Standard Galerkin method results in numerical instabilities when applied to the convection-dominated convection-diffusion equations. One approach to address this issue is the variational multiscale (VMS) stabilization technique. However, in the VMS in practice, geometry transformations of the corresponding operators are required, and the diffusion term in the stabilization part involves Christoffel symbols, which not appear in the classical weak form of the 2nd-order differential equations. Furthermore, the residual-driven stabilized finite element equation in the VMS method requires integration over multiple terms with different orders of polynomials. Therefore, intensive computational resources are needed to evaluate these terms, which makes the application of this method computationally expensive, especially when high-order elements are used. Optimum parallelization is therefore desirable. This work demonstrates the implementation and verification the Galerkin approach stabilized using the VMS technique on a hybrid parallel framework with simultaneous use of different parallelization paradigms including shared memory (OpenMP), distributed memory (MPI), and GPGPUs. Load balancing on one heterogeneous computing platform is achieved by offloading the calculations to multiple GPUs, using shared memory parallelism for loops, and distributed memory for linear solvers. Verification of this implementation includes the convergence rate analysis using problems with manufactured solutions, and a benchmark case is solved to compare the convergence rate with other published work. The speed-up data are reported.

The numerical solution of the two–dimensional, time–dependent, incompressible Navier–Stokes equations (NSE) in curvilinear coordinates using the nodal integral method (NIM) is presented in this paper. The developed numerical scheme is applied to solve fluid flow problems in domains discretized using quadrilateral cells. Three test problems are numerically solved to assess the accuracy and efficiency of the method. The scheme is second-order accurate in space for all considered deformation levels and Re numbers. The results of the current scheme are compared with the other numerical results in the literature, and good agreement is obtained for all considered cases. The NIM demonstrates superior accuracy per cell compared to other second–order finite–volume schemes considered in this work. New benchmark results obtained from the solution on fine meshes are presented.

Digital twins (DT) are synchronized clones that mirror their physical counterparts at all times, enabling real-time monitoring, analysis, decision-making, and planning for optimized operations. Digital twins can transcend traditional two-dimensional interfaces by incorporating VR/AR/MR (XR) technology, providing immersive, intuitive, and interactive representations of systems, assets, and processes. Furthermore, real-time data from sensors, simulations, and other sources can be integrated into the XR-enabled digital twins, leading to better and more intuitive understanding and to more efficiently monitor, analyze, and maintain complex systems such as nuclear assets. Immersive, interactive, multi-player XR capabilities embedded in DTs will allow spatially accurate and more realistic representation, leading to improved risk assessment, optimized and predictive maintenance scheduling, enhanced situational awareness, and more effective communication among interdisciplinary teams. By combining the strengths of XR and digital twins, nuclear facilities can achieve heightened safety, operational efficiency, and decision-making accuracy. Marrying XR technology with digital twins is also likely to extend the utilization of digital twins to optimize those aspects of the design and operation of nuclear assets that involve human beings–specifically in human factors engineering. Training can also be significantly enhanced if DTs are linked with XR technology. These systems may also be used to assess human performance through human factors engineering for the safety analysis of nuclear assets. A specific example is assessing human performance in semi-autonomous nuclear assets or operating multiple nuclear assets. After briefly reviewing digital twins of nuclear systems from the perspective of XR technology, this paper summarises our work in the nuclear energy space on VR/AR/MR and how these can be integrated into the newly evolving DTs of nuclear assets. The paper also describes the potential use of such systems in optimizing the design and operations of nuclear systems. As XR technology advances, its symbiotic relationship with digital twins can significantly reshape the landscape of nuclear operations and asset management.

Virtual Reality (VR) is increasingly being used in the gaming and entertainment sectors. It has also been gradually finding its way in the industrial training sector. But its potential in the education field has yet to be fully recognized. Recent breakthroughs have made head-mounted displays (HMDs) for virtual reality (VR) accessible to the educational institutions and to the public at a low cost. Similar breakthroughs are on the horizon in the haptic gloves technology, which are likely to lead to low cost and functional haptic gloves. Combination of VR with haptic gloves is expected to significantly change the user experience for the better. This paper seeks to explore the untapped potential of VR technology in education, specifically focusing on its applications in STEM laboratories. After a review of our past work in the development of VR labs for STEM subjects, we will describe recent advances made in the development and implementation of VR STEM labs in engineering.

Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). These ODEs or their approximations are analytically solved, and the solutions are used to develop the set of discrete equations. Since this method depends on transverse averaging, the standard application of NIM gets restricted to domains with boundaries parallel to one of the coordinate axes (2D) or coordinate planes (3D). The hybrid nodal-integral/finite-element method (NI-FEM) has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the bulk of the domain and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) are discretized using coarse NIM cells (NIM subdomains), and the rest of the domain is discretized using FEM elements (FEM subdomains). The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM and the FEM subdomains. Since the discrete variables in the two numerical approaches are different, this requires special treatment of the discrete quantities on the interfaces. We here report the development of hybrid NI-FEM--applicable to problems in arbitrary domains--for the 3D, incompressible Navier-Stokes equations (NSE) that are coupled to the energy equation via the Boussinesq approximation. The approach is compared with standard FEM and NEK to study its efficiency.

Heat conduction problems with convection and/or radiation boundary conditions arise in nuclear and other engineering fields. This paper presents a finite element approach to solve transient, three-dimensional (3-D), nonlinear heat conduction problems using quadratic tetrahedral elements with area/volume coordinates and quadrature integrations. Boundary conditions modeled include surface convection, to-ambient radiation, and surface-to-surface radiation (using the radiosity matrix method, RMM). Instead of the commonly used hemi-cube method, L′Huilier’s theorem is used here to calculate view factors. Contact constraints are also imposed. Interface condition is set up for the interfaces of different material. The one- and two-step backward differentiation are used for time integration. Nonlinearities introduced by temperature-dependent material thermal properties and the Stefan–Boltzmann law for radiation are handled using the Picard and/or Newton–Raphson methods. The convergence rate of the numerical scheme is verified using an analytical solution obtained using the method of manufactured solutions (MMS). Several practical cases are simulated, including that of two spheres in contact with each other, and results are compared to those obtained using commercial codes.

The natural convection phenomena inside cylindrical enclosures with three different internal objects-cylinder, cuboid, and hexagonal prism-are studied in this paper. The effect of the geometry of the inner object on the flow characteristic and heat transfer process is studied. The simulations are carried out using a recently developed, coarse-mesh nodal integral method. The model is validated using the numerical results reported in the literature.

Nodal integral method (NIM) is derived in general, 2D, curvilinear coordinates and applied to solve the convection-diffusion equation in domains discretized using quadrilateral elements. The quadrilateral ele-ments in the Cartesian system are transformed into rectangular elements in curvilinear coordinates using bi-linear Lagrangian interpolation functions. An approximation for the transverse-integration operator is developed in curvilinear coordinates. The convection-diffusion equation and the continuity conditions at the common edge between two adjacent nodes are derived in curvilinear coordinates. A new approxi-mation of the cross-derivative terms resulting from transforming the governing equation to curvilinear coordinates is developed using the discrete unknowns of NIM (line-averaged variables). Three numerical test problems are solved to assess the accuracy and efficiency of the newly developed scheme. The re-sults show that the scheme developed in this paper is second-order accurate in space and time, which is the same order of accuracy for traditional NIM for regularly-shaped elements. The results show that the accuracy of NIM for quadrilateral elements is maintained even for coarse mesh and highly distorted elements . (c) 2022 Elsevier Ltd. All rights reserved.

Nodal integral methods (NIM) are a class of efficient coarse mesh method that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE), and these ODEs or their approximations are analytically solved. Since this method depends on transverse averaging, the standard application of this approach gets restricted to domains that have boundaries that are parallel to one of the coordinate axes (2D) or coordinate planes (3D). The hybrid nodal-integral/finite-element method (NI-FEM) has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM and the rest of the domain can be solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the regions solved by the NIM and the FEM. Since the discrete variables in the two numerical approaches are different, this requires special treatment of the discrete quantities on the interface between the two different types of discretized elements. We here report the development of hybrid NI-FEM in a parallel framework in Fortran using PETSc for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is efficient compared to standalone conventional numerical schemes like FEM.

