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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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DIANA, a digital-analog simulation program for IBM 1620 2 computer, simulates physical systems by solving the simultaneous differential equations describing the systems. It expands and optimizes the input-output capabilities, permits additional flexibility in midstream program alternation, and minimizes the computational time.
Concepts and results are presented in the fields of mathematical modeling, economics, and stability analysis. A coupled renewal-differential equation structure is presented as a modeling form for systems possessing hereditary characteristics, and this structure is applied to a model of the Austrian theory of business cycles. For realistic conditions, the system is shown to have an infinite number of poles, and conditions are presented which are both necessary and sufficient for all poles to lie strictly in the left half plane.
A digital local linearization technique was used to solve a system of stiff differential equations which simulate a magnetic bearing assembly. The results prove the technique to be accurate, stable, and efficient when compared to a general purpose variable order Adams method with a stiff option.
An explicit finite difference scheme consisting of a predictor and a corrector has been developed and applied to solve some hyperbolic partial differential equations (PDEs). The corrector is a convex-type function which is applied at each time level and at each mesh point. It consists of a parameter which may be estimated such that for larger time steps the algorithm should remain stable and generate a fast speed of convergence to the steady-state solution. Some examples have been given.
The finite element methods (FEM) have proved to be a powerful technique for the solution of boundary value problems associated with partial differential equations of either elliptic, parabolic, or hyperbolic type. They also have a good potential for utilization on parallel computers particularly in relation to the concept of domain decomposition. This report is intended as an introduction to the FEM for the nonspecialist. It contains a survey which is totally nonexhaustive, and it also contains as an illustration, a report on some new results concerning two specific applications, namely a free boundary fluid-structure interaction problem and the Euler equations for inviscid flows.
New Runge-Kutta-Nystrom formulas of the eighth, seventh, sixth, and fifth order are derived for the special second-order (vector) differential equation x = f (t,x). In contrast to Runge-Kutta-Nystrom formulas of an earlier NASA report, these formulas provide a stepsize control procedure based on the leading term of the local truncation error in x. This new procedure is more accurate than the earlier Runge-Kutta-Nystrom procedure (with stepsize control based on the leading term of the local truncation error in x) when integrating close to singularities. Two central orbits are presented as examples. For these orbits, the accuracy and speed of the formulas of this report are compared with those of Runge-Kutta-Nystrom and Runge-Kutta formulas of earlier NASA reports.
New coarse grid multigrid operators for problems with highly oscillatory coefficients are developed. These types of operators are necessary when the characters of the differential equations on coarser grids or longer wavelengths are different from that on the fine grid. Elliptic problems for composite materials and different classes of hyperbolic problems are practical examples. The new coarse grid operators can be constructed directly based on the homogenized differential operators or hierarchically computed from the finest grid. Convergence analysis based on the homogenization theory is given for elliptic problems with periodic coefficients and some hyperbolic problems. These are classes of equations for which there exists a fairly complete theory for the interaction between shorter and longer wavelengths in the problems. Numerical examples are presented.
This paper deals with the identification of spatially varying parameters in systems of finite spatial extent which can be described by second order hyperbolic differential equations. Two questions have been addressed. The first deals with 'partial identification' and inquires into the possibility of retrieving all the eigenvalues of the system from response data obtained at one location x-asterisk epsilon (0, 1). The second deals with the identification of the distributed coefficients rho(x), a(x) and b(x). Sufficient conditions for unique identification of all the eigenvalues of the system are obtained, and conditions under which the coefficients can be uniquely identified using suitable response data obtained at one point in the spatial domain are determined. Application of the results and their usefulness is demonstrated in the identification of the properties of tall building structural systems subjected to dynamic load environments.
Linear fractional transformation for obtaining rational approximations to response of physical system defined by second order nonlinear differential equation with constant coefficients
A study is made of the mathematical solution of the differential equation of motion of a test particle in the equatorial plane of the Kerr gravitational field, using S (Schwarzschild-like) coordinates. A qualitative solution of this equation leads to the conclusion that there can only be 25 different types of orbits. For each value of a, the results are presented in a master diagram for which h and e are the parameters. A master diagram divides the h, e parameter space into regions such that at each point within one of these regions the types of admissible orbits are qualitatively the same. A pictorial representation of the physical orbits in the r, phi plane is also given.
