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Viewgraphs on accuracy assessment for grid adaptation are presented. Topics covered include: uniform refinement, grid convergence, adapted grids, oblique shock wave, supersonic flow, and sonic flow.
Self-adaptive mesh refinement dynamically matches the computational demands of a solver for partial differential equations to the activity in the application's domain. In this paper we present two C++ class libraries, P++ and AMR++, which significantly simplify the development of sophisticated adaptive mesh refinement codes on (massively) parallel distributed memory architectures. The development is based on our previous research in this area. The C++ class libraries provide abstractions to separate the issues of developing parallel adaptive mesh refinement applications into those of parallelism, abstracted by P++, and adaptive mesh refinement, abstracted by AMR++. P++ is a parallel array class library to permit efficient development of architecture independent codes for structured grid applications, and AMR++ provides support for self-adaptive mesh refinement on block-structured grids of rectangular non-overlapping blocks. Using these libraries, the application programmers' work is greatly simplified to primarily specifying the serial single grid application and obtaining the parallel and self-adaptive mesh refinement code with minimal effort. Initial results for simple singular perturbation problems solved by self-adaptive multilevel techniques (FAC, AFAC), being implemented on the basis of prototypes of the P++/AMR++ environment, are presented. Singular perturbation problems frequently arise in large applications, e.g. in the area of computational fluid dynamics. They usually have solutions with layers which require adaptive mesh refinement and fast basic solvers in order to be resolved efficiently.
There is a need to develop fine pitch x-ray collimator grids as an enabling technology for planned future missions. The grids consist of an array of thin parallel strips of x-ray absorbing material, such as tungsten, with pitches ranging from 34 microns to 2.036 millimeters. The grids are the key components of a new class of spaceborne instruments known as 'x-ray modulation collimators.' These instruments are the first to produce images of celestial sources in the hard x-ray and gamma-ray spectral regions.
A parallel grid-generation algorithm and its implementation on the Intel iPSC/860 computer are described. The grid-generation scheme is based on an algebraic formulation of homotopic relations. Methods for utilizing the inherent parallelism of the grid-generation scheme are described, and implementation of multiple levELs of parallelism on multiple instruction multiple data machines are indicated. The algorithm is capable of providing near orthogonality and spacing control at solid boundaries while requiring minimal interprocessor communications. Results obtained on the Intel hypercube for a blended wing-body configuration are used to demonstrate the effectiveness of the algorithm. Fortran implementations bAsed on the native programming model of the iPSC/860 computer and the Express system of software tools are reported. Computational gains in execution time speed-up ratios are given.
A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses.
An alternative method for domain connectivity among systems of overset grids is presented. Reference uniform Cartesian systems of points are used to achieve highly efficient domain connectivity, and form the basis for a future fully automated system. The Cartesian systems are used to approximated body surfaces and to map the computational space of component grids. By exploiting the characteristics of Cartesian Systems, Chimera type hole-cutting and identification of donor elements for intergrid boundary points can be carried out very efficiently. The method is tested for a range of geometrically complex multiple-body overset grid systems.
Algebraic procedures are described for the automatic generation of structured, single-block flow computation grids for relatively simple configurations (wing, fuselage, and fin). For supersonic flows, a quasi two-dimensional grid for Euler-marching codes is developed, and some sample results in graphical form are included. A type of grid for subsonic flow calculation is also described. The techniques are algebraic and are based on a generalization of the method of transfinite interpolation.
Component adaptive grid (CAG) methods for solving hyperbolic partial differential equations (PDE's) are discussed in this paper. Applying recent stability results for a class of numerical methods on uniform grids. The convergence of these methods for linear problems on component adaptive grids is established here. Furthermore, the computational error can be estimated on CAG's using the stability results. Using these estimates, the error can be controlled on CAG's. Thus, the solution can be computed efficiently on CAG's within a given error tolerance. Computational results for time dependent linear problems in one and two space dimensions are presented.
The multidimensional self-adaptive grid code, SAGE, has proven to be a flexible and useful tool in the solution of complex flow problems. Both 2- and 3-D examples given in this report show the code to be reliable and to substantially improve flowfield solutions. Since the adaptive procedure is a marching scheme the code is extremely fast and uses insignificant CPU time compared to the corresponding flow solver. The SAGE program is also machine and flow solver independent. Significant effort was made to simplify user interaction, though some parameters still need to be chosen with care. It is also difficult to tell when the adaption process has provided its best possible solution. This is particularly true if no experimental data are available or if there is a lack of theoretical understanding of the flow. Another difficulty occurs if local features are important but missing in the original grid; the adaption to this solution will not result in any improvement, and only grid refinement can result in an improved solution. These are complex issues that need to be explored within the context of each specific problem.
