An overlapped grid method for multigrid, finite volume/difference flow solvers: MaGGiE

The objective is to develop a domain decomposition method via overlapping/embedding the component grids, which is to be used by upwind, multi-grid, finite volume solution algorithms. A computer code, given the name MaGGiE (Multi-Geometry Grid Embedder) is developed to meet this objective. MaGGiE takes independently generated component grids as input, and automatically constructs the composite mesh and interpolation data, which can be used by the finite volume solution methods with or without multigrid convergence acceleration. Six demonstrative examples showing various aspects of the overlap technique are presented and discussed. These cases are used for developing the procedure for overlapping grids of different topologies, and to evaluate the grid connection and interpolation data for finite volume calculations on a composite mesh. Time fluxes are transferred between mesh interfaces using a trilinear interpolation procedure. Conservation losses are minimal at the interfaces using this method. The multi-grid solution algorithm, using the coaser grid connections, improves the convergence time history as compared to the solution on composite mesh without multi-gridding

Image Segmentation with Eigenfunctions of an Anisotropic Diffusion Operator

We propose the eigenvalue problem of an anisotropic diffusion operator for image segmentation. The diffusion matrix is defined based on the input image. The eigenfunctions and the projection of the input image in some eigenspace capture key features of the input image. An important property of the model is that for many input images, the first few eigenfunctions are close to being piecewise constant, which makes them useful as the basis for a variety of applications such as image segmentation and edge detection. The eigenvalue problem is shown to be related to the algebraic eigenvalue problems resulting from several commonly used discrete spectral clustering models. The relation provides a better understanding and helps developing more efficient numerical implementation and rigorous numerical analysis for discrete spectral segmentation methods. The new continuous model is also different from energy-minimization methods such as geodesic active contour in that no initial guess is required for in the current model. The multi-scale feature is a natural consequence of the anisotropic diffusion operator so there is no need to solve the eigenvalue problem at multiple levels. A numerical implementation based on a finite element method with an anisotropic mesh adaptation strategy is presented. It is shown that the numerical scheme gives much more accurate results on eigenfunctions than uniform meshes. Several interesting features of the model are examined in numerical examples and possible applications are discussed

Multi-Level Adaptive Techniques (MLAT) for singular-perturbation problems

The multilevel (multigrid) adaptive technique, a general strategy of solving continuous problems by cycling between coarser and finer levels of discretization is described. It provides very fast general solvers, together with adaptive, nearly optimal discretization schemes. In the process, boundary layers are automatically either resolved or skipped, depending on a control function which expresses the computational goal. The global error decreases exponentially as a function of the overall computational work, in a uniform rate independent of the magnitude of the singular-perturbation terms. The key is high-order uniformly stable difference equations, and uniformly smoothing relaxation schemes

Wavelet based Adaptive RBF Method for Nearly Singular Poisson-Type Problems on Irregular Domains

We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs over irregularly shaped domains. For a problem defined over Ω ∈ ℜd, the boundary of an irregularly shaped domain, Γ, is defined as a boundary curve that is a product of a Heaviside function along the normal direction and a piecewise continuous tangential curve. The link between the original wavelet based adaptive method presented in Libre, Emdadi, Kansa, Shekarchi, and Rahimian (2008, 2009) or LEKSR method and the generalized one is given through the use of simple Heaviside masking procedure. In addition level dependent thresholding were introduced to improve the efficiency and convergence rate of the solution. We will show how the generalized wavelet based adaptive method can be applied for detecting nearly singularities in Poisson type PDEs over irregular domains. The numerical examples have illustrated that the proposed method is powerful to analyze the Poisson type PDEs with rapid changes in gradients and nearly singularities

Workshop on Grid Generation and Related Areas

A collection of papers given at the Workshop on Grid Generation and Related Areas is presented. The purpose of this workshop was to assemble engineers and scientists who are currently working on grid generation for computational fluid dynamics (CFD), surface modeling, and related areas. The objectives were to provide an informal forum on grid generation and related topics, to assess user experience, to identify needs, and to help promote synergy among engineers and scientists working in this area. The workshop consisted of four sessions representative of grid generation and surface modeling research and application within NASA LeRC. Each session contained presentations and an open discussion period

Finite element analysis of transonic flows in cascades: Importance of computational grids in improving accuracy and convergence

The finite element method is applied for the solution of transonic potential flows through a cascade of airfoils. Convergence characteristics of the solution scheme are discussed. Accuracy of the numerical solutions is investigated for various flow regions in the transonic flow configuration. The design of an efficient finite element computational grid is discussed for improving accuracy and convergence

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Structured grid generation for gas turbine combustion systems

Commercial pressures to reduce time-scales encourage innovation in the design and analysis cycle of gas turbine combustion systems. The migration of Computational Fluid Dynamics (CFD) from the purview of the specialist into a routine analysis tool is crucial to achieve these reductions and forms the focus of this research. Two significant challenges were identified: reducing the time-scale for creating and solving a CFD prediction and reducing the level of expertise required to perform a prediction. The commercial pressure for the rapid production of CFD predictions, coupled with the desire to reduce the risk associated with adopting a new technology led, following a review of available techniques, to the identification of structured grids as the current optimum methodology. It was decided that the task of geometry definition would be entirely performed within commercial Computer Aided Design (CAD) systems. A critical success factor for this research was the adoption of solid models for the geometry representation. Solids ensure consistency, and accuracy, whilst eliminating the need for the designer to undertake difficult, and time consuming, geometry repair operations. The versatility of parametric CAD systems were investigated on the complex geometry of a combustion system and found to be useful in reducing the overhead in altering the geometry for a CFD prediction. Accurate and robust transfer between CAD and CFD systems was achieved by the use of direct translators. Restricting the geometry definition to solid models allowed a novel two stage grid generator to be developed. In stage one an initial algebraic grid is created. This reduces user interaction to a minimum, by the employment of a series of logical rules based on the solid model to fill in any missing grid boundary condition data. In stage two the quality of the grid is improved by redistributing nodes using elliptical partial differential equations. A unique approach of improving grid quality by simultaneously smoothing both internal and surface grids was implemented. The smoothing operation was responsible for quality, and therefore reduced the level of grid generation expertise required. The successful validation of this research was demonstrated using several test cases including a CFD prediction of a complete combustion system

Computational fluid dynamics

An overview of computational fluid dynamics (CFD) activities at the Langley Research Center is given. The role of supercomputers in CFD research, algorithm development, multigrid approaches to computational fluid flows, aerodynamics computer programs, computational grid generation, turbulence research, and studies of rarefied gas flows are among the topics that are briefly surveyed