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At least 19 records

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NASA NTRS1993

Time-dependent grid adaptation for meshes of triangles and tetrahedra

This paper presents in viewgraph form a method of optimizing grid generation for unsteady CFD flow calculations that distributes the numerical error evenly throughout the mesh. Adaptive meshing is used to locally enrich in regions of relatively large errors and to locally coarsen in regions of relatively small errors. The enrichment/coarsening procedures are robust for isotropic cells; however, enrichment of high aspect ratio cells may fail near boundary surfaces with relatively large curvature. The enrichment indicator worked well for the cases shown, but in general requires user supervision for a more efficient solution.

Russ D Rausch↗

NASA NTRS2011

Image Segmentation, Registration, Compression, and Matching

A novel computational framework was developed of a 2D affine invariant matching exploiting a parameter space. Named as affine invariant parameter space (AIPS), the technique can be applied to many image-processing and computer-vision problems, including image registration, template matching, and object tracking from image sequence. The AIPS is formed by the parameters in an affine combination of a set of feature points in the image plane. In cases where the entire image can be assumed to have undergone a single affine transformation, the new AIPS match metric and matching framework becomes very effective (compared with the state-of-the-art methods at the time of this reporting). No knowledge about scaling or any other transformation parameters need to be known a priori to apply the AIPS framework. An automated suite of software tools has been created to provide accurate image segmentation (for data cleaning) and high-quality 2D image and 3D surface registration (for fusing multi-resolution terrain, image, and map data). These tools are capable of supporting existing GIS toolkits already in the marketplace, and will also be usable in a stand-alone fashion. The toolkit applies novel algorithmic approaches for image segmentation, feature extraction, and registration of 2D imagery and 3D surface data, which supports first-pass, batched, fully automatic feature extraction (for segmentation), and registration. A hierarchical and adaptive approach is taken for achieving automatic feature extraction, segmentation, and registration. Surface registration is the process of aligning two (or more) data sets to a common coordinate system, during which the transformation between their different coordinate systems is determined. Also developed here are a novel, volumetric surface modeling and compression technique that provide both quality-guaranteed mesh surface approximations and compaction of the model sizes by efficiently coding the geometry and connectivity/topology components of the generated models. The highly efficient triangular mesh compression compacts the connectivity information at the rate of 1.5-4 bits per vertex (on average for triangle meshes), while reducing the 3D geometry by 40-50 percent. Finally, taking into consideration the characteristics of 3D terrain data, and using the innovative, regularized binary decomposition mesh modeling, a multistage, pattern-drive modeling, and compression technique has been developed to provide an effective framework for compressing digital elevation model (DEM) surfaces, high-resolution aerial imagery, and other types of NASA data.

Yadegar, Jacob↗

NASA NTRS2000

High Order Discontinuous Gelerkin Methods for Convection Dominated Problems with Application to Aeroacoustics

This project is about the investigation of the development of the discontinuous Galerkin finite element methods, for general geometry and triangulations, for solving convection dominated problems, with applications to aeroacoustics. On the analysis side, we have studied the efficient and stable discontinuous Galerkin framework for small second derivative terms, for example in Navier-Stokes equations, and also for related equations such as the Hamilton-Jacobi equations. This is a truly local discontinuous formulation where derivatives are considered as new variables. On the applied side, we have implemented and tested the efficiency of different approaches numerically. Related issues in high order ENO and WENO finite difference methods and spectral methods have also been investigated. Jointly with Hu, we have presented a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method. Jointly with Hu, we have constructed third and fourth order WENO schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. The third order schemes are based on a combination of linear polynomials with nonlinear weights, and the fourth order schemes are based on combination of quadratic polynomials with nonlinear weights. We have addressed several difficult issues associated with high order WENO schemes on unstructured mesh, including the choice of linear and nonlinear weights, what to do with negative weights, etc. Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations. Jointly with P. Montarnal, we have used a recently developed energy relaxation theory by Coquel and Perthame and high order weighted essentially non-oscillatory (WENO) schemes to simulate the Euler equations of real gas. The main idea is an energy decomposition under the form epsilon = epsilon(sub 1) + epsilon(sub 2), where epsilon(sub 1) is associated with a simpler pressure law (gamma)-law in this paper) and the nonlinear deviation epsilon(sub 2) is convected with the flow. A relaxation process is performed for each time step to ensure that the original pressure law is satisfied. The necessary characteristic decomposition for the high order WENO schemes is performed on the characteristic fields based on the epsilon(sub l) gamma-law. The algorithm only calls for the original pressure law once per grid point per time step, without the need to compute its derivatives or any Riemann solvers. Both one and two dimensional numerical examples are shown to illustrate the effectiveness of this approach.

