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DOE OSTI2022

Probing for the Trace Estimation of a Permuted Matrix Inverse Corresponding to a Lattice Displacement

We report thatpProbing is a general technique that is used to reduce the variance of the Hutchinson stochastic estimator for the trace of the inverse of a large, sparse matrix A. The variance of the estimator is the sum of the squares of the off-diagonal elements of A -1 . Therefore, this technique computes probing vectors that when used in the estimator annihilate the largest off-diagonal elements. For matrices that display decay of the magnitude of |A$^{-1}_{ij}$| with the graph distance between nodes i and j, this is achieved through graph coloring of increasing powers A k . Equivalently, when a matrix stems from a lattice discretization, it is computationally beneficial to find a distance-k coloring of the lattice. Previously, a hierarchical coloring was proposed so that k can be increased at runtime as needed without discarding previous work. In this work, we study probing for the more general problem of computing the trace of a permutation of A -1 , say PA -1 . The motivation comes from lattice quantum chromodynamics (QCD), where we need to construct “disconnected diagrams” to extract flavor-separated generalized parton functions. In lattice QCD, where the matrix has a four-dimensional toroidal lattice structure, these nonlocal operators correspond to a PA -1 , where P is the permutation relating to some displacement $\vec{p}$ in one or more dimensions. We focus on a single dimension displacement (p), but our methods are general. We show that probing on A k or (PA) k does not annihilate the largest magnitude elements. To resolve this issue, our displacement-based probing works on PA k using a new coloring scheme that works directly on appropriately displaced neighborhoods on the lattice. We prove lower bounds on the number of colors needed and study the effect of this scheme on variance reduction, both theoretically and experimentally on a real-world lattice QCD calculation. We achieve orders of magnitude speedup over the unprobed or the naively probed methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1972

Plasticity and the crack opening displacement in shells.

The plastic deformations and the crack opening displacement, delta, in cylindrical shells with an axial or circumferential crack and spherical shells with a meridional crack are considered. It is assumed that outside the perturbation zone of the crack the shell is subjected to uniform membrane loads perpendicular to the crack. The plastic strip model is used to calculate the plastic zone size. The crack opening displacement is calculated as the crack surface displacement at the crack tips by using the conventional superposition technique. In cylindrical shells with an axial crack the crack surface displacement perpendicular to the shell surface (i.e., the bulging) is also evaluated. The results are applied to a set of existing experimental data on the fracture of cylindrical shells. The tentative conclusion is that in dealing with the fracture of thin-walled structures, among various fracture models delta = constant appears to be the most satisfactory criterion.

Erdogan, F.↗

NASA NTRS1974

Effects of Mach number on pitot-probe displacement in a turbulent boundary layer

Experimental pitot-probe-displacement data have been obtained in a turbulent boundary layer at a local free-stream Mach number of 4.63 and unit Reynolds number of 6.46 million meter. The results of this study were compared with lower Mach number results of previous studies. It was found that small probes showed displacement only, whereas the larger probes showed not only displacement but also distortion of the shape of the boundary-layer profile. The distortion pattern occurred lower in the boundary layer at the higher Mach number than at the the lower Mach number. The maximum distortion occurred when the center of the probe was about one probe diameter off the test surface. For probes in the wall contact position, the indicated Mach numbers were, for all probes tested, close to the true profile. Pitot-probe displacement was found to increase significantly with increasing Mach number.

Allen, J. M.↗

NASA NTRS1980

Crack-opening displacement transducer

Crack-opening displacement transducer consists of 30 deg cone, coil spring, and linear-displacement transducer. Conical probe is used to measure crack opening. Cone is pressed firmly into crack by spring. As applied load causes crack to open up, cone is pushed further into it. Movement of cone, and thus crack growth, is monitored by linear-displacement transducer. Method gives more accurate measurement of crack-opening displacement of very narrow slots.

