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The vibration in a flexible robot arm modeled by a moving slender prismatic beam is considered. It is found that the extending and contracting motions have destabilizing and stabilizing effects on the vibratory motions, respectively. The vibration analysis is based on a Galerkin approximation with time-dependent basis functions. Typical numerical results are presented to illustrate the qualitative features of vibrations.
The present computer algorithm for the time-accurate flow analysis of rotating turbomachines is based on the finite-volume method and employs a high-resolution approximate Riemann solver for interface flux definitions and an implicit numerical scheme that possesses apparent unconditional stability. Block-block interfaces, including dynamic ones, are treated in such a way as to mimic interior block communication. The turbomachine configurations treated by way of illustration are 8-8-bladed and 11-9-bladed versions of a contrarotating unducted fan engine.
A method for the approximate calculation of transonic flow over airfoils, including shock waves and viscous effects, is described. Numerical solutions are obtained by use of a computer program which is discussed in the appendix. The importance of including the boundary layer in the analysis is clearly demonstrated, as well as the need to improve on existing procedures near the trailing edge. Comparisons between calculations and experimental data are presented for both conventional and supercritical airfoils, emphasis being on the surface pressure distribution, and good agreement is indicated.
Optimal control of distributed parameter systems using multilevel techniques
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.
An analysis is presented of the effect of torsional flexibility on the rolling characteristics at supersonic speeds of tapered unswept wings with partial-span constant-percent-chord ailerons extending inboard from the wing tip. The geometric variables considered are aspect ratio, taper ratio, aileron span, and aileron chord. The shape of the wing-torsional-stiffness curve is assumed and the twisting moment is considered to result solely from the pressure distribution caused by aileron deflection, so that the necessity of using a successive-approximation method is avoided. Because of the complexity of the equations resulting from the analysis, numerical calculations from the equations are presented in a series of figures. A computational form is provided to be used in conjunction with these figures so that calculations can be made without reference to the analysis.
Deep neural networks have been extensively used for the solution of both the forward and the inverse problem for dynamical systems. However, their implementation necessitates optimizing a high-dimensional space of parameters and hyperparameters. This fact, along with the requirement of substantial computational resources, pose a barrier to achieving high numerical accuracy, but also interpretability. Here, to address the above challenges, we present Random Projection-based Operator Networks (RandONets): shallow networks with random projections and tailor-made numerical analysis methods that learn accurately and fast linear and nonlinear operators. Building on previous works, we prove that RandOnets are universal approximators of linear and nonlinear operators. Due to their simplicity, RandONets provide a one-step transformation of the input space, facilitating interpretability. For the evaluation of their performance, we focus on operators of PDEs. We show, that RandONets outperform by several orders of magnitude, both in terms of numerical approximation accuracy and computational cost, the “vanilla” DeepONets. Hence, we believe that our method will trigger further developments in the field of scientific machine learning, for the development of new ‘’light”schemes that will provide high accuracy while reducing dramatically the computational cost. A MATLAB toolbox for RandONets, including demos, is available on GitHub at https://github.com/GianlucaFabiani/RandONets.
An energy-conserving and an energy-and-enstrophy conserving numerical schemes are derived by approximating the Hamiltonian formulation of the inviscid shallow water flows based on the vorticity-divergence variables. These schemes also conserve the first-order moments such as mass and vorticity, as usual. The conservative properties of the schemes stem from the skew-symmetry and singularities of the Poisson brackets, which are carefully retained in the discrete approximations. Here, the schemes operate on unstructured orthogonal dual meshes, over bounded or unbounded domains, and they are also shown to possess the same optimal dispersive wave relations as those of the Z-grid scheme, which is a consequence of the use of the vorticity and divergence variables.
Analytic error bounds are presented for the solutions of approximate models for self-excited near-symmetric rigid bodies. The error bounds are developed for analytic solutions to Euler's equations of motion. The results are applied to obtain a simplified analytic solution for Eulerian rates and angles. The results of a sample application of the range and error bound expressions for the case of the Galileo spacecraft experiencing transverse torques demonstrate the use of the bounds in analyses of rigid body spin change maneuvers.
