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At least 811 records · Page 45
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

Design and convergence analysis of the conforming virtual element method for polyharmonic problems

In the recent years, there has been a tremendous interest to numerical methods that approximate partial differential equations (PDEs) on computational meshes with arbitrarily-shaped polygonal/polyhedral (polytopal, for short) elements. A nonexhaustive list of such methods include the Mimetic Finite Difference method [4, 9, 21, 24, 28, 33, 35, 62, 64–67], the Polygonal Finite Element Method [68, 73, 74], the polygonal Discontinuous Galerkin Finite Element Methods [5, 7, 8, 10, 13–15, 18, 36, 38, 39, 41, 50], the Hybridizable Discontinuous Galerkin and Hybrid High-Order Methods [46–48, 51], the Gradient Discretization method [49, 53, 54], the Finite Volume Method [52], and the BEM-based FEM [71]. An alternative approach that also proved to be very successful is provided by the Virtual Element method (VEM), which was originally proposed in [19] for the numerical treatment of second-order elliptic problems [43, 44], and readily extended to linear and nonlinear elasticity [22, 25, 56], plate bending problems [37], Cahn-Hilliard equation [3], Stokes equations [2], Laplace-Beltrami equation [55], Darcy-Brinkam equation [76], discrete topology optimization problems [6], fracture networks problems [29], eigenvalue problems [59, 75]. The mixed virtual element formulation was proposed in [20, 34]; the nonconforming Virtual element formulations was proposed for second-order elliptic problems in [16], and later extended to general advection-reaction-diffusion problems, Stokes equation, the biharmonic problems, the iegenvalue problem, and the Schrodinger equation in [11, 30, 42, 45, 58, 80]. The p- and hp-version of the VEM were developed in [26, 27] and efficient multigrid methods for the resulting linear system of equations in [12]. A posteriori error estimates can be found in [40].

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

Dynamics of a split torque helicopter transmission

Split torque designs, proposed as alternatives to traditional planetary designs for helicopter main rotor transmissions, can save weight and be more reliable than traditional designs. This report presents the results of an analytical study of the system dynamics and performance of a split torque gearbox that uses a balance beam mechanism for load sharing. The Lagrange method was applied to develop a system of equations of motion. The mathematical model includes time-varying gear mesh stiffness, friction, and manufacturing errors. Cornell's method for calculating the stiffness of spur gear teeth was extended and applied to helical gears. The phenomenon of sidebands spaced at shaft frequencies about gear mesh fundamental frequencies was simulated by modeling total composite gear errors as sinusoid functions. Although the gearbox has symmetric geometry, the loads and motions of the two power paths differ. Friction must be considered to properly evaluate the balance beam mechanism. For the design studied, the balance beam is not an effective device for load sharing unless the coefficient of friction is less than 0.003. The complete system stiffness as represented by the stiffness matrix used in this analysis must be considered to precisely determine the optimal tooth indexing position.

Krantz, Timothy L.↗
DOE OSTI2014

Triggering collapse of the presolar dense cloud core and injecting short-lived radioisotopes with a shock wave. III. Rotating three-dimensional cloud cores

A key test of the supernova triggering and injection hypothesis for the origin of the solar system's short-lived radioisotopes is to reproduce the inferred initial abundances of these isotopes. We present here the most detailed models to date of the shock wave triggering and injection process, where shock waves with varied properties strike fully three-dimensional, rotating, dense cloud cores. The models are calculated with the FLASH adaptive mesh hydrodynamics code. Three different outcomes can result: triggered collapse leading to fragmentation into a multiple protostar system; triggered collapse leading to a single protostar embedded in a protostellar disk; or failure to undergo dynamic collapse. Shock wave material is injected into the collapsing clouds through Rayleigh-Taylor fingers, resulting in initially inhomogeneous distributions in the protostars and protostellar disks. Cloud rotation about an axis aligned with the shock propagation direction does not increase the injection efficiency appreciably, as the shock parameters were chosen to be optimal for injection even in the absence of rotation. For a shock wave from a core-collapse supernova, the dilution factors for supernova material are in the range of ∼10{sup –4} to ∼3 × 10{sup –4}, in agreement with recent laboratory estimates of the required amount of dilution for {sup 60}Fe and {sup 26}Al. We conclude that a type II supernova remains as a promising candidate for synthesizing the solar system's short-lived radioisotopes shortly before their injection into the presolar cloud core by the supernova's remnant shock wave.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

