Search NASA⌕ Search

SEARCH · Search NASA

Results for “Cauchy”

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.

171 records · Page 10
All sourcesNASA NTRSDOE OSTI
NASA NTRS2001

Energy Stable Flux Formulas For The Discontinuous Galerkin Discretization Of First Order Nonlinear Conservation Laws

We consider the discontinuous Galerkin (DG) finite element discretization of first order systems of conservation laws derivable as moments of the kinetic Boltzmann equation. This includes well known conservation law systems such as the Euler For the class of first order nonlinear conservation laws equipped with an entropy extension, an energy analysis of the DG method for the Cauchy initial value problem is developed. Using this DG energy analysis, several new variants of existing numerical flux functions are derived and shown to be energy stable.

Barth, Timothy↗
NASA NTRS2004

High Order And High Resolution Methods For a Model CAA Problem

The initial value problem for the first order linear wave equation in one space dimension is treated for two cases with specified initial data and grid, and data from solutions at t = 400 and t = 800 are presented, as prescribed for Problem 1 in Category 1. Results are shown from computations with a sequence of recently developed high order and high resolution methods which combine Hermite interpolation, Cauchy-Kowaleskya recursion for time derivatives, and Taylor series time advancement. These methods have the same order of accuracy in time as in space. Results are shown from methods that range from third to nineteenth order. The stated problems with the prescribed coarse grid can be simulated with errors that are at the level of machine accuracy if the method is sufficiently high order. In addition, the growth of the maximum absolute error out to t = 100,000 is given for simulations with the stated problem data.

Goodrich, John W.↗
NASA NTRS2007

A Time-Accurate Upwind Unstructured Finite Volume Method for Compressible Flow with Cure of Pathological Behaviors

A time-accurate, upwind, finite volume method for computing compressible flows on unstructured grids is presented. The method is second order accurate in space and time and yields high resolution in the presence of discontinuities. For efficiency, the Roe approximate Riemann solver with an entropy correction is employed. In the basic Euler/Navier-Stokes scheme, many concepts of high order upwind schemes are adopted: the surface flux integrals are carefully treated, a Cauchy-Kowalewski time-stepping scheme is used in the time-marching stage, and a multidimensional limiter is applied in the reconstruction stage. However even with these up-to-date improvements, the basic upwind scheme is still plagued by the so-called "pathological behaviors," e.g., the carbuncle phenomenon, the expansion shock, etc. A solution to these limitations is presented which uses a very simple dissipation model while still preserving second order accuracy. This scheme is referred to as the enhanced time-accurate upwind (ETAU) scheme in this paper. The unstructured grid capability renders flexibility for use in complex geometry; and the present ETAU Euler/Navier-Stokes scheme is capable of handling a broad spectrum of flow regimes from high supersonic to subsonic at very low Mach number, appropriate for both CFD (computational fluid dynamics) and CAA (computational aeroacoustics). Numerous examples are included to demonstrate the robustness of the methods.

Loh, Ching Y.↗
NASA NTRS2012

Causal Correlation Functions and Fourier Transforms: Application in Calculating Pressure Induced Shifts

By adopting a concept from signal processing, instead of starting from the correlation functions which are even, one considers the causal correlation functions whose Fourier transforms become complex. Their real and imaginary parts multiplied by 2 are the Fourier transforms of the original correlations and the subsequent Hilbert transforms, respectively. Thus, by taking this step one can complete the two previously needed transforms. However, to obviate performing the Cauchy principal integrations required in the Hilbert transforms is the greatest advantage. Meanwhile, because the causal correlations are well-bounded within the time domain and band limited in the frequency domain, one can replace their Fourier transforms by the discrete Fourier transforms and the latter can be carried out with the FFT algorithm. This replacement is justified by sampling theory because the Fourier transforms can be derived from the discrete Fourier transforms with the Nyquis rate without any distortions. We apply this method in calculating pressure induced shifts of H2O lines and obtain more reliable values. By comparing the calculated shifts with those in HITRAN 2008 and by screening both of them with the pair identity and the smooth variation rules, one can conclude many of shift values in HITRAN are not correct.

correlation function↗
NASA NTRS2016

Irregular Mare Patches (IMPs): 100 Ma or 3 Ga?

IMPs exhibit a perplexing combination of characteristics that are consistent with either an approximately 100 Ma or 3 Ga formation. Dozens of small-area IMPs have crisp morphologies and crater size-frequency distributions (SFDs) that denote relatively recent geologic activity (less than 100 Ma); however, the apparently well-developed regolith on portions of the IMPs are in conflict with such a young age [1]. To test possible formation hypotheses (e.g., [1-5]), which range from ancient volcanism to contemporary outgassing, we examined IMP morphology at the meter-scale with LROC NAC images and derived elevation models. We focused on the largest IMPs (Ina, Sosigenes, Cauchy, Maskelyne, and Nubium), where contacts between deposits are best developed. Most of our observations are consistent with multiple generations of inflation and breakouts (or squeeze-ups) of basaltic lavas that were affected by local slopes. Some of the extrusions coalesced into larger mounds or filled pre-existing craters. We did not observe evidence of large-scale void space (e.g., fissures, fractures, linear depressions, or pits) within or beneath the mounds or rougher deposits (e.g., [5]). But, small-scale voids may be signified by isolated pitted textures. We also did not detect evidence of the cooling fractures or lava plates expected in young lava flows and observed in lunar impact melt deposits. The smooth texture of the mounds is enigmatic. Block-less craters suggest at least 5 m of friable or poorly-cohesive material (such as regolith), yet mound margins exhibit slopes greater than 30 deg requiring significant material strength. Blocks are not common on the mounds, but are sometimes excavated by impacts (usually excavated from beneath the mounds). The uneven deposits are equally enigmatic and texturally varied (blocky, pitted, and crenulated). They are deficient in superposed craters compared to the mounds. If the mounds are indeed of similar age to the rougher units, then their different superposed crater morphologies and SFDs need to be explained by factors other than their ages. Any mounds originally composed of friable surface materials would evolve differently from more coherent deposits (e.g., [6-7]).