In this work, we will construct a simple method to be utilized in the interpretation of the properties of a thin plasma from a study of the transport of charged particles through it. We do so by first demonstrating how charged particle fluxes encode local plasma information in their spectra for sufficiently thick plasmas and then propose how to extend that analysis to the thin plasma limit where the extensive geometry cannot be neglected through arguments from scale separation. We provide a numerical treatment of the transport problem and demonstrate that it is in good agreement with the approximation we present. Finally, we utilize both the numerical solution and our novel approximation to study the impact of the extensive scale of the plasma on the yield and flux normalized high-energy neutron spectra resulting from the upscattering of charged fuel ions. Using this analysis, we show that (after controlling for fusion yield) for the same uniform densities and temperatures, larger plasmas have higher magnitude but softer reaction-in-flight (RIF) neutron spectra relative to smaller plasmas. This is because larger plasmas retain more knocked-on suprathermal ions within their bulk and can downscatter them to lower average energies, while smaller plasmas allow a larger fraction of high energy particles to “range out” of the system prior to substantial downscattering or inducing an RIF reaction.
The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations for initial-boundary-value problems of the hyperbolic type. For the numerical experiments, select the one-dimensional inviscid gas-dynamic equations in conservation-law form is selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit boundary schemes is emphasized. Some numerical examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit versus implicit boundary schemes, unconditionally stable schemes, and underspecification or overspecification of boundary conditions.
When measuring velocities in turbulent gas flow, approximation signal analysis with hot wire anemometers having one and two wire probes are used. A numeric test of standard analyses shows the resulting systemmatic error increases quickly with increasing turbulent intensity. Since it also depends on the turbulence structure, it cannot be corrected. The use of such probes is thus restricted to low turbulence. By means of three wire probes (in two dimensional flows with X wire probes) in principle, instantaneous values of velocity can be determined, and an asymmetric arrangement of wires has a theoretical advantage.
A research program is currently underway with the final objective of developing a fundamental understanding of the controlling mechanisms underlying the flammability diagrams of solid combustible materials and their derived fire properties. Given that there is a high possibility of an accidental fire occurring in a space-based facility, understanding the fire properties of materials that will be used in such facilities is of critical importance. With this purpose, the flammability diagrams of the materials, as those produced by the Lateral Ignition and Flame Spread Test (LIFT) apparatus and by a new forced flow device, the Forced Flow Ignition and Flame Spread Test (FIST) apparatus, will be obtained. The specific objective of the program is to apply the new flammability apparatus, which will more accurately reflect the potential ambient conditions of space-based environments, to the characterization of the materials for space applications. This paper presents a parametric study of oxidizer flow effects on the ignition curve of the flammability diagrams of PMMA. The dependence of the ignition delay time on the external radiant flux and either the sample width (LIFT) or the flow velocity (FIST) has been studied. Although preliminary, the results indicate that natural and forced convection flow changes, affect the characteristics of the ignition curves of the flammability diagrams. The major effect on the ignition time appears to be due to convective transfer variations at the fuel surface. At high radiant fluxes or high flow velocities, however, it appears that gas phase processes become increasingly important, affecting the overall ignition delay time. A numerical analysis of the solid fuel heating and pyrolysis has also been developed. The theoretical predictions approximate the experiments well for conditions in which the gas phase induction time is negligible.