Simonds, R. A.↗

NASA NTRS1979

Mode 1 crack surface displacements for a round compact specimen subject to a couple and force

Mode I displacement coefficients along the crack surface are presented for a radially cracked round compact specimen, treated as a plane elastostatic problem, subjected to two types of loading; a uniform tensile stress and a nominal bending stress distribution across the net section. By superposition the resultant displacement coefficient or the corresponding influence coefficient can be obtained for any practical load location. Load line displacements are presented for A/D ratios ranging from 0.40 to 0.95, where A is the crack length measured from the crack mouth to the crack tip and D is the specimen diameter. Through a linear extrapolation procedure crack mouth displacements are also obtained. Experimental evidence shows that the results are valid over the range of A/D ratios analyzed for a practical pin loaded round compact specimen.

Gross, B.↗

NASA NTRS1979

Mode I crack surface displacements for a round compact specimen subject to a couple and force

Mode I displacement coefficients along the crack surface are presented for a radially cracked round compact specimen, treated as a plane elastostatic problem, subjected to two types of loading; a uniform tensile stress and a nominal bending stress distribution across the net section. By superposition the resultant displacement coefficient or the corresponding influence coefficient can be obtained for any practical load location. Load line displacements are presented for A/D ratios ranging from 0.40 to 0.95, where A is the crack length measured from the crack mouth to the crack tip and D is the specimen diameter. Through a linear extrapolation procedure crack mouth displacements are also obtained. Experimental evidence shows that the results of this study are valid over the range of A/D ratios analyzed for a practical pin loaded round compact specimen.

Gross, B.↗

NASA NTRS1979

Stress-intensity factors and crack-opening displacements for round compact specimens

A two dimensional, boundary collocation stress analysis was used to analyze various round compact specimens. The influence of the round external boundary and of pin-loaded holes on stress intensity factors and crack opening displacements was determined as a function of crack-length-to-specimen-width ratios. A wide-range equation for the stress intensity factors was developed. Equations for crack-surface displacements and load-point displacements were also developed. In addition, stress intensity factors were calculated from compliance methods to demonstrate that load-displacement records must be made at the loading points and not along the crack line for crack-length-to-specimen-width ratios less than about 0.4.

Newman, J. C., Jr.↗

NASA NTRS1981

Existence of axisymmetric modes of thin cylindrical shells with axial displacement restraint

A necessary and sufficient condition is derived for the existence of the axisymmetric mode of cylindrical shells with a radial displacement having one half wave along the axis and axial displacement restrained at both ends. The condition is purely geometric and consists of an upper bound on the mean radius of thickness ratio for a given fixed value of the length to mean radius ratio, above which the mode with one half wave in radial displacement along the axis ceases to exist. The proof is based on enforcing two basic lemmas concerned with the simplicity of the eigenvalues of the shell and the uniform ordering of these eigenvalues with the number of nodes of their corresponding radial displacement eigenfunctions.

El-Raheb, M.↗

NASA NTRS1981

Mixed models and reduced/selective integration displacement models for nonlinear analysis of curved beams

Simple mixed models are developed for use in the geometrically nonlinear analysis of deep arches. A total Lagrangian description of the arch deformation is used, the analytical formulation being based on a form of the nonlinear deep arch theory with the effects of transverse shear deformation included. The fundamental unknowns comprise the six internal forces and generalized displacements of the arch, and the element characteristic arrays are obtained by using Hellinger-Reissner mixed variational principle. The polynomial interpolation functions employed in approximating the forces are one degree lower than those used in approximating the displacements, and the forces are discontinuous at the interelement boundaries. Attention is given to the equivalence between the mixed models developed herein and displacement models based on reduced integration of both the transverse shear and extensional energy terms. The advantages of mixed models over equivalent displacement models are summarized. Numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed and to permit a comparison of their performance with that of other mixed models reported in the literature.

Noor, A. K.↗

NASA NTRS1982

Crack displacements for J/I/ testing with compact specimens

The suggestion is made that the standard compact specimen (with opening displacement measured at the crack mouth) may be entirely suitable for J-integral determinations if a very simple conversion factor is used. Experimental determination of J-integral values requires the measurement of displacements at the points of load application. For the compact specimen this is a difficult task. On the basis of studies reported by Newman (1979) and Fisher and Buzzard (1980), it is suggested that for any J-based test the standard compact specimen can be used. A very good approximation to the load point displacement (within 3.4 percent) can be obtained by measuring the crack mouth displacement and multiplying by 0.773.

