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Results for “Discontinuous finite element method”

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 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

A Verification Suite of Test Cases for the Barotropic Solver of Ocean Models

Abstract The development of any atmosphere or ocean model warrants a suite of test cases (TCs) to verify its spatial and temporal discretizations, order of accuracy, stability, reproducibility, portability, scalability, etc. In this paper, we present a suite of shallow water TCs designed to verify the barotropic solver of atmosphere and ocean models. These include the non‐dispersive coastal Kelvin wave; the dispersive inertia‐gravity wave; the dispersive planetary and topographic Rossby waves; the barotropic tide; and a non‐linear manufactured solution. These TCs check the implementation of the linear pressure gradient term; the linear constant or variable‐coefficient Coriolis and bathymetry terms; and the non‐linear advection terms. Simulation results are presented for a variety of time‐stepping methods as well as two spatial discretizations: a mimetic finite volume method based on the TRiSK scheme, and a high‐order discontinuous Galerkin spectral element method. The experimental procedure for conducting these numerical experiments is detailed. It underscores several key considerations that vary depending on the chosen spatial discretization method. Finally, convergence studies of every TC are conducted with refinement in both space and time, only in space, and only in time. The convergence slopes match the expected theoretical predictions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

A peridynamic failure analysis of fiber-reinforced composite laminates using finite element discontinuous Galerkin approximations

Here, this paper presents a discontinuous Galerkin weak form for bond-based peridynamic models to predict the damage of fiber-reinforced composite laminates. To represent the anisotropy of a laminate in a peridynamic model, a lamina is simplified as a transversely isotropic medium under a plane stress condition. The laminated structure is modeled by stacking the surface mesh layers along the thickness direction according to the laminate sequence. To avoid a mesh dependence on either the fiber orientation or the discretization, the spherical harmonic expansion theory is employed to construct a function for the micro-elastic modulus in terms of the bond-fiber angle. The laminate material is decomposed into an isotropic matrix material part and a transversely isotropic material part. The bond stiffness can be evaluated using the engineering material constants, based on the equivalence between the elastic energy density in the peridynamic theory and the elastic energy density in the classic continuum mechanics theory. Benchmark tests are conducted to verify the proposed model. Numerical results illustrate that the convergence of simulations with different horizon sizes and meshes can be achieved. In terms of damage analysis, the proposed model can capture the dynamic process of the complex coupling of the inner-layer and delamination damage modes.

42 ENGINEERING↗
DOE OSTI2023

Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

Here, we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in H 1 , H(curl), and H(div) and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for Nédélec and Raviart–Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any “black-box” preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implementation in the finite element library MFEM. A range of challenging three-dimensional problems are used to corroborate the theoretical properties and demonstrate the flexibility and scalability of the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Performance and Reliability of Bonded Interfaces for High-temperature Packaging: Annual Progress Report

As maximum device temperatures approach 200 °Celsius, continuous operation, sintered silver materials promise to maintain bonds at these high temperatures without excessive degradation rates. A detailed characterization of the thermal performance and reliability of sintered silver materials and processes has been initiated for the next year. Future steps in crack modeling include efforts to simulate crack propagation directly using the extended finite element method (X-FEM), a numerical technique that uses the partition of unity method for modeling discontinuities such as cracks in a system.

30 DIRECT ENERGY CONVERSION↗
DOE OSTI2017

Implementation of CMFD Acceleration Scheme in PROTEUS-MOC

The PROTEUS-MOC code is a three-dimensional (3D) transport code, which is being developed at Argonne National Laboratory under the DOE-NE's Nuclear Energy Advanced Modeling and Simulation (NEAMS) program. It provides a fully consistent 3D discretization of the transport equation by combining the two-dimensional (2D) method of characteristics (MOC) for the planar direction with the discontinuous Galerkin method for the treatment of the axial variable. PROTEUS-MOC uses the unstructured finite element method (FEM) and thus allows the explicit representation of any extruded geometries. It also has a capability of massively parallel computation in energy group, angle, plane, and space using a top-level GMRES linear algebra solver. For parallel computation in space, it uses an unstructured domain decomposition by adapting METIS. The computational burden for direct heterogeneous core calculations based on unstructured geometries is excessive for practical reactor problems, and hence it is essential to implement an efficient acceleration scheme in PROTEUS-MOC. Recently, the coarse mesh finite difference (CMFD) method, which was originally applied to the nodal diffusion methods, has been successfully applied to the MOC transport solvers, resulting in a remarkable reduction in the number of fission source iterations. In addition, a modified CMFD method has been developed to overcome the limitation of the conventional CMFD that is applicable only to neutronics solvers with a conservative discretization. This CMFD method was derived to be consistent with the FEM, which is based on a nonconservative discretization. In this method, the neutron balance in each coarse mesh is enforced by introducing a pseudo absorption cross section (PAXS), and the well-established alternating solution process of CMFD and transport calculations is employed to accelerate source convergence. This modified CMFD acceleration scheme was adopted in PROTEUS-MOC to accelerate the fission source iteration. Pin-cell and assembly based CMFD schemes have been implemented by defining individual pin cells or assemblies as a coarse mesh. For this, an algorithm to map the finite elements to coarse meshes was developed, and the routines were developed to determine the coupling coefficients and PAXS of the CMFD equations from the transport solution. Preliminary performance tests were carried out using the C5G7 benchmark problem. This paper presents the main features of this CMFD acceleration scheme and the preliminary test results.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS2004