Shu, Chi-Wang↗

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.↗

NASA NTRS2000

Higher Order Modeling In the BEM/FEM Hybrid Formulation

Hybrid formulations using low order curl-conforming bases to represent the total electric field within a finite element region and low order divergence-conforming bases to represent equivalent electric and magnetic currents on the boundary are well known. However, higher-order divergence and curl-conforming bases have been shown to provide significant benefits in convergence rates and accuracy when employed in strictly integral equation and strictly finite element formulations. In this paper, a hybrid electric field formulation employing higher order bases is presented. The paper addresses benefits and issues associated with using higher order divergence-and curl-conforming bases in the hybrid finite element/boundary element electric field formulation. The method of singularity subtraction may be used to compute the self terms of the boundary integral when the bases are of low order. But this method becomes laborious and requires great care when the divergence conforming bases are of higher order. In order to handle these singularities simply and accurately, a generalized Gaussian quadrature method is employed in which the expansion functions account for the singularity. In preliminary tests of the higher order hybrid formulation, the equivalent electric current induced by scattering of a plane wave from a square dielectric cylinder is examined. Accurate results are obtained using only a two-triangle mesh when the current basis is of order 4 or 5. Additional results are presented comparing the error obtained using higher order bases to that obtained using lower order bases when the number of unknowns is approximately equal. Also, convergence rates obtained with higher order bases are compared to those obtained with lower order bases for selected sample problems.

Fink, Patrick W.↗

NASA NTRS1992

Spatial and temporal adaptive procedures for the unsteady aerodynamic analysis of airfoils using unstructured meshes

An algorithm which combines spatial and temporal adaption for the time integration of the two dimensional Euler equations on unstructured meshes of triangles is presented. Spatial adaption involves mesh enrichment to add elements in high gradient regions of the flow and mesh coarsening to remove elements where they are no longer needed. Temporal adaption is a time accurate, local time stepping procedure which integrates the flow equations in each cell according to the local numerical stability constraint. The flow solver utilizes a four stage Runge-Kutta time integration scheme with an upwind flux-split spatial discretization. Results obtained using spatial and temporal adaption indicate that highly accurate solutions can be obtained with a significant savings of computing time over global time stepping.

Hooker, John R.↗

NASA NTRS1992

Spatial and temporal adaptive procedures for the unsteady aerodynamic analysis of airfoils using unstructured meshes

An algorithm which combines spatial and temporal adaption for the time integration of the two-dimensional Euler equations on unstructured meshes of triangles is presented. Spatial adaption involves mesh enrichment to add elements in high gradient regions of the flow and mesh coarsening to remove elements where they are no longer needed. Temporal adaption is a time accurate, local time stepping procedure which integrates the flow equations in each cell according to the local numerical stability constraint. The flow solver utilizes a four-stage Runge-Kutta time integration scheme with an upwind flux-split spatial discretization. Results obtained using spatial and temporal adaption indicate that highly accurate solutions can be obtained with a significant savings of computing time over global time stepping.

