Derivatives¶
This module houses functions used to convert the derivatives with respect to nuclides and or specific regions to derivatives with respect to the optimization parameters. Often the output of a given MC Transport/sensitivity calculation is the derivative of some figure of merit to the macroscopic cross section of each isotope in the problem. Macroscopic cross secitons are proportional to the number density of that isotope by a factor equivalent to the unchanging microscopic cross seciton. .. math:: deltak_{eff}/deltaN_{i}*sigma Where i is specific to a given isotope in the problem. What we can reasonable control in a system is the density of some material or compound, which is a function of the densities of each constituent isotope. This require a chain rule for derivatives, i.e., .. math:: deltak_{eff}/deltarho = sum(deltak_{eff}/deltaN_i * deltaN_i/deltarho) Furthermore, the ADAM solver is generally not setup to optimized directly over the density parameter $rho$, rather, some transformation funciton is applied to contrain the domain of this parameter without constraining the algorithm itself. This transformation function and the execution of its chain rule is actually housed in the :doc:’Problem Definition’ object since it is thought of as a problem constraint. test link is :ref:`ADAM Control Module <ADAM_control_module>’
Created on Mon Sep 5 10:59:10 2022
@author: noahwalton
- ADAM.derivatives.combine_nuclide_derivatives(nuclide_derivatives)[source]¶
Function used to combine derivatives from individual nuclides into a derivative wrt the material.
Because ADAM wants the derivative with respect to the number density of a given material, and that is a function of individual isotopes’ number density, the chain rule for derivatives must be used. In this case, it is a simple sum over all nuclide derivatives because the total atom density for a material is just a sum of all constituent atom densities.
- Parameters
nuclide_derivatives (dictionary) – This is a dictionary with keys corresponding to nuclides and values corresponding to the absolute sensitivity/derivative of k-effective with respect to the keyed nuclide.
- Returns
combined_derivatives – Combined derivative with respect to the material composed of the individual nuclides.
- Return type
float