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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Treatment of singularities in cracked bodies

Three-dimensional finite-element analyses of middle-crack tension (M-T) and bend specimens subjected to mode I loadings were performed to study the stress singularity along the crack front. The specimen was modeled using 20-node isoparametric elements. The displacements and stresses from the analysis were used to estimate the power of singularities using a log-log regression analysis along the crack front. The analyses showed that finite-sized cracked bodies have two singular stress fields of the form rho = C sub o (theta, z) r to the -1/2 power + D sub o (theta, phi) R to the lambda rho power. The first term is the cylindrical singularity with the power -1/2 and is dominant over the middle 96 pct (for Poisson's ratio = 0.3) of the crack front and becomes nearly zero at the free surface. The second singularity is a vertex singularity with the vertex point located at the intersection of the crack front and the free surface. The second term is dominant at the free surface and becomes nearly zero away from the the boundary layer. The thickness of the boundary layer depends on Poisson's ratio of the material and is independent of the specimen type. The thickness of the boundary layer varied from 0 pct to about 5 pct of the total specimen thickness as Poisson's ratio varied from 0.0 to 0.45. Because there are two singular stress fields near the free surface, the strain energy release rate (G) is an appropriate parameter to measure the severity of the crack.

Shivakumar, K. N.↗
NASA NTRS1990

Treatment of singularities in cracked bodies

Three-dimensional finite-element analyses of middle-crack tension (M-T) and bend specimens subjected to mode I loadings were performed to study the stress singularity along the crack front. The specimen was modeled using 20-node isoparametric elements. The displacements and stresses from the analysis were used to estimate the power of singularities using a log-log regression analysis along the crack front. The analyses showed that finite-sized cracked bodies have two singular stress fields of the form rho = C sub o (theta, z) r to the -1/2 power + D sub o (theta, phi) R to the lambda rho power. The first term is the cylindrical singularity with the power -1/2 and is dominant over the middle 96 pct (for Poisson's ratio = 0.3) of the crack front and becomes nearly zero at the free surface. The second singularity is a vertex singularity with the vertex point located at the intersection of the crack front and the free surface. The second term is dominant at the free surface and becomes nearly zero away from the boundary layer. The thickness of the boundary layer depends on Poisson's ratio of the material and is independent of the specimen type. The thickness of the boundary layer varied from 0 pct to about 5 pct of the total specimen thickness as Poisson's ratio varied from 0.0 to 0.45. Because there are two singular stress fields near the free surface, the strain energy release rate (G) is an appropriate parameter to measure the severity of the crack.

Shivakumar, K. N.↗
NASA NTRS1992

Singularities in the classical Rayleigh-Taylor flow: Formation and subsequent motion

The creation and subsequent motion of singularities of solution to classical Rayleigh-Taylor flow (two dimensional inviscid, incompressible fluid over a vacuum) are discussed. For a specific set of initial conditions, we give analytical evidence to suggest the instantaneous formation of one or more singularities at specific points in the unphysical plane, whose locations depend sensitively on small changes in initial conditions in the physical domain. One-half power singularities are created in accordance with an earlier conjecture; however, depending on initial conditions, other forms of singularities are also possible. For a specific initial condition, we follow a numerical procedure in the unphysical plane to compute the motion of a one-half singularity. This computation confirms our previous conjecture that the approach of a one-half singularity towards the physical domain corresponds to the development of a spike at the physical interface. Under some assumptions that appear to be consistent with numerical calculations, we present analytical evidence to suggest that a singularity of the one-half type cannot impinge the physical domain in finite time.

Tanveer, S.↗
NASA NTRS1993

Singularities in the classical Rayleigh-Taylor flow - Formation and subsequent motion

The creation and subsequent motion of singularities of solution to classical Rayleigh-Taylor flow (two dimensional inviscid, incompressible fluid over a vacuum) are discussed. For a specific set of initial conditions, we give analytical evidence to suggest the instantaneous formation of one or more singularities at specific points in the unphysical plane, whose locations depend sensitively on small changes in initial conditions in the physical domain. One-half power singularities are created in accordance with an earlier conjecture; however, depending on initial conditions, other forms of singularities are also possible. For a specific initial condition, we follow a numerical procedure in the unphysical plane to compute the motion of a one-half singularity. This computation confirms our previous conjecture that the approach of a one-half singularity towards the physical domain corresponds to the development of a spike at the physical interface. Under some assumptions that appear to be consistent with numerical calculations, we present analytical evidence to suggest that a singularity of the one-half type cannot impinge the physical domain in finite time.

