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DOE OSTI2018

Singularly Perturbed System of Parabolic Equations in the Critical Case

We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudary-layer functions. The construction of the asymptotics is based on the regularization method for singularly perturbed problems developed by S. A. Lomov and adapted to singularly perturbed parabolic equations with two viscous boundaries by A. S. Omuraliev.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1977

Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domain

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eignevalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: the interior of a spherical triangle, and the exterior of a great circle arc extending for less than pi radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.

Walden, H.↗

NASA NTRS1987

A new formulation of electromagnetic wave scattering using an on-surface radiation boundary condition approach

A new formulation of electromagnetic wave scattering by convex, two-dimensional conducting bodies is reported. This formulation, called the on-surface radiation condition (OSRC) approach, is based upon an expansion of the radiation condition applied directly on the surface of a scatterer. It is now shown that application of a suitable radiation condition directly on the surface of a convex conducting scatterer can lead to substantial simplification of the frequency-domain integral equation for the scattered field, which is reduced to just a line integral. For the transverse magnetic case, the integrand is known explicitly. For the transverse electric case, the integrand can be easily constructed by solving an ordinary differential equation around the scatterer surface contour. Examples are provided which show that OSRC yields computed near and far fields which approach the exact results for canonical shapes such as the circular cylinder, square cylinder, and strip. Electrical sizes for the examples are ka = 5 and ka = 10. The new OSRC formulation of scattering may present a useful alternative to present integral equation and uniform high-frequency approaches for convex cylinders larger than ka = 1. Structures with edges or corners can also be analyzed, although more work is needed to incorporate the physics of singular currents at these discontinuities. Convex dielectric structures can also be treated using OSRC.

Kriegsmann, Gregory A.↗

DOE OSTI2025

Extended Galerkin Neural Network Approximation of Singular Variational Problems with Error Control

We present extended Galerkin neural networks, a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of new weighted least squares variational formulations suitable for use in neural network approximation of general BVPs, and (2) an “extended” feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus greatly improving approximability of singular solutions. Furthermore, numerical results are presented for several problems, including steady Stokes flow around reentrant corners and in convex corners with Moffatt eddies in order to demonstrate efficacy of the method.

a posteriori error estimate↗

DOE OSTI2022

Nanoscale Mapping and Defect-Assisted Manipulation of Surface Plasmon Resonances in 2D Bi 2 Te 3 /Sb 2 Te 3 In-Plane Heterostructures

Here, the Bi 2 Te 3 /Sb 2 Te 3 in-plane heterostructure is reported as a low-dimensional tunable chalcogenide well suited as plasmonic building block for the visible-UV spectral range. Electron-driven plasmon excitations of low-dimensional Bi 2 Te 3 /Sb 2 Te 3 are investigated by monochromated electron energy loss spectroscopy spectrum imaging. To resolve the nanoscale spatial distribution of various local plasmonic resonances, singular value decomposition is used to disentangle the spectral data and identify the individual spectral contributions of various corner, edge, and face modes. Furthermore, defect-plasmon interactions are investigated both for nanoscale intrinsic and thermally induced extrinsic polygonal defects (in situ sublimation). Signature of defect-induced red shift ranging from a several hundreds of millielectronvolts to a few electronvolts, broadening of various plasmon response, together with selective enhancement and significant variations in their intensity are detected. This study highlights the presence of a heterointerface and identifies defects as physical tuning pathways to modulate the plasmonic response over a broad spectral range. Finally, the experimental observations are compared qualitatively and validated with numerical simulations using the electron-driven discrete dipole approximation. Low-dimensional Bi 2 Te 3 /Sb 2 Te 3 as a less explored plasmonic system holds great promises as emerging platform for integrated plasmonics. Furthermore, introducing controlled structural defects can open the door for nanoengineering of plasmonic properties in such systems.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Odd-viscosity-induced passivation of Moffatt vortices

