In this paper, we present a reduced-order modeling approach to study the Molten Salt Fast Reactor (MSFR). Our approach is nonintrusive and based on the proper orthogonal decomposition method. We include adaptivity in selecting the sampling points both in time and parameter space. Steady-state and transient analysis were both performed using the developed models. In the steady-state analysis, we capture the effect of 30 model parameters on the spatial distributions of fission power and temperature, and on the multiplication factor. The dimensionality of the fission power was reduced from the 104288 nominal dimensions in the physical space to 10 dimensions in the reduced space, whereas the temperature was reduced from 220972 dimensions to 3. The reduced model was then used for uncertainty and sensitivity study of the maximum temperature in the reactor and the multiplication factor. In the transient analysis, the reduced model captured the effect of perturbations in the flow rate of salt in the intermediate circuit on the fission power density and temperature. The reduced models were successfully tested on a set of points that were not part of the snapshots used during the construction stage.

@en

We present an approach to build a reduced-order model for nonlinear, time-dependent, parametrized partial differential equations in a nonintrusive manner. The approach is based on combining proper orthogonal decomposition (POD) with a Smolyak hierarchical interpolation model for the POD coefficients. The sampling of the high-fidelity model to generate the snapshots is based on a locally adaptive sparse grid method. The novelty of the work is in the adaptive sampling of time, which is treated as an additional parameter. The goal is to have a robust and efficient sampling strategy that minimizes the risk of overlooking important dynamics of the system while disregarding snapshots at times when the dynamics are not contributing to the construction of the reduced model. The developed algorithm was tested on three numerical tests. The first was an advection problem parametrized with a five-dimensional space. The second was a lid-driven cavity test, and the last was a neutron diffusion problem in a subcritical nuclear reactor with 11 parameters. In all tests, the algorithm was able to detect and include more snapshots in important transient windows, which produced accurate and efficient representations of the high-fidelity models.

@en

Large-scale complex systems require high-fidelity models to capture the dynamics of the system accurately. For example, models of nuclear reactors capture multiphysics interactions (e.g., radiation transport, thermodynamics, heat transfer, and fluid mechanics) occurring at various scales of time (prompt neutrons to burn-up calculations) and space (cell and core calculations). The complexity of thesemodels, however, renders their use intractable for applications relying on repeated evaluations, such as control, optimization, uncertainty quantification, and sensitivity studies.@en

We use a novel nonintrusive adaptive Reduced Order Modeling method to build a reduced model for a molten salt reactor system. Our approach is based on Proper Orthogonal Decomposition combined with locally adaptive sparse grids. Our reduced model captures the effect of 27 model parameters on k_eff of the system and the spatial distribution of the neutron flux and salt temperature. The reduced model was tested on 1000 random points. The maximum error in multiplication factor was found to be less than 50 pcm and the maximum L₂ error in the flux and temperature were less than 1%. Using 472 snapshots, the reduced model was able to simulate any point within the defined range faster than the high-fidelity model by a factor of 5×10⁶. We then employ the reduced model for uncertainty and sensitivity analysis of the selected parameters on k_eff and the maximum temperature of the system.

@en

Large-scale complex systems require high fidelity models to capture the dynamics of the system accurately. The complexity of these models, however, renders their use to be expensive for applications relying on repeated evaluations, such as control, optimization, and uncertainty quantification. Proper Orthogonal Decomposition (POD) is a powerful Reduced Order Modelling (ROM) technique developed to reduce the computational burden of high fidelity models. In cases where the model is inaccessible, POD can be used in a nonintrusive manner. The accuracy and efficiency of the nonintrusive reduced model are highly dependent on the sampling scheme, especially for high dimensional problems. To that end, we study integrating the locally adaptive sparse grids with the POD method to develop a novel nonintrusive POD-based reduced order model. In our proposed approach, the locally adaptive sparse grid is used to adaptively control the sampling scheme for the POD snapshots, and the hierarchical interpolant is used as a surrogate model for the POD coefficients. An approach to efficiently update the surpluses of the sparse grids with each POD snapshots update is also introduced. The robustness and efficiency of the locally adaptive algorithm are increased by introducing a greediness parameter, and a strategy to validate the reduced model after convergence. The proposed validation algorithm can also enrich the reduced model around regions of detected discrepancies. Three numerical test cases are presented to demonstrate the potential of the proposed POD-Adaptive algorithm. The first is a nuclear reactor point kinetics, the second is a general diffusion problem, and the last is a variation of the analytical Morris function. The results show that the developed algorithm reduced the number of model evaluations compared to the classical sparse grid approach. The built reduced models captured the dynamics of the reference systems with the desired tolerances. The non-intrusiveness and simplicity of the method provide great potential for a wide range of practical large scale applications.

@en

Reduced order models are effective in reducing the computational burden of large-scale complex systems. Proper Orthogonal Decomposition (POD) is one of the most important methods for such application. Nevertheless, problems parametrized on high dimensional spaces require computations of an enormous number of simulations in the offline phase. In this paper, the use of sparse grids is suggested to select the sampling points in an efficient manner. The method exploits the hierarchical nature of the Smolyak algorithm to select the sparse grid level based on the singular values of the POD basis. Then, a nonintrusive reduced order model is built using Smolyak’s combination technique. The proposed method was tested and compared with Radial Basis Functions in two nuclear applications. The first was a one-dimensional slab solved as a diffusion eigenvalue problem and the second was the two-dimensional IAEA benchmark problem. In both cases, the results showed that while Radial Basis Functions resulted in a faster reduced order model, Smolyak’s model provided superior accuracy. @en