Control of time variable, nonlinear multivariable systems using Liapunov's direct method
Controlling time variable nonlinear multivariable systems using Lipunous direct method
MATRIX ALGEBRA↗
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Microscopic Lagrangian description of warm plasmas. I - Linear wave propagation. II - Nonlinear wave interactions
It is pointed out that the conventional iterative analysis of nonlinear plasma wave phenomena, which involves a direct use of Maxwell's equations and the equations describing the particle dynamics, leads to formidable theoretical and algebraic complexities, especially for warm plasmas. As an effective alternative, the Lagrangian method may be applied. It is shown how this method may be used in the microscopic description of small-signal wave propagation and in the study of nonlinear wave interactions. The linear theory is developed for an infinite, homogeneous, collisionless, warm magnetoplasma. A summary is presented of a perturbation expansion scheme described by Galloway and Kim (1971), and Lagrangians to third order in perturbation are considered. Attention is given to the averaged-Lagrangian density, the action-transfer and coupled-mode equations, and the general solution of the coupled-mode equations.
Kim, H.↗
Celestial mechanics with geometric algebra
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
Hestenes, D.↗
Physical optics analysis of a four-reflector antenna, part 1
Concern has been raised for the 64-m to 70-m antenna upgrade project that the 70-m system may experience greater S-band beam-pointing perturbations than the 64-m system. The S-band perturbations are due to minor (higher order) mode generation, causing subtle cross-polarization fields affecting beam pointing direction, as described herein. For the antennas in their present configuration (64 m), a slight S-band gain degradation of about 0.05 dB can be attributed to these effects. Therefore, a full physical optics analysis was performed for the present-day 64-m system, as described herein. The results were compared with past analyses and experimental observations in order to verify the algebra and computer code with the intent of deriving a valid analysis method for accurately analyzing the 70-m shaped dual reflector Cassegrainian antenna. The results of the new analysis appear to be in excellent agreement with previous analyses and experimental data.
Cha, A. G.↗
Physical optics analysis of a four-reflector antenna
Concern has been raised for the 64-m to 70-m antenna upgrade project that the 70-m system may experience greater S-band beam-pointing perturbations than the 64-m system. The S-band perturbations are due to minor (higher order) mode generation, causing subtle cross-polarization fields affecting beam pointing direction, as described herein. For the antennas in their present configuration (64 m), a slight S-band gain degradation of about 0.05 dB can be attributed to these effects. Therefore, a full physical optics analysis was performed for the present-day 64-m system, as described herein. The results were compared with past analyses and experimental observations in order to verify the algebra and computer code with the intent of deriving a valid analysis method for accurately analyzing the 70-m shaped dual reflector Cassegrainian antenna. The results of the new analysis appear to be in excellent agreement with previous analyses and experimental data.
Cha, A. G.↗
Workshop on Engineering Turbulence Modeling
Discussed here is the future direction of various levels of engineering turbulence modeling related to computational fluid dynamics (CFD) computations for propulsion. For each level of computation, there are a few turbulence models which represent the state-of-the-art for that level. However, it is important to know their capabilities as well as their deficiencies in order to help engineers select and implement the appropriate models in their real world engineering calculations. This will also help turbulence modelers perceive the future directions for improving turbulence models. The focus is on one-point closure models (i.e., from algebraic models to higher order moment closure schemes and partial differential equation methods) which can be applied to CFD computations. However, other schemes helpful in developing one-point closure models, are also discussed.
Povinelli, Louis A.↗
Comparison Between Experiment and Prediction for a Transonic Turbulent Separated Flow
Solutions of the time-dependent, mass-averaged Navier-Stokes equations are compared In detail with experimental results obtained on an axisymmetric "bump" model at a transonic Mach number that produced an extensive separated now region. In addition, an inverse boundary method is evaluated for this type of flow. The Cebeci-Smith algebraic and the Wilcox-Rubesin two-equation turbulence models used in the Navier-Stokes calculations both predict the maximum boundary-layer displacement thickness generated by the interaction reasonably well, with the details of the now best described with the two-equation formulation. However, both models predict a shock location substantially farther aft on the bump than observed experimentally. This error in shock location was slightly less with the two-equation model (0.12 chord compared with 0.16 chord). In the vicinity of the shock, the calculations predict a more rapid increase in turbulent shear stress than observed in the experimental results; this more rapid increase is believed to be the cause or the poor predictions in shock position.