CFD Simulation of Helium Flow Loop Test Section

A helium flow loop is being assembled at Oak Ridge National Laboratory to analyze heat transfer enhancement for systems such as blanket and divertor components. To efficiently identify optimum geometries for heat transfer enhancement in these applications, simulation work is performed to optimize test section designs that are built and tested in the helium flow loop that operates at 4 MPa and a mass flow rate of 100 g/s. Different ribbed geometries that examine rib shape, rib height, rib orientation, rib spacing, and three dimensional orientation are modeled and simulated in STAR-CCM+ to compare their ability to remove heat and mitigate pressure drop. Following the simulations, models are selected and manufactured for the helium flow loop tests. Simulations initially focus on a hydrodynamic study to determine the appropriate mesh and physics models and then add a heat flux to analyze the heat transfer abilities of the models. The simulations are run in steady state and use a Reynolds-averaged Navier-Stokes k-ε turbulence model. The helium is modeled as an ideal gas. The simulation explores models of geometries that enhance the heat transfer and decrease pressure drop with an overall goal of increasing fluid collision with the wall. Enhanced geometries are simulated to select appropriate designs for manufacturing, and preliminary experimental results are used to validate the simulations. Furthermore, the factors that are being analyzed in the comparison between the experimental and the simulated results include matching thermocouple temperatures, pressure drop, roughness, and fluid velocity.

42 ENGINEERING↗
DOE OSTI2024

Graph-Based Representations and Applications to Process Simulation

Rapid and robust convergence of a process flowsheet is critical to enable large-scale simulations that address core scientific questions related to process design, optimization, and sustainability. However, due to the highly coupled and nonlinear nature of chemical processes, efficiently solving a flowsheet remains a challenge. In this work, we show that graph representations of the underlying physical phenomena in unit operations may help identify potential avenues to systematically reformulate the network of equations and enable more robust topology-based convergence of flowsheets. To this end, we developed graph abstractions of the governing equations of vapor-liquid and liquid-liquid equilibrium separation equipment. These graph abstractions consist of a mesh of interconnected variable nodes and equation nodes that are systematically generated through PhenomeNode, a new open-source library in Python developed in this study. We show that partitioning the graph into separate mass, energy, and equilibrium subgraphs can help decouple nonlinearities and guide decomposition algorithms. By employing the graph abstraction on an industrial separation process for separating glacial acetic acid from water, we implemented a new block decomposition scheme in BioSTEAM and demonstrated that this can accelerate convergence over a traditional sequential modular approach.

Distillation↗
NASA NTRS1995

Domain decomposition for aerodynamic and aeroacoustic analyses, and optimization

The overarching theme was the domain decomposition, which intended to improve the numerical solution technique for the partial differential equations at hand; in the present study, those that governed either the fluid flow, or the aeroacoustic wave propagation, or the sensitivity analysis for a gradient-based optimization. The role of the domain decomposition extended beyond the original impetus of discretizing geometrical complex regions or writing modular software for distributed-hardware computers. It induced function-space decompositions and operator decompositions that offered the valuable property of near independence of operator evaluation tasks. The objectives have gravitated about the extensions and implementations of either the previously developed or concurrently being developed methodologies: (1) aerodynamic sensitivity analysis with domain decomposition (SADD); (2) computational aeroacoustics of cavities; and (3) dynamic, multibody computational fluid dynamics using unstructured meshes.