Orange, T. W.↗

NASA NTRS1984

Nonlinear displacement analysis of advanced propeller structures using NASTRAN

The steady state displacements of a rotating advanced turboprop are computed using the geometrically nonlinear capabilities of COSMIC NASTRAN Rigid Format 4 and MSC NASTRAN Solution 64. A description of the modified Newton-Raphson algorithm used by Solution 64 and the iterative scheme used by Rigid Format 4 is provided. A representative advanced turboprop, SR3, was used for the study. Displacements for SR3 are computed for rotational speeds up to 10,000 rpm. The results show Solution 64 to be superior for computating displacements of flexible rotating structures. This is attributed to its ability to update the displacement dependent centrifugal force during the solution process.

Lawrence, C.↗

NASA NTRS1985

TM Geometric Performance: Line to Line Displacement Analysis (LLDA)

An early acquired TM data frame (track 23, frame 35 of 22 August 1982) was used to assess the geometric performance of the LANDSAT-4 thematic mapper instrument. The developed method consists in determining the displacements of adjacent lines by locating the maximum of the correlation profile between the two lines. Then the line to line displacement is used to characterize the TM instrument in respect to: scan to scan jitter and image skew, scan profile non-linearity, along scan jitter, detector to detector displacement, and band to band displacement. To verify the quality of the reported performance parameters, some extracted geometric parameters were used to correct the original raw data. Comparison between P-tape data and results of reprocessed data shows no significant differences in an area of 1024 x 1024 pixels.

Fusco, L.↗

NASA NTRS1986

Displacement thresholds in central and peripheral vision during tracking

The effects of stimulus duration and angular velocity on a subject's judgment of displacement threshold are examined. Twenty-six male subjects between 17-45 years with uncorrected 20:20 distance acuity and normal color perception and stereopsis studied a series of forced choice, paired comparison trials in which a long, thin, collimated horizontally oriented line moved downward through 12 angles ranging from 0.6-60 arcmin and judged which stimulus moved in each pair. The displacements were produced by 0.25, 0.5, 1, 2, and 4 sec stimulus duration and 2.5, 5, 10, and 15 deg/sec angular rates. Stimulus velocity, stimulus duration, mean threshold displacement, and mean confidence results are analyzed. It is observed that displacement judgment accuracy is increased with increasing stimulus duration. The data are compared with the results of Johnson and Leibowitz (1976) and Johnson and Scobey (1982), and good correlation with the Johnson and Leibowitz data is detected. The data reveal that threshold is based on a constant stimulus velocity over this range of durations and velocities. The data are applicable to the study of the final approach to landing of medium and large commercial jet aircraft.

Haines, R. F.↗

NASA NTRS1987

Calculations of the displacement damage and short-circuit current degradation in proton irradiated (AlGa)As-GaAs solar cells

A theoretical model for computing the displacement damage defect density and the short-circuit current (I sub sc) degradation in proton-irradiated (AlGa)As-GaAs p-n junction solar cells is presented. Assumptions were made with justification that the radiation induced displacement defects form an effective recombination center which controls the electron and hole lifetimes in the junction space charge region and in the n-GaAs active layer of the irradiated GaAs p-n junction cells. The degradation of I sub sc in the (AlGa)As layer was found to be negligible compared to the total degradation. In order to determine the I sub sc degradation, the displacement defect density, path length, range, reduced energy after penetrating a distance x, and the average number of displacements formed by one proton scattering event were first calculated. The I sub sc degradation was calculated by using the electron capture cross section in the p-diffused layer and the hole capture cross section in the n-base layer as well as the wavelength dependent absorption coefficients. Excellent agreement was found between the researchers calculated values and the measured I sub sc in the proton irradiated GaAs solar cells for proton energies of 100 KeV to 10 MeV and fluences from 10 to the 10th power p/square cm to 10 to the 12th power p/square cm.

Yeh, C. S.↗

NASA NTRS1987

Displacement surface calculations for a hypersonic aircraft

A method is presented to calculate the three-dimensional displacement surface about a hypersonic aircraft. This calculation requires a flow-field-solution to at least the thin-layer Navier-Stokes equations about the aircraft as input. An upwind, finite-volume code developed at NASA Langley Research Center was used to obtain the flow-field solution. The displacement surface is the three-dimensional counterpart to the two-dimensional displacement thickness. Flow-field solutions, along with the displacement surface calculations, are presented for a generic hypersonic aircraft at a Mach number of 24.5 and angle of attack of zero and one degree.