Experiences on p-Version Time-Discontinuous Galerkin's Method for Nonlinear Heat Transfer Analysis and Sensitivity Analysis

The focus of this research is on the development of analysis and sensitivity analysis equations for nonlinear, transient heat transfer problems modeled by p-version, time discontinuous finite element approximation. The resulting matrix equation of the state equation is simply in the form ofA(x)x = c, representing a single step, time marching scheme. The Newton-Raphson's method is used to solve the nonlinear equation. Examples are first provided to demonstrate the accuracy characteristics of the resultant finite element approximation. A direct differentiation approach is then used to compute the thermal sensitivities of a nonlinear heat transfer problem. The report shows that only minimal coding effort is required to enhance the analysis code with the sensitivity analysis capability.

Hou, Gene↗
DOE OSTI2017

Level set methods for detonation shock dynamics using high-order finite elements

Level set methods are a popular approach to modeling evolving interfaces. We present a level set ad- vection solver in two and three dimensions using the discontinuous Galerkin method with high-order nite elements. During evolution, the level set function is reinitialized to a signed distance function to maintain ac- curacy. Our approach leads to stable front propagation and convergence on high-order, curved, unstructured meshes. The ability of the solver to implicitly track moving fronts lends itself to a number of applications; in particular, we highlight applications to high-explosive (HE) burn and detonation shock dynamics (DSD). We provide results for two- and three-dimensional benchmark problems as well as applications to DSD.

finite element methods↗
DOE OSTI2020

Fourier Analyses of High-Order Continuous and Discontinuous Galerkin Methods

In this paper, we present a Fourier analysis of wave propagation problems subject to a class of continuous and discontinuous discretizations using high-degree Lagrange polynomials. This allows us to obtain explicit analytical formulas for the dispersion relation and group velocity and, for the first time to our knowledge, characterize analytically the emergence of gaps in the dispersion relation at specific wavenumbers, when they exist, and compute their specific locations. Wave packets with energy at these wavenumbers will fail to propagate correctly, leading to significant numerical dispersion. We also show that the Fourier analysis generates mathematical artifacts, and we explain how to remove them through a branch selection procedure conducted by analysis of eigenvectors and associated reconstructed solutions. The higher frequency eigenmodes, named erratic in this study, are also investigated analytically and numerically.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A finite element method for angular discretization of the radiation transport equation on spherical geodesic grids

Discrete ordinate (S N ) and filtered spherical harmonics (FP N ) based schemes have been proven to be robust and accurate in solving the Boltzmann transport equation but they have their own strengths and weaknesses in different physical scenarios. We present a new method based on a finite element approach in angle that combines the strengths of both methods and mitigates their disadvantages. The angular variables are specified on a spherical geodesic grid with functions on the sphere being represented using a finite element basis. A positivity-preserving limiting strategy is employed to prevent non-physical values from appearing in the solutions. Here, the resulting method is then compared with both S N and FP N schemes using four test problems and is found to perform well when one of the other methods fail.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2002

Finite Element A Posteriori Error Estimation for Heat Conduction

This research investigates residual-based a posteriori error estimates for finite element approximations of heat conduction in single-layer and multi-layered materials. The finite element approximation, based upon hierarchical modelling combined with p-version finite elements, is described with specific application to a two-dimensional, steady state, heat-conduction problem. Element error indicators are determined by solving an element equation for the error with the element residual as a source, and a global error estimate in the energy norm is computed by collecting the element contributions. Numerical results of the performance of the error estimate are presented by comparisons to the actual error. Two methods are discussed and compared for approximating the element boundary flux. The equilibrated flux method provides more accurate results for estimating the error than the average flux method. The error estimation is applied to multi-layered materials with a modification to the equilibrated flux method to approximate the discontinuous flux along a boundary at the material interfaces. A directional error indicator is developed which distinguishes between the hierarchical modeling error and the finite element error. Numerical results are presented for single-layered materials which show that the directional indicators accurately determine which contribution to the total error dominates.