Baysal, Oktay↗
DOE OSTI2025

High order interpolation of magnetic fields with vector potential reconstruction for particle simulations

We propose a method for interpolating divergence-free continuous magnetic fields via vector potential reconstruction using Hermite interpolation, which ensures high-order continuity for applications requiring adaptive, high-order ordinary differential equation (ODE) integrators, such as the Dormand-Prince method. The method provides C(m) continuity and achieves high-order accuracy, making it particularly suited for particle trajectory integration and Poincaré section analysis under optimal integration order and timestep adjustments. Through numerical experiments, we demonstrate that the Hermite interpolation method preserves volume and continuity, which are critical for conserving toroidal canonical momentum and magnetic moment in guiding center simulations, especially over long-term trajectory integration. Furthermore, we analyze the impact of insufficient derivative continuity on Runge-Kutta schemes and show how it degrades accuracy at low error tolerances, introducing discontinuity-induced truncation errors. Lastly, we demonstrate performant Poincaré section analysis in two relevant settings of field data collocated from finite element meshes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Elucidating metal–organic framework structures using synchrotron serial crystallography

Metal organic frameworks (MOFs) are porous crystalline materials that display a wide variety of physical and chemical properties. Their single crystal structure determination is often challenging because in most cases micro- or nano-sized crystals spontaneously form upon MOF synthesis, which cannot be recrystallized. The production of larger single crystals for structure determination involves optimizing, and thus modifying, the conditions of synthesis, in which success cannot be guaranteed. Failure to produce crystals suitable for single-crystal X-ray diffraction leaves the 3D structure of the MOF compound unknown, and scientists must resort to more challenging structure solution methods based on X-ray powder or electron diffraction data. These laborious tasks can be avoided by using serial crystallography techniques which merge data collected on many micro-crystals. Here, we report the application of three synchrotron serial crystallography methods. We call these “mesh”, “grid” and “mesh&collect” scans. “Still” images (no rotation) are collected in the mesh scan approach, whereas small rotational wedges are collected in the grid scan method. The third protocol, mesh&collect, combines the acquisition of still images and rotational wedges. Using these means, we determine the ab initio structure of benchmark MOFs, MIL-100(Fe) and ZIF-8, that differ largely in unit cell size. These methods are expected to be widely applicable and facilitate structure determination of many MOF microcrystalline systems.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Coupling multi-fidelity xRAGE with machine learning for graded inner shell design optimization in double shell capsules

Bayesian optimization has shown promise for the design optimization of inertial confinement fusion targets. Specifically, in Vazirani et al. [Phys. Plasmas 28 , 122709 (2021)], optimal designs for double shell capsules with graded inner shells were identified using one-dimensional xRAGE simulation yield calculations. While the machine learning models were able to accurately learn and predict one-dimensional simulation target performance, using simulations with higher fidelity would improve design optimization and better match with the expected experimental performance. However, higher fidelity physics modeling, i.e., two-dimensional xRAGE simulations, requires significantly larger computational time/cost, usually at least an order of magnitude, in comparison with one-dimensional simulations. This study presents a multi-fidelity Bayesian optimization, in which the machine learning model leverages low-fidelity (one-dimensional xRAGE) and high-fidelity (two-dimensional xRAGE) simulations to more accurately predict “pre-shot” target performance with respect to the expected experimental performance. By building a multi-fidelity Bayesian optimization framework coupled with xRAGE, the low-fidelity and high-fidelity simulations are able to inform one another, such that we have: (1) improved physics modeling in comparison with using low-fidelity simulations alone, (2) reduced computational time/cost in comparison with using high-fidelity simulations alone, and (3) more confidence in the expected performance of optimized targets during real-world experiments. In the future, we plan to use this robust multi-fidelity Bayesian optimization methodology to expedite the design of graded inner shells further and eventually full capsules as a part of the current double shell campaign at the National Ignition Facility.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

ASU’s DAC polymer-enhanced cyanobacterial bioproductivity (AUDACity)

