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798 records · Page 45
All sourcesNASA NTRSDOE OSTI
NASA NTRS1997

The Topology of Symmetric Tensor Fields

Combinatorial topology, also known as "rubber sheet geometry", has extensive applications in geometry and analysis, many of which result from connections with the theory of differential equations. A link between topology and differential equations is vector fields. Recent developments in scientific visualization have shown that vector fields also play an important role in the analysis of second-order tensor fields. A second-order tensor field can be transformed into its eigensystem, namely, eigenvalues and their associated eigenvectors without loss of information content. Eigenvectors behave in a similar fashion to ordinary vectors with even simpler topological structures due to their sign indeterminacy. Incorporating information about eigenvectors and eigenvalues in a display technique known as hyperstreamlines reveals the structure of a tensor field. The simplify and often complex tensor field and to capture its important features, the tensor is decomposed into an isotopic tensor and a deviator. A tensor field and its deviator share the same set of eigenvectors, and therefore they have a similar topological structure. A a deviator determines the properties of a tensor field, while the isotopic part provides a uniform bias. Degenerate points are basic constituents of tensor fields. In 2-D tensor fields, there are only two types of degenerate points; while in 3-D, the degenerate points can be characterized in a Q'-R' plane. Compressible and incompressible flows share similar topological feature due to the similarity of their deviators. In the case of the deformation tensor, the singularities of its deviator represent the area of vortex core in the field. In turbulent flows, the similarities and differences of the topology of the deformation and the Reynolds stress tensors reveal that the basic addie-viscosity assuptions have their validity in turbulence modeling under certain conditions.

Levin, Yingmei↗
DOE OSTI2022

Imposition of local boundary conditions in peridynamics without a fictitious layer and unphysical stress concentrations

Here, this study introduces a general approach for the imposition of local boundary conditions in non- ordinary state-based peridynamics (NOSB PD) to eliminate the displacement kinks near the boundary without a fictitious layer under quasi-static loading conditions. It identifies the underlying reason for the unphysical displacement kinks. Under an imposed linear displacement field, the NOSB PD equilibrium equation is not satisfied near the boundary due to the unsymmetric horizon of material points. However, the equilibrium equation derived by using the PD differential operator is satisfied at such material points. Therefore, the material body is divided into three regions to satisfy the equilibrium equations and to impose displacement and tractions boundary conditions. This approach does not deviate from the original NOSB PD; however, it provides a simple solution to eliminate the displacement kink near the boundary, which leads to unphysical stress concentrations. Its efficacy is demonstrated by considering elastic rectangular and square plates subjected to various types of boundary conditions leading to homogeneous as well as nonhomogeneous deformations. The creep response of a rectangular plate further proves the robustness of the present approach. Also, a quasi-static crack propagation from a pre-existing crack in a square plate under mode-I, mode-II and mixed-mode loading conditions demonstrates its capability for failure prediction based on the critical stretch criteria. Finally, its applicability for 3D analysis is demonstrated by considering a rectangular prism under applied stretch and normal stress.

42 ENGINEERING↗
NASA NTRS2001

Spectral Measurements of Meteorite Powders: Implications for 433 Eros

One of the goals of the NEAR-Shoemaker mission to 433 Eros was to determine if it has a meteoritic analog. The primary means of making such a link are the X-ray/gamma-ray spectrometers, which measure elemental compositions of the surface, and the multi-spectral imager (MSI) and near-infrared spectrometer (NIS), which measure spectral reflectance. For determining meteoritic analogs using the X-ray/gamma-ray spectrometer data, the primary data used for comparison is the set of bulk chemical analyses of meteorites done by Jarosewich. These bulk chemical analyses were done on samples now found in the Smithsonian's Analyzed Meteorite Powder collection (USNM 7073). For determining meteoritic analogs using MSI/NIS spectral data, the primary data used for comparison is the set of meteoritic spectra compiled by Gaffey. To expand the set of meteoritic spectra available to the scientific community, we have initiated a spectral study of over 70 samples (primarily ordinary chondrites) found in the Smithsonian's Analyzed Meteorite Powder collection and an electron microprobe study of their corresponding thin sections. This set of spectral and compositional data should allow for better constraints on the distribution of meteorites in plots of band area ratios versus Band I centers and the usefulness of equations for deriving mineralogic compositions from band parameters. These spectral data can also be combined with previous spectral studies of other meteorite types such as the primitive achondrites, eucrites, and angrites to determine how useful the derived band parameters are for differentiating between different meteorite classes. These spectral data can also be used for testing the Modified Gaussian Model (MGM) for determining modal abundances and mafic mineral chemistries from reflectance spectra.

