Search NASA⌕ Search

SEARCH · Search NASA

Results for “derivative discontinuity”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Introduction to Generalized Functions with Applications in Aerodynamics and Aeroacoustics

Generalized functions have many applications in science and engineering. One useful aspect is that discontinuous functions can be handled as easily as continuous or differentiable functions and provide a powerful tool in formulating and solving many problems of aerodynamics and acoustics. Furthermore, generalized function theory elucidates and unifies many ad hoc mathematical approaches used by engineers and scientists. We define generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support, then introduce the concept of generalized differentiation. Generalized differentiation is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some results of classical analysis, are derived with the generalized function theory. Other applications of the generalized function theory in aerodynamics discussed here are the derivations of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of the finite part of divergent integrals, the derivation of the Oswatitsch integral equation of transonic flow, and the analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics include the derivation of the Kirchhoff formula for moving surfaces, the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

Farassat, F.↗
NASA NTRS2016

Unexpected Nonlinear Effects in Superconducting Transition-Edge Sensors

When a normal metal transitions into the superconducting state the DC resistance drops from a finite value to zero over some finite transition width in temperature, current, and magnetic field. Superconducting transition-edge sensors (TESs) operate within this transition region and uses resistive changes to measure deposited thermal energy. This resistive transition is not perfectly smooth and a wide range of TES designs and materials show sub-structure in the resistive transition (as seen in smooth nonmonotonic behavior, jump discontinuities, and hysteresis in the devices current-voltage relation and derivatives of the resistance with respect to temperature, bias current, and magnetic field). TES technology has advanced to the point where for many applications this structure is the limiting factor in performance and optimization consists of finding operating points away from these structures. For example, operating at or near this structure can lead to nonlinearity in the detectors response and gain scale, limit the spectral range of the detector by limiting the usable resistive range, and degrade energy resolution. The origin of much of this substructure is unknown. This presentation investigates a number of possible sources in turn. First we model the TES as a superconducting weak-link and solve for the characteristic differential equations current and voltage time dependence. We find:(1) measured DC biased current-voltage relationship is the time-average of a much higher frequency limit cycle solution.(2) We calculate the fundamental frequency and estimate the power radiated from the TES treating the bias leads as an antennae.(3) The solution for a set of circuit parameters becomes multivalued leading to current transitions between levels.(4)The circuit parameters can change the measure resistance and mask the true critical current. As a consequence the TES resistance surface is not just a function of temperature, current, and magnetic field but is also a function of the circuit elements (such as shunt resistor, SQUID inductance, and capacitor values). In other words, same device measured in different electrical circuits will have a different resistive surface in temperature, current, and magnetic field. Next we consider that at the transition temperature of a superconductor both the magnetic penetration depth and coherence length are divergent. As a consequence these important characteristic length scales are changing with operating point. We present measurements on devices showing commensurate behavior between these characteristic lengths and the length scale of added normal metal structures. Reordering of proximity vortices leads to discontinuities and irreversibility of the current-voltage curves. Last we consider a weak-link TES including both thermal activated resistance effects and the effect of the magnetic penetration depth being a function of temperature and magnetic field. We derive its impact on the resistive transition surface and the important device parameters a and b.

Superconducting Detectors↗
NASA NTRS1973

Further results on cooling mechanisms for soft X-ray flares

Analysis of data from the soft X-ray proportional counter spectrometer on OSO-4 has been continued along the lines of earlier work. It was noticed that the time profiles of X-ray counts from the OSO-4 instrument's 1 to 3 Angstrom detectors for two events showed a very rapid initial decay after maximum, followed by a much more slowly falling section. At least in the case of the larger of the two events, the change of slope seems quite definitely to be discontinuous. Temperatures and emission measures for both flares have been derived, by fitting an emission function to eight-channel count histograms which form the output of the 1 to 3 Angstrom detectors, analyzed into eight energy intervals by pulse-height analysis. The computer program is described. It employs temperature and continuum emission measures in the Culhane-Acton formula and the flux in the 6.7 KeV iron-line feature as free parameters, adjusting them by small amounts in successive iterations until the original historgram is approximately reproduced. A chi-squared is used to examine the agreement between histograms and terminates the iteration accordingly. Account is taken of the proportional counters' energy resolution in the fitting process.

