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.

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.

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.