Odd viscosity is associated with a number of striking physical phenomena: a disk rotating in a viscous liquid experiences a normal compressive or tensile stress depending on its sense of rotation; chiral active matter interfacial deformations level off in a manner akin to surface tension; and unstable thermocapillary films become stabilized. In this article we show that odd viscosity decreases the size and intensity of Moffatt vortices in a corner formed by a solid substrate and a free interface. The effect is present because odd viscosity creates a normal-stress-induced shear stress at the liquid–gas interface. The new shear stress is out of phase and oppositely directed with respect to the interfacial liquid velocity. It squeezes the vortices and damps their rotation. As a consequence, Moffatt vortices can now be observed with greater facility in an experiment. Odd viscosity also increases the range of corner angles over which Moffatt vortices can be observed, and, in general, it shows a tendency to moderate singular behaviour.

Mechanics↗

DOE OSTI2015

Dirac points and van Hove singularities of silicene under uniaxial strain

First-principles calculations have been performed to investigate the low energy electronic properties and van Hove singularities (VHSs) of silicene under uniaxial strain. The Dirac points (DPs) persist when silicene is stretched uniaxially, while they are shifted away from the corners (K points) of the first Brillouin zone (FBZ). The relative positions of DPs with respect to the K points for silicene strained along the armchair (AC) or zigzag (ZZ) direction show opposite tendency compared with strained graphene, which is due to the larger deformation of the unit cell of strained silicene than that of strained graphene. Moreover, for silicene under AC or ZZ strain, the Fermi velocities around DPs along the positive and negative directions of the FBZ show rather significant difference. The nature of the VHS just above the Fermi energy undergoes a transition from the π* band to the σ* band for silicene under increasing AC or ZZ strain. These observations suggest uniaxial strain as an effective route to tune the electronic properties of silicene for potential applications in future electronic devices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Layer-dependent spin-resolved electronic structure of ferromagnetic triple-layered ruthenate Sr4Ru3O10

High-resolution angle- and spin-resolved photoemission spectroscopy (ARPES) of the triple-layered ruthenate Sr4Ru3O10 reveals features of the electronic structure that extend our understanding of the layered strontium ruthenates. The spectra near the Fermi energy are very different from the nonmagnetic analogues Sr2RuO4 and Sr3Ru2O7 with distinct Fermi surfaces for wide electronlike minority spin bands around the zone center and narrow holelike majority spin Fermi surface contours around the zone corners. The most dramatic results are two narrow spectral peaks ∼30 meV below the Fermi level, a spin-minority holelike band at the Brillouin zone center, and a spin-majority saddle-band van Hove singularity at the zone edge, which exhibits almost 100% spin polarization at low temperature, and a strong temperature dependent coherence-incoherence crossover attributed to Hund metal correlations. Quantitative comparison of the ARPES to spin-polarized density functional theory (DFT) calculations identify the specific antibonding and nonbonding orbital origins of the narrow bands, with a prediction of different spatial localization in the central and outer layers. This is shown to be consistent with experimental ARPES multizone matrix element intensity variations, and implicates outer-layer-specific control of the in-plane metamagnetism. The renormalization of the bands relative to the mean-field DFT, the demonstration of spin-polarized oxygen bands, and of spin-minority and spin-majority band-crossing hybridization provide a more complete picture of the magnetism which displays aspects of both delocalized and local moment behavior.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1982

Altitude transitions in energy climbs

The aircraft energy-climb trajectory for configurations with a sharp transonic drag rise is well known to possess two branches in the altitude/Mach-number plane. Transition in altitude between the two branches occurs instantaneously, a 'corner' in the minimum-time solution obtained with the energy-state model. If the initial and final values of altitude do not lie on the energy-climb trajectory, then additional jumps (crude approximations to dives and zooms) are required at the initial and terminal points. With a singular-perturbation approach, a 'boundary-layer' correction is obtained for each altitude jump, the transonic jump being a so-called 'internal' boundary layer, different in character from the initial and terminal layers. The determination of this internal boundary layer is examined and some computational results for an example presented.