Hooker, John R.↗

NASA NTRS2006

Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains

Borrowing from techniques developed for conservation law equations, we have developed both monotone and higher order accurate numerical schemes which discretize the Hamilton-Jacobi and level set equations on triangulated domains. The use of unstructured meshes containing triangles (2D) and tetrahedra (3D) easily accommodates mesh adaptation to resolve disparate level set feature scales with a minimal number of solution unknowns. The minisymposium talk will discuss these algorithmic developments and present sample calculations using our adaptive triangulation algorithm applied to various moving interface problems such as etching, deposition, and curvature flow.

Barth, Timothy J.↗

NASA NTRS1999

Improvements to the Unstructured Mesh Generator MESH3D

The AIRPLANE process starts with an aircraft geometry stored in a CAD system. The surface is modeled with a mesh of triangles and then the flow solver produces pressures at surface points which may be integrated to find forces and moments. The biggest advantage is that the grid generation bottleneck of the CFD process is eliminated when an unstructured tetrahedral mesh is used. MESH3D is the key to turning around the first analysis of a CAD geometry in days instead of weeks. The flow solver part of AIRPLANE has proven to be robust and accurate over a decade of use at NASA. It has been extensively validated with experimental data and compares well with other Euler flow solvers. AIRPLANE has been applied to all the HSR geometries treated at Ames over the course of the HSR program in order to verify the accuracy of other flow solvers. The unstructured approach makes handling complete and complex geometries very simple because only the surface of the aircraft needs to be discretized, i.e. covered with triangles. The volume mesh is created automatically by MESH3D. AIRPLANE runs well on multiple platforms. Vectorization on the Cray Y-MP is reasonable for a code that uses indirect addressing. Massively parallel computers such as the IBM SP2, SGI Origin 2000, and the Cray T3E have been used with an MPI version of the flow solver and the code scales very well on these systems. AIRPLANE can run on a desktop computer as well. AIRPLANE has a future. The unstructured technologies developed as part of the HSR program are now targeting high Reynolds number viscous flow simulation. The pacing item in this effort is Navier-Stokes mesh generation.

Thomas, Scott D.↗

NASA NTRS1993

Line relaxation methods for the solution of 2D and 3D compressible flows

An implicit finite element based algorithm for the compressible Navier-Stokes equations is outlined, and the solution of the resulting equation by a line relaxation on general meshes of triangles or tetrahedra is described. The problem of generating and adapting unstructured meshes for viscous flows is reexamined, and an approach for both 2D and 3D simulations is proposed. An efficient approach appears to be the use of an implicit/explicit procedure, with the implicit treatment being restricted to those regions of the mesh where viscous effects are known to be dominant. Numerical examples demonstrating the computational performance of the proposed techniques are given.

Hassan, O.↗

NASA NTRS1988

Finite element solution of the Euler equations in two and three dimensions

The paper presents a finite element procedure for solving the equations of compressible flow over bodies of arbitrary geometry. The numerical solution algorithm employed is an explicit two-step version of a second order Taylor-Galerkin scheme. The discretization of the computational domain into unstructured meshes of triangles in two dimensions and tetrahedra in three dimensions is performed by an automatic mesh generator. In the approach, the mesh generator is coupled to the finite element solver to produce an adaptive remeshing procedure.

Peiro, J.↗

NASA NTRS1990

Unstructured finite element mesh generation and adaptive procedures for CFD

A method is described for generating unstructured meshes of triangles or tetrahedra for computational domains of complex geometrical shape. To illustrate the power of the approach, it is applied to the solution of flows past several complete aircraft configurations. The advocated approach allows for the natural incorporation of mesh adaptivity and this is demonstrated for both inviscid and viscous computations in two and three dimensions.

