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All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

Solution adaptivity using a triangular mesh

Solution adaptivity is discussed first from a general perspective and then from the specific viewpoint of triangular meshes. The use of a general connectivity triangular mesh is emphasized. The development of monitor surfaces and their geometric properties is discussed. Mesh point movement is addressed, as is the dynamic restructuring of the connectivity pattern among moving modes. Changes in the number of mesh nodes to obtain suitable refinement for a physical simulation is examined, and the use of locally regular structures to offset the data structure limitation of a general connectivity triangular mesh and thus to obtain an enhanced range of application is considered. To illustrate the basic features of the adaptive triangular mesh strategy, an application to the study of plasma equilibrium is briefly considered.

Eiseman, P. R.↗
NASA NTRS1989

Multigrid solution of the Navier-Stokes equations on triangular meshes

A Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite element Galerkin approximation, which can be shown to be equivalent to a finite volume approximation for regular equilateral triangular meshes. Integration steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. Directional scaling of the artificial dissipation and the implicit residual smoothing operator is achieved for unstructured meshes by considering local mesh stretching vectors at each point. The accuracy of the scheme for highly stretched triangular meshes is validated by comparing computed flat-plate laminar boundary layer results with the well known similarity solution, and by comparing laminar airfoil results with those obtained from various well-established structured quadrilateral-mesh codes. The convergence efficiency of the present method is also shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.

Mavriplis, Dimitri J.↗
NASA NTRS1990

Multigrid solution of the Navier-Stokes equations on triangular meshes

A Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite element Galerkin approximation, which can be shown to be equivalent to a finite volume approximation for regular equilateral triangular meshes. Integration steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. Directional scaling of the artificial dissipation and the implicit residual smoothing operator is achieved for unstructured meshes by considering local mesh stretching vectors at each point. The accuracy of the scheme for highly stretched triangular meshes is validated by comparing computed flat-plate laminar boundary layer results with the well known similarity solution, and by comparing laminar airfoil results with those obtained from various well-established structured quadrilateral-mesh codes. The convergence efficiency of the present method is also shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.

Mavriplis, D. J.↗
NASA NTRS2016

Addressing the Big-Earth-Data Variety Challenge with the Hierarchical Triangular Mesh

We have implemented an updated Hierarchical Triangular Mesh (HTM) as the basis for a unified data model and an indexing scheme for geoscience data to address the variety challenge of Big Earth Data. We observe that, in the absence of variety, the volume challenge of Big Data is relatively easily addressable with parallel processing. The more important challenge in achieving optimal value with a Big Data solution for Earth Science (ES) data analysis, however, is being able to achieve good scalability with variety. With HTM unifying at least the three popular data models, i.e. Grid, Swath, and Point, used by current ES data products, data preparation time for integrative analysis of diverse datasets can be drastically reduced and better variety scaling can be achieved. In addition, since HTM is also an indexing scheme, when it is used to index all ES datasets, data placement alignment (or co-location) on the shared nothing architecture, which most Big Data systems are based on, is guaranteed and better performance is ensured. Moreover, our updated HTM encoding turns most geospatial set operations into integer interval operations, gaining further performance advantages.

SciDB↗
DOE OSTI2024

Gyrokinetic electromagnetic particle simulations in triangular meshes with C1 finite elements

The triangular mesh-based gyrokinetic scheme enables comprehensive axis-to-edge studies across the entire plasma volume. Our approach employs triangular finite elements with first-derivative continuity (C1), building on previous work to facilitate gyrokinetic simulations. Additionally, we have adopted the mixed variable/pullback scheme for gyrokinetic electromagnetic particle simulations. The filter-free treatment in the poloidal cross-section with triangular meshes introduces unique features and challenges compared to previous treatments using structured meshes. Our implementation has been validated through benchmarks using ITPA-toroidicity-induced Alfvén eigenmode parameters, showing its capability in moderate to small electron skin depth regimes. Additional examinations using experimental parameters confirm its applicability to realistic plasma conditions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1989

Multigrid solution of the Navier-Stokes equations on triangular meshes

A new Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite-element Galerkin approximation, which can be shown to be equivalent to a finite-volume approximation for regular equilateral triangular meshes. Integration to steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. The convergence efficiency of the present method is shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.