Weston, A. R.↗

NASA NTRS1994

Equilibrium fluid interface behavior under low- and zero-gravity conditions

We describe here some of our recent mathematical work, which forms a basis for the Interface Configuration Experiment scheduled for USML-2. The work relates to the design of apparatus that exploits microgravity conditions for accurate determination of contact angle. The underlying motivation for the procedures rests on a discontinuous dependence of the capillary free surface interface S on the contact angle gamma, in a cylindrical capillary tube whose section (base) omega contains a protruding corner with opening angle 2 alpha. Specifically, in a gravity-free environment, omega can be chosen so that, for all sufficiently large fluid volume, the height of S is uniquely determined as a (single-valued) function mu(x,y) entirely covering the base; the height mu is bounded over omega uniformly in gamma throughout the range absolute value of (gamma -(pion/2)) less than or equal to alpha, while for absolute value of (gamma - (pion/2)) greater than alpha fluid will necessarily move to the corner and uncover the base, rising to infinity (or falling to negative infinity) at the vertex, regardless of volume. We mention here only that procedures based on the phenomenon promise excellent accuracy when gamma is close pion/2 but may be subject to experimental error when gamma is close to zero (orpion), as the 'singular' part of the domain over which the fluid accumulates (or disappears) when a critical angle gamma theta is crossed then becomes very small and may be difficult to observe. We ignore the trivial case gamma is equal to pion/2 (planar free surface), to simplify the discussion.

Concus, Paul↗

NASA NTRS1977

Mathematical and computational studies of equilibrium capillary free surfaces

The results of several independent studies are presented. The general question is considered of whether a wetting liquid always rises higher in a small capillary tube than in a larger one, when both are dipped vertically into an infinite reservoir. An analytical investigation is initiated to determine the qualitative behavior of the family of solutions of the equilibrium capillary free-surface equation that correspond to rotationally symmetric pendent liquid drops and the relationship of these solutions to the singular solution, which corresponds to an infinite spike of liquid extending downward to infinity. The block successive overrelaxation-Newton method and the generalized conjugate gradient method are investigated for solving the capillary equation on a uniform square mesh in a square domain, including the case for which the solution is unbounded at the corners. Capillary surfaces are calculated on the ellipse, on a circle with reentrant notches, and on other irregularly shaped domains using JASON, a general purpose program for solving nonlinear elliptic equations on a nonuniform quadrilaterial mesh. Analytical estimates for the nonexistence of solutions of the equilibrium capillary free-surface equation on the ellipse in zero gravity are evaluated.

Albright, N.↗

NASA NTRS1976

Computation of the inviscid supersonic flow over an external axial corner

A second-order finite-difference procedure is used to evaluate the inviscid supersonic flowfield surrounding an external axial corner composed of swept planar compression surfaces and representing the inlets on existing high-speed aircraft. The governing partial differential equations in conservation-law form are hyperbolic with respect to the axial coordinate and are solved iteratively by means of MacCormack's algorithm. The procedure treats both the peripheral shock wave and vortical singularities as discontinuities. Numerical results are presented for two parametric studies regarding the effects on the flowfield of varying the free-stream Mach number and the leading edge sweep of the horizontal wedge. Results of parametric Mach number study agree with the Mach number independence principle in that as the Mach number increases, such characteristics as shock shape, cross-flow sonic line location, and vortical singularity position approach an asymptote.

