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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Spectral Element Applications in Complex Nuclear Reactor Geometries: Tet-to-Hex Meshing

The spectral element code Nek5000 is an open-source, higher-order computational fluid dynamics code developed at Argonne National Laboratory. It is designed to solve incompressible Navier-Stokes equations, but it also has a low-Mach-number approximation feature available. Large eddy simulation is approached by explicit filtering of the velocity field (and other fields) to mimic the effect of dissipation due to the unresolved scale. The computational domain is decomposed into second-order hexahedral elements that conform to the boundaries. However, generating a high-quality pure-hexahedral mesh can be challenging for some problems. For simple geometries, traditional blocking methods can be used to decompose the domain into smaller blocks to generate a so-called structural mesh. A structural mesh can maintain good orthogonality but can have a highly skewed mesh to conform to the geometry, as well as unnecessary refinement in the far field. Moreover, for geometries with relative complexity, blocking the geometry becomes impossible. To address these issues, we adopted a let-to-hex strategy to generate a pure hexahedral mesh for Nek5000. First, we generate a pure tetrahedral mesh for the geometry; then we divide one tetrahedral element into four hexahedral elements. A pure tetrahedral mesh could be easily generated for complex geometries by using many current meshing codes. In this paper, we use the commercial codes ANSYS meshing and ANSYSICEM to generate the pure tetrahedral mesh and then convert it to a pure hexahedral mesh. Boundary layers are extruded in ANSYSICEM to maintain near-wall resolution.

ANSYS ICEM↗
DOE OSTI2019

A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements

Multiphase flow can be difficult to simulate with high accuracy due to the wide range of scales associated with various multiphase phenomena. These scales may range from the size of individual particles to the entire domain of interest. Traditionally, large scale systems can only be simulated using averaging approaches that filter out the locations of individual particles. In this work, the Euler–Lagrange method is used to simulate large-scale dense particle systems in which each individual particle is tracked. In order to accomplish this, the highly scalable spectral element code nek5000 has been extended to handle the multiple levels of multiphase coupling in these systems. These levels include what has been called one-, two-, and four-way coupling. Here, each level has been separated to detail the computational impact of each stage. A binned ghost particle algorithm has also been developed to efficiently handle the challenges of two- and four-way coupling in a parallel processing context. The algorithms and their implementations are then shown to scale to 65,536 Message Passing Interface (MPI) ranks in both the strong and weak limits. After this, validation is performed through simulation of a small-scale fluidized bed. Lastly, a large-scale fluidized bed is simulated with 65,536 MPI ranks and is able to capture the unique physics of the onset of fluidization.

Computer Science↗
DOE OSTI2024

Preserving Superconvergence of Spectral Elements for Curved Domains [Slides]

Finite Element Methods (FEM) and Spectral Element Methods (SEM) are crucial for solving partial differential equations (PDEs) on complex geometries. SEM offers superior accuracy due to potential superconvergence for simple domains. Challenges persist for domains with curved boundaries, restricting SEM’s advantages in real-world applications. A proposed solution is the introduction of a novel strategy to enhance accuracy and maintain superconvergence of SEM in curved domains. The strategy includes a mesh-generation procedure with geometrically refined elements near curved boundaries and a post-processing phase using the Adaptive Extended Stencil Finite Element Method (AES-FEM). The method, named AES-FEM post-processed Spectral Element Method (ApSEM), aligns the accuracy of non-tensor-product elements with superconvergent spectral elements.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

Here, a mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.

