In this paper, we derive a mathematical formulation of the higher order adjoint NEM-M2B2 equations by preconditioning the nodal interface neutron currents equations of the forward equations system, and by using the Lagrangian Multipliers analysis method. In the NEM-M2B2 system of equations, the quadratic transverse leakage approximation is used to model the leakage of neutrons between each node in the system. The solution of the adjoint equation can be used to perform adjoint-based predictive sensitivity/perturbation analysis. As an example, we use the mathematical adjoint solution as sensitivity weighting for predicting the response of the IAEA-3D benchmark’s eigenvalue to a perturbation in the independent parameters of the system (i.e., cross-sections). We also derive perturbation equations associated with the particular NEM-M2B2 model we are using. These perturbation-equations are used in predicting the model eigenvalue change without resorting to recalculating the forward NEM-M2B2 system of equations again (labeled as exact calculations). They also enabled construction of a reactivity sensitivity map showing the importance of each calculation node of the benchmark depending on its spatial and spectral coordinates. Perturbations were imposed on both the absorption cross-sections (fast and thermal) and the scattering cross-section of the IAEA-3D benchmark problem. Several verification steps were taken to ensure that the developed mathematical adjoint solver is adequate for adjoint analysis (e.g., commutativity checks, and comparison against exact calculations).

在本文中，我们通过对正方程系统的节点界面中子电流方程进行预处理，并使用拉格朗日乘子分析方法，推导出高阶伴随NEM-M2B2方程的数学公式。在 NEM-M2B2 方程组中，二次横向泄漏近似用于模拟系统中每个节点之间的中子泄漏。伴随方程的解可用于执行基于伴随的预测灵敏度/扰动分析。例如，我们使用数学伴随解作为灵敏度加权来预测 IAEA-3D 基准的特征值对系统独立参数（即横截面）扰动的响应。我们还推导出与我们正在使用的特定 NEM-M2B2 模型相关的扰动方程。这些扰动方程用于预测模型特征值变化，而无需再次重新计算正向 NEM-M2B2 方程组（标记为精确计算）。他们还能够构建反应性敏感性图，显示基准的每个计算节点的重要性，具体取决于其空间和光谱坐标。对 IAEA-3D 基准问题的吸收截面（快速和热）和散射截面都施加了扰动。采取了几个验证步骤以确保开发的数学伴随求解器足以进行伴随分析（例如，交换性检查和与精确计算的比较）。这些扰动方程用于预测模型特征值变化，而无需再次重新计算正向 NEM-M2B2 方程组（标记为精确计算）。他们还能够构建反应性敏感性图，显示基准的每个计算节点的重要性，具体取决于其空间和光谱坐标。对 IAEA-3D 基准问题的吸收截面（快速和热）和散射截面都施加了扰动。采取了几个验证步骤以确保开发的数学伴随求解器足以进行伴随分析（例如，交换性检查和与精确计算的比较）。这些扰动方程用于预测模型特征值变化，而无需再次重新计算正向 NEM-M2B2 方程组（标记为精确计算）。他们还能够构建反应性敏感性图，显示基准的每个计算节点的重要性，具体取决于其空间和光谱坐标。对 IAEA-3D 基准问题的吸收截面（快速和热）和散射截面都施加了扰动。采取了几个验证步骤以确保开发的数学伴随求解器足以进行伴随分析（例如，交换性检查和与精确计算的比较）。