Tanveer, S.↗
DOE OSTI2022

Linking the singularities of cosmological correlators

Much of the structure of cosmological correlators is controlled by their singularities, which in turn are fixed in terms of flat-space scattering amplitudes. An important challenge is to interpolate between the singular limits to determine the full correlators at arbitrary kinematics. This is particularly relevant because the singularities of correlators are not directly observable, but can only be accessed by analytic continuation. In this paper, we study rational correlators — including those of gauge fields, gravitons, and the inflaton — whose only singularities at tree level are poles and whose behavior away from these poles is strongly constrained by unitarity and locality. We describe how unitarity translates into a set of cutting rules that consistent correlators must satisfy, and explain how this can be used to bootstrap correlators given information about their singularities. We also derive recursion relations that allow the iterative construction of more complicated correlators from simpler building blocks. In flat space, all energy singularities are simple poles, so that the combination of unitarity constraints and recursion relations provides an efficient way to bootstrap the full correlators. In many cases, these flat-space correlators can then be transformed into their more complex de Sitter counterparts. As an example of this procedure, we derive the correlator associated to graviton Compton scattering in de Sitter space, though the methods are much more widely applicable.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Direct geometric probe of singularities in band structure

A quantum system’s energy landscape may have points where multiple energy surfaces are degenerate and that exhibit singular geometry of the wave function manifold, with major consequences for the system’s properties. Ultracold atoms in optical lattices have been used to indirectly characterize such points in the band structure. Here we measured the non-Abelian transformation produced by transport directly through the singularities. We accelerated atoms along a quasi-momentum trajectory that enters, turns, and then exits the singularities at linear and quadratic band-touching points of a honeycomb lattice. Measurements after transport identified the topological winding numbers of these singularities to be 1 and 2, respectively. Our work introduces a distinct method for probing singularities that enables the study of non-Dirac singularities in ultracold-atom quantum simulators.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1975

Finite element techniques applied to cracks interacting with selected singularities

The finite-element method for computing the extensional stress-intensity factor for cracks approaching selected singularities of varied geometry is described. Stress-intensity factors are generated using both displacement and J-integral techniques, and numerical results are compared to those obtained experimentally in a photoelastic investigation. The selected singularities considered are a colinear crack, a circular penetration, and a notched circular penetration. Results indicate that singularities greatly influence the crack-tip stress-intensity factor as the crack approaches the singularity. In addition, the degree of influence can be regulated by varying the overall geometry of the singularity. Local changes in singularity geometry have little effect on the stress-intensity factor for the cases investigated.

Conway, J. C.↗
NASA NTRS1991

Singularities in water waves and Rayleigh-Taylor instability

Singularities in inviscid two-dimensional finite-amplitude water waves and inviscid Rayleigh-Taylor instability are discussed. For the deep water gravity waves of permanent form, through a combination of analytical and numerical methods, results describing the precise form, number, and location of singularities in the unphysical domain as the wave height is increased are presented. It is shown how the information on the singularity in the unphysical region has the same form as for deep water waves. However, associated with such a singularity is a series of image singularities at increasing distances from the physical plane with possibly different behavior. Furthermore, for the Rayleigh-Taylor problem of motion of fluid over a vacuum and for the unsteady water wave problem, integro-differential equations valid in the unphysical region are derived, and how these equations can give information on the nature of singularities for arbitrary initial conditions is shown.

Tanveer, S.↗
NASA NTRS1992

The double universal joint wrist on a manipulator: Solution of inverse position kinematics and singularity analysis

This paper presents three methods to solve the inverse position kinematics position problem of the double universal joint attached to a manipulator: (1) an analytical solution for two specific cases; (2) an approximate closed form solution based on ignoring the wrist offset; and (3) an iterative method which repeats closed form position and orientation calculations until the solution is achieved. Several manipulators are used to demonstrate the solution methods: cartesian, cylindrical, spherical, and an anthropomorphic articulated arm, based on the Flight Telerobotic Servicer (FTS) arm. A singularity analysis is presented for the double universal joint wrist attached to the above manipulator arms. While the double universal joint wrist standing alone is singularity-free in orientation, the singularity analysis indicates the presence of coupled position/orientation singularities of the spherical and articulated manipulators with the wrist. The cartesian and cylindrical manipulators with the double universal joint wrist were found to be singularity-free. The methods of this paper can be implemented in a real-time controller for manipulators with the double universal joint wrist. Such mechanically dextrous systems could be used in telerobotic and industrial applications, but further work is required to avoid the singularities.