Johnson, D. A.↗
OWL for space mission systems development at JPL with semantic architecture styles
This paper presents a perspective on the challenges involved in using OWL for representing the diverse heterogeneous viewpoints involved in the engineering and operation of space missions.
algebraic specification↗
Sparse Linear Algebra Toolkit for Computational Aerodynamics
Finding solutions to sparse linear systems of equations is an essential step in Computational Engineering applications of interest to NASA. Linear systems of equations are composed and solved in almost every computational engineering application. The characteristics of linear systems vary greatly from one application to another. Accordingly, there are a wide variety of methods for the solution of linear systems of equations. The operations and methods prepared by the authors are focused on linear systems of interest to NASA, primarily those associated with Computational Fluid Dynamics (CFD), Aeroelasticity, and Aeroacoustics. The Sparse Linear Algebra Toolkit (SLAT) is a coordinated collection of software featuring operations, methods, and data structures that are useful when solving sparse linear systems of equations on modern computer architectures. The implemented operations and methods are designed and tuned for parallelism in shared memory, in distributed memory, and across the hybrid combination of distributed-shared memory. The toolkit includes novel methods and implementations for modern architectures and facilitates development of new approaches for meeting NASA’s evolving computational engineering challenges using evolving computer architectures that are not available in vendor libraries. In this paper, significant features and interfaces within SLAT are presented and verified for simulations performed with NASA’s CFD solver, FUN3D. The runtime and scaling performance of the Generalized Minimum Residual (GMRES) method implemented in SLAT is analyzed for the linear subproblems within the solution of turbulent Navier-Stokes equations employed in the simulation of high-lift configurations. Prior to this work, the SPARSKIT GMRES implementation was the only Krylov subspace method available within FUN3D. A strong scaling study shows the SLAT GMRES implementation facilitates accurate Reynolds-averaged Navier-Stokes CFD solutions between 15% and 56% faster than the SPARSKIT GMRES implementation.
Stephen L Wood↗
Finite-analytic numerical solution of heat transfer in two-dimensional cavity flow
Heat transfer in cavity flow is numerically analyzed by a new numerical method called the finite-analytic method. The basic idea of the finite-analytic method is the incorporation of local analytic solutions in the numerical solutions of linear or nonlinear partial differential equations. In the present investigation, the local analytic solutions for temperature, stream function, and vorticity distributions are derived. When the local analytic solution is evaluated at a given nodal point, it gives an algebraic relationship between a nodal value in a subregion and its neighboring nodal points. A system of algebraic equations is solved to provide the numerical solution of the problem. The finite-analytic method is used to solve heat transfer in the cavity flow at high Reynolds number (1000) for Prandtl numbers of 0.1, 1, and 10.
Chen, C.-J.↗
Multigrid Methods for Fully Implicit Oil Reservoir Simulation
In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.
Molenaar, J.↗
A finite element procedure for nonlinear prebuckling and initial postbuckling analysis
A procedure cast in a form appropriate to the finite element method is presented for geometrically nonlinear prebuckling and postbuckling structural analysis, including the identification of snap-through type of buckling. The principal features of this procedure are the use of direct iteration for solution of the nonlinear algebraic equations in the prebuckling range, an interpolation scheme for determination of the initial bifurcation point, a perturbation method in definition of the load-displacement behavior through the postbuckling regime, and extrapolation in determination of the limit point for snap-through buckling. Three numerical examples are presented in illustration of the procedure and in comparison with alternative approaches.
Mau, S. T.↗
Equation solving program for aerodynamic lifting surface theory
A description of and user's manual are presented for one of a group of FORTRAN programs which, together, can be used for the analysis and design of wings in steady, subsonic flow according to a kernel function method lifting surface theory. This particular program is the one which solves the sets of simultaneous, linear, algebraic equations arising from the thin wing analysis. This program has the capability of striking out rows and columns of the aerodynamic influence matrix and rows of the associated boundary condition vectors (right hand sides). This capability significantly enhances the effectiveness of the kernel function method of lifting surface theory because studies of the convergence of solutions with the number of control points can be done with the calculation of only a single influence matrix.