Mavriplis, D. J.↗
NASA NTRS2003

Analysis and Improvement of Upwind and Centered Schemes on Quadrilateral and Triangular Meshes

Second-order accurate upwind and centered schemes are presented in a framework that facilitates their analysis and comparison. The upwind scheme employed consists of a reconstruction step (MUSCL approach) followed by an upwind step (Roe's flux-difference splitting). The two centered schemes are of Lax-Friedrichs (L-F) type. They are the nonstaggered versions of the Nessyahu-Tadmor (N-T) scheme and the CE/SE method with epilson = 1/2. The upwind scheme is extended to the case of two spatial dimensions (2D) in a straightforward manner. The N-T and CE/SE schemes are extended in a manner similar to the 2D extensions of the CE/SE scheme by Wang and Chang for a triangular mesh and by Zhang, Yu, and Chang for a quadrilateral mesh. The slope estimates, however, are simplified. Fourier stability and accuracy analyses are carried out for these schemes for the standard 1D and the 2D quadrilateral mesh cases. In the nonstandard case of a triangular mesh, the triangles must be paired up when analyzing the upwind and N-T schemes. An observation resulting in an extended N-T scheme which is faster and uses only one-third of the storage for flow data compared with the CE/SE method is presented. Numerical results are shown. Other improvements to the schemes are discussed.

Huynh, H. T.↗
NASA NTRS1983

Numerical simulations of fuel droplet flows using a Lagrangian triangular mesh

The incompressible, Lagrangian, triangular grid code, SPLISH, was converted for the study of flows in and around fuel droplets. This involved developing, testing and incorporating algorithms for surface tension and viscosity. The major features of the Lagrangian method and the algorithms are described. Benchmarks of the algorithms are given. Several calculations are presented for kerosene droplets in air. Finally, extensions which make the code compressible and three dimensional are discussed.

Fritts, M. J.↗
NASA NTRS2004

A Well-Balanced Central-Upwind Scheme for the 2D Shallow Water Equations on Triangular Meshes

We are interested in approximating solutions of the two-dimensional shallow water equations with a bottom topography on triangular meshes. We show that there is a certain flexibility in choosing the numerical fluxes in the design of semi-discrete Godunov-type central schemes. We take advantage of this fact to generate a new second-order, central-upwind method for the two-dimensional shallow water equations that is well-balanced. We demonstrate the accuracy of our method as well as its balance properties in a variety of examples.

Bryson, Steve↗
NASA NTRS1993

An upwind multigrid method for solving viscous flows on unstructured triangular meshes

A multigrid algorithm is combined with an upwind scheme for solving the two dimensional Reynolds averaged Navier-Stokes equations on triangular meshes resulting in an efficient, accurate code for solving complex flows around multiple bodies. The relaxation scheme uses a backward-Euler time difference and relaxes the resulting linear system using a red-black procedure. Roe's flux-splitting scheme is used to discretize convective and pressure terms, while a central difference is used for the diffusive terms. The multigrid scheme is demonstrated for several flows around single and multi-element airfoils, including inviscid, laminar, and turbulent flows. The results show an appreciable speed up of the scheme for inviscid and laminar flows, and dramatic increases in efficiency for turbulent cases, especially those on increasingly refined grids.

Bonhaus, Daryl Lawrence↗
NASA NTRS1989

Two-dimensional Euler computations on a triangular mesh using an upwind, finite-volume scheme

A numerical procedure was developed for the finite-volume solution of the Euler equations on unstructured triangular meshes based on a flux-difference split upwind method. Techniques for implementing Roe's (1985) approximate Reimann solver together with the preprocessing MUSCL differencing on unstructured grids are presented. Applications and comparisons with structured grid problems are carried out for a supersonic shock reflection problem, the supersonic flow over a blunt body, the transonic flow over NACA 0012 and RAE 2822 airfoils, and the flow about a double element Karman-Trefftz airfoil.