Power spectra play an important role in the theory of inflation, and their ability to reproduce current observational data to high accuracy is often considered a triumph of inflation, largely because of a lack of credible alternatives. In previous work we introduced an alternative picture for the cosmological power spectra based on the nonperturbative features of the quantum version of Einstein’s gravity, instead of currently popular inflation models based on scalar fields. The key ingredients in this new picture are the appearance of a nontrivial gravitational vacuum condensate (directly related to the observed cosmological constant), and a calculable renormalization group running of Newton’s G on cosmological scales. More importantly, one notes the absence of any fundamental scalar fields in this approach. Results obtained previously were largely based on a semi-analytical treatment, and thus, while generally transparent in their implementation, often suffered from the limitations of various approximations and simplifying assumptions. In this work, we extend and refine our previous calculations by laying out an updated and extended analysis, which now utilizes a set of suitably modified state-of-the-art numerical programs (ISiTGR, MGCAMB and MGCLASS) developed for observational cosmology. As a result, we are able to remove some of the approximations employed in our previous studies, leading to a number of novel and detailed physical predictions. These should help in potentially distinguishing the vacuum condensate picture of quantum gravity from that of other models such as scalar field inflation. Here, besides the matter power spectrum P m (k) , we work out, in detail, predictions for what are referred to as the TT, TE, EE, BB angular spectra, as well as their closely related lensing spectra. However, the current limited precision of observational data today (especially on large angular scales) does not allow us yet to clearly prove or disprove either set of ideas. Nevertheless, by exploring in more details the relationship between gravity and cosmological matter and radiation both analytically and numerically, together with an expected future influx of increasingly accurate observational data, one can hope that the new quantum gravitational picture can be subjected to further stringent tests in the near future.
A general method of error analysis and correction is investigated for the discrete finite-element results for cylindrical shell structures. The method for error analysis is an adaptation of the method of successive approximation. When applied to the equilibrium equations of shell theory, successive approximations derive an approximate continuous solution from the discrete finite-element results. The advantage of this continuous solution is that it contains continuous partial derivatives of an order higher than the basis functions of the finite-element solution. Preliminary numerical results are presented in this paper for the error analysis of finite-element results for a postbuckled stiffened cylindrical panel modeled by a general purpose shell code. Numerical results from the method have previously been reported for postbuckled stiffened plates. A procedure for correcting the continuous approximate solution by Newton's method is outlined.
Nontrivial benchmark solutions are developed for the galactic ion transport (GIT) equations in the straight-ahead approximation. These equations are used to predict potential radiation hazards in the upper atmosphere and in space. Two levels of difficulty are considered: (1) energy independent, and (2) spatially independent. The analysis emphasizes analytical methods never before applied to the GIT equations. Most of the representations derived have been numerically implemented and compared to more approximate calculations. Accurate ion fluxes are obtained (3 to 5 digits) for nontrivial sources. For monoenergetic beams, both accurate doses and fluxes are found. The benchmarks presented are useful in assessing the accuracy of transport algorithms designed to accommodate more complex radiation protection problems. In addition, these solutions can provide fast and accurate assessments of relatively simple shield configurations.
An experimental study of low Reynolds number nozzle flow was performed. A brief comparison was made between some of the experimental performance data and performance predicted by a viscous flow code. The performance of 15, 20, and 25 deg conical nozzles, bell nozzles, and trumpet nozzles was evaluated with unheated nitrogen and hydrogen. The numerical analysis was applied to the conical nozzles only, using an existing viscous flow code that was based on a slender-channel approximation. Although the trumpet and 25 deg conical nozzles had slightly better performance at lower Reynolds numbers, it is unclear which nozzle is superior as all fell within the experimental error band. The numerical rssults were found to agree with experimental results for nitrogen and for some of the hydrogen data. Some code modification is recommended to improve confidence in the performance prediction.
Finite element analysis, applied to computational fluid dynamics (CFD) problem classes, presents a formal procedure for establishing the ingredients of a discrete approximation numerical solution algorithm. A classical Galerkin weak-statement formulation, formed on a Taylor series extension of the conservation law system, is developed herein that embeds a set of parameters eligible for constraint according to specification of suitable norms. The derived family of Taylor weak statements is shown to contain, as special cases, over one dozen independently derived CFD algorithms published over the past several decades for the high speed flow problem class. A theoretical analysis is completed that facilitates direct qualitative comparisons. Numerical results for definitive linear and nonlinear test problems permit direct quantitative performance comparisons.