Stopar, Julie↗
NASA NTRS2018

Acoustic Radiation Pressure

The theoretical foundation of acoustic radiation pressure in plane wave beams is reexamined. It is shown from finite deformation theory and the Boltzmann-Ehrenfest Adiabatic Principle that the Brillouin stress tensor (BST) is the radiation stress in Lagrangian coordinates (not Eulerian coordinates) and that the terms in the BST are not the momentum flux density and mean excess Eulerian stress but are simply contributions to the variation in the wave oscillation period resulting from changes in path length and true wave velocity, respectively, from virtual variations in the strain. It is shown that the radiation stress in Eulerian coordinates is the mean Cauchy stress (not the momentum flux density, as commonly assumed) and that Langevin's second relation does not yield an assessment of the mean Eulerian pressure, since the enthalpy used in the traditional derivations is a function of the thermodynamic tensions - not the Eulerian pressure. It is shown that the transformation between Lagrangian and Eulerian quantities cannot be obtained from the commonly-used expansion of one of the quantities in terms of the particle displacement, since the expansion provides only the difference between the value of the quantity at two different points in Cartesian space separated by the displacement. The proper transformation is obtained only by employing the transformation coefficients of finite deformation theory, which are defined in terms of the displacement gradients. Finite deformation theory leads to the result that for laterally unconfined, plane waves the Lagrangian and Eulerian radiation pressures are equal with the value (1/4)(2K) along the direction of wave propagation, where (K) is the mean kinetic energy density, and zero in directions normal to the propagation direction. This is contrary to the Langevin result that the Lagrangian radiation pressure in the propagation direction is equal to (2K) and the BST result that the Eulerian radiation pressure in that direction is the momentum flux density.

Finite deformation theory↗
NASA NTRSDate not supplied

Explicit Discontinuous Galerkin Methods for Conservation Laws

The two explicit DG methods in this study are based on a ‘predictor-corrector’ formulation, the first introduced by Lörcher, Gassner, and Munz (2007, 2008) called space–time expansion discontinuous Galerkin or STE-DG scheme, and the second, introduced independently by the author (Huynh 2006, 2013) called the upwind moment scheme. The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky (CK) procedure, which involves no interaction of the data among neighboring cells. The corrector step also shares the same space-time integration formulation and is where interaction of the data among neighboring cells takes place; the difference, however, is in how the resulting space-time volume integral is estimated. As a consequence of the different estimates, for the case of advection in one spatial dimension (1D), the moment scheme has a CFL (Courant-Friedrichs-Lewy) condition of 1 for all p and is accurate to order 2p+1, i.e., it possesses the super accuracy property, whereas the STE-DG method has a more restrictive CFL condition and is accurate to the expected order of p+1. For 1D advection, compared with the CFL conditions of 1/(2p+1) of standard RK-DG (Runge-Kutta) scheme where space and time discretization are of the same order, the moment scheme allows a significantly larger time step size. It also turns out that the scheme yields a result identical to Van Leer’s scheme III (1977), which amounts to shifting the data a distance of advection corresponding to the time step and projecting the result onto the space of polynomial solutions. Contrary to Van Leer’s approach, however, the space-time ‘predictor-corrector’ formulation facilitates extensions to the case of systems of equations. Concerning 2D extensions, in the case of advection, when the flow is along the diagonal direction, the CFL conditions for the moment schemes become restrictive as will be shown by Fourier (Von Neumann) stability and accuracy analyses. Since the moment scheme employs the right Radau points as collocation points in time, the method is closely related to the implicit Radau IIA scheme, which is stable for any time step size. The role of Radau IIA in relieving stability restriction for these explicit DG schemes remains to be explored

Discontinuous Galerkin↗
NASA NTRSDate not supplied

Comparison of Pre- and Post-Flight Estimations of Pitch Damping Coefficients for Unguided Entry Vehicles

Pre-flight predictions of pitch damping coefficient (C M q + C m ȧ ) curves are used in entry simulations to determine flight readiness and if any dynamic remediation strategies are required. Accurate characterization of these pitch damping coefficient curves is crucial for mission success. Historically, pre-flight estimations of these curves are derived experimentally either via ballistic range testing or with 1-degree-of-freedom (DOF) free-or forced-oscillation wind tunnel testing. With post-flight best-estimated trajectories, a fitting procedure as described in Karlgaard, et. al. is used to compare pre-and post-flight pitch damping coefficients. This technique relies on an assumed analytical form of the trajectory, a limited 3-DOF representation described in Schoenenberger and Queen, which results in an Euler-Cauchy solution form. This solution assumes constant density, thus limited amplitude multi-peak windows of the trajectory are considered to populate the pitch damping-amplitudes pace. The estimated post-flight pitch damping approximations and the pre-flight predictions shown are effective representations of pitch damping, or the integrated effect of the pitch damping coefficient over one pitch cycle at the corresponding cycle’s peak amplitude. Effective pitch damping coefficient curves provide more intuitive insights about where the vehicle is stable and permits readily processed comparisons between data-sets. It is important to note that most pitch damping curves, such as those reported in aero databases and utilized in flight mechanics trajectory simulations, are represented in instantaneous angle-of-attack space distinct from what is presented in the paper.

pitch damping coefficient↗