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All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given.

Eiseman, Peter R.↗
NASA NTRS1989

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given.

Eiseman, Peter R.↗
NASA NTRSDate not supplied

Extending an Economical Third-Order Inviscid Nodal-Gradient Cell-Centered Finite-Volume Method to Mixed-Element Grids

In this paper, we extend an economical third-order nodal-gradient cell-centered finite-volume method, originally developed for tetrahedral grids, to mixed-element grids. It is shown that the efficient quadratic interpolation formula essential to eliminating second derivatives from a third-order accurate discretization can be easily extended to an arbitrary cell type. For the surface flux integration, we consider a split-face approach, where a quadrilateral face is split into two triangles, and the third-order method for tetrahedra is directly applied. Also, we derive a second-derivative-free, high-order, volume quadrature formula for an arbitrary cell. Numerical results are presented for accuracy verification and applications with three-dimensional nontetrahedral grids.

Computational Fluid Dynamics↗
NASA NTRS1996

Research in Parallel Algorithms and Software for Computational Aerosciences

Phase 1 is complete for the development of a computational fluid dynamics CFD) parallel code with automatic grid generation and adaptation for the Euler analysis of flow over complex geometries. SPLITFLOW, an unstructured Cartesian grid code developed at Lockheed Martin Tactical Aircraft Systems, has been modified for a distributed memory/massively parallel computing environment. The parallel code is operational on an SGI network, Cray J90 and C90 vector machines, SGI Power Challenge, and Cray T3D and IBM SP2 massively parallel machines. Parallel Virtual Machine (PVM) is the message passing protocol for portability to various architectures. A domain decomposition technique was developed which enforces dynamic load balancing to improve solution speed and memory requirements. A host/node algorithm distributes the tasks. The solver parallelizes very well, and scales with the number of processors. Partially parallelized and non-parallelized tasks consume most of the wall clock time in a very fine grain environment. Timing comparisons on a Cray C90 demonstrate that Parallel SPLITFLOW runs 2.4 times faster on 8 processors than its non-parallel counterpart autotasked over 8 processors.

Domel, Neal D.↗
NASA NTRS1996

Research in Parallel Algorithms and Software for Computational Aerosciences

Phase I is complete for the development of a Computational Fluid Dynamics parallel code with automatic grid generation and adaptation for the Euler analysis of flow over complex geometries. SPLITFLOW, an unstructured Cartesian grid code developed at Lockheed Martin Tactical Aircraft Systems, has been modified for a distributed memory/massively parallel computing environment. The parallel code is operational on an SGI network, Cray J90 and C90 vector machines, SGI Power Challenge, and Cray T3D and IBM SP2 massively parallel machines. Parallel Virtual Machine (PVM) is the message passing protocol for portability to various architectures. A domain decomposition technique was developed which enforces dynamic load balancing to improve solution speed and memory requirements. A host/node algorithm distributes the tasks. The solver parallelizes very well, and scales with the number of processors. Partially parallelized and non-parallelized tasks consume most of the wall clock time in a very fine grain environment. Timing comparisons on a Cray C90 demonstrate that Parallel SPLITFLOW runs 2.4 times faster on 8 processors than its non-parallel counterpart autotasked over 8 processors.

Domel, Neal D.↗
NASA NTRS1996

DSMC Grid Methodologies for Computing Low-Density, Hypersonic Flows About Reusable Launch Vehicles

Two different grid methodologies are studied for application to DSMC simulations about reusable launch vehicles. One method uses an unstructured, tetrahedral grid while the other uses a structured, variable-resolution Cartesian grid. The relative merits of each method are discussed in terms of accuracy, computational efficiency, and overall ease of use. Both methods are applied to the computation of a low-density, hypersonic flow about a winged single-stage-to-orbit reusable launch vehicle concept at conditions corresponding to an altitude of 120 km. Both methods are shown to give comparable results for both surface and flowfield quantities as well as for the overall aerodynamic behavior. For the conditions simulated, the flowfield about the vehicle is very rarefied but the DSMC simulations show significant departure from free-molecular predictions for the surface friction and heat transfer as well as certain aerodynamic quantities.