Kutler, P.↗

NASA NTRS1976

Diffraction of a shock wave by a compression corner. I - Regular reflection

The unsteady, two-dimensional flowfield resulting from the interaction of a moving planar shock wave with a compression corner is determined using a second-order, discontinuity-fitting, finite-difference approach. The time-dependent Euler equations are transformed to normalize the distance between the body and peripheral shock and to include the existing self-similar property of the flow. The resulting set of partial differential equations in conservation-law form is then solved in a time-dependent fashion using MacCormack's scheme. The vortical singularity, which lies on the body surface, and the single reflected shock are both treated as discontinuities in the numerical procedure. The results of the numerical simulation compare quite favorably with existing experimental interferograms and yield better flowfield resolution than previous first-order, shock-capturing, numerical solutions.

Kutler, P.↗

NASA NTRS2003

Gibbs-Thomson Law for Singular Step Segments: Thermodynamics Versus Kinetics

Classical Burton-Cabrera-Frank theory presumes that thermal fluctuations are so fast that at any time density of kinks on a step is comparable with the reciprocal intermolecular distance, so that the step rate is about isotropic within the crystal plane. Such azimuthal isotropy is, however, often not the case: Kink density may be much lower. In particular, it was recently found on the (010) face of orthorhombic lysozyme that interkink distance may exceed 500-600 intermolecular distances. Under such conditions, Gibbs-Thomson law (GTL) may not be applicable: On a straight step segment between two corners, communication between the comers occurs exclusively by kink exchange. Annihilation between kinks of opposite sign generated at the comers results in the grain in step energy entering GTL. If the step segment length l much greater than D/v, where D and v are the kink diffusivity and propagation rate, respectively, the opposite kinks have practically no chance to annihilate and GTL is not applicable. The opposite condition of the GTL applicability, l much less than D/v, is equivalent to the requirement that relative supersaturation Delta(sub mu)/kT much less than alpha/l, where alpha is molecular size. Thus, GTL may be applied to a segment of 10(exp 3)alpha approx. 3 x 10(exp -5)cm approx 0.3 micron only if supersaturation is less than 0.1%, while practically used driving forces for crystallization are much larger. Relationships alternative to the GTL for different, but low, kink density have been discussed. They confirm experimental evidences that the Burton-Cabrera-Frank theory of spiral growth is growth rates twice as low as compared to the observed figures. Also, application of GTL results in unrealistic step energy while suggested kinetic law give reasonable figures.

Chernov, A. A.↗

DOE OSTI2022

Discovery of charge density wave in a kagome lattice antiferromagnet

A hallmark of strongly correlated quantum materials is the rich phase diagram resulting from competing and intertwined phases with nearly degenerate ground state energies. A well-known example is the copper oxides, where a charge density wave (CDW) is ordered well above and strongly coupled to the magnetic order to form spin-charge separated stripes that compete with superconductivity. Recently, such rich phase diagrams have also been revealed in correlated topological materials. In two-dimensional kagome lattice metals consisting of corner-sharing triangles, the geometry of the lattice can produce flat bands with localized electrons, non-trivial topology, chiral magnetic order, superconductivity and CDW order. While CDW has been found in weakly electron correlated nonmagnetic AV 3 Sb 5 (A = K, Rb, Cs), it has not yet been observed in correlated magnetic ordered kagome lattice metals. Here we report the discovery of CDW within the antiferromagnetic (AFM) ordered phase of kagome lattice FeGe. The CDW in FeGe occurs at wavevectors identical to that of AV 3 Sb 5 , enhances the AFM ordered moment, and induces an emergent anomalous Hall effect. Furthermore, our findings suggest that CDW in FeGe arises from the combination of electron correlations-driven AFM order and van Hove singularities-driven instability possibly associated with a chiral flux phase, in stark contrast to strongly correlated copper oxides and nickelates, where the CDW precedes or accompanies the magnetic order.