Whitaker, D. L.↗
NASA NTRS1984

Adaptive triangular mesh generation

A general adaptive grid algorithm is developed on triangular grids. The adaptivity is provided by a combination of node addition, dynamic node connectivity and a simple node movement strategy. While the local restructuring process and the node addition mechanism take place in the physical plane, the nodes are displaced on a monitor surface, constructed from the salient features of the physical problem. An approximation to mean curvature detects changes in the direction of the monitor surface, and provides the pulling force on the nodes. Solutions to the axisymmetric Grad-Shafranov equation demonstrate the capturing, by triangles, of the plasma-vacuum interface in a free-boundary equilibrium configuration.

Erlebacher, G.↗
NASA NTRS1990

Galerkin finite difference Laplacian operators on isolated unstructured triangular meshes by linear combinations

The Galerkin weighted residual technique using linear triangular weight functions is employed to develop finite difference formulae in Cartesian coordinates for the Laplacian operator on isolated unstructured triangular grids. The weighted residual coefficients associated with the weak formulation of the Laplacian operator along with linear combinations of the residual equations are used to develop the algorithm. The algorithm was tested for a wide variety of unstructured meshes and found to give satisfactory results.

Baumeister, Kenneth J.↗
NASA NTRS1995

Adaptive unstructured triangular mesh generation and flow solvers for the Navier-Stokes equations at high Reynolds number

A method for generating high quality unstructured triangular grids for high Reynolds number Navier-Stokes calculations about complex geometries is described. Careful attention is paid in the mesh generation process to resolving efficiently the disparate length scales which arise in these flows. First the surface mesh is constructed in a way which ensures that the geometry is faithfully represented. The volume mesh generation then proceeds in two phases thus allowing the viscous and inviscid regions of the flow to be meshed optimally. A solution-adaptive remeshing procedure which allows the mesh to adapt itself to flow features is also described. The procedure for tracking wakes and refinement criteria appropriate for shock detection are described. Although at present it has only been implemented in two dimensions, the grid generation process has been designed with the extension to three dimensions in mind. An implicit, higher-order, upwind method is also presented for computing compressible turbulent flows on these meshes. Two recently developed one-equation turbulence models have been implemented to simulate the effects of the fluid turbulence. Results for flow about a RAE 2822 airfoil and a Douglas three-element airfoil are presented which clearly show the improved resolution obtainable.

Ashford, Gregory A.↗
NASA NTRS1985

Finite difference operators on unstructured triangular meshes

A new form of the Laplace operator is derived which is shown to be second-order accurate when calculated at mesh triangle nodes if the cells corresponding to the nodes have a high degree of symmetry. It is demonstrated that the Laplace operator directly derived from Stokes' integral theorem is only zeroth-order accurate on these same cells. The construction of gradient and Laplace operators on unstructured triangular grids are discussed from the variational point of view. Two discrete representations of the Laplacian are compared to each other, and conclusions are drawn regarding their accuracy on a pointwise basis. Finally, geometric relations valid on arbitrary polygons are given for completeness in an appendix.

Erlebacher, G.↗
DOE OSTI2024

T RI M E ++: Multi-threaded triangular meshing in two dimensions

We present T RI M E ++, a multi-threaded software library designed for generating two-dimensional meshes for intricate geometric shapes using the Delaunay triangulation. Multi-threaded parallel computing is implemented throughout the meshing procedure, making it suitable for fast generation of large-scale meshes. Three iterative meshing algorithms are implemented: the DistMesh algorithm, the centroidal Voronoi diagram meshing, and a hybrid of the two. We compare the performance of the three meshing methods in T RI M E ++, and show that the hybrid method retains the advantages of the other two. The software library achieves significant parallel speedup when generating large-scale meshes containing between 10 4 to 10 7 points. T RI M E ++ can handle complicated geometries and generates adaptive meshes of high quality.

97 MATHEMATICS AND COMPUTING↗