ASU’s DAC polymer-enhanced cyanobacterial bioproductivity (AUDACity) project aims to demonstrate a novel, scalable method for removing carbon dioxide (CO 2 ) directly from ambient air and delivering it to cyanobacterial cultures to produce commodity biofuel, mid-value protein for supplements, and high value phycocyanin (PC), a natural blue colorant (Figure A). This approach uses low-cost, reusable anion exchange polymers embedded in modular mesh packets, which capture CO 2 during drying cycles when exposed to ambient air, and release concentrated CO 2 into aqueous cultivation systems. The project addresses a critical challenge in energy research needed for developing sustainable, economically viable methods of Direct Air Capture (DAC) that can be integrated with bio-based systems for fuel and chemical production. AUDACity contributes to scientific understanding by integrating materials chemistry, cyanobacterial biology, and system engineering to create a distributed CO 2 delivery platform. Key insights have emerged around the design of biocompatible sorbents, optimization of CO 2 capture-release cycles, and durability of packet-based delivery systems under outdoor conditions. Notably, the team has synthesized and tested a range of polymer sorbents, identified mechanisms of material degradation and fouling, and advanced both lab- and pilot-scale cultivation systems to evaluate performance. From a technical and economic standpoint, AUDACity shows promise for achieving cost-effective CO 2 capture and delivery into aqueous media and biofuel production. Preliminary techno-economic analysis (TEA) indicates that the DAC system based on current performance can reach $\$$680/tonne CO 2 delivered into aqueous solution; with reasonable improvements to sorbent lifetime, sorbent capacity, reducing water uptake the approach could reach $\$$66/tonne by avoiding the need for energy-intensive sorbent regeneration and CO 2 compression, making it more feasible for decentralized deployment. With these costs for CO 2 and by extracting and selling high-value PC ($\$$50/kg) and mid-value protein supplement ($\$$6/kg), the remaining biomass can be hydrothermally treated into biofuel for $\$$2.50/gallon, and would support a small first-of-a-kind biorefinery capable of producing 500 barrels per day of biofuel. The project offers meaningful public benefits by advancing carbon removal technologies that are low-energy, modular, and adaptable to non-arable land and brackish water use. It aligns with national goals to develop advanced biotechnology and supports future pathways for bio-based fuels and products. By enabling direct coupling of CO 2 transfer into aqueous medium and biological carbon utilization, AUDACity lays the groundwork for effective algae cultivation without wasteful CO 2 delivery and is a promising and innovative solution for low-carbon fuel and bioproduct generation contributing to a vigorous bioeconomy.

09 BIOMASS FUELS↗
NASA NTRS1996

3D Structured Grid Adaptation

Grid adaptation for structured meshes is the art of using information from an existing, but poorly resolved, solution to automatically redistribute the grid points in such a way as to improve the resolution in regions of high error, and thus the quality of the solution. This involves: (1) generate a grid vis some standard algorithm, (2) calculate a solution on this grid, (3) adapt the grid to this solution, (4) recalculate the solution on this adapted grid, and (5) repeat steps 3 and 4 to satisfaction. Steps 3 and 4 can be repeated until some 'optimal' grid is converged to but typically this is not worth the effort and just two or three repeat calculations are necessary. They also may be repeated every 5-10 time steps for unsteady calculations.

Banks, D. W.↗
DOE OSTI2016

The application of MAMMOTH for a detailed tightly coupled fuel pin simulation with a station blackout

The accurate calculation of desired quantities to predict fuel behavior in a nuclear reactor requires the solution of inter linked equations representing different physics phenomena. Traditional fuels performance codes often rely on internal empirical models for the pin power density and a simplified boundary condition on the cladding outer surface. These simplifications are performed because of the difficulty of coupling applications or codes on differing domains and mapping the required data. In this work the MAMMOTH application is employed in the demonstration of an approach that more consistent with first principles, in a simulation that couples the neutronics application Rattlesnake, the thermal fluids application RELAP-7 and the fuels performance application BISON. A single fuel pin was modeled based on the dimensions of a fuel rod from a Westinghouse 17*17 optimized PWR fuel assembly. The simulation consisted of a depletion period of 1343 days, representing three full operating cycles, followed by a station blackout (SBO) event. The fuel rod was depleted with a total pin power of 65.81 kW. After 1343 days the fission power was reduced to zero (simulating a reactor shutdown). Decay heat calculations provided the time-varying energy source after this time. For this problem, Rattlesnake, BISON, and RELAP-7 are coupled under MAMMOTH in a tightly coupled approach. Each system solves its physics on a separate mesh and, for RELAP-7 and BISON, on only a subset of the full problem domain. Rattlesnake solves the neutronics over the whole domain, which includes the fuel, cladding, gaps, water, and top and bottom rod holders. Here BISON is applied to the fuel and cladding with a 2-D axisymmetric domain, and RELAP-7 is applied to the flow of the circular outer water channel with a set of 1-D flow equations. For this work, the mesh on the Rattlesnake side is either be 3-D (for low order PN transport or diffusion) or 2-D axisymmetric (for diffusion). BISON has a matching ring structure mesh for the fuel so both the power density and local burnup are transferred accurately from Rattlesnake. At each depletion time step, Rattlesnake calculates a power density, fission density rate, burnup distribution and fast flux based on the current water density and fuel temperature. These are then mapped to the BISON mesh to perform a fuels performance solution. BISON calculates the fuel temperature and cladding surface temperature based upon the current power density and bulk fluid temperature. RELAP-7 then calculates the fluid temperature, water density fraction and water phase velocity based upon the cladding surface temperature. The fuel temperature and the fluid density are then passed back to Rattlesnake for another neutronics calculation. For this paper, sets of results from the detailed calculation are provided for both depletion and the SBO event. We demonstrate that a detailed simulation akin to first principles can be achieved using MAMMOTH and the various MOOSE application on differing domains. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2023

