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Report discusses spurious numerical solutions sometimes produced by explicit Runge-Kutta numerical integrations. Includes steady-state solutions, periodic orbits, and chaotic phenomena.
Limited word size in contemporary microprocessors causes numerical problems in autonomous satellite navigation applications. Numerical error introduced in navigation computations performed on small wordlength machines can cause divergence of sequential estimation algorithms. To insure filter reliability, square root algorithms have been adopted in many applications. The optimal navigation algorithm requires a careful match of the estimation algorithm, dynamic model, and numerical integrator. In this investigation, the relationship of several square root filters and numerical integration methods is evaluated to determine their relative performance for satellite navigation applications. The numerical simulations are conducted using the Phase I GPS constellation to determine the orbit of a LANDSAT-D type satellite. The primary comparison is based on computation time and relative estimation accuracy.
The analysis of lunar laser ranging data requires very accurate calculations of the lunar physical librations. Libration terms are given which arise from the additive and planetary terms in the lunar theory. The large size of the recently discovered terms due to third degree gravitational harmonics will allow some of these harmonics to be measured by laser ranging to the moon. Numerical integration promises to be an effective method of calculating librations. Comparison of numerical integrations with analytic series indicates that the calculation of the series due to third and fourth degree harmonics is not yet as accurate as the more extensively developed second degree terms.-
We describe how orbital tunnels could be used to transport payloads through the Earth. If you use a brachistochrone for the tunnel, the body forces in the tunnel become overwhelmingly large for small angular distances traveled. Projectiles move along an orbital tunnel faster than they would along abrachistochrone connecting the same points but the body force components cancel. We describe how parabolic Keplerian orbits outside the object merge onto quasi-Keplerian orbits inside the object. We use models of the interior of the Earth with three values of the polytropic index (n) to calculate interior or bits that travel between surface points. The n¼3 results are also scaled to the Sun. Numerical integrations of the equations describing polytropes were used to generate the initial models. Numerical integration of the equations of motion are then used to calculate the angular distance you can travel along the surface and the traversal time as a function of the parabolic periaps is distance for each model. Trajectories through objects of low central condensation show a focusing effect that decreases as the central condensation increases. Analytic solutions for the trajectories in a homogeneous sphere are derived and compared to the numeric results. The results can be scaled to other planets, stars, or even globular clusters.
Techniques are developed for the design of discrete highly distributed sensor systems for use in intelligent structures. First the functional requirements for such a system are presented. Discrete spatially averaging strain sensors are then identified as satisfying the functional requirements. A variety of spatial weightings for spatially averaging sensors are examined, and their wave number characteristics are determined. Preferable spatial weightings are identified. Several numerical integration rules used to integrate such sensors in order to determine the global deflection of the structure are discussed. A numerical simulation is conducted using point and rectangular sensors mounted on a cantilevered beam under static loading. Gage factor and sensor position uncertainties are incorporated to assess the absolute error and standard deviation of the error in the estimated tip displacement found by numerically integrating the sensor outputs. An experiment is carried out using a statically loaded cantilevered beam with five point sensors. It is found that in most cases the actual experimental error is within one standard deviation of the absolute error as found in the numerical simulation.
A numerical method for the determination of gravity anomalies from geoid heights is described using the inverse Stokes formula. This discrete form of the inverse Stokes formula applies a numerical integration over the azimuth and an integration over a cubic interpolatory spline function which approximates the step function obtained from the numerical integration. The main disadvantage of the procedure is the lack of a reliable error measure. The method was applied on geoid heights derived from GEOS-3 altimeter measurements in the calibration area of the GEOS-3 satellite.
Predictions of a free-piston Stirling engine model are compared with RE1000 engine test data taken at NASA-Lewis Research Center. The model validation and the engine testing are being done under a joint interagency agreement between the Department of Energy's Oak Ridge National Laboratory and NASA-Lewis. A kinematic code developed at Lewis was upgraded to permit simulation of free-piston engine performance; it was further upgraded and modified at Lewis and is currently being validated. The model predicts engine performance by numerical integration of equations for each control volume in the working space. Piston motions are determined by numerical integration of the force balance on each piston or can be specified as Fourier series. In addition, the model Fourier analyzes the various piston forces to permit the construction of phasor force diagrams. The paper compares predicted and experimental values of power and efficiency and shows phasor force diagrams for the RE1000 engine displacer and piston. Further development plans for the model are also discussed.
Predictions of a free-piston Stirling engine model are compared with RE1000 engine test data taken at NASA-Lewis Research Center. The model validation and the engine testing are being done under a joint interagency agreement between the Department of Energy's Oak Ridge National Laboratory and NASA-Lewis. A kinematic code developed at Lewis was upgraded to permit simulation of free-piston engine performance; it was further upgraded and modified at Lewis and is currently being validated. The model predicts engine performance by numerical integration of equations for each control volume in the working space. Piston motions are determined by numerical integration of the force balance on each piston or can be specified as Fourier series. In addition, the model Fourier analyzes the various piston forces to permit the construction of phasor force diagrams. The paper compares predicted and experimental values of power and efficiency and shows phasor force diagrams for the RE1000 engine displacer and piston. Further development plans for the model are also discussed.
A numerical integration is undertaken of the integral expression for the thermal conductivity due to collisions of relativistic degenerate electrons and compared to the limiting-case analytic formula. The integration is designed to encompass all temperature/electron-plasma-frequency temperature ratios between the melting temperature and the Fermi temperature. High accuracy fits are demonstrated by interpolating the values of the integrals of the function and by using an asymptotic function by Urpin and Yakovlev (1980). The numerically integrated expression compares favorably to the limiting-case analytic asymptotic formula by Urpin and Yakovlev, and the results are valid for temperatures above and below the electron-plasma-frequency temperature. The present techniques can be used in stellar opacity calculations and in the study of the propagation of deflagration fronts of compact stellar remnants.