58 GEOSCIENCES↗
DOE OSTI2023

Demonstration of RANS models with wall functions in the spectral element code Nek5000

The spectral element based computational fluid dynamics (CFD) code Nek5000 has been traditionally used for high-fidelity applications, such as direct numerical simulation (DNS) and large eddy simulation (LES). These techniques require very fine numerical resolution to accurately capture turbulent fluctuations which can be prohibitively expensive for users without access to leadership class computing facilities. For broader application and adoption, significant effort has been invested to develop Reynolds-averaged Navier–Stokes (RANS) capabilities in Nek5000. Here, this work presents details of the implementation and demonstration of the standard wall functions for the κ–τ model in Nek5000. Results using the wall-modeled approach are compared to a wall-resolved approach for cases with negligible pressure gradient, viz., channel flow, pipe flow and flow in a reactor subchannel. Results show reasonably good agreement between the two approaches for friction factor and Nusselt number. Some expected differences are identified near the wall. These cases demonstrate the potential for significant computational savings by using much coarser meshes for the wall-modeled approach, with only minor differences between the predicted result. Additionally, several Reynolds numbers up to 1,000,000 are demonstrated for pipe flow and predicted friction factors and Nusselt numbers compared well to available correlations, with the worst below 10%. As the Reynolds number is increased, better agreement is observed between the correlations and the wall-modeled approach. In addition, flow in a molten salt fast reactor (MSFR) core is considered which features an adverse pressure gradient and flow separation. It showcases the inability of standard wall functions to accurately predict flows with adverse pressure gradients. The results, however, match reasonably well in trend in regions of the flow where the boundary layer is attached. Ongoing research is dedicated to include a pressure gradient correction to wall functions to improve the accuracy of flows with separation or reattachment and adverse or favorable pressure gradients.

42 ENGINEERING↗
DOE OSTI2024

Preserving Superconvergence of Spectral Elements for Curved Domains

Spectral element methods (SEM), extensions of finite element methods (FEM), have emerged as significant techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver superior accuracy due to the potential superconvergence in nodal solutions for well-shaped tensor-product elements. However, the accuracy of SEM often degrades in complex geometries due to geometric inaccuracies near curved boundaries and the loss of superconvergence with simplicial or non-tensor-product elements. To overcome the first issue, we propose using geometric refinement, which both refines the mesh near high-curvature regions and increases the degree of geometric basis functions. We show that when using mixed-element meshes with tensor-product elements in the interior of the domain, curvature-based geometric refinement near boundaries can improve the accuracy of the interior elements by reducing pollution errors and preserving the superconvergence in nodal solutions. To address the second issue, we introduce ApSEM, a post-processing technique using the adaptive extended stencil finite element method (AES-FEM) to recover the accuracy near the curved boundaries. The combination of curvature-based geometric refinement and accurate post-processing offers an effective and easier-to-implement alternative to methods reliant on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 2D and 3D and show up to two orders of magnitude of improvement in the solution accuracy, even when the elements are poorly shaped near boundaries. We also show the efficiency of ApSEM as it can recover superconvergence in nodal solutions without drastically increasing the computational cost.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A face-upwinded spectral element method

Here we present a new high-order accurate discretisation on unstructured meshes of quadrilateral elements. Our Face Upwinded Spectral Element (FUSE) method uses the same node distribution as a high-order continuous Galerkin (CG) method, but with a particular choice of node locations within each element and an upwinded stencil on the face nodes. This results in a number of benefits, including fewer degrees of freedom and straight-forward integration with CG. We present the derivation of the scheme and the analysis of its properties, in particular showing stability using von Neumann analysis. We show numerical evidence for its accuracy and efficiency on multiple classes of problems including convection-dominated flows, Poisson's equation, and the incompressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

AMIP Simulation with the CAM4 Spectral Element Dynamical Core

We evaluate the climate produced by the Community Earth System Model, version 1, running with the new spectral-element atmospheric dynamical core option. The spectral-element method is congured to use a cubed-sphere grid, providing quasi-uniform resolution over the sphere, increased parallel scalability and removing the need for polar filters. It uses a fourth order accurate spatial discretization which locally conserves mass and moist total energy. Using the Atmosphere Model Intercomparison Project protocol, we compare the results from the spectral-element dy- namical core with those produced by the default nite-volume dynamical core and with observations.

