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Results for “Hodge decomposition”

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.

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9 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Imposing equilibrium on experimental 3-D stress fields using Hodge decomposition and FFT-based optimization

Here, we present a methodology to impose micromechanical constraints, i.e. stress equilibrium at grain and sub-grain scale, to an arbitrary (non-equilibrated) voxelized stress field obtained, for example, by means of synchrotron X-ray diffraction techniques. The method consists in finding the equilibrated stress field closest (in L 2 -norm sense) to the measured non-equilibrated stress field, via the solution of an optimization problem. The extraction of the divergence-free (equilibrated) part of a general (non-equilibrated) field is performed using the Hodge decomposition of a symmetric matrix field, which is the generalization of the Helmholtz decomposition of a vector field into the sum of an irrotational field and a solenoidal field. The combination of: a) the Euler–Lagrange equations that solve the optimization problem, and b) the Hodge decomposition, gives a differential expression that contains the bi-harmonic operator and two times the curl operator acting on the experimental stress field. These high-order derivatives can be efficiently performed in Fourier space. The method is applied to filter the non-equilibrated parts of a synthetic piecewise constant stress fields with a known ground truth, and stress fields in Gum Metal, a beta-Ti-based alloy measured in-situ using Diffraction Contrast Tomography (DCT). In both cases, the largest corrections were obtained near grain boundaries.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Vector Field Decompositions Using Multiscale Poisson Kernel

Extraction of multiscale features using scale-space is one of the fundamental approaches to analyze scalar fields. However, similar techniques for vector fields are much less common, even though it is well known that, for example, turbulent flows contain cascades of nested vortices at different scales. The challenge is that the ideas related to scale-space are based upon iteratively smoothing the data to extract features at progressively larger scale, making it difficult to extract overlapping features. Instead, we consider spatial regions of influence in vector fields as scale, and introduce a new approach for the multiscale analysis of vector fields. Rather than smoothing the flow, we use the natural Helmholtz-Hodge decomposition to split it into small-scale and large-scale components using progressively larger neighborhoods. Our approach creates a natural separation of features by extracting local flow behavior, for example, a small vortex, from large-scale effects, for example, a background flow. We demonstrate our technique on large-scale, turbulent flows, and show multiscale features that cannot be extracted using state-of-the-art techniques.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Unitarization from geometry

We study the perturbative unitarity of scattering amplitudes in general dimensional reductions of Yang–Mills theories and general relativity on closed internal manifolds. For the tree amplitudes of the dimensionally reduced theory to have the expected high-energy behavior of the higher-dimensional theory, the masses and cubic couplings of the Kaluza–Klein states must satisfy certain sum rules that ensure there are nontrivial cancellations between Feynman diagrams. These sum rules give constraints on the spectra and triple overlap integrals of eigenfunctions of Laplacian operators on the internal manifold and can be proven directly using Hodge and eigenfunction decompositions. One consequence of these constraints is that there is an upper bound on the ratio of consecutive eigenvalues of the scalar Laplacian on closed Ricci-flat manifolds with special holonomy. This gives a sharp bound on the allowed gaps between Kaluza–Klein excitations of the graviton that also applies to Calabi–Yau compactifications of string theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1987

Nonlinear beam kinematics by decomposition of the rotation tensor

A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion, extension, shear deformation, and warping of an initially curved and twisted beam.

Danielson, D. A.↗
NASA NTRS1987

Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor

A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion extension, shear deformation, and warping of an initially curved and twisted beam.

Danielson, D. A.↗
NASA NTRS1997

Sample Diluter for Detecting Hypergolic Propellants and Other Toxic or Hazardous Gases

