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A unified approach based on h-p approximation in space variables and various time discretization schemes for solving both incompressible and compressible flows is presented. Two different techniques for a posteriori error estimation and accompanying h-p adaptive strategies are discussed.
There are three basic versions of the finite element method, called the h, p and h-p versions. They are characterized by the way in which the finite element meshes and polynomial degree of elements are chosen. They differ in computer implementation (program architecture) and mathematical analysis. The manner in which the meshes and polynomial degree of the elements affect the accuracy of finite element solutions is examined. The approach is to fix certain parameters and increase the number of degrees of freedom so that the finie element solutions converge to the exact solution. Such a systematic increase of the number of degrees of freedom is called extension because it can be interpreted as a systematic extension of finite element spaces.
A new finite element method solving compressible Navier-Stokes equations is proposed. The method is based on a version of Strang's operator splitting and an h-p adaptive finite element approximation in space. This paper contains the formulation of the method with a detailed discussion of boundary conditions, a sample adaptive strategy and numerical examples involving compressible viscous flow over a flat plate with Reynolds number Re = 1000 and Re = 10,000.
An extension of the familiar Taylor-Galerkin method to arbitrary h-p spatial approximations is proposed. Boundary conditions are analyzed, and a linear stability result for arbitrary meshes is given, showing the unconditional stability for the parameter of implicitness alpha not less than 0.5. The wedge and blunt body problems are solved with both linear, quadratic, and cubic elements and h-adaptivity, showing the feasibility of higher orders of approximation for problems with shocks.
The principal ideas of h-p adaptive finite element methods for fluid dynamics problems are discussed. Applications include acoustics, compressible Euler and both compressible and incompressible Navier-Stokes equations. Several numerical examples illustrate the presented concepts.
The commitment to develop the National Aerospace Plane and Maneuvering Reentry Vehicles has generated resurgent interest in the technology required to design structures for hypersonic flight. The principal objective of this research and development effort has been to formulate and implement a new class of computational methodologies for accurately predicting fine scale phenomena associated with this class of problems. The initial focus of this effort was to develop optimal h-refinement and p-enrichment adaptive finite element methods which utilize a-posteriori estimates of the local errors to drive the adaptive methodology. Over the past year this work has specifically focused on two issues which are related to overall performance of a flow solver. These issues include the formulation and implementation (in two dimensions) of an implicit/explicit flow solver compatible with the hp-adaptive methodology, and the design and implementation of computational algorithm for automatically selecting optimal directions in which to enrich the mesh. These concepts and algorithms have been implemented in a two-dimensional finite element code and used to solve three hypersonic flow benchmark problems (Holden Mach 14.1, Edney shock on shock interaction Mach 8.03, and the viscous backstep Mach 4.08).
The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.
Hydrogen masers used as aids in meeting the routine frequency and time operational requirements within the 64 m antenna Deep Space Network. Both the operational syntonation (frequency synchronization) and the the clock (epoch) synchronization requirements were established through the use of specifically calibrated H-P E215061A flying clock. The sync/synt to UTC was maintained using LORAN and TV in simultaneous reception mode. The sync/synt within the 64 m net was maintained through the use of very long base interferometry. Results indicate that the hydrogen masers perform well within the required specifications.
Some fundamental principles for developing computer programs which are compatible with array-oriented computers are presented. The emphasis is on basic techniques for structuring computer codes which are applicable in FORTRAN and do not require a special programming language or exact a significant penalty on a scalar computer. Researchers who are using numerical techniques to solve problems in engineering can apply these basic principles and thus develop transportable computer programs (in FORTRAN) which contain much vectorizable code. The vector architecture of the ASC is discussed so that the requirements of array processing can be better appreciated. The "vectorization" of a finite-difference viscous shock-layer code is used as an example to illustrate the benefits and some of the difficulties involved. Increases in computing speed with vectorization are illustrated with results from the viscous shock-layer code and from a finite-element shock tube code. The applicability of these principles was substantiated through running programs on other computers with array-associated computing characteristics, such as the Hewlett-Packard (H-P) 1000-F.