36 MATERIALS SCIENCE↗

NASA NTRS1988

Stress-intensity factors for small surface and corner cracks in plates

Three-dimensional finite-element and finite-alternating methods were used to obtain the stress-intensity factors for small surface and corner cracked plates subjected to remote tension and bending loads. The crack-depth-to-crack-length ratios (a/c) ranged from 0.2 to 1 and the crack-depth-to-plate-thickness ratios (a/t) ranged from 0.05 to 0.2. The performance of the finite-element alternating method was studied on these crack configurations. A study of the computational effort involved in the finite-element alternating method showed that several crack configurations could be analyzed with a single rectangular mesh idealization, whereas the conventional finite-element method requires a different mesh for each configuration. The stress-intensity factors obtained with the finite-element-alternating method agreed well (within 5 percent) with those calculated from the finite-element method with singularity elements.

Raju, I. S.↗

NASA NTRS1989

Stress-intensity factors for small surface and corner cracks in plates

Raju, Ivatury S.↗

DOE OSTI2013

The new Hg-rich barium indium mercurides BaIn{sub x}Hg{sub 7−x} (x=3.1) and BaIn{sub x}Hg{sub 11−x} (x=0–2.8)

The title compounds BaIn{sub x}Hg{sub 7−x} (x=3.1(1)) and BaIn{sub x}Hg{sub 11−x} (x=0–2.8) were synthesized from stoichiometric ratios of the elements in Ta crucibles. Their crystal structures have been determined using single crystal X-ray data. BaIn{sub x}Hg{sub 7−x} (x=3.1(1)) crystallizes in a new structure type (orthorhombic, oC16, space group Cmmm: a=512.02(1), b=1227.68(3), c=668.61(2) pm, Z=2, R1=0.0311). In the structure, the atoms of the three crystallographically different mixed In/Hg positions form planar nets of four-, six- and eight-membered rings. These nets are shifted against each other such that the four-membered rings form empty distorted cubes. The cubes are connected via common edges, corners and folded ladders, which are also found in BaIn{sub 2}/BaHg{sub 2} (KHg{sub 2} structure type) and BaIn (α-NaHg type). The Ba atoms are centered in the eight-membered rings and exhibit an overall coordination number of 20. The [BaM{sub 20}] polyhedra and twice as many distorted [M{sub 8}] cubes tesselate the space. BaIn{sub 2.8}Hg{sub 8.2} (cubic, cP36, space group Pm3{sup ¯}m, a=961.83(1) pm, Z=3, R1=0.0243) is the border compound of the phase width BaIn{sub x}Hg{sub 11−x} of the rare BaHg{sub 11} structure type. In the structure, ideal [M{sub 8}] cubes (at the corners of the unit cell) and BaM{sub 20} polyhedra (at the edges of the unit cell) represent the building blocks comparable to the other new In mercuride. In accordance with the increased In/Hg content, additional M-pure regions appear: the center of the unit cell contains a huge [Hg(1)M(2){sub 12}M(3,4){sub 32}] polyhedron, a Hg-centered cuboctahedron of In/Hg atoms surrounded by a capped cantellated cube of 32 additional M atoms. For both structure types, the bonding situation and the ‘coloring’, i.e. the In/Hg distribution of the polyanionic network, are discussed considering the different sizes of the atoms and the charge distribution (Bader AIM charges), which have been calculated within the framework of FP-LAPW density functional theory. - Graphical abstract: BaIn{sub 2.6}Hg{sub 4.4}: distorted cubes [(In/Hg){sub 8}] (green, like in BaHg{sub 11}), folded ladders (violet, like in BaIn, BaHg{sub 2} and BaIn{sub 2}) and Ba coordination polyhedra [Ba(In/Hg){sub 20}] (blue, like in BaHg{sub 11}). - Highlights: • The Hg-rich In-mercuride BaIn{sub 3.1}Hg{sub 3.9} crystallizes with a singular structure type. • The phase width of the BaHg{sub 11} structure in BaIn{sub x}Hg{sub 11-x} ends at x=2.8. • The relations of both compounds with other alkaline-earth mercurides are outlined. • The Hg/In coloring of the polyanion is discussed considering the structure features. • Bonding aspects are explored using band structure calculations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