This book is based on a course presented at the Lewis Research Center for engineers and scientists who were interested in increasing their knowledge of differential equations. Those results which can actually be used to solve equations are therefore emphasized; and detailed proofs of theorems are, for the most part, omitted. However, the conclusions of the theorems are stated in a precise manner, and enough references are given so that the interested reader can find the steps of the proofs.
A gradient-based approach to multidisciplinary design optimization enables efficient scalability to large numbers of design variables. However, the need for derivatives causes difficulties when integrating ordinary differential equations (ODEs) in models. To simplify this, we propose the use of the general linear methods framework, which unifies all Runge-Kutta and linear multistep methods. This approach enables rapid implementation of integration methods without the need to differentiate each one, even in a gradient-based optimization context. We also develop a new parallel time integration algorithm that enables vectorization across time steps. We present a set of benchmarking results using a stiff ODE, a non-stiff nonlinear ODE, and an orbital dynamics ODE, and compare integration methods. In a modular gradient-based multidisciplinary design optimization context, we find that the new parallel time integration algorithm with high-order implicit methods, especially Gauss-Legendre collocation, is the best choice for a broad range of problems.
An algorithm for solving a large class of two- and three-dimensional nonseparable elliptic partial differential equations (PDE's) is developed and tested. It uses a modified D'Yakanov-Gunn iterative procedure in which the relaxation factor is grid-point dependent. It is easy to implement and applicable to a variety of boundary conditions. It is also computationally efficient, as indicated by the results of numerical comparisons with other established methods. Furthermore, the current algorithm has the advantage of possessing two important properties which the traditional iterative methods lack; that is: (1) the convergence rate is relatively insensitive to grid-cell size and aspect ratio, and (2) the convergence rate can be easily estimated by using the coefficient of the PDE being solved.
Two efforts to numerically simulate the flow about the Augmentor-Wing airfoil in the cruise configuration using the GRAPE elliptic partial differential equation grid generator algorithm are discussed. The Augmentor-Wing consists of a main airfoil with a slotted trailing edge for blowing and two smaller airfoils shrouding the blowing jet. The airfoil and the algorithm are described, and the application of GRAPE to an unsteady viscous flow simulation and a transonic full-potential approach is considered. The procedure involves dividing a complicated flow region into an arbitrary number of zones and ensuring continuity of grid lines, their slopes, and their point distributions across the zonal boundaries. The method for distributing the body-surface grid points is discussed.
Algorithms previously developed by the author give formulas which can be used for the efficient symbolic computation of series expansions to solutions of nonlinear systems of ordinary differential equations. As a by product of this analysis, formulas are derived which relate to trees to the coefficients of the series expansions, similar to the work of Leroux and Viennot, and Lamnabhi, Leroux and Viennot.
Techniques have been developed to determine in a systematic way the local behavior near constant solutions. Local integral manifolds play a very important role in this development, as they have also for ordinary differential equations. An attempt is made to indicate a few more applications of these methods to some problems in bifurcation in the spirit of Sotomayor (to appear) and to a growth model of Cooke and Yorke (to appear). It is also shown how to prove a theorem on stability under constantly acting disturbances using these methods.
Workload characterization is used for modeling and evaluating of computing systems at different levels of detail. We present workload characterization for a class of Computational Fluid Dynamics (CFD) applications that solve Partial Differential Equations (PDEs). This workload characterization focuses on three high performance computing platforms: SGI Origin2000, EBM SP-2, a cluster of Intel Pentium Pro bases PCs. We execute extensive measurement-based experiments on these platforms to gather statistics of system resource usage, which results in workload characterization. Our workload characterization approach yields a coarse-grain resource utilization behavior that is being applied for performance modeling and evaluation of distributed high performance metacomputing systems. In addition, this study enhances our understanding of interactions between PDE solver workloads and high performance computing platforms and is useful for tuning these applications.
The response of a clamped panel to supersonically convected turbulence is considered. A theoretical model in the form of an integro-differential equation is employed that takes into account the coupling between the panel motion and the surrounding acoustic medium. The kernels of the integrals, which represent induced pressures due to the panel motion, are Green's functions for sound radiations under various moving and stationary sources. An approximate analysis is made by following a finite-element Ritz-Galerkin procedure. Preliminary numerical results, in agreement with experimental findings, indicate that the acoustic damping is the controlling mechanism of the response.