Williams, Robert L., III↗
NASA NTRS1991

Singularities in water waves and Rayleigh-Taylor instability

Singularities in inviscid two-dimensional finite-amplitude water waves and inviscid Rayleigh-Taylor instability are discussed. For the deep water gravity waves of permanent form, through a combination of analytical and numerical methods, results describing the precise form, number, and location of singularities in the unphysical domain as the wave height is increased are presented. It is shown how the information on the singularity in the unphysical region has the same form as for deep water waves. However, associated with such a singularity is a series of image singularities at increasing distances from the physical plane with possibly different behavior. Furthermore, for the Rayleigh-Taylor problem of motion of fluid over a vacuum and for the unsteady water wave problem, integro-differential equations valid in the unphysical region are derived, and how these equations can give information on the nature of singularities for arbitrary initial conditions is shown.

Tanveer, S.↗
NASA NTRS1992

Large-scale sparse singular value computations

Four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture are presented. Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left and right-singular vectors) for sparse matrices arising from two practical applications: information retrieval and seismic reflection tomography are emphasized. The target architectures for implementations are the CRAY-2S/4-128 and Alliant FX/80. The sparse SVD problem is well motivated by recent information-retrieval techniques in which dominant singular values and their corresponding singular vectors of large sparse term-document matrices are desired, and by nonlinear inverse problems from seismic tomography applications which require approximate pseudo-inverses of large sparse Jacobian matrices.

Berry, Michael W.↗
DOE OSTI2023

Spatially quasi-periodic bifurcations from periodic traveling water waves and a method for detecting bifurcations using signed singular values

We present a method of detecting bifurcations by locating zeros of a signed version of the smallest singular value of the Jacobian. This enables the use of quadratically convergent root-bracketing techniques or Chebyshev interpolation to locate bifurcation points. Only positive singular values have to be computed, though the method relies on the existence of an analytic or smooth singular value decomposition (SVD). The sign of the determinant of the Jacobian, computed as part of the bidiagonal reduction in the SVD algorithm, eliminates slope discontinuities at the zeros of the smallest singular value. We use the method to search for spatially quasi-periodic traveling water waves that bifurcate from large-amplitude periodic waves. The water wave equations are formulated in a conformal mapping framework to facilitate the computation of the quasi-periodic Dirichlet-Neumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravity-capillary waves. In both cases, the waves have two spatial quasi-periods whose ratio is irrational. We follow the secondary branches via numerical continuation beyond the realm of linearization about solutions on the primary branch to obtain traveling water waves that extend over the real line with no two crests or troughs of exactly the same shape. The pure gravity wave problem is of relevance to ocean waves, where capillary effects can be neglected. Such waves can only exist through secondary bifurcation as they do not persist to zero amplitude. The gravity-capillary wave problem demonstrates the effectiveness of using the signed smallest singular value as a test function for multi-parameter bifurcation problems. This test function becomes mesh independent once the mesh is fine enough.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Altermagnetism from coincident Van Hove singularities: application to κ -Cl

Realizing two-dimensional (2D) altermagnets is important for spintronics applications. Here we propose a microscopic template for stabilizing 2D altermagnetism through Van Hove singularities that are coincident in both energy and momentum. These coincident Van Hove singularities are a generic consequence of non-symmorphic symmetries in nine 2D space groups. Due to nontrivial symmetry properties of the Hamiltonian, these coincident Van Hove singularities allow new hopping interactions between the Van Hove singularities that do not appear in analogous Van Hove singularity based patch models for cuprates and graphene. We show these new interactions can give rise to various weak coupling, and BCS-based instabilities, including altermagnetism, nematicity, inter-band d-wave superconductivity, and orbital altermagnetic order. We apply our results to quasi-2D organic κ -Cl in which altermagnetism is known to appear.

electronic properties and materials↗
DOE OSTI2025

A Class of Sparse Johnson–Lindenstrauss Transforms and Analysis of their Extreme Singular Values