Peraire, J.↗

NASA NTRS1990

Temporal-adaptive Euler/Navier-Stokes algorithm for unsteady aerodynamic analysis of airfoils using unstructured dynamic meshes

A temporal adaptive algorithm for the time-integration of the two-dimensional Euler or Navier-Stokes equations is presented. The flow solver involves an upwind flux-split spatial discretization for the convective terms and central differencing for the shear-stress and heat flux terms on an unstructured mesh of triangles. The temporal adaptive algorithm is a time-accurate integration procedure which allows flows with high spatial and temporal gradients to be computed efficiently by advancing each grid cell near its maximum allowable time step. Results indicate that an appreciable computational savings can be achieved for both inviscid and viscous unsteady airfoil problems using unstructured meshes without degrading spatial or temporal accuracy.

Kleb, William L.↗

NASA NTRS1990

Temporal-adaptive Euler/Navier-Stokes algorithm for unsteady aerodynamic analysis of airfoils using unstructured dynamic meshes

Kleb, William L.↗

NASA NTRS1990

Shock interference studies on a circular cylinder at Mach 16

This paper presents the results of a computational study on the shock interference problems on a cylindrical body typical of an engine inlet cowl leading edge at a nominal Mach number of 16. The two-dimensional Navier-Stokes equations are solved assuming the flow to be in chemical and thermal equilibrium and using a finite element method. The algorithm employs a cell-centered fully implicit upwind scheme. Adaptively generated unstructured meshes of triangles and quadrilaterals are employed. Under certain conditions the finite element code resulted in oscillatory solutions for shock interference at Mach 16. Some of the causes of the unsteady behavior are identified, and to the extent possible, such situations are avoided in the present application. Two shock interference conditions involving a Type IV (supersonic jet) and a Type III (attaching shear layer) are solved. The results are compared with available experimental data and reasonable agreement is observed. A semi-empirical method is also used to estimate the maximum surface heat flux and static pressure.

Prabhu, Ramadas K.↗

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 NTRS1999

Self-Avoiding Walks Over Adaptive Triangular Grids

Space-filling curves is a popular approach based on a geometric embedding for linearizing computational meshes. We present a new O(n log n) combinatorial algorithm for constructing a self avoiding walk through a two dimensional mesh containing n triangles. We show that for hierarchical adaptive meshes, the algorithm can be locally adapted and easily parallelized by taking advantage of the regularity of the refinement rules. The proposed approach should be very useful in the runtime partitioning and load balancing of adaptive unstructured grids.

Heber, Gerd↗

NASA NTRS2003

Three-Dimensional Morphological Analysis of ALH84001 Magnetite Using Electron Tomography

We report here the crystal morphologies of MV-1 and ALH84001 magnetites as calculated by back-projection using electron tomography. In the present study, we used a 300 keV TEM with a field emission gun (Tecnai F-30 from FEI Inc.), equipped with a 2048 x 2048 pixel CCD camera from Gatan Inc. to image magnetite crystals over tilt ranges of approx. +/- 72 deg in 2 deg tilt intervals. The images were aligned for back-projection, either manually, or through the use of fiducial 5 nm Au spheres affixed to the specimen prior to microscopy. Three-dimensional (3-D) reconstructions were computed using weighted back-projection of the tilted views. The tomograms were viewed and analyzed as a series of slices 1.0 nm thick, taken parallel to the specimen-supporting grid, using the IMOD software package. The shape of each magnetite crystal was determined by defining the external contour of a given magnetite in each slice and assembling a stack of these contours in 3-D. To aid in visualization, the stacked contour array was reduced to an optimal mesh by Delaunay triangulation. The surface normal to each of the triangles in the mesh was calculated and the triangle faces colored according to the orientation of that surface normal relative to the principal crystallographic axis of magnetite. Green surfaces correspond to {111} orientations, blue surfaces to {100} orientations, and red surfaces to {110} orientations. Triangles whose surface normal did not correspond to one of the principal axes were colored gray. Within the experimental and numerical uncertainties of the deconvolution, the tomographic reconstruction of both MV-1 and ALH84001 magnetites are equivalent and correspond to a truncated hexa-octahedral morphology.

Thomas-Keprta, Kathie L.↗