Spectral methods for partial differential equations

Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid dynamical applications are emphasized

Development of semi-coarsening techniques

Departing from Mulder's semi-coarsening technique for first order PDEs, the notion of a grid of grids is introduced and a multi-level finite-volume technique for second order elliptic PDEs is developed. Various grid transfer operators are investigated, in combination with damped Jacobi relaxation. Convergence rates as they are predicted by Fourier local mode analysis are compared with practical measurements. The wide variety of grids at our disposal leads to the notion of coherent representations of a function on different grids. A sawtooth multi-level algorithm is proposed for the case of multiple semi-coarsening. A hierarchical set of basis functions for finite volumes on sparse grids is briefly discussed

Spectral methods for inviscid, compressible flows

Report developments in the application of spectral methods to two dimensional compressible flows are reviewed. A brief introduction to spectral methods -- their history and especially their implementation -- is provided. The stress is on those techniques relevant to transonic flow computation. The spectral multigrid iterative methods are discussed with application to the transonic full potential equation. Discontinuous solutions of the Euler equations are considered. The key element is the shock fitting technique which is briefly explained

Turbomachinery CFD on parallel computers

The role of multistage turbomachinery simulation in the development of propulsion system models is discussed. Particularly, the need for simulations with higher fidelity and faster turnaround time is highlighted. It is shown how such fast simulations can be used in engineering-oriented environments. The use of parallel processing to achieve the required turnaround times is discussed. Current work by several researchers in this area is summarized. Parallel turbomachinery CFD research at the NASA Lewis Research Center is then highlighted. These efforts are focused on implementing the average-passage turbomachinery model on MIMD, distributed memory parallel computers. Performance results are given for inviscid, single blade row and viscous, multistage applications on several parallel computers, including networked workstations

Volume 2: Explicit, multistage upwind schemes for Euler and Navier-Stokes equations

The objective of this study was to develop a high-resolution-explicit-multi-block numerical algorithm, suitable for efficient computation of the three-dimensional, time-dependent Euler and Navier-Stokes equations. The resulting algorithm has employed a finite volume approach, using monotonic upstream schemes for conservation laws (MUSCL)-type differencing to obtain state variables at cell interface. Variable interpolations were written in the k-scheme formulation. Inviscid fluxes were calculated via Roe's flux-difference splitting, and van Leer's flux-vector splitting techniques, which are considered state of the art. The viscous terms were discretized using a second-order, central-difference operator. Two classes of explicit time integration has been investigated for solving the compressible inviscid/viscous flow problems--two-state predictor-corrector schemes, and multistage time-stepping schemes. The coefficients of the multistage time-stepping schemes have been modified successfully to achieve better performance with upwind differencing. A technique was developed to optimize the coefficients for good high-frequency damping at relatively high CFL numbers. Local time-stepping, implicit residual smoothing, and multigrid procedure were added to the explicit time stepping scheme to accelerate convergence to steady-state. The developed algorithm was implemented successfully in a multi-block code, which provides complete topological and geometric flexibility. The only requirement is C degree continuity of the grid across the block interface. The algorithm has been validated on a diverse set of three-dimensional test cases of increasing complexity. The cases studied were: (1) supersonic corner flow; (2) supersonic plume flow; (3) laminar and turbulent flow over a flat plate; (4) transonic flow over an ONERA M6 wing; and (5) unsteady flow of a compressible jet impinging on a ground plane (with and without cross flow). The emphasis of the test cases was validation of code, and assessment of performance, as well as demonstration of flexibility

Research in applied mathematics, numerical analysis, and computer science

Research conducted at the Institute for Computer Applications in Science and Engineering (ICASE) in applied mathematics, numerical analysis, and computer science is summarized and abstracts of published reports are presented. The major categories of the ICASE research program are: (1) numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; (2) control and parameter identification; (3) computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, and structural analysis; and (4) computer systems and software, especially vector and parallel computers