Burbine, T. H.↗
DOE OSTI2025

Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble

In this work we explore the capability of physics-informed neural networks (PINNs) to discover multiple solutions. Many real-world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty lies in providing appropriate initial conditions or guesses, as widely used time-marching schemes and Newton’s method are highly sensitive to these choices when solving complex computational problems. While machine learning models, particularly PINNs, have shown promise in solving DEs, their ability to capture multiple solutions remains underexplored. In this work, we propose a simple and practical approach using PINNs to learn and discover multiple solutions. We first demonstrate that PINNs, when combined with random initialization and deep ensemble method—originally developed for uncertainty quantification—can effectively uncover multiple solutions to nonlinear ordinary and partial DEs. Although training large ensembles of PINNs may appear computationally demanding, this can be done efficiently using vectorization techniques supported by modern deep learning frameworks, allowing many networks to be trained simultaneously. Our approach highlights the critical role of initialization in shaping solution diversity, addressing an often-overlooked aspect of machine learning for scientific computing. Furthermore, we propose utilizing PINN-generated solutions as initial conditions or initial guesses for conventional numerical solvers to enhance accuracy and efficiency in capturing multiple solutions. Extensive numerical experiments, including the Allen–Cahn equation and cavity flow, where our approach successfully identifies both stable and unstable solutions, validate the effectiveness of our method. These findings establish a general and efficient framework for addressing solution multiplicity in nonlinear DEs.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

An introduction to generalized functions with some applications in aerodynamics and aeroacoustics

In this paper, we start with the definition of generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support. The concept of generalization differentiation is introduced next. This is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some of the results of classical analysis, such as Leibniz rule of differentiation under the integral sign and the divergence theorem, are derived using the generalized function theory. It is shown that the divergence theorem remains valid for discontinuous vector fields provided that the derivatives are all viewed as generalized derivatives. This implies that all conservation laws of fluid mechanics are valid as they stand for discontinuous fields with all derivatives treated as generalized deriatives. Once these derivatives are written as ordinary derivatives and jumps in the field parameters across discontinuities, the jump conditions can be easily found. For example, the unsteady shock jump conditions can be derived from mass and momentum conservation laws. By using a generalized function theory, this derivative becomes trivial. Other applications of the generalized function theory in aerodynamics discussed in this paper are derivation of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of finite part of divergent integrals, derivation of Oswatiitsch integral equation of transonic flow, and analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics presented here include the derivation of the Kirchoff formula for moving surfaces,the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

Farassat, F.↗
NASA NTRS2003

Time as an Observable in Nonrelativistic Quantum Mechanics

The argument follows from the viewpoint that quantum mechanics is taken not in the usual form involving vectors and linear operators in Hilbert spaces, but as a boundary value problem for a special class of partial differential equations-in the present work, the nonrelativistic Schrodinger equation for motion of a structureless particle in four- dimensional space-time in the presence of a potential energy distribution that can be time-as well as space-dependent. The domain of interest is taken to be one of two semi-infinite boxes, one bounded by two t=constant planes and the other by two t=constant planes. Each gives rise to a characteristic boundary value problem: one in which the initial, input values on one t=constant wall are given, with zero asymptotic wavefunction values in all spatial directions, the output being the values on the second t=constant wall; the second with certain input values given on both z=constant walls, with zero asymptotic values in all directions involving time and the other spatial coordinates, the output being the complementary values on the z=constant walls. The first problem corresponds to ordinary quantum mechanics; the second, to a fully time-dependent version of a problem normally considered only for the steady state (time-independent Schrodinger equation). The second problem is formulated in detail. A conserved indefinite metric is associated with space-like propagation, where the sign of the norm of a unidirectional state corresponds to its spatial direction of travel.

Hahne, G. E.↗