@article{altahhan_geemert_avramova_ivanov_2022, title={Extending a low-order inhomogeneous adjoint equations model to a higher-order model with verification on integral applications}, volume={177}, ISSN={["1873-2100"]}, DOI={10.1016/j.anucene.2022.109277}, abstractNote={• Development and verification of a NEM-M2B2 mathematical inhomogeneous-adjoint nodal diffusion solver. • Application of the Lagrangian multipliers method to derive the nodal mathematical inhomogeneous-adjoint. • Derived the local linear prediction formula, specific for the forward NEM-M2B2 model, and utilized it to study the repercussions of perturbations in the IAEA-3D benchmark on the Axial Offset (AO). • Verified the generalized adjoint code developed while showing detailed steps of how inhomogeneous adjoint codes can be verified. • Compared between the low-order inhomogeneous NEM-M0 adjoint and the developed higher order inhomogeneous NEM-M2B2 model for the AO as a RoI. • Introduced the Mantissa theory to explain the behavior of the linear adjoint models and prediction formulas. A higher-order nodal mathematical inhomogeneous adjoint model conjugate to the NEM-M2B2 nodal diffusion forward model is developed and introduced in this research. Verification of the developed model is presented through applications in perturbation analysis and the IAEA-3D benchmark including adjusted forms of it. This paper’s objective is to explore ways of extending and optimizing a mathematical adjoint capability suitable for use in an industrial reactor code, such that it becomes not merely an approximate but rather the exact adjoint counterpart to the typically used higher-order nodal forward solvers used in mature industrial reactor codes. Specifically, it is investigated how to upgrade an already available lower-order nodal mathematical adjoint solver towards higher-order accuracy. An example of the latter is the lower-order nodal adjoint solver used in the ARTEMIS reactor code, in the technical context of stabilization and acceleration of embedded control rod search mechanisms. Though the latter adjoint solver proved suitable for the needed preconditioning purposes, while also enabling the benefit of computationally very lean adjoint iterations, several future developments could benefit from having a higher-order adjoint nodal solver available as well. By using a preconditioned form of the base NEM-M2B2 nodal diffusion forward model and by using variational analysis, we have obtained a higher-order nodal mathematical adjoint that can have a physical interpretation associated with it as a Lagrangian multiplier. The nodal mathematical adjoint is then developed for the Axial Offset (AO) as a Response of Interest (RoI) which leads to an inhomogeneous adjoint system of equations. A solution verification of the adjoint developed is done through analyzing the effects coming from perturbations in the absorption and the scattering cross-sections. The applications investigated include axially and radially traveling perturbations along the reactor’s core. Several locations for the traveling perturbations are chosen to represent important locations in the core. Comparison between the low-order and the higher-order adjoint models is conducted. The forward model is set to the NEM-M2B2 nodal diffusion equation for both adjoints during the comparison. The higher-order adjoint model developed show consistent results in comparison to its lower-order sibling, suggesting the preference of using the developed higher-order model for adjoint computations.}, journal={ANNALS OF NUCLEAR ENERGY}, author={Altahhan, Muhammad Ramzy and Geemert, Rene and Avramova, Maria and Ivanov, Kostadin}, year={2022}, month={Nov} }
@article{altahhan_geemert_avramova_ivanov_2021, title={Development and verification of a higher-order mathematical adjoint nodal diffusion solver}, volume={163}, ISSN={["1873-2100"]}, DOI={10.1016/j.anucene.2021.108548}, abstractNote={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).}, journal={ANNALS OF NUCLEAR ENERGY}, author={Altahhan, Muhammad Ramzy and Geemert, Rene and Avramova, Maria and Ivanov, Kostadin}, year={2021}, month={Dec} }
@article{altahhan_aboanber_abou-gabal_2017, title={Analytical solution of the Telegraph Point Reactor Kinetics model during the cold start-up of a nuclear reactor}, volume={109}, ISSN={["0306-4549"]}, DOI={10.1016/j.anucene.2017.06.001}, abstractNote={We solve the new model of the Point Reactor Kinetics (PRK) equations developed based on the Telegraph approximation of the neutron transport equation analytically for a linear insertion of reactivity typically introduced by lifting the control rods discontinuously and manually during the cold start-up of a subcritical nuclear reactor. The Telegraph model introduces a new parameter called the Relaxation Time (τ) and we study its impact on the analytical solution for this case of reactivity insertion and for several speeds of lifting the control rods. We find that as the Relaxation time increases, the solution response is relaxed behind that of the diffusion model which was solved earlier in the literature for the same case. On the other hand, as the Relaxation time tends to zero, we find that the response of the neutron density tends to that of the diffusion case yielding a verification for the new proposed solution. Moreover, when reducing the Control Rod lifting speed, the effect of the relaxation time and hence the Telegraph approximation is reduced and it approaches that of the diffusion. We discuss Both mathematical and physical reasons for the cause of these behaviors.}, journal={ANNALS OF NUCLEAR ENERGY}, author={Altahhan, Muhammad Ramzy and Aboanber, Ahmed E. and Abou-Gabal, Hanaa H.}, year={2017}, month={Nov}, pages={574–582} }
@article{altahhan_aboanber_abou-gabal_nagy_2017, title={Response of the point-reactor telegraph kinetics to time varying reactivities}, volume={98}, ISSN={["0149-1970"]}, DOI={10.1016/j.pnucene.2017.03.008}, abstractNote={The new model of the Point Reactor Kinetics (PRK) equations developed based on the Telegraph approximation of the neutron transport equation, is solved for several cases of time varying Reactivities insertions and Temperature feedback while comparing it to that of the diffusion PRK model in an infinite Thermal Homogenous Nuclear Reactor. Diffusion PRK is based on the Neutron Diffusion Equation which is a parabolic differential equation and hence it assumes an infinite velocity of propagation, while neutrons propagate with a finite velocity. By the introduction of the hyperbolic type Telegraph equation which is a more accurate representation of the neutron transport than the diffusion equation and in which neutrons propagate with a finite velocity, one could overcome this paradox that contradicts causality. The new model introduces a new parameter called the relaxation time (τ), which is not present in the diffusion approximation, and affects the neutron density calculations. Both Ramp insertions of reactivity and Sinusoidal insertions of reactivity were studied, as well as the effect of The Adiabatic Temperature feedback. The general phenomena in the solution of the new model is a Relaxation in the time response of the solution. It is found that the Telegraph model with its extra second order time derivative, will give observable different values than that of the diffusion even when we used small (τ) especially for the cases at which the neutron density changes rapidly.}, journal={PROGRESS IN NUCLEAR ENERGY}, author={Altahhan, Muhammad Ramzy and Aboanber, Ahmed E. and Abou-Gabal, Hanaa H. and Nagy, Mohamed S.}, year={2017}, month={Jul}, pages={109–122} }