Lang, Christapher G.↗
DOE OSTI2022

Subcell limiting strategies for discontinuous Galerkin spectral element methods

Here, we present a general family of subcell limiting strategies to construct robust high-order accurate nodal discontinuous Galerkin (DG) schemes. The main strategy is to construct compatible low order finite volume (FV) type discretizations that allow for convex blending with the high-order variant with the goal of guaranteeing additional properties, such as bounds on physical quantities and/or guaranteed entropy dissipation. For an implementation of this main strategy, four main ingredients are identified that may be combined in a flexible manner: (i) a nodal high-order DG method on Legendre–Gauss–Lobatto nodes, (ii) a compatible robust subcell FV scheme, (iii) a convex combination strategy for the two schemes, which can be element-wise or subcell-wise, and (iv) a strategy to compute the convex blending factors, which can be either based on heuristic troubled-cell indicators, or using ideas from flux-corrected transport methods. By carefully designing the metric terms of the subcell FV method, the resulting methods can be used on unstructured curvilinear meshes, are locally conservative, can handle strong shocks efficiently while directly guaranteeing physical bounds on quantities such as density, pressure or entropy. We further show that it is possible to choose the four ingredients to recover existing methods such as a provably entropy dissipative subcell shock-capturing approach or a sparse invariant domain preserving approach. We test the versatility of the presented strategies and mix and match the four ingredients to solve challenging simulation setups, such as the KPP problem (a hyperbolic conservation law with non-convex flux function), turbulent and hypersonic Euler simulations, and MHD problems featuring shocks and turbulence.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A higher-order finite element reactive transport model for unstructured and fractured grids

Abstract This work presents a new reactive transport framework that combines a powerful geochemistry engine with advanced numerical methods for flow and transport in subsurface fractured porous media. Specifically, the PhreeqcRM interface (developed by the USGS) is used to take advantage of a large library of equilibrium and kinetic aqueous and fluid-rock reactions, which has been validated by numerous experiments and benchmark studies. Fluid flow is modeled by the Mixed Hybrid Finite Element (FE) method, which provides smooth velocity fields even in highly heterogenous formations with discrete fractures. A multilinear Discontinuous Galerkin FE method is used to solve the multicomponent transport problem. This method is locally mass conserving and its second order convergence significantly reduces numerical dispersion. In terms of thermodynamics, the aqueous phase is considered as a compressible fluid and its properties are derived from a Cubic Plus Association (CPA) equation of state. The new simulator is validated against several benchmark problems (involving, e.g., Fickian and Nernst-Planck diffusion, isotope fractionation, advection-dispersion transport, and rock-fluid reactions) before demonstrating the expanded capabilities offered by the underlying FE foundation, such as high computational efficiency, parallelizability, low numerical dispersion, unstructured 3D gridding, and discrete fraction modeling.

58 GEOSCIENCES↗
NASA NTRS1989

Finite element formulations for compressible flows

Researchers started their studies on the development and application of computational methods for compressible flows. Particular attention was given to proper numerical treatment of sharp layers occurring in such problems and to general mesh generation capabilities for intricate computational geometries. Mainly finite element methods enhanced with several state-of-the art techniques (such as the streamline-upwind/Petrov-Galerkin, discontinuity capturing, adaptive implicit-explicit, and trouped element-by-element approximate factorization schemes) were employed.

Tezduyar, Tayfun E.↗
NASA NTRS1989

A new family of stable elements for the Stokes problem based on a mixed Galerkin/least-squares finite element formulation

Adding to the classical Hellinger-Reissner formulation, a residual form of the equilibrium equation, a new Galerkin/least-squares finite element method is derived. It fits within the framework of a mixed finite element method and is stable for rather general combinations of stress and velocity interpolations, including equal-order discontinuous stress and continuous velocity interpolations which are unstable within the Galerkin approach. Error estimates are presented based on a generalization of the Babuska-Brezzi theory. Numerical results (not presented herein) have confirmed these estimates as well as the good accuracy and stability of the method.

Franca, Leopoldo P.↗
DOE OSTI2020

Neutronics Enhanced Meshing Operations (NEMO) Users Manual V.1.0

The is a Python framework developed at Idaho National Laboratory that uses the Cubit Python to build Cubit models and export them in EXODUS II format. These EXODUS mesh files can be used by any -based application, but NEMO is specialized for coupled neutronics calculations. Therefore, the primary purpose of NEMO is to prepare mesh files for reactor geometries and generate a homogenization overlay file for the Serpent Monte Carlo code. This homogenization overlay file ensures the consistency of the mesh with the generation of homogenized cross sections and reference fluxes for the Superhomogenization method. NEMO also supports the preparation of mesh files for thermal and thermo-mechanical models, particularly the discontinuous finite element gap heat transfer model in Griffin. This manual introduces the user to the basic capabilities in NEMO for prismatic geometries.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS2018

A Higher-Order Space-Time Finite-Element Method for Moving-Body and Fluid-Structure Interaction Problems

We present a high-order finite-element method for moving body and fluid/structure interaction problems. Our solution strategy is based on a space-time discontinuous Galerkin (DG) spectral-element discretization which extends to arbitrary order of accuracy. The space-time DG discretization is a natural choice for moving body and fluid-structure interaction problems as moving surfaces are incorporated simply by considering curved space-time elements whose space-time faces align with the moving body. We present a discontinuous-Galerkin in time discretization for six-degree of motion modeling of rigid bodies, and a continuous-Galerkin discretization for equations of linear elasticity to generate curved space-time meshes. Numerical results for several simple 2D test cases are presented in order to verify the implementation of the different models. Finally we present a preliminary dynamic simulation of a parachute.

High-Order↗
DOE OSTI2019

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Here, we propose an extension of the embedded boundary method known as “shifted boundary method” to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded.

97 MATHEMATICS AND COMPUTING↗