Slam Simulations: An Application of Computational Fluid Dynamics

The ability of ships and other marine vehicles to maintain forward speed in heavy weather is often limited by the phenomenon of slamming. The hydrodynamics of this impact problem are poorly understood in a quantative sense, though previous research has identified various physical mechanisms important to ship slamming. This thesis describes the development of a number of numerical tools designed to study the non-linear free surface flow problems caused by slamming. The research focused on two major areas of study. Firstly, a generalised simulation method for the solution of the mass and momentum conservation equations over a fluid domain bounded by a free surface and containing dynamic boundaries was developed. This technique was used to study the hydrodynamic impact of an arbitrary shaped body. A finite difference time marching solution to the continuity coupled Navier Stokes equations was employed as the basis of the simulation technique. A novel application of the well known source distribution method was used to model moving solid boundaries of arbitrary shape within the confines of the regular finite difference mesh. This particular aspect of the research allowed a marriage of traditional hydrodynamic theory and the more recent developments in computational fluid dynamics. The second area of study examined the behaviour of the compressible air layer formed between the free surface and an approaching bluff body. Again, the techniques of computational fluid dynamics were employed to solve the equations of mass and momentum conservation in the air layer and the associated free surface motions caused by the build up of pressure beneath the falling body. The thesis presents a number of computational examples in order to illustrate the development and final levels of accuracy achieved by the two classes of numerical algorithm mentioned above. Simulations of steady viscous free surface flow, wavemaker modelling, vehicle motions and added mass computations are employed to test the numerical algorithms. Results from hydrodynamic impact simulations are presented and compared with existing numerical and experimental data, for a range of hull shapes. The effects of air entrapment on impact geometry is discussed via a comparison of results obtained by the present method and data from both computations and experiment reported in the open literature. Conclusions concerning the quantitative importance of the various physical parameters involved in slamming on the ship scale are presented at the end of the thesis

CES-515 Towards Localization and Mapping of Autonomous Underwater Vehicles: A Survey

Autonomous Underwater Vehicles (AUVs) have been used for a huge number of tasks ranging from commercial, military and research areas etc, while the fundamental function of a successful AUV is its localization and mapping ability. This report aims to review the relevant elements of localization and mapping for AUVs. First, a brief introduction of the concept and the historical development of AUVs is given; then a relatively detailed description of the sensor system used for AUV navigation is provided. As the main part of the report, a comprehensive investigation of the simultaneous localization and mapping (SLAM) for AUVs are conducted, including its application examples. Finally a brief conclusion is summarized

Final report on the Copper Mountain conference on multigrid methods

The Copper Mountain Conference on Multigrid Methods was held on April 6-11, 1997. It took the same format used in the previous Copper Mountain Conferences on Multigrid Method conferences. Over 87 mathematicians from all over the world attended the meeting. 56 half-hour talks on current research topics were presented. Talks with similar content were organized into sessions. Session topics included: fluids; domain decomposition; iterative methods; basics; adaptive methods; non-linear filtering; CFD; applications; transport; algebraic solvers; supercomputing; and student paper winners

Development of a Navier-Stokes algorithm for parallel-processing supercomputers

An explicit flow solver, applicable to the hierarchy of model equations ranging from Euler to full Navier-Stokes, is combined with several techniques designed to reduce computational expense. The computational domain consists of local grid refinements embedded in a global coarse mesh, where the locations of these refinements are defined by the physics of the flow. Flow characteristics are also used to determine which set of model equations is appropriate for solution in each region, thereby reducing not only the number of grid points at which the solution must be obtained, but also the computational effort required to get that solution. Acceleration to steady-state is achieved by applying multigrid on each of the subgrids, regardless of the particular model equations being solved. Since each of these components is explicit, advantage can readily be taken of the vector- and parallel-processing capabilities of machines such as the Cray X-MP and Cray-2