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Results for “Polygonal finite elements”

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 OSTI2018

Finite deformation cohesive polygonal finite elements for modeling pervasive fracture

In this work, we introduce a framework for modeling dynamic fracture problems using cohesive polygonal finite elements. Random polygonal meshes provide a robust, efficient method for generating an unbiased network of fracture surfaces. Further, these meshes have more facets per element than standard triangle or quadrilateral meshes, providing more possible facets per element to insert cohesive surfaces. This property of polygonal meshes is advantageous for the modeling of pervasive fracture. We use both Wachspress and maximum entropy shape functions to form a finite element basis over the polygons. Fracture surfaces are captured through dynamically inserted cohesive zone elements at facets between the polygons in the mesh. Contact is enforced through a penalty method that is applied to both closed cohesive surfaces and general interpenetration of two polygonal elements. Finally, several numerical examples are presented that illustrate the capabilities of the method and demonstrate convergence of solutions.

97 MATHEMATICS AND COMPUTING↗
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↗
DOE OSTI2018

PolyMat: an efficient Matlab code for multi-material topology optimization

We introduce a Matlab implementation of topology optimization for compliance minimization on unstructured polygonal finite element meshes that efficiently accommodates many materials and many volume constraints. Leveraging the modular structure of the educational code, PolyTop, we extend it to the multi-material version, PolyMat, with only a few modifications. First, a design variable for each candidate material is defined in each finite element. Next, we couple a Discrete Material Optimization interpolation with the existing penalization and introduce a new parameter such that we can employ continuation and smoothly transition from a convex problem without any penalization to a non-convex problem in which material mixing and intermediate densities are penalized. Mixing that remains due to the density filter operation is eliminated via continuation on the filter radius. To accommodate flexibility in the volume constraint definition, the constraint function is modified to compute multiple volume constraints and the design variable update is modified in accordance with the Zhang-Paulino-Ramos Jr. (ZPR) update scheme, which updates the design variables associated with each constraint independently. The formulation allows for volume constraints controlling any subset of the design variables, i.e., they can be defined globally or locally for any subset of the candidate materials. Borrowing ideas for mesh generation on complex domains from PolyMesher, we conclude which design variables are associated with each local constraint of arbitrary geometry. A number of examples are presented to demonstrate the many material capability, the flexibility of the volume constraint definition, the ease with which we can accommodate passive regions, and how we may use local constraints to break symmetries or achieve graded geometries.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method (FEM) for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular and square meshes where the method coincides with the P 1 - P 0 Scott-Vogelius scheme, which is well-known to be unstable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. Finally, we numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the $\mathbb{P}_1$ – $\mathbb{P}_0$ Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Improving climate model coupling through a complete mesh representation: a case study with E3SM (v1) and MOAB (v5.x)

One of the fundamental factors contributing to the spatiotemporal inaccuracy in climate modeling is the mapping of solution field data between different discretizations and numerical grids used in the coupled component models. The typical climate computational workflow involves evaluation and serialization of the remapping weights during the preprocessing step, which is then consumed by the coupled driver infrastructure during simulation to compute field projections. Tools like Earth System Modeling Framework (ESMF) and TempestRemap offer capability to generate conservative remapping weights, while the Model Coupling Toolkit (MCT) that is utilized in many production climate models exposes functionality to make use of the operators to solve the coupled problem. However, such multistep processes present several hurdles in terms of the scientific workflow and impede research productivity. In order to overcome these limitations, we present a fully integrated infrastructure based on the Mesh Oriented datABase (MOAB) library, which allows for a complete description of the numerical grids and solution data used in each submodel. Through a scalable advancing-front intersection algorithm, the supermesh of the source and target grids are computed, which is then used to assemble the high-order, conservative, and monotonicity-preserving remapping weights between discretization specifications. The Fortran-compatible interfaces in MOAB are utilized to directly link the submodels in the Energy Exascale Earth System Model (E3SM) to enable online remapping strategies in order to simplify the coupled workflow process. We demonstrate the superior computational efficiency of the remapping algorithms in comparison with other state-of-the-science tools and present strong scaling results on large-scale machines for computing remapping weights between the spectral element atmosphere and finite volume discretizations on the polygonal ocean grids.