Craig, I. J. D.↗
NASA NTRS1975

Heat transfer in laminar wall boundary layer within noncentered unsteady expansion wave

The paper summarizes results of theoretical and experimental investigations of wall temperature change and heat transfer in the laminar boundary layer formed within noncentered unsteady expansion waves in shock tubes. The study is restricted to a class of noncentered plane waves which have one key feature in common with centered waves, i.e., the first derivatives of the inviscid flow quantities have values or are discontinuous at the wavehead. The wall temperature is calculated implicitly by matching the local heat transfer rates of gas and wall thermal boundary layers. Experimentally, the wall temperature was measured using a thin-film resistance thermometer and heat transfer was then deduced using a one-dimensional unsteady heat conduction equation.

Srinivasan, G.↗
NASA NTRS1983

Sound diffraction at wall impedance discontinuities in a circular cylinder, investigated using Wiener-Hopf technique

Rigorous solutions are presented for sound diffraction in a circular cylinder with axial discontinuities of the wall admittance (or impedance). Analytical expressions are derived for the reflection and the transmission coefficients for duct modes. The results are discussed quantitatively in the limits of small admittance shifts (delta) and of low frequencies (ka). One of the results is the low frequency behavior of the reflection coefficient R(o) sub 00 of the fundamental mode. For the mode of a hardwall duct reflected from the junction with a softwall duct, (R(o) sub oo yields - (1-square root of (ka) square root of (2/i delta)); this result is in contrast to the frequency dependence of the reflection from the open end of a hardwall duct, for which R(o) sub oo yields - 1-(ka) squared/2 .

Cho, Y. C.↗
NASA NTRS1986

Necessary conditions for maximax problems with application to aeroglide of hypervelocity vehicles

This paper presents the necessary conditions for solving Chebyshev minimax (or maximax) problems with bounded control. The jump conditions obtained are applicable to problems with single or multiple maxima. By using Contensou domain of maneuverability, it is shown that when the maxima are isolated single points the control is generally continuous at the jump point in the minimax problems and discontinuous in the maximax problems in which the first time derivative of the maximax function contains the control variable. The theory is applied to the problem of maximizing the flight radius in a closed circuit glide of a hypervelocity vehicle and to a maximax optimal control problem in which the control appears explicitly with the first time derivative of the maximax function.

Vinh, N. X.↗
NASA NTRS1998

An Astronomical Test of CCD Photometric Precision

This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques. we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

Koch, David↗
NASA NTRS1999

A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Hyperbolic Systems

This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques, we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

Larson, Mats G.↗
NASA NTRS1984

Canonical solutions for unsteady flow fields

The initial value problem of one-dimensional gas-dynamics involving discontinuous, nonuniform initial data is discussed. Canonical solutions which are valid in a small x, t region aroung a discontinuity, and which include the first order effects of nonuniformities in the data, are derived explicitly. The theory is derived by considering a group of elementary piston problems. Solutions with a shock or with a centered expansion wave are worked out individually in order to relate initial flow properties and their gradients to the speed and acceleration of the discontinuity waves. They are then combined to represent the solution of a general initial value problem by regarding the piston path as a contact line. In addition, problems with chemical reaction are discussed in terms of elementary piston problems which involve strong detonation waves, Chapman-Jouguet detonation waves, and deflagration waves.

Liu, G. C.↗
NASA NTRS1983

Algebraic grid generation with corner singularities

A simple noniterative algebraic procedure is presented for generating smooth computational meshes on a quadrilateral topology. Coordinate distribution and normal derivative are provided on all boundaries, one of which may include a slope discontinuity. The boundary conditions are sufficient to guarantee continuity of global meshes formed of joined patches generated by the procedure. The method extends to 3-D. The procedure involves a synthesis of prior techniques stretching functions, cubic blending functions, and transfinite interpolation - to which is added the functional form of the corner solution. The procedure introduces the concept of generalized blending, which is implemented as an automatic scaling of the boundary derivatives for effective interpolation. Some implications of the treatment at boundaries for techniques solving elliptic PDE's are discussed in an Appendix.