Hardware was developed to dilute vapor samples of purged hypergolic propellants (with air) into the range of existing instruments for detection of such toxic vapors. Since these detectors are normally used to monitor at the threshold limit value (TLV), most do not have quantitative capability at percent levels which relate to lower explosion limit (LEL) and fire hazards. For example, the upper limits of Energetic Sciences (ESI) 6000 series detectors used at KSC are 200 parts per million (ppm) for monomethyl hydrazine (MMH) and 500 ppm for nitrogen dioxide (NO2) arising from decomposition of nitrogen tetroxide (N2O4). Orbiter Processing Facility (OPF) personnel servicing Shuttle thrusters need to measure up to 250 ppm MMH and 7500 ppm NO2 with portable, intrinsically safe instruments. Our objective was to quickly fabricate a sample diluter out of existing materials as a temporary measure while other parallel efforts were conducted to provide a commercial or in-house-developed instrument to detect high propellant levels. A 3 to 1 diluter would bring 500 ppm MMH into the range of the existing fuel ESI, and a 30 to 1 diluter would do the same for NO2. In this way, familiar equipment already available would be used, resulting in minimal paperwork, safety, and training impacts and low cost. An MMH vapor sample-diluter was constructed from a 1/4-inch Kynar tee, along with specially designed lengths of sample and dilution tubing. The sample line was 3 feet of Bev-A-Line 4, 1/4 inch tube leading to the straight run of the tee. The side run of the tee had a 17-inch length of Bev-A-Line 4, 1/4-inch tube, for nominal 3 to 1 dilution. A gas sample bag was prepared and assayed at 113 ppm MMH, and diluted vapor sarnples were assayed at 39.5 ppm, or a measured dilution of 2.9 to 1. For NO2, a 316 stainless steel (SS) 1/8-inch tee with 49.5 inches of coiled, 1/8-inch outside diameter (OD) 316 SS tubing was used as the sarnpling end of the dilution system. The side run of the tee was open. The measured dilution ratio, based on the input value of 6,480 ppm NO2 and the average output value of 233 ppm, was 28 to 1. Thus, sample-diluters were successful in diluting concentrated hypergolic propellant vapors, both MMH and N2O4, into the ranges of existing TLV detectors.

Barile, R. G.↗
NASA NTRS1997

Sample Diluter for Detecting Hypergolic Propellants and Other Toxic or Hazardous Gases

Hardware was developed to dilute vapor samples of purged hypergolic propellants (with air) into the range of existing instruments for detection of such toxic vapors. Since these detectors are normally used to monitor at the threshold limit value (TLV), most do not have quantitative capability at percent levels which relate to lower explosion limit (LEL) and fire hazards. For example, the upper limits of Energetic Sciences (ESI) 6000 series detectors used at KSC are 200 parts per million (ppm) for monomethyl hydrazine (MMH) and 500 ppm for nitrogen dioxide (NO2) arising from decomposition of nitrogen tetroxide (N2O4). Orbiter Processing Facility (OPF) personnel servicing Shuttle thrusters need to measure up to 250 ppm MMH and 7,500 ppm NO2 with portable, intrinsically safe instruments. Our objective was to quickly fabricate a sample diluter out of existing materials as a temporary measure while other parallel efforts were conducted to provide a commercial or in-house-developed instrument to detect high propellant levels. A 3 to 1 diluter would bring 500 ppm MMH into the range of the existing fuel ESI, and a 30 to 1 diluter would do the same for NO2. In this way, familiar equipment already available would be used, resulting in minimal paperwork, safety, and training impacts and low cost. An MMH vapor sample-diluter was constructed from a 1/4-inch Kynar tee, along with specially designed lengths of sample and dilution tubing. The sample line was 3 feet of Bev-A-Line 4, 1/4-inch tube leading to the straight run of the tee. The side run of the tee had a 17-inch length of Bev-A-Line 4, 1/4-inch tube, for nominal 3 to 1 dilution. A gas sample bag was prepared and assayed at 113 ppm ppm MMH, and diluted vapor samples were assayed at 39.5 ppm, or a measured dilution of 2.9 to 1. For NO2, a 316 stainless steel (SS) 1/8-inch tee with 49.5 inches of coiled, 1/8-inch outside diameter (OD) 316 SS tubing was used as the sampling end of the dilution system. The side run of the tee was open. The measured dilution ratio, based on the input value of 6,480 ppm NO2, and the average output value of 233 ppm, was 28 to 1. Thus, sample-diluters were successful in diluting concentrated hypergolic propellant vapors, both MMH and N2O4 into the ranges of existing TLV detectors.

Barile, R. G.↗