The Johnson–Lindenstrauss (JL) lemma is a powerful tool for dimensionality reduction in modern algorithm design. The lemma states that any set of high-dimensional points in a Euclidean space can be projected into lower dimensions while approximately preserving pairwise Euclidean distances. Random matrices satisfying this lemma are called JL transforms (JLTs). Inspired by existing $s$-hashing JLTs with exactly $s$ nonzero elements on each column, the present work introduces an ensemble of sparse matrices encompassing so-called $s$-hashing-like matrices whose expected number of nonzero elements on each column is $s$. The independence of the sub-Gaussian entries of these matrices and the knowledge of their exact distribution play an important role in their analyses. Using properties of independent sub-Gaussian random variables, these matrices are demonstrated to be JLTs, and their smallest nontrivial singular values and largest singular values are estimated nonasymptotically using a technique from geometric functional analysis. As the dimensions of the matrix grow to infinity, these singular values are proved to converge almost surely to fixed quantities (by using the universal Bai–Yin law) and in distribution to the Gaussian orthogonal ensemble Tracy–Widom law after proper rescalings. Understanding the behaviors of extreme singular values is important in general because they are often used to define a measure of stability of matrix algorithms. For example, JLTs were recently used in derivative-free optimization algorithmic frameworks to select random subspaces in which are constructed random models or poll directions to achieve scalability, and hence estimating their smallest singular value in particular helps determine the dimension of these subspaces.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Quantum fate of the Choptuik naked singularity

Classical critical collapse provides a dynamical route from smooth initial data to a naked singularity, representing a sharper violation of predictability than ordinary black hole singularities. We argue that this distinction is erased by quantum backreaction. Building on the semiclassical interior analysis, where quantum self-energy of the collapsing matter generates a universal growing mode and a finite mass gap, we study the exterior naked singularity region that determines global visibility in the Einstein-scalar system. We analyze controlled exterior models in both $2+1$ and $3+1$ dimensions. In the former, smooth matching and physical boundary conditions analytically select a vacuum polarization state, whose backreaction cloaks the classically naked region by a quantum trapped branch. In the latter, numerical horizon tracing shows that near a quantum-shifted threshold the exterior develops finite-mass marginally trapped surfaces rather than a zero-mass naked endpoint. These results suggest a global quantum picture in which the Choptuik naked singularity shares the fate of an ordinary black hole singularity: quantum effects push the putative Cauchy horizon behind a quantum-generated horizon, thereby reducing the loss of predictability to the standard black hole evaporation problem.

FOS: Physical sciences↗
NASA NTRS1979

Singular perturbations and the sounding rocket problem

In this paper, Goddard's problem of maximizing the final altitude of a sounding rocket (a singular problem of optimal control) is analyzed using singular perturbation methods. The problem is first cast in singular perturbation form and then solved to zero order by adding boundary-layer corrections to the reduced solution. For a quadratic drag law, a closed-form solution is obtained, although consideration of a numerical example indicates that this solution is not useful for practical sounding rockets. However, use of state variable transformations allows a very accurate numerical approximation to be constructed. It is concluded that application of singular perturbation methods to the well-known sounding rocket problem indicates that these methods may have utility in dealing with singular problems of optimal control.

Ardema, M. D.↗
NASA NTRS1980

Stress singularities at the vertex of a cylindrically anisotropic wedge

The plane elasticity problem for a cylindrically anisotropic solid is formulated. The form of the solution for an infinite wedge shaped domain with various homogeneous boundary conditions is derived and the nature of the stress singularity at the vertex of the wedge is studied. The characteristic equations giving the stress singularity and the angular distribution of the stresses around the vertex of the wedge are obtained for three standard homogeneous boundary conditions. The numerical examples show that the singular behavior of the stresses around the vertex of an anisotropic wedge may be significantly different from that of the isotropic material. Some of the results which may be of practical importance are that for a half plane the stress state at r = 0 may be singular and for a crack the power of stress singularity may be greater or less than 1/2.

Delale, F.↗
NASA NTRS1982

Application of matrix singular value properties for evaluating gain and phase margins of multiloop systems

A stability margin evaluation method in terms of simultaneous gain and phase changes in all loops of a multiloop system is presented. A universal gain-phase margin evaluation diagram is constructed by generalizing an existing method using matrix singular value properties. Using this diagram and computing the minimum singular value of the system return difference matrix over the operating frequency range, regions of guaranteed stability margins can be obtained. Singular values are computed for a wing flutter suppression and a drone lateral attitude control problem. The numerical results indicate that this method predicts quite conservative stability margins. In the second example if the eigenvalue magnitude is used instead of the singular value, as a measure of nearness to singularity, more realistic stability margins are obtained. However, this relaxed measure generally cannot guarantee global stability.

Mukhopadhyay, V.↗