Wilmoth, Richard G.↗
NASA NTRS2002

The Overgrid Interface for Computational Simulations on Overset Grids

Computational simulations using overset grids typically involve multiple steps and a variety of software modules. A graphical interface called OVERGRID has been specially designed for such purposes. Data required and created by the different steps include geometry, grids, domain connectivity information and flow solver input parameters. The interface provides a unified environment for the visualization, processing, generation and diagnosis of such data. General modules are available for the manipulation of structured grids and unstructured surface triangulations. Modules more specific for the overset approach include surface curve generators, hyperbolic and algebraic surface grid generators, a hyperbolic volume grid generator, Cartesian box grid generators, and domain connectivity: pre-processing tools. An interface provides automatic selection and viewing of flow solver boundary conditions, and various other flow solver inputs. For problems involving multiple components in relative motion, a module is available to build the component/grid relationships and to prescribe and animate the dynamics of the different components.

Chan, William M.↗
NASA NTRS2014

Aerostructural Level Set Topology Optimization for a Common Research Model Wing

The purpose of this work is to use level set topology optimization to improve the design of a representative wing box structure for the NASA common research model. The objective is to minimize the total compliance of the structure under aerodynamic and body force loading, where the aerodynamic loading is coupled to the structural deformation. A taxi bump case was also considered, where only body force loads were applied. The trim condition that aerodynamic lift must balance the total weight of the aircraft is enforced by allowing the root angle of attack to change. The level set optimization method is implemented on an unstructured three-dimensional grid, so that the method can optimize a wing box with arbitrary geometry. Fast matching and upwind schemes are developed for an unstructured grid, which make the level set method robust and efficient. The adjoint method is used to obtain the coupled shape sensitivities required to perform aerostructural optimization of the wing box structure.

Dunning, Peter D.↗
NASA NTRSDate not supplied

Comparison of Two Approaches to Constructing Second- and Third-Order Nodal-Gradient Cell-Centered Finite-Volume Methods for Mixed-Element Grids

In this paper, we introduce a third-order nodal-gradient cell-centered finite volume scheme applicable to mixed-element grids based on a single numerical flux per face. Previously, we considered a split-quadrilateral-face approach, where we split each quadrilateral face of a non-tetrahedral cell into two triangles and then apply second- and third-order schemes designed for tetrahedral grids. This approach is relatively simple to implement but expensive because it requires two numerical flux evaluations per quadrilateral face. In this study, we consider another approach, where the surface flux quadrature is performed with a single numerical flux per face with flux corrections. It requires a 3×3 correction matrix to be stored at each face but is more economical than the split-quadrilateral-face approach. The two approaches are compared for three-dimensional flows on irregular hexahedral grids, and their relative merits are discussed.

Third-order scheme↗
NASA NTRS1992

Interferograms, Schlieren, and Shadowgraphs Constructed from Real- and Ideal-Gas, Two- and Three-Dimensional Computed Flowfields

The construction of interferograms, schlieren, and shadowgraphs from computed flowfield solutions permits one-to-one comparisons of computed and experimental results. A method for constructing these images from both ideal- and real-gas, two- and three-dimensional computed flowfields is described. The computational grids can be structured or unstructured, and multiple grids are an option. Constructed images are shown for several types of computed flows including nozzle, wake, and reacting flows; comparisons to experimental images are also shown. In addition, the sensitivity of these images to errors in the flowfield solution is demonstrated, and the constructed images can be used to identify problem areas in the computations.

Yates, Leslie A.↗
NASA NTRS1992

Images constructed from computed flowfields

A method for constructing interferograms, schlieren, and shadowgraphs from ideal- and real-gas, two- and three-dimensional computed flowfileds is described. The computational grids can be structured or unstructured, and multiple grids are an option. The constructed images are compared to experimental images for several types of flow, including a ramp, a blunt-body, a nozzle, and a reacting flow. The constructed images simulate the features observed in the experimental images. They are sensitive to errors in the flowfield solutions and can be used to identify solution errors. In addition, techniques for obtaining phase shifts from experimental finite-fringe interferograms and for removing experimentally induced phase-shift errors are discussed. Both the constructed images and calculated phase shifts can be used for validation of computational fluid dynamics (CFD) codes.

Yates, Leslie A.↗
NASA NTRS1992

Images constructed from computed flow fields

A method for constructing interferograms, schlieren, and shadowgraphs from ideal- and real-gas, two- and three-dimensional computed flow fields is described. The computational grids can be structured or unstructured, and multiple grids are an option. The constructed images are compared to experimental images for several types of flow, including a ramp, a blunt-body, a nozzle, and a reacting flow. The constructed images simulate the features observed in the experimental images. They are sensitive to errors in the flow-field solutions and can be used to identify solution errors. In addition, techniques for obtaining phase shifts from experimental finite-fringe interferograms and for removing experimentally induced phase-shift errors are discussed. Both the constructed images and calculated phase shifts can be used for validation of computational fluid dynamics (CFD) codes.