Comparing the Performance of Julia on CPUs versus GPUs and Julia-MPI versus Fortran-MPI: a case study with MPAS-Ocean (Version 7.1)

Abstract. Some programming languages are easy to develop at the cost of slow execution, while others are fast at runtime but much more difficult to write. Julia is a programming language that aims to be the best of both worlds – a development and production language at the same time. To test Julia's utility in scientific high-performance computing (HPC), we built an unstructured-mesh shallow water model in Julia and compared it against an established Fortran-MPI ocean model, the Model for Prediction Across Scales–Ocean (MPAS-Ocean), as well as a Python shallow water code. Three versions of the Julia shallow water code were created: for single-core CPU, graphics processing unit (GPU), and Message Passing Interface (MPI) CPU clusters. Comparing identical simulations revealed that our first version of the Julia model was 13 times faster than Python using NumPy, where both used an unthreaded single-core CPU. Further Julia optimizations, including static typing and removing implicit memory allocations, provided an additional 10–20× speed-up of the single-core CPU Julia model. The GPU-accelerated Julia code was almost identical in terms of performance to the MPI parallelized code on 64 processes, an unexpected result for such different architectures. Parallelized Julia-MPI performance was identical to Fortran-MPI MPAS-Ocean for low processor counts and ranges from 2× faster to 2× slower for higher processor counts. Our experience is that Julia development is fast and convenient for prototyping but that Julia requires further investment and expertise to be competitive with compiled codes. We provide advice on Julia code optimization for HPC systems.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

A General Framework for Error-controlled Unstructured Scientific Data Compression

Data compression plays a key role in reducing storage and I/O costs. Traditional lossy methods primarily target data on rectilinear grids and cannot leverage the spatial coherence in unstructured mesh data, leading to suboptimal compression ratios. We present a multi-component, error-bounded compression framework designed to enhance the compression of floating-point unstructured mesh data, which is common in scientific applications. Our approach involves interpolating mesh data onto a rectilinear grid and then separately compressing the grid interpolation and the interpolation residuals. This method is general, independent of mesh types and typologies, and can be seamlessly integrated with existing lossy compressors for improved performance. We evaluated our framework across twelve variables from two synthetic datasets and two real-world simulation datasets. The results indicate that the multi-component framework consistently outperforms state-of-the-art lossy compressors on unstructured data, achieving, on average, a 2.3 − 3.5× improvement in compression ratios, with error bounds ranging from 1 × 10 the −6 to 1×10−2. We further investigate impact of hyperparameters, such as grid spacing and error allocation, to deliver optimal compression ratios in diverse datasets.