Richardson, Pamela F.↗

NASA NTRS1989

Integrated control and health monitoring capacitive displacement sensor development task. Orbit transfer rocket engine technology program

The development of a miniature multifunction turbomachinery shaft displacement sensor using state-of-the-art non-contract capacitive sensing technology is described. Axial displacement, radial displacement, and speed are sensed using a single probe within the envelope normally required for a single function. A survey of displacement sensing technology is summarized including inductive, capacitive, optical and ultrasonic techniques. The design and operation of an experimental triple function sensor is described. Test results are included showing calibration tests and simultaneous dynamic testing of multiple functions. Recommendations for design changes are made to improve low temperature performance, reliability, and for design of a flight type signal conditioning unit.

Collamore, Frank N.↗

NASA NTRS1993

Alternating prism exposure causes dual adaptation and generalization to a novel displacement

In two experiments, we examined the hypothesis that repeatedly adapting and readapting to two mutually conflicting sensory environments fosters the development of a separate adaptation to each situation (dual adaptation) as well as an increased ability to adapt to a novel displacement (adaptive generalization). In the preliminary study, subjects alternated between adapting their visuomotor coordination to 30-diopter prismatic displacement and readapting to normal vision. Dual adaptation was observed by the end of 10 alternation cycles. However, an unconfounded test of adaptive generalization was prevented by an unexpected prism-adaptive shift in preexposure baselines for the dual-adapted subjects. In the primary experiment, the subjects adapted and readapted to opposite 15-diopter displacements for a total of 12 cycles. Both dual adaptation and adaptive generalization to a 30-diopter displacement were obtained. These findings may be understood in terms of serial reversal learning and 'learning to learn'.

Welch, Robert B.↗

NASA NTRS1994

Analysis of coseismic surface displacement gradients using radar interferometry: New insights into the Landers earthquake

The map of the coseismic displacement field generated by interferometric processing of synthetic aperture radar (SAR) images taken before and after the June 28, 1992, Landers earthquake sequence brings new insights into the nature of deformation caused by these earthquakes. We use the interferometric map generated by Massonnet et al. (1993) to analyze the surface displacement field in the vicinity of the fault trace. Complexities in the fringe pattern near the fault reflect short-wavelength variations of the surface rupture and slip distribution, and attest to large displacement gradients. Along two sections of the fault, characteristic fringe patterns can be recognized, contrasting in density and direction with patterns observed away from the rupture. In order to understand the observed fringe patterns, we compute synthetic interferograms in three simple cases: (1) rigid-body rotations about a vertical axis, (2) about a horizontal axis (tilt), and (3) distributed, simple shear. The orientation and spatial separation of interferometric fringes predicted by these models help constrain near-field deformation and rupture parameters. Where the Kickapoo fault connects with the Homestead Valley fault, the interferogram shows a clear pattern of parallel N20 deg W fringes separated by about 160 m. This pattern and vertical offsets measured along the Kickapoo fault suggest that the block between this fault and the Johnson Valley fault may have been tilted, down to the west. A 5-km block lifted by 1 m on one side would be tilted by an angle of 0.01 deg (190 microrad), producing fringes separated by about 160 m, parallel to the tilt axis. Such a tilt, parallel to a N20 deg W direction, would account for the gradual, northward increase of the vertical slip component observed along the Kickapoo fault. This tilt may also explain the 1 m of reverse slip observed along the 'slip gap' section of the Homestead Valley break. Between the southern end of the Johnson Valley fault and the Eureka Peak fault, where no surface rupture has been mapped, the dense pattern of fringes implies distributed shear, probably resulting from fault slip at depth. The density and direction of the fringes in the gap are consistent with a right-lateral slip of 1.2-3.8 m on a blind fault locked above the depth of 1.5-2 km. Such observations of small wavelength features in the SAR interferogram bring new insights into the near-field displacement gradient and thus on response of the uppermost crust to seismic rupture.

Peltzer, Gilles↗