Medan, R. T.↗
Algebraic grid generation with boundary orthogonality control
This paper presents a new approach in applying the blending function method to the generation of structured finite-difference meshes. It is shown that the surface equation developed by Coons (1967) can be easily adapted to generate two-dimensional meshes, and its extension to hyperspace allows the construction of three-dimensional meshes. The robust method in its basic form blends among the boundary curves/surfaces conforming to their intrinsic slopes. When parameter-controlled boundary-slope corrections are introduced, it is possible to establish varying degrees of orthogonality along the boundaries. Additional controls are made possible by moving points in the parameter space, which affects the movement of points in the physical space. The combination of these controls provides a simple and yet powerful means for exploiting the blending-function method to satisfy the needs of practical CFD applications. Results from a validation program for two-dimensions are shown for a variety of geometries.
Luh, Raymond Ching-Chung↗
Application of the implicit MacCormack scheme to the parabolized Navier-Stokes equations
MacCormack's implicit finite-difference scheme was used to solve the two-dimensional parabolized Navier-Stokes (PNS) equations. This method for solving the PNS equations does not require the inversion of block tridiagonal systems of algebraic equations and permits the original explicit MacCormack scheme to be employed in those regions where implicit treatment is not needed. The advantages and disadvantages of the present adaptation are discussed in relation to those of the conventional Beam-Warming scheme for a flat plate boundary layer test case. Comparisons are made for accuracy, stability, computer time, computer storage, and ease of implementation. The present method was also applied to a second test case of hypersonic laminar flow over a 15% compression corner. The computed results compare favorably with experiment and a numerical solution of the complete Navier-Stokes equations.
Lawrence, J. L.↗
Applications of symbolic computing methods to the dynamic analysis of large systems
Since the symbolic computing language is very well suited to the operations with algebraic equations, techniques use the transfer function concept as a tool for the analysis of large linear dynamic systems. Techniques were coded in the experimental symbolic computer language FORMAC. The first of these approaches, REDUCE 1, establishes the techniques and a computer program to symbolically reduce arbitrary block diagrams associated with large systems for desired transfer functions. Symbolic closed form solutions are determined in several forms including an expanded form in terms of the driving frequencies and system constants. Programs are also written to numerically evaluate the symbolic solutions. A second computer program, REDUCE 2, is also based on the use of symbolic computing methods and was written to accommodate large engineering systems.
Lorenzo, C. F.↗
Simulation of a steady-state integrated human thermal system.
The mathematical model of an integrated human thermal system is formulated. The system consists of an external thermal regulation device on the human body. The purpose of the device (a network of cooling tubes held in contact with the surface of the skin) is to maintain the human body in a state of thermoneutrality. The device is controlled by varying the inlet coolant temperature and coolant mass flow rate. The differential equations of the model are approximated by a set of algebraic equations which result from the application of the explicit forward finite difference method to the differential equations. The integrated human thermal system is simulated for a variety of combinations of the inlet coolant temperature, coolant mass flow rate, and metabolic rates. Two specific cases are considered: (1) the external thermal regulation device is placed only on the head and (2) the devices are placed on the head and the torso. The results of the simulation indicate that when the human body is exposed to hot environment, thermoneutrality can be attained by localized cooling if the operating variables of the external regulation device(s) are properly controlled.
Hsu, F. T.↗
Simulation of a steady-state integrated human thermal system.
The mathematical model of an integrated human thermal system is formulated. The system consists of an external thermal regulation device on the human body. The purpose of the device (a network of cooling tubes held in contact with the surface of the skin) is to maintain the human body in a state of thermoneutrality. The device is controlled by varying the inlet coolant temperature and coolant mass flow rate. The differential equations of the model are approximated by a set of algebraic equations which result from the application of the explicit forward finite difference method to the differential equations. The integrated human thermal system is simulated for a variety of combinations of the inlet coolant temperature, coolant mass flow rate, and metabolic rates.
Hsu, F. T.↗