Yates, Leslie A.↗
NASA NTRS1993

Interferograms, schlieren, and shadowgraphs constructed from real- and ideal-gas, two- and three-dimensional computed flowfields

The construction of interferograms, schlieren, and shadowgraphs from computed flowfield solutions permits one-to-one comparisons of computed and experimental results. A method of constructing these images from both ideal- and real-gas, two and three-dimensional computed flowfields is described. The computational grids can be structured or unstructured, and multiple grids are an option. Constructed images are shown for several types of computed flows including nozzle, wake, and reacting flows; comparisons to experimental images are also shown. In addition, th sensitivity of these images to errors in the flowfield solution is demonstrated, and the constructed images can be used to identify problem areas in the computations.

Yates, Leslie A.↗
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 NTRS2006

More About the Tetrahedral Unstructured Software System

TetrUSS is a comprehensive suite of computational fluid dynamics (CFD) programs that won the Software of the Year award in 1996 and has found increasing use in government, academia, and industry for solving realistic flow problems (especially in aerodynamics and aeroelastics of aircraft having complex shapes). TetrUSS includes not only programs for solving basic equations of flow but also programs that afford capabilities for efficient generation and utilization of computational grids and for graphical representation of computed flows (see figure). The 2004 version of the Tetrahedral Unstructured Software System (TetrUSS), which is one of two software systems reported in "NASA s 2004 Software of the Year," NASA Tech Briefs, Vol. 28, No. 10 (October 2004), page 18, has been improved greatly since 1996. These improvements include (1) capabilities to simulate viscous flow by solving the Navier-Stokes equations on unstructured grids, (2) portability to personal computers from diverse manufacturers, (3) advanced models of turbulence, (4) a parallel-processing version of one of the unstructured-grid Navier-Stokes-equation-solving programs, and (5) advanced programs for generating unstructured grids.

Abdol-Hamid, Khaled S.↗
NASA NTRS2017

Statistical Analysis of CFD Solutions from the 6th AIAA CFD Drag Prediction Workshop

A graphical framework is used for statistical analysis of the results from an extensive N-version test of a collection of Reynolds-averaged Navier-Stokes computational fluid dynamics codes. The solutions were obtained by code developers and users from North America, Europe, Asia, and South America using both common and custom grid sequences as well as multiple turbulence models for the June 2016 6th AIAA CFD Drag Prediction Workshop sponsored by the AIAA Applied Aerodynamics Technical Committee. The aerodynamic configuration for this workshop was the Common Research Model subsonic transport wing-body previously used for both the 4th and 5th Drag Prediction Workshops. This work continues the statistical analysis begun in the earlier workshops and compares the results from the grid convergence study of the most recent workshop with previous workshops.

Statistical analysis↗
NASA NTRS2005

Wind-US Unstructured Flow Solutions for a Transonic Diffuser

The Wind-US Computational Fluid Dynamics flow solver computed flow solutions for a transonic diffusing duct. The calculations used an unstructured (hexahedral) grid. The Spalart-Allmaras turbulence model was used. Static pressures along the upper and lower wall agreed well with experiment, as did velocity profiles. The effect of the smoothing input parameters on convergence and solution accuracy was investigated. The meaning and proper use of these parameters are discussed for the benefit of Wind-US users. Finally, the unstructured solver is compared to the structured solver in terms of run times and solution accuracy.

Mohler, Stanley R., Jr.↗
NASA NTRSDate not supplied

Updates to Implicit Edge-Based Gradient Methods

In this paper, we report updates to the implicit edge-based gradient methods originally introduced in [H. Nishikawa, AIAA Paper 2020-3048, 2020]. First, we clarify the relationship between gradient accuracy and truncation error and show that the quadratic method involves a free parameter. Then, we provide a complete description and a simplified matrix form of the implicit gradient systems including a consistent boundary treatment, and derive a set of parameters for achieving fourth-order gradient accuracy on regular tetrahedral grids. A stability analysis is performed for a relaxation scheme used to solve the implicit gradient systems, and the result serves as a guide for choosing parameters. Numerical results are shown for accuracy verification and also for realistic inviscid flow problems in three dimensions, including flows with shock waves.

Weighted Least-Squares↗