CAM-SE↗
DOE OSTI2012

CAM-SE: A scalable spectral element dynamical core for the Community Atmosphere Model.

The Community Atmosphere Model (CAM) version 5 includes a spectral element dynamical core option from NCAR's High-Order Method Modeling Environment. It is a continuous Galerkin spectral finite element method designed for fully unstructured quadrilateral meshes. The current configurations in CAM are based on the cubed-sphere grid. The main motivation for including a spectral element dynamical core is to improve the scalability of CAM by allowing quasi-uniform grids for the sphere that do not require polar filters. In addition, the approach provides other state-of-the-art capabilities such as improved conservation properties. Spectral elements are used for the horizontal discretization, while most other aspects of the dynamical core are a hybrid of well tested techniques from CAM's finite volume and global spectral dynamical core options. Here we first give a overview of the spectral element dynamical core as used in CAM. We then give scalability and performance results from CAM running with three different dynamical core options within the Community Earth System Model, using a pre-industrial time-slice configuration. We focus on high resolution simulations of 1/4 degree, 1/8 degree, and T340 spectral truncation.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2017

Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling

The mass-lumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points, known as cubature points. To date, such points are only known analytically in tensor domains, such as quadrilateral or hexahedral elements. Thus, the diagonal-mass-matrix spectral element method (SEM) in non-tensor domains always relies on numerically computed interpolation points or quadrature points. However, only the cubature points for degrees 1 to 6 are known, which is the reason that we have developed a p-norm-based optimization algorithm to obtain higher-order cubature points. In this way, we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9. The dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration. Simultaneously, the Lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties. As a result, such points provide both good interpolation properties and integration accuracy. The Courant–Friedrichs–Lewy (CFL) numbers are tabulated for the conventional Fekete-based triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubature-based TSEM (OTSEM). A complementary study demonstrates the spectral convergence of the OTSEM. A numerical example conducted on a half-space model demonstrates that the OTSEM improves the accuracy by approximately one order of magnitude compared to the conventional Fekete-based TSEM. In particular, the accuracy of the 7th-order OTSEM is even higher than that of the 14th-order Fekete-based TSEM. Furthermore, the OTSEM produces a result that can compete in accuracy with the quadrilateral SEM (QSEM). The high accuracy of the OTSEM is also tested with a non-flat topography model. In terms of computational efficiency, the OTSEM is more efficient than the Fekete-based TSEM, although it is slightly costlier than the QSEM when a comparable numerical accuracy is required. - Highlights: • Higher-order cubature points for degrees 7 to 9 are developed. • The effects of quadrature rule on the mass and stiffness matrices has been conducted. • The cubature points have always positive integration weights. • Freeing from the inversion of a wide bandwidth mass matrix. • The accuracy of the TSEM has been improved in about one order of magnitude.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Separating Physics and Dynamics Grids for Improved Computational Efficiency in Spectral Element Earth System Models

Previous studies have shown that atmospheric models with a spectral element grid can benefit from putting physics calculations on a relatively coarse finite volume grid. Here we demonstrate an alternative high-order, element-based mapping approach used to implement a quasi-equal-area, finite volume physics grid in E3SM. Unlike similar methods, the new method in E3SM requires topology data purely local to each spectral element, which trivially allows for regional mesh refinement. Simulations with physics grids defined by 2 × 2, 3 × 3, and 4 × 4 divisions of each element are shown to verify that the alternative physics grid does not qualitatively alter the model solution. The model performance is substantially affected by the reduction of physics columns when using the 2 × 2 grid, which can increase the throughput of physics calculations by roughly 60%–120% depending on whether the computational resources are configured to maximize throughput or efficiency. A pair of regionally refined cases are also shown to highlight the refinement capability.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Solving the Grad–Shafranov equation using spectral elements for tokamak equilibrium with toroidal rotation