Vinokur, M.↗
NASA NTRSDate not supplied

Development of a Consistent MODIS and VIIRS Cloud Detection Approach for CERES

A consistent cloud fraction record across various satellite platforms is essential for maintaining a long-term and stable climate data record of Earth's energy budget. With the Aqua satellite nearing the end of its operational lifetime, the continuation of this record relies on utilizing VIIRS observations from NOAA20 for cloud detection in NASA’s Clouds and Earth’s Radiant Energy System (CERES) project. However, integrating data from VIIRS and MODIS instruments poses challenges due to their distinct characteristics, such as varying spatial resolutions and different spectral channels. As a result, deriving consistent cloud properties from these two sensors without introducing artificial discontinuities in the time series remains a complex and challenging task. This paper will present progress toward developing a unified MODIS and VIIRS cloud mask using common channels to produce consistent cloud properties for CERES next edition (Ed5) Earth radiation budget data products. The fundamental approach taken in the CERES cloud mask is to compare the observed radiances to the expected background clear sky radiances. Therefore, one vital step is to compute clear sky radiances with a radiative transfer model that accurately accounts for satellite-specific, spectrally dependent surface reflectance, surface emission, and atmospheric absorption. Refined radiative transfer models and updated ancillary data inputs including surface emissivity maps, IGBP, snow and ice maps are incorporated into the processing framework to improve cloud detection consistency and accuracy. Pixel level cloud mask results and monthly global cloud fraction comparisons between MODIS and VIIRS will be presented to evaluate their consistency. Remaining challenges will be discussed. It is expected that this work will contribute consistent cloud properties for CERES that adequately bridges the MODIS and VIIRS imager data records.

CERES↗
NASA NTRS2013

High-Order Space-Time Methods for Conservation Laws

Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a Courant-Friedrichs-Lewy (CFL) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree p, then the time step sizes are typically proportional to 1/p(exp 2)). Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown

Huynh, H. T.↗
NASA NTRS2004

Error Analysis for Discontinuous Galerkin Method for Parabolic Problems

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.

Kaneko, Hideaki↗
NASA NTRS1993

Radiative transfer modelling of discontinuous tree canopies at microwave frequencies

Development of a first-order radiative transfer model for predicting backscatter from tree canopies has been underway at the University of Michigan Radiation Laboratory for some time. This model is known as the Michigan Microwave Canopy Scattering (MIMICS) model. This article presents the second-generation MIMICS model (MIMICS II) which accounts for canopies with discontinuous (open) crown layer geometries. MIMICS II models open crown layers by treating the location, size, and shape of the individual tree crowns as random variables. The backscattering coefficients for the canopy are then determined by introducing statistics derived from these parameters into the radiative transfer solution. Application of the radiative transfer equations to the discontinuous canopy geometry is presented. The resulting model is a robust fully polarimetric solution that is applicable over a wide variety of canopy architectures. Model simulations are compared to results generated with the continuous canopy model. The effect of the open crown geometry is found to be most significant at shallow incidence angles and at high frequencies for trees with well-developed crowns. Under these conditions, the gaps in the crown layer give rise to a notable increase in crown layer transmissivity which allows the radar to see through to the lower layers of the canopy more easily, thereby directly affecting the backscatter contribution of the trunks and ground.

Mcdonald, K. C.↗
NASA NTRS1985

Pulsar magnetospheres in binary systems

The criterion for stability of a tangential discontinuity interface in a magnetized, perfectly conducting inviscid plasma is investigated by deriving the dispersion equation including the effects of both gravitational and centrifugal acceleration. The results are applied to neutron star magnetospheres in X-ray binaries. The Kelvin-Helmholtz instability appears to be important in determining whether MHD waves of large amplitude generated by instability may intermix the plasma effectively, resulting in accretion onto the whole star as suggested by Arons and Lea and leading to no X-ray pulsar behavior.

Ershkovich, A. I.↗
NASA NTRS1991

Finite element computation of compressible flows with the SUPG formulation

Finite element computation of compressible Euler equations is presented in the context of the streamline-upwind/Petrov-Galerkin (SUPG) formulation. The SUPG formulation, which is based on adding stabilizing terms to the Galerkin formulation, is further supplemented with a shock capturing operator which addresses the difficulty in maintaining a satisfactory solution near discontinuities in the solution field. The shock capturing operator, which has been derived from work done in entropy variables for a similar operator, is shown to lead to an appropriate level of additional stabilization near shocks, without resulting in excessive numerical diffusion. An implicit treatment of the impermeable wall boundary condition is also presented. This treatment of the no-penetration condition offers increased stability for large Courant numbers, and accelerated convergence of the computations for both implicit and explicit applications. Several examples are presented to demonstrate the ability of this method to solve the equations governing compressible fluid flow.

Le Beau, G. J.↗
NASA NTRS1993

A thermodynamic analysis of propagating subcritical cracks with cohesive zones

The results of the so-called energetic approach to fracture with particular attention to the issue of energy dissipation due to crack propagation are applied to the case of a crack with cohesive zone. The thermodynamic admissibility of subcritical crack growth (SCG) is discussed together with some hypotheses that lead to the derivation of SCG laws. A two-phase cohesive zone model for discontinuous crack growth is presented and its thermodynamics analyzed, followed by an example of its possible application.

Allen, David H.↗