Two viscous-flow computer codes compared by applying them to five test cases of steady-state transonic viscous flows about transonic airfoils. Two codes were: FLOMG, which solves Navier-Stokes differential equations of flow by implementing explicit, Runge-Kutta, finite-volume, multigrid numerical-integration algorithm; and ARC2D, which implements implicit, finite-difference, approximate-factorization, eigenvector-diagonalization numerical-integration algorithm.
This paper introduces the implementation of electromagnetic transient (EMT) simulations of a photovoltaic (PV) inverter module using the Implicit-Explicit (ImEx) solver in the Suite of Nonlinear and Differential/Algebraic Equation Solvers (SUNDIALS). This study demonstrates the effectiveness of the ImEx solver in overcoming the challenges inherent in simulating the complex dynamics of PV inverter modules. Furthermore, using SUNDIALS’ ImEx solver module ARKODE for EMT simulation automates key aspects of the process, such as numerical integration, providing substantial benefits including enhanced consistency, faster implementation, reduced human error, and the capability to handle the complexities of advanced numerical integration. By conducting comparative simulations with an implicit method used in commercial software, the research showcases the ImEx solver’s capability in achieving high accuracy and reliability. Results indicate that leveraging the ImEx approach significantly enhances modeling fidelity and reduces simulation setup times, offering a promising tool for the EMT analysis of PV inverter systems in power electronics-dominated power grids.
The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.
Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical solution to the problem on N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The derivation of the second-order solution is summarized by showing the essential steps, some in functional form. The general asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-earth, and interplanetary solutions. The results show that the accuracies of the asymptotic solutions range from an order of magnitude better than conic approximations to that of numerical integration itself. Also, since no iterations are required, the asymptotic boundary value solutions are obtained in a fraction of the time required for comparable numerically integrated solutions. The subject of minimizing the second-order error is discussed, and recommendations made for further work directed toward achieving a uniform accuracy in all applications.
The numerical integration of ordinary differential equations (ODEs) is common in trajectory optimization. Previous studies have shown that the use of adaptive-step Runge-Kutta integration schemes can introduce errors in the derivatives due to non-differentiable behavior within the integration scheme itself. The authors previously demonstrated the use of an implicit adaptation to the Runge-Kutta methods to mitigate this issue, but at a significant computational cost. In this work, the authors demonstrate an explicit adaptation to the Runge-Kutta methods to address the issue that does not incur a significant computational cost.
This paper identifies and analyzes frozen orbits for use by the Topography Experiment (TOPEX) mission. Frozen orbits are characterized by almost no long-term change in eccentricity or argument of periapsis. The standard method of frozen orbit prediction is shown to be inadequate for TOPEX inclinations due to the effect of higher degree zonal harmonic gravity terms. A method is described from which long-term motion in mean eccentricity, argument of periapsis, and inclination can be predicted without numerical integration and is used to locate frozen orbits. A zonal gravity field of degree 13 is shown to be necessary and sufficient for TOPEX frozen orbit prediction. Results are verified by numerical integration methods. Frozen orbits are available for most TOPEX orbits under consideration and short-term altitude variations and maximum altitude rates are specified. Alternatives to the frozen orbit which may result in lower altitude variations and rates are briefly examined.
Preliminary work about intrinsic numerical integrators evolving on groups is described. Fix a finite dimensional Lie group G; let g denote its Lie algebra, and let Y(sub 1),...,Y(sub N) denote a basis of g. A class of numerical algorithms is presented that approximate solutions to differential equations evolving on G of the form: dot-x(t) = F(x(t)), x(0) = p is an element of G. The algorithms depend upon constants c(sub i) and c(sub ij), for i = 1,...,k and j is less than i. The algorithms have the property that if the algorithm starts on the group, then it remains on the group. In addition, they also have the property that if G is the abelian group R(N), then the algorithm becomes the classical Runge-Kutta algorithm. The Cayley algebra generated by labeled, ordered trees is used to generate the equations that the coefficients c(sub i) and c(sub ij) must satisfy in order for the algorithm to yield an rth order numerical integrator and to analyze the resulting algorithms.
Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical boundary value solution applicable to a broad class of trajectory problems. In addition, the earlier first-order solutions have been extended to second-order to determine if improved accuracy is possible. Comparisons between the asymptotic solution and numerical integration for several lunar and interplanetary trajectories show that the asymptotic solution is generally quite accurate. Also, since no iterations are required, a solution to the boundary value problem is obtained in a fraction of the time required for numerically integrated solutions.
A software library has been developed to enable high-fidelity computational simulation of the dynamics of multiple spacecraft distributed over a region of outer space and acting with a common purpose. All of the modeling capabilities afforded by this software are available independently in other, separate software systems, but have not previously been brought together in a single system. A user can choose among several dynamical models, many high-fidelity environment models, and several numerical-integration schemes. The user can select whether to use models that assume weak coupling between spacecraft, or strong coupling in the case of feedback control or tethering of spacecraft to each other. For weak coupling, spacecraft orbits are propagated independently, and are synchronized in time by controlling the step size of the integration. For strong coupling, the orbits are integrated simultaneously. Among the integration schemes that the user can choose are Runge-Kutta Verner, Prince-Dormand, Adams-Bashforth-Moulton, and Bulirsh- Stoer. Comparisons of performance are included for both the weak- and strongcoupling dynamical models for all of the numerical integrators.