutron spectrum uncertainties. The current version of the random sampling method does not include uncertainties in the fission neutron spectrum. If the ALL mode is selected, there is no need to fill in the Nuclide and Reaction option cards.
GroupOption <EnergySize>

Where:

  • GroupOption is the keyword for specifying the number of energy groups (which must match the number of energy groups used in the disturbance factor database). EnergySize is the corresponding parameter. Currently, EnergySize only supports three parameters: 44, 56, and 252.

The output file for uncertainty analysis in .uncertainty format is as follows:

================================ Uncertainty Information ================================
the relative standard deviation of General response (% delta-R/R) due to cross-section covariance data is:
2.1822E+01 +/- 2.6232E+01 %delta-R/R   //Represents the total uncertainty caused by nuclear data
                covariance matrix
nuclide-reaction      with       nuclide-reaction          %delta-R/R due to this matrix
    92235,  4                       92235,   4             1.4745E-01  +/- 6.8564E-04
    92235,  16                      92235,   16            0.0000E+00  +/- 0.0000E+00
    92235,  18                      92235,   18            3.3984E-01  +/- 3.4711E-02
    ...

Where the lines below the covariance matrix indicate the contributions of the covariance between reaction 1 and reaction 2 to the uncertainty of the response quantity as represented in the covdir file’s covariance matrix.