A generalized adjoint framework for sensitivity and global error estimation in time-dependent nuclear reactor simulations
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We develop a general framework for computing the adjoint variable to nuclear engineering problems governed by a set of differential-algebraic equations (DAEs). The nuclear engineering community has a rich history of developing and applying adjoints for sensitivity calculations; many such formulations, however, are specific to a certain set of equations, variables, or solution techniques. Any change or addition to the physics model would require a reformulation of the adjoint problem and substantial difficulties in its software implementation. In this work we propose an abstract framework that allows for the modification and expansion of the governing equations, leverages the existing theory of adjoint formulation for DAEs, and results in adjoint equations that can be used to efficiently compute sensitivities for parametric uncertainty quantification. Moreover, as we justify theoretically and demonstrate numerically, the same framework can be used to estimate global time discretization error. We first motivate the framework and show that the coupled Bateman and transport equations, which govern the time-dependent neutronic behavior of a nuclear reactor, may be formulated as a DAE system with a power constraint. We then use a variational approach to develop the parameter-dependent adjoint framework and apply existing theory to give formulations for sensitivity and global time discretization error estimates using the adjoint variable. We apply the framework to two problems: a simple pendulum problem with known solution, which serves to verify our global error estimation procedure, and a 1D model of a traveling wave reactor. We then emphasize the utility of the framework by adding heat transfer physics to the reactor problem and showing that no reformulation of the adjoint framework is required for application to the new physics. We conclude that the abstraction of the adjoint approach into a general framework will facilitate multiphysics reactor modeling in large-scale software projects. 2012 Elsevier Ltd. All rights reserved.