58 GEOSCIENCES↗
DOE OSTI2019

Development of a P 2 element with optimal L 2 convergence for biharmonic equation

It is well known that convergence rate of finite element approximation is suboptimal in the L 2 norm for solving biharmonic equations when P 2 or Q 2 element is used. The goal of this paper is to derive a weak Galerkin (WG) P 2 element with the L 2 optimal convergence rate by assuming the exact solution sufficiently smooth. In addition, our new WG finite element method can be applied to general mesh such as hybrid mesh, polygonal mesh or mesh with hanging node. The numerical experiments have been conducted on different meshes including hybrid meshes with mixed of pentagon and rectangle and mixed of hexagon and triangle.

Mu, Lin↗
DOE OSTI2018

Tetrahedral Mesh Adaptation for Lagrangian Shock Hydrodynamics

Lagrangian shock hydrodynamics simulations will fail to proceed past a certain time if the mesh is approaching tangling. A common solution is an Arbitrary Lagrangian Eulerian (ALE) form, in which the mesh is improved (remeshing) and the solution is remapped onto the improved mesh. The simplest remeshing techniques involve moving only the nodes of the mesh. More advanced remeshing techniques involve altering the mesh connectivity in portions of the domain in order to prevent tangling. Work has been done using Voronoi-based polygonal mesh generators and 2D quad/triangle mesh adaptation. Here, this paper presents the use of tetrahedral mesh adaptation methods as the remeshing step in an otherwise Lagrangian finite element shock hydrodynamics code called Alexa.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Memory-Aware External Facelist Calculation: A Data-Parallel Atomic Hash Counting Approach

Unstructured volumetric meshes serve as fundamental data representations in various scientific simulations and analyses. They play a crucial role in representing complex computational domains and are essential for important numerical techniques, such as finite element analysis. Whenever such a mesh is read from a file, streamed in-situ, or generated by algorithms, scientific visualization libraries rely on calculating the external surface of a geometry, named “external facelist”, to produce a polygonal mesh for rendering. Consequently, external facelist calculation has become one of the most widely used algorithms in the scientific visualization domain, necessitating optimal performance. In this paper, we explore relevant work on external facelist calculation algorithms in two common visualization libraries, VTK and Viskores, assess their performance and memory constraints, and introduce a novel memory-aware external facelist calculation algorithm employing an atomic hash counting approach. This algorithm fully leverages Viskores' data-parallel primitive operations, facilitating its execution across diverse many-core architectures. Our algorithm features the lowest memory footprint on the GPU and the second-lowest on the CPU among all evaluated methods, and it also delivers the fastest performance on both CPU and GPU. It has been made available under an open-source license in the VTK and Viskores visualization systems.

Tsalikis, Spiros [Kitware] (ORCID:0000000151137195↗
DOE OSTI2021

MOOSE Framework Meshing Enhancements to Support Reactor Analysis

MOOSE-based physics codes require an input finite element mesh on which the physics solution is calculated, reported, and transferred to other physics codes. The use of difficult-touse, external licensed software is often required to generate high quality meshes for reactor geometries. High-fidelity geometry modeling also requires elaborate tracking of groups of elements for material property assignment and output reporting which can be considerably complex for the user to identify and maintain. Under the U.S. Department of Energy Office of Nuclear Energy Advanced Modeling and Simulation (NEAMS) program, several meshingrelated enhancements have been developed for the MOOSE framework to address user challenges in creating finite element meshes for advanced reactor geometries. MOOSE mesh generators have been developed to mesh hexagonal geometries (pins, ducted assemblies, and cores) commonly found in liquid-metal cooled fast reactor concepts. The mesh generator used for hexagonal pin cells is generic for regular polygons and therefore may also be used for Cartesian pin cells. Hexagonal pin cells can be stitched into ducted assemblies, and assemblies can be stitched together into a core. The user may specify region ids, region names, and other preferences on the mesh. This control is useful for later material mapping in the MOOSE-based physics codes input. A capability was also developed for meshing rotating control drums including determination of material volume fractions in each mesh element as a function of time. Control drum meshes may be stitched to other hexagonal assemblies to create a core configuration. Additional mesh generators were developed that wrap around the hexagonal meshing capabilities and utilize “extra element integer” ID values on each element. In regular Cartesian or hexagonal assemblies or cores, the bookkeeping of element groups for both material assignment and output reporting can now be automated through assignment of pin, assembly, core, axial and depletion id values stored as extra element integers. The extra element tags on the mesh greatly speed the reactor analyst’s efforts to map materials to meshes, track depletion zones, and parse output such as axial pin power distributions. At the highest level, pin, assembly, and core mesh generators (with this reactor terminology) have also been developed to easily generate regular Cartesian and hexagonal cores, including axial extrusion. These reactor geometry builders call upon the previously mentioned capabilities to produce analysis-ready 3D meshes including material assignments. Open source mesh triangulation capabilities were also investigated for integration into the MOOSE framework to address the need for meshing the core periphery region which extends from the irregular outer assembly border to a cylindrical boundary. Options are limited due to licensing constraints, and the recommendation is pursue building a native MOOSE Delaunay triangulator routine with full functionality. Finally, a series of verification problems were performed with NEAMS physics tools. All developed capabilities will be available in the new open-source “Reactor” module of the MOOSE framework, which is accessible to any MOOSE-based NEAMS physics tool.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2018