Many CFD grid generation systems are in use nationally, but few comparative studies have been performed to quantify their relative merits. A study was undertaken to systematically evaluate and select the best CFD grid generation codes available. Detailed evaluation criteria were established as the basis for the evaluation conducted. Descriptions of thirty-four separate criteria, grouped into eight general categories are provided. Benchmark test cases, developed to test basic features of selected codes, are described in detail. Scoring guidelines were generated to establish standards for measuring code capabilities, ensuring uniformity of ratings, and minimizing personal bias among the three code evaluators. Ten candidate codes were identified from government, industry, universities, and commercial software companies. A three phase evaluation was conducted. In Phase 1, ten codes identified were screened through conversations with code authors and other industry experts. Seven codes were carried forward into a Phase 2 evaluation in which all codes were scored according to the predefined criteria. Two codes emerged as being significantly better than the others: RAGGS and GRIDGEN. Finally, these two codes were carried forward into a Phase 3 evaluation in which complex 3-D multizone grids were generated to verify capability.
Stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations. These new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions. Specifically, we show how to implement Legendre Galerkin, Legendre collocation, and Laguerre Galerkin methodology on arbitrary grids.
A method has been developed to accurately compute the viscous flow in three-dimensional (3-D) enclosures. This method is the 3-D extension of a two-dimensional (2-D) method developed for the calculation of flow over airfoils. The 2-D method has been tested extensively and has been shown to accurately reproduce experimental results. As in the 2-D method, the 3-D method provides for the non-iterative solution of the incompressible Navier-Stokes equations by means of a fully coupled implicit technique. The solution is calculated on a body fitted computational mesh incorporating a staggered grid methodology. In the staggered grid method, the three components of vorticity are defined at the centers of the computational cell sides, while the velocity components are defined as normal vectors at the centers of the computational cell faces. The staggered grid orientation provides for the accurate definition of the vorticity components at the vorticity locations, the divergence of vorticity at the mesh cell nodes and the conservation of mass at the mesh cell centers. The solution is obtained by utilizing a fractional step solution technique in the three coordinate directions. The boundary conditions for the vorticity and velocity are calculated implicitly as part of the solution. The method provides for the non-iterative solution of the flow field and satisfies the conservation of mass and divergence of vorticity to machine zero at each time step. To test the method, the calculation of simple driven cavity flows have been computed. The driven cavity flow is defined as the flow in an enclosure driven by a moving upper plate at the top of the enclosure. To demonstrate the ability of the method to predict the flow in arbitrary cavities, results will he shown for both cubic and curved cavities.
The past ten years have seen steady progress in surface modeling procedures, and wholesale changes in grid generation technology. Today, it seems fair to state that a satisfactory grid can be developed to model nearly any configuration of interest. The issues at present focus on operational concerns such as cost and quality. Continuing evolution of the engineering process is placing new demands on the technologies of surface modeling and grid generation. In the evolution toward a multidisciplinary analysis-bascd design environment, methods developed for Computational Fluid Dynamics are finding acceptance in many additional applications. These two trends, the normal evolution of the process and a watershed shift toward concurrent and multidisciplinary analysis, will be considered in assessing current capabilities and needed technological improvements.
The status of CFD methods based on the use of block-structured grids for analyzing viscous flows over complex configurations is examined. The objective of the present study is to make a realistic assessment of the usability of such grids for routine computations typically encountered in the aerospace industry. It is recognized at the very outset that the total turnaround time, from the moment the configuration is identified until the computational results have been obtained and postprocessed, is more important than just the computational time. Pertinent examples will be cited to demonstrate the feasibility of solving flow over practical configurations of current interest on block-structured grids.
Two different approaches to the fully automatic generation of structured multi-block grids in two dimensions are presented. The work aims to simplify the user interactivity necessary for the definition of a multiple block grid topology. The first approach is based on an advancing front method commonly used for the generation of unstructured grids. The original algorithm has been modified toward the generation of large quadrilateral elements. The second method is based on the divide-and-conquer paradigm with the global domain recursively partitioned into sub-domains. For either method each of the resulting blocks is then meshed using transfinite interpolation and elliptic smoothing. The applicability of these methods to practical problems is demonstrated for typical geometries of fluid dynamics.
With examples, we illustrate how implicitly specified surfaces can be used for grid generation with GridPro/az3000. The particular examples address two questions: (1) How do you model intersecting tubes with fillets? and (2) How do you generate grids inside the intersected tubes? The implication is much more general. With the results in a forthcoming paper which develops an easy-to-follow procedure for implicit surface modeling, we provide a powerful means for rapid prototyping in grid generation.
The development of implicit schemes for obtaining steady state solutions to the Euler and Navier-Stokes equations on unstructured grids is outlined. Applications are presented that compare the convergence characteristics of various implicit methods. Next, the development of explicit and implicit schemes to compute unsteady flows on unstructured grids is discussed. Next, the issues involved in parallelizing finite volume schemes on unstructured meshes in an MIMD (multiple instruction/multiple data stream) fashion are outlined. Techniques for partitioning unstructured grids among processors and for extracting parallelism in explicit and implicit solvers are discussed. Finally, some dynamic load balancing ideas, which are useful in adaptive transient computations, are presented.
This paper presents two partitioning schemes that guarantee connected components given a connected initial grid. Connected components are important for convergence of methods such as domain decomposition or multigrid. For many of the grids tested, the schemes produce partitions as good (in terms of number of cut edges) or better than spectral partitioning and require only modest computational resources. This paper describes the two schemes in detail and presents comparison results from a number of two and three dimensional unstructured grids.