Collection of computer programs assembled for use in water-analysis laboratories. First program calculates quality-control parameters used in routine water analysis. Second calculates line of best fit for standard concentrations and absorbances entered. Third calculates specific conductance from conductivity measurement and temperature at which measurement taken. Fourth calculates any one of four types of residue measured in water. Fifth, sixth, and seventh calculate results of titrations commonly performed on water samples. Eighth converts measurements, to actual dissolved-oxygen concentration using oxygen-saturation values for fresh and salt water. Ninth and tenth perform calculations of two other common titrimetric analyses. Eleventh calculates oil and grease residue from water sample. Last two use spectro-photometric measurements of absorbance at different wavelengths and residue measurements. Programs included in collection written for Hewlett-Packard 2647F in H-P BASIC.
Recent advances in FEM techniques and applications are discussed in reviews and reports presented at the ICASE/LaRC workshop held in Hampton, VA in July 1986. Topics addressed include FEM approaches for partial differential equations, mixed FEMs, singular FEMs, FEMs for hyperbolic systems, iterative methods for elliptic finite-element equations on general meshes, mathematical aspects of FEMS for incompressible viscous flows, and gradient weighted moving finite elements in two dimensions. Consideration is given to adaptive flux-corrected FEM transport techniques for CFD, mixed and singular finite elements and the field BEM, p and h-p versions of the FEM, transient analysis methods in computational dynamics, and FEMs for integrated flow/thermal/structural analysis.
New developments in a posteriori error estimates, smart algorithms, and h- and h-p adaptive finite element methods are discussed in the context of two- and three-dimensional compressible and incompressible flow simulations. Applications to rotor-stator interaction, rotorcraft aerodynamics, shock and viscous boundary layer interaction and fluid-structure interaction problems are discussed.
Currently the Pioneer 10 and Voyager 1 and 2 spacecraft are beyond the orbit of Pluto, traveling outward from the Sun. Each is capable of detecting ultraviolet radiation resonantly scattered from hydrogen and helium atoms in the heliosphere and local interstellar medium. These observations are particularly well suited for the investigation of the large-scale heliospheric H and He distributions because the Voyager spacecraft are heading upstream, into the direction of local interstellar flow, whereas Pioneer 10 is heading downstream. Observations of the brightest resonance line, H Lyman-alpha, reveals that beyond about 20 AU from the Sun, upstream intensities decrease less quickly as a function of solar distance than downstream intensities. This implies that the heliospheric H distributions in the upstream and downstream directions are significantly different. Heliospheric H atoms originate in the local interstellar flow, and must penetrate through the heliospheric interface, where they are subject to charge exchange collisions with solar wind and interstellar protons. Models indicate that this process is probably responsible for the upstream/downstream difference in H Lyman-alpha. In addition, a recent spectroscopic determination of the H atom velocity distribution in the inner heliosphere implies a significant deceleration in the bulk flow speed of the heliospheric hydrogen gas relative to the helium flow, an effect that is also likely due to H-p charge exchange occurring in the upstream heliospheric interface region. In this presentation, recent heliospheric resonance line observations and their interpretations will be reviewed, focusing on their sensitivity to large-scale heliospheric structure.
Finite element and finite volume methods are used in a variety of design simulations when it is necessary to compute fields throughout regions that contain varying materials or geometry. Convergence of the simulation can be assessed by uniformly increasing the mesh density until an observable quantity stabilizes. Depending on the electrical size of the problem, uniform refinement of the mesh may be computationally infeasible due to memory limitations. Similarly, depending on the geometric complexity of the object being modeled, uniform refinement can be inefficient since regions that do not need refinement add to the computational expense. In either case, convergence to the correct (measured) solution is not guaranteed. Adaptive mesh refinement methods attempt to selectively refine the region of the mesh that is estimated to contain proportionally higher solution errors. The refinement may be obtained by decreasing the element size (h-refinement), by increasing the order of the element (p-refinement) or by a combination of the two (h-p refinement). A successful adaptive strategy refines the mesh to produce an accurate solution measured against the correct fields without undue computational expense. This is accomplished by the use of a) reliable a posteriori error estimates, b) hierarchal elements, and c) automatic adaptive mesh generation. Adaptive methods are also useful when problems with multi-scale field variations are encountered. These occur in active electronic devices that have thin doped layers and also when mixed physics is used in the calculation. The mesh needs to be fine at and near the thin layer to capture rapid field or charge variations, but can coarsen away from these layers where field variations smoothen and charge densities are uniform. This poster will present an adaptive mesh refinement package that runs on parallel computers and is applied to specific microelectronic device simulations. Passive sensors that operate in the infrared portion of the spectrum as well as active device simulations that model charge transport and Maxwell's equations will be presented.