The Grad–Shafranov equation is solved using spectral elements for tokamak equilibrium with toroidal rotation. The Grad–Shafranov solver builds upon and extends the NIMEQ code (Howell and Sovinec, 2014) previously developed for static tokamak equilibria. Both geometric and algebraic convergence are achieved as the polynomial degree of the spectral-element basis increases. A new analytical solution to the Grad–Shafranov equation is obtained for Solov’ev equilibrium in presence of rigid toroidal rotation, in addition to a previously obtained analytical solution for a different set of equilibrium and rotation profiles. The numerical solutions from the extended NIMEQ are benchmarked with the analytical solutions, with good agreements. Besides, the extended NIMEQ code is benchmarked with the FLOW code (Guazzotto et al., 2004). The modification of pressure profile induced by toroidal flow is investigated. The relative change of pressure profile is found significant around the edge.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Electromagnetic wave propagation based upon spectral-element methodology in dispersive and attenuating media

SUMMARY We build on mathematical equivalences between Maxwell’s wave equations for an electromagnetic medium and elastic seismic wave equations. This allows us to readily model Maxwell’s wave propagation in the spectral-element codes SPECFEM2D and SPECFEM3D, written for acoustic, viscoelastic and poroelastic seismic wave propagation, providing the ability to handle complex geometries, inherent to finite-element methods and retaining the strength of exponential convergence and accuracy due to the use of high-degree polynomials to interpolate field functions on the elements, characteristic to spectral-element methods (SEMs). Attenuation and dispersion processes related to the frequency dependence of dielectric permittivity and conductivity are also included using a Zener model, similar to shear attenuation in viscoelastic media or viscous diffusion in poroelastic media, and a Kelvin–Voigt model, respectively. Ability to account for anisotropic media is also discussed. Here, we limit ourselves to certain dielectric permittivity tensor geometries, in order to conserve a diagonal mass matrix after discretization of the system of equations. Doing so, simulation of Maxwell’s wave equations in the radar frequency range based on SEM can be solved using explicit time integration schemes well suited for parallel computation. We validate our formulation with analytical solutions. In 2-D, our implementation allows for the modelling of both a transverse magnetic (TM) mode, suitable for surface based reflection ground penetration radar type of applications, and a transverse electric (TE) mode more suitable for crosshole and vertical radar profiling setups. Two 2-D examples are designed to demonstrated the use of the TM and TE modes. A 3-D example is also presented, which allows for the full TEM solution, different antenna orientations, and out-of-plane variations in material properties.

58 GEOSCIENCES↗
DOE OSTI2023

Preserving Superconvergence of Spectral Elements for Curved Domains via $h$ and $p$-Geometric Refinement

Spectral element methods (SEM), extensions of finite element methods (FEM), have emerged as significant techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver superior accuracy due to the potential superconvergence in nodal solutions for well-shaped tensor-product elements. However, the accuracy of SEM often degrades in complex geometries due to geometric inaccuracies near curved boundaries and the loss of superconvergence with simplicial or non-tensor-product elements. To overcome the first issue, we propose using h- and p-geometric refinement, which refines the mesh near high-curvature regions and increases the degree of geometric basis functions, respectively. We show that when using mixed-element meshes with tensor-product elements in the interior of the domain, curvature-based geometric refinement near boundaries can improve the accuracy of the interior elements by reducing pollution errors and preserving the superconvergence in nodal solutions. To address the second issue, we introduce a post-processing technique using the adaptive extended stencil finite element method (AES-FEM) to recover the accuracy near the curved boundaries. The combination of curvature-based geometric refinement and accurate post-processing offers an effective and easier-to-implement alternative to methods reliant on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 2D and show up to two orders of magnitude of improvement in the solution accuracy, even when the elements are poorly shaped near boundaries.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Stabilization of numerical interchange in spectral-element magnetohydrodynamics