Gong, Qian↗
DOE OSTI2022

Enabling Parallel Execution of System-level Simulations in SAM

This report summarizes the recent code updates related to “element ghosting” in SAM to enable the parallel execution of system-level simulations using multiple processors/cores. Unlike typical MOOSE-based applications, for system-level simulations, SAM mostly deals with a collection of discrete small pieces of meshes, and the connection of physics on these meshes are realized by using “connector” types of components/code structures, such as conjugate heat transfer and flow junctions. The required code implementation is to correctly mark the necessary ghost elements for each type of such components/code structures; thus, the lower-level libraries can correctly perform the necessary data transfer between processors (CPUs) when executed in parallel mode. After the code updates, SAM can now run system-level simulations in the parallel mode. The parallel execution capability was then tested with an ABTR input model with 23k DOFs. Significant speedup was demonstrated when the optimal number of CPUs were used in parallel mode. Future systematic studies on parallelization performance using additional test cases covering different physics/scenarios will be needed to provide additional insights into the scalability of SAM parallelization.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2015

Soft Core/Shell Packages for Stretchable Electronics

This paper presents materials and core/shell architectures that provide optimized mechanical properties in packages for stretchable electronic systems. Detailed experimental and theoretical studies quantitatively connect the geometries and elastic properties of the constituent materials to the overall mechanical responses of the integrated systems, with a focus on interfacial stresses, effective modulus, and maximum extent of elongation. Specific results include core/shell designs that lead to peak values of the shear and normal stresses on the skin that remain less than 10 kPa even for applied strains of up to 20%, thereby inducing minimal somatosensory perception of the device on the human skin. Additional, strain-limiting mesh structures embedded in the shell improve mechanical robustness by protecting the active components from strains that would otherwise exceed the fracture point. Demonstrations in precommercial stretchable electronic systems illustrate the utility of these concepts.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Conservative Numerical Schemes with Optimal Dispersive Wave Relations: Part II. Numerical Evaluations

A new energy and enstrophy conserving scheme (EEC) for the shallow water equations is proposed and evaluated using a suite of test cases over the global spherical or bounded domain. The evaluation is organized around a set of pre-defined properties: accuracy of individual operators, accuracy of the whole scheme, conservation of key quantities, control of the divergence variable, representation of the energy and enstrophy spectra, and simulation of nonlinear dynamics. The results confirm that the scheme is between the first and second order accurate, and conserves the total energy and potential enstrophy up to the time truncation errors. Here, the scheme is capable of producing more physically realistic energy and enstrophy spectra, indicating that it can help prevent the unphysical energy cascade towards the finest resolvable scales. With an optimal representation of the dispersive wave relations, the scheme is able to keep the flow close to being non-divergent, and maintain the geostrophically balanced structures with large-scale geophysical flows over long-term simulations.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2016

Assessment of SFR Wire Wrap Simulation Uncertainties

Predictive modeling and simulation of nuclear reactor performance and fuel are challenging due to the large number of coupled physical phenomena that must be addressed. Models that will be used for design or operational decisions must be analyzed for uncertainty to ascertain impacts to safety or performance. Rigorous, structured uncertainty analyses are performed by characterizing the model’s input uncertainties and then propagating the uncertainties through the model to estimate output uncertainty. This project is part of the ongoing effort to assess modeling uncertainty in Nek5000 simulations of flow configurations relevant to the advanced reactor applications of the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program. Three geometries are under investigation in these preliminary assessments: a 3-D pipe, a 3-D 7-pin bundle, and a single pin from the Thermal-Hydraulic Out-of-Reactor Safety (THORS) facility. Initial efforts have focused on gaining an understanding of Nek5000 modeling options and integrating Nek5000 with Dakota. These tasks are being accomplished by demonstrating the use of Dakota to assess parametric uncertainties in a simple pipe flow problem. This problem is used to optimize performance of the uncertainty quantification strategy and to estimate computational requirements for assessments of complex geometries. A sensitivity analysis to three turbulent models was conducted for a turbulent flow in a single wire wrapped pin (THOR) geometry. Section 2 briefly describes the software tools used in this study and provides appropriate references. Section 3 presents the coupling interface between Dakota and a computational fluid dynamic (CFD) code (Nek5000 or STARCCM+), with details on the workflow, the scripts used for setting up the run, and the scripts used for post-processing the output files. In Section 4, the meshing methods used to generate the THORS and 7-pin bundle meshes are explained. Sections 5, 6 and 7 present numerical results for the 3-D pipe, the single pin THORS mesh, and the 7-pin bundle mesh, respectively.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