Nodal Integral Methods (NIM) are a class of coarse-mesh numerical methods developed to solve partial differential equations (PDEs). They are more accurate and efficient than the conventional finite-difference, finite-volume, and finite-element methods because of the use of approximate analytical solutions to the governing differential equations in the scheme's development. The transverse integration process, which reduces the PDE into a set of ODEs, restricts the application of NIM to domains discretized by rectangular elements in 2D and cuboid elements in 3D. Two ap-proaches presented in this paper relax this restriction and extend NIM's efficiency to arbitrary geometries in 3D. In the first approach, NIM is derived in general 3D curvilinear coordinates and applied to solve problems in domains discretized by hexahedral elements. The hexahedral elements in the Cartesian system are transformed into cubes in curvilinear coordinates, where the transverse integration procedure can be applied. The second approach is a hybrid nodal-integral/finite-element approach. In this approach, the bulk of the domain is discretized into regular cuboid elements, and the regions adjacent to curve boundaries are discretized by tetrahedral elements. The standard NIM is applied to the cuboid elements, while the finite-element is used for the tetrahedral elements. The two approaches are used to numerically solve the convection-diffusion equation (CDE) in four different computational domains. The method of manufactured solution (MMS) is used to construct exact temperature profiles for all test cases. The compu-tational domains of the first two test cases are cylindrical annulus and solid cylinder. The first two test cases are solved using both approaches, where the accuracy and efficiency of both methods are compared. The third test case is solved in a spherical domain using approach I only, while the fourth is solved in a cuboid domain with a hemispherical cavity using approach II only.

The main advantage of using Nodal Integral Methods (NIM) in solving partial differential equations (PDEs) is that they lead to an accurate solution over relatively coarse meshes. Due to the use of the transverse integration procedure in the derivation, most of the NIMs developed were restricted to PDEs with isotropic diffusion terms only. Recently, the NIM has been extended to arbitrary geometries using isoparametric mapping approach, which resulted in the transformation of isotropic diffusion to anisotropic diffusion. This required the development of a method to approximate the cross-derivative terms that is consistent with the order of accuracy of traditional NIMs. In this paper, the 3D, time-dependent, anisotropic convection-diffusion equation is solved numerically using the NIM. A new method to approximate the cross-derivative terms based on the actual discrete unknowns of the NIM - namely, the line-averaged or surface-averaged variables - is developed. Also, the previously developed approximation for the cross derivative terms in 2D (Kumar et al., 2013) that is based on the corner point values is extended to 3D. Six numerical test cases are solved to test the accuracy and efficiency of both approaches. The accuracy of NIM for the anisotropic diffusion equation is maintained even for coarse meshes for both cross derivative approximations. However, the surface-averaged-based approximation is more accurate and efficient than the point-value-based approximation in all cases. The NIM using the surface-averaged based approximation is second-order accurate in space and time. On the other hand, the NIM using the point-value-based approximation is between first and second-order accurate in space, and second order accurate in time. Published by Elsevier Ltd.

This NEUP project 16-10579 entitled “A Computational-Experimental Study to Simulate Mixing and Thermal Stratification in SFRs”, has investigated important phenomena of thermal stratification and mixing in Sodium-cooled Faster reactors (SFRs). A liquid metal experimental set-up was built at Kansas State University to generate high fidelity experimental data and validate computational models. The scope of the work involved - scaled down design of experimental facility, construction and commissioning of the new facility, conducting thermal-hydraulics experiments relevant to the physics of thermal stratification and mixing in the plena, developing 3D CFD models of the experiments, developing SAS4A/SASSYS-1 system model of the liquid metal experimental facility , survey of previous data in literature on stratification experiments and recommendations for improved models.