In this study, auxiliary numerical projections of the divergence of flow velocity and vorticity parallel to magnetic field are developed and tested for the purpose of suppressing unphysical interchange instability in magnetohydrodynamic simulations. The numerical instability arises with equal-order C 0 finite- and spectral-element expansions of the flow velocity, magnetic field, and pressure and is sensitive to behavior at the limit of resolution. The auxiliary projections are motivated by physical field-line bending, and coercive responses to the projections are added to the flow-velocity equation. Their incomplete expansions are limited to the highest-order orthogonal polynomial in at least one coordinate of the spectral elements. Cylindrical eigenmode computations show that the projections induce convergence from the stable side with first-order ideal-MHD equations during h-refinement and p-refinement. Hyperbolic and parabolic projections and responses are compared, together with different methods for avoiding magnetic divergence error. Lastly, the projections are also shown to be effective in linear and nonlinear time-dependent computations with the NIMROD code [C. R. Sovinec, et al., J. Comput. Phys. 195 (2004) 355-386], provided that the projections introduce numerical dissipation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2017

Turbulence statistics in a spectral element code: a toolbox for High-Fidelity Simulations

In the present document we describe a toolbox for the spectral-element code Nek5000, aimed at computing turbulence statistics. The toolbox is presented for a small test case, namely a square duct with L x = 2h, L y = 2h and L z = 4h, where x, y and z are the horizontal, vertical and streamwise directions, respectively. The number of elements in the xy-plane is 16 X 16 = 256, and the number of elements in z is 4, leading to a total of 1,204 spectral elements. A polynomial order of N = 5 is chosen, and the mesh is generated using the Nek5000 tool genbox. The toolbox presented here allows to compute mean-velocity components, the Reynolds-stress tensor as well as turbulent kinetic energy (TKE) and Reynolds-stress budgets. Note that the present toolbox allows to compute turbulence statistics in turbulent flows with one homogeneous direction (where the statistics are based on time-averaging as well as averaging in the homogeneous direction), as well as in fully three-dimensional flows (with no periodic directions, where only time-averaging is considered).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

CAM-SE–CSLAM: Consistent Coupling of a Conservative Semi-Lagrangian Finite-Volume Method with Spectral Element Dynamics

An algorithm to consistently couple a conservative semi-Lagrangian finite-volume transport scheme with a spectral element (SE) dynamical core is presented. The semi-Lagrangian finite-volume scheme is the Conservative Semi-Lagrangian Multitracer (CSLAM), and the SE dynamical core is the National Center for Atmospheric Research (NCAR)’s Community Atmosphere Model–Spectral Elements (CAM-SE). The primary motivation for coupling CSLAM with CAM-SE is to accelerate tracer transport for multitracer applications. The coupling algorithm result is an inherently mass-conservative, shape-preserving, and consistent (for a constant mixing ratio, the CSLAM solution reduces to the SE solution for air mass) transport that is efficient and accurate. This is achieved by first deriving formulas for diagnosing SE airmass flux through the CSLAM control volume faces. Thereafter, the upstream Lagrangian CSLAM areas are iteratively perturbed to match the diagnosed SE airmass flux, resulting in an equivalent upstream Lagrangian grid that spans the sphere without gaps or overlaps (without using an expensive search algorithm). This new CSLAM algorithm is not specific to airmass fluxes provided by CAM-SE but applies to any airmass fluxes that satisfy the Lipshitz criterion and for which the Courant number is less than one.

Lauritzen, Peter Hjort↗
DOE OSTI2023

Preserving Superconvergence of Spectral Elements for Curved Domains via h and p-Geometric Refinement [Slides]

Spectral element methods (SEM) are extensions of finite element methods (FEM) that employ Gauss-Lobatto or similar nodes instead of equidistant nodes for high-order elements. SEM can deliver superior accuracy compared to equidistant FEM due to potential superconvergence. However, significant challenges remain for domains with curved boundaries, which have limited the advantages of SEM for real-world applications. In this work, we propose a novel approach to bolster the overall accuracy and preserve the superconvergence of SEM over curved domains.

97 MATHEMATICS AND COMPUTING↗