A unique exercise facility has been developed and used to perform tipper body ergometry tests for space applications. Originally designed to simulate the muscular, cardiovascular and thermoregulatory responses to working in zero gravity, this facility may be used to conduct basic thermoregulatory investigations applicable to multiple sclerosis patients. An environmental chamber houses the tipper body ergometer and permits control of temperature, air now and humidify. The chamber is a closed system and recirculate-s air after conditioning if. A Cybex Lipper body ergometer has been mounted horizontally on the wall of the environmental chamber. In this configuration, the subject lies underneath the arm crank on a supine seat in order to turn the crank. The supine seat can be removed in order to introduce other equipment into the chamber such as a stool to allow upright arm cranking, or a treadmill to allow walk-run experiments. Physiological and environmental signals are fed into a Strawberry Tree data acquisition system while being monitored and logged using the Workbench software program. Physiological monitoring capabilities include 3-lead EKG using an H-P patient monitor, 5 site skin temperature and core temperature using YSI thermistors, and O2 consumption and CO2 production using AMFTFK Applied Electrochemistry analyzers and sensors. This comprehensive data acquisition set tip allows for calculation of various thermoregulatory indices including heat storage, evaporative heat loss, latent heat loss, and metabolic rate. The current system is capable of adding more data acquisition channels if needed. Some potential studies that could be carried out using the facility include: 1) An investigation into the efficiency of cooling various segments of the body to lower Tc 1-2 F. 2) A series of heat and mass balance studies comparing various LCG configurations.
The Parallel Tools Consortium (or PTOOLS) was established in late 1993 in response to concerns that effective parallel tool development requires more input from the user community. The group consists of real users and real developers who actually talk to each other and who are completing the development of 100% user-specified tools. We have found that developers -- some from academia, others from the major HPC vendors (H-P/Convex, IBM, Intel, Meiko, SGI/Cray Research, etc.) and the government -- are very open to end-users' needs and opinions. The main activities of PTOOLS include the PTOOLS Annual Meeting and various sponsored projects.
In the proposal, the following three objectives are stated: (1) A p-version of the discontinuous Galerkin method for a one dimensional parabolic problem will be established. It should be recalled that the h-version in space was used for the discontinuous Galerkin method. An a priori error estimate as well as a posteriori estimate of this p-finite element discontinuous Galerkin method will be given. (2) The parameter alpha that describes the behavior double vertical line u(sub t)(t) double vertical line 2 was computed exactly. This was made feasible because of the explicitly specified initial condition. For practical heat transfer problems, the initial condition may have to be approximated. Also, if the parabolic problem is proposed on a multi-dimensional region, the parameter alpha, for most cases, would be difficult to compute exactly even in the case that the initial condition is known exactly. The second objective of this proposed research is to establish a method to estimate this parameter. This will be done by computing two discontinuous Galerkin approximate solutions at two different time steps starting from the initial time and use them to derive alpha. (3) The third objective is to consider the heat transfer problem over a two dimensional thin plate. The technique developed by Vogelius and Babuska will be used to establish a discontinuous Galerkin method in which the p-element will be used for through thickness approximation. This h-p finite element approach, that results in a dimensional reduction method, was used for elliptic problems, but the application appears new for the parabolic problem. The dimension reduction method will be discussed together with the time discretization method.
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