Reducing spurious mesh motion in Lagrangian finite volume and discontinuous Galerkin hydrodynamic methods

The Lagrangian finite volume (FV) cell-centered hydrodynamic (CCH) method and the Lagrangian discontinuous Galerkin (DG) CCH method have been demonstrated to be quite stable and capable of producing very accurate solutions on many mesh topologies. However, some challenges can arise with higher-order elements and polygonal elements that have many deformational degrees of freedom. With these types of meshes, elements can deform in unphysical ways and the mesh can tangle. In this study, we present methods for obtaining more robust Lagrangian solutions on polygonal and higher-order elements. The robustness is achieved by (1) incorporating a new iterative method that modifies the velocity reconstructions in the corners of the elements, and (2) a new multidirectional approximate Riemann solver that, when coupled with the iterative method, reduces spurious mesh motion. Lastly, the details of the numerical methods are discussed and their utility is demonstrated on a diverse suite of test problems using higher-order and polygonal elements.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Design, analysis and numerical experiments for the virtual element p and hp approximations of elliptic eigenvalue problems

We discuss the p- and the hp-versions of the virtual element method for the approximation of the eigenvalues of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schr¨odinger equation with a pseudo-potential term. We present in details the analysis of the p-version of the method, proving the exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing hp-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the hp-spaces.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

A higher order approximate static condensation method for multi-material diffusion problems

The paper studies an approximate static condensation method for the diffusion problem with discontinuous diffusion coefficients. The method allows for a general polygonal mesh which is unfitted to the material interfaces. Moreover, the interfaces can be discontinuous across the mesh edges as typical for numerical reconstructions using the volume or moment-of-fluid methods. We apply a mimetic finite difference method to solve local diffusion problems and use P 1 (mortar) edge elements to couple local problems into the global system. The condensation process and the properties of the resulting algebraic system are discussed. It is demonstrated that the method is second order accurate on smooth solutions and performs well for problems with high contrast in diffusion coefficients. Experiments also show the robustness with respect to position of the material interfaces against the underlying mesh.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A Low Order Mimetic Finite Difference Method for Resistive Magnetohydrodynamics in 2D

We developed a low order Mimetic Finite Difference (MFD) discretization for the equations of magnetohydrodynamics (MHD) in two space dimensions. These equations describe the evolution of the electric and magnetic fields in the presence of prescribed velocity field.The method is designed to work on general polygonal meshes and preserves the divergence-free condition on the magnetic field. The electric field is discretized at the vertices/nodes and the magnetic field uses edge-based discretization. The method reconstructs the magnetic field to extract nodal values necessary to approximate some terms present in Ohm’s law. We test the robustness of our numerical scheme on three different types of meshes: with triangular elements, quadrilateral and unstructured polyhedrons obtain from a Voronoi tesselation. Analysis of the convergence for each of the aforementioned mesh types is presented. We finish with a test problem that shows the method is capable of modelling magnetic reconnection.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Coupling methods between finite element-based Boussinesq-type wave and particle-based free-surface flow models [Coupling methods between finite element-based Boussinesq-typewave and particle-based free-surface flow models]

We develop one-way coupling methods between a Boussinesq-type wave model based on the discontinuous Galerkin finite element method and a free-surface flow model based on a mesh-free particle method to strike a balance between accuracy and computational cost. In our proposed model, computation of the wave model in the global domain is conducted first, and the nonconstant velocity profiles in the vertical direction are reproduced by using its results. Computation of the free-surface flow is performed in a local domain included within the global domain with interface boundaries that move along the reproduced velocity field in a Lagrangian fashion. To represent the moving interfaces, we used a polygon wall boundary model for mesh-free particle methods. Verification and validation tests of our proposed model are performed, and results obtained by the model are compared with theoretical values and experimental results to show its accuracy and applicability.

97 MATHEMATICS AND COMPUTING↗