In multi-reactor and multi-turbine nuclear power plants (MMNPP), several reactor units working in parallel provide steam to turbines that generate power for different purposes. Due to a common steam header that all turbines are connected to, changes in the power output of one turbine can influence the performance of other turbines (and their power output) when operating in load following scenarios. This may threaten the stable and safe operation of the plant. To avoid the coupling effect between the power outputs of turbines, this paper presents a composite control method to decouple turbine power from each other. This (power) decoupling control strategy is composed of a neural network based inverse system (NNIS) and a robust controller. The neural network inverse system approaches the inverse system of the original nonlinear system. The robust controller, containing PI controllers and integrators, is added to the NNIS to construct a closed loop decoupling controller with strong robustness. Simulation results indicate that the proposed decoupling controller has excellent power decoupling capabilities and robust performance as well as fast tracking capability.

Stability analyses are carried out for a natural circulation lead-cooled fast reactor which is at its design stage. Because of its natural circulation feature, flow stability as well as nuclear-coupled thermal hydraulics stability is of some concern. Modal Expansion Method (MEM) and Reduced Order Model (ROM) are used for neutronics and thermal hydraulics part, respectively. These two separate models are coupled through feedback effects. Different simulation scenarios on the separate and coupled models are reported.

This article is an introduction and motivational contribution for the special issue "Nuclear Reactors and related nonlinear systems: Aspects of efficient modeling and stability analysis". The authors aim to demonstrate the performance of the systematic bifurcation analysis for the stability analysis of nonlinear dynamic systems such as nuclear and thermal-hydraulic systems. In a first part, the motivation of this approach is explained in detail (part 1: Theory) and the authors provide some mathematical basics, which are necessary in order to understand the results of 3 examples of the application of bifurcation theory which will be discussed in more detail in part 2 (Applications, Progress in Nuclear Energy 113 (2019) 263-280). In this context, we would also like to point to the importance of modern methods of model order reduction (MOR) and the relationship between mathematically optimized reduced order models (ROMs) and simplified dynamical models. To represent the benefits of the bifurcation theory for our purposes compactly, already published results of earlier works were selected and partly new interpreted and revised on the advanced knowledge level. The conclusions are the result of work in this field in recent years and should be a basis of discussion for the future work of the community.

This paper is regarded as a continuation of the paper "Principles for the application of bifurcation theory for the systematic analysis of nuclear reactor stability, Part1" with the intention to provide examples demonstrating the application of the bifurcation analysis method in the framework of reactor stability analysis. Hence, we continue with chapter 5 which is devoted to three examples: (1) two-phase flow stability analysis, (2) occurrence of a generalized Hopf bifurcation (GHB) during a real nuclear reactor stability test and (3) existence of a complex stability behaviour in an environment of a double Hopf bifurcation point (Hopf-Hopf bifurcation, HHB). The efficiency of the RAM-ROM method is demonstrated for an operating point of NPP Leibstadt for which a sufficient experimental and system code database is available. These three examples of system dynamics demonstrate the partly very complex stability behaviour of nonlinear systems which cannot be explained by the application of linear stability analysis methods such as the estimation of the decay ratio (as a linear stability indicator). The consequences of the found bifurcation types in examples 2 and 3 on the particular solution structure of the underlying dynamic system will be discussed by using their respective normal forms in order to provide the reader a more clear access to the complex system behaviour around these bifurcation points. In case of the Hopf-Hopf bifurcation, we only present a selected part of solutions in this paper and refer the reader to a future paper, where more details of this bifurcation type are summarized and consequences to the full system are interpreted.

Stability analyses of nuclear-coupled thermal hydraulics are carried out for SUPERSTAR – a natural circulation heavy metal (lead-cooled) fast reactor. Attributed to its natural circulation characteristic, flow stability as well as nuclear-coupled thermal hydraulics stability are of some concern. Modal expansion method as an extension for point reactor kinetics is used for neutronics and reduced order model for thermal hydraulic part. These two separate models are coupled through feedback effects. Different scenarios initiated by reactivity and thermal hydraulics conditions are simulated using the coupled models in time domain. It is shown that SUPERSTAR design has considerable stability margin at the nominal operating point. Design becomes less stable as transit time between core exit and core inlet is increased.