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NASA NTRS1987

Covariance and the hierarchy of frame bundles

This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite Lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and for G-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.

Estabrook, Frank B.↗

DOE OSTI2019

A fast Fourier transform-based mesoscale field dislocation mechanics study of grain size effects and reversible plasticity in polycrystals

A numerical implementation of a non-local polycrystal plasticity theory based on a mesoscale version of the field dislocation mechanics theory (MFDM) of Acharya and Roy (2006) is presented using small-strain elasto-viscoplastic fast Fourier transform-based (EVPFFT) algorithm developed by Lebensohn et al. (2012). In addition to considering plastic flow and hardening only due to SSDs (statistically stored dislocations) as in the classic EVPFFT framework, the proposed method accounts for the evolution of GND (geometrically necessary dislocations) densities solving a hyperbolic-type partial differential equation, and GND effects on both plastic flow and hardening. This allows consideration of an enhanced strain-hardening law that includes the effect of the GND density tensor. The numerical implementation of a reduced version of the MFDM is presented in the framework of the FFT-based augmented Lagrangian procedure of Michel et al. (2001). A Finite Differences scheme combined with discrete Fourier transforms is applied to solve both incompatibility and equilibrium equations. The numerical procedure named MFDM-EVPFFT is used to perform full field simulations of polycrystal plasticity considering different grain sizes and their mechanical responses during monotonic tensile and reversible tension-compression tests. Using Voronoi tessellation and periodic boundary conditions, voxelized representative volume elements (RVEs) with different grain sizes are generated. With MFDM-EVPFFT, a Hall-Petch type scaling law is obtained in contrast with the conventional crystal plasticity EVPFFT. In the case of reversible plasticity, a stronger Bauschinger effect is observed with the MFDM-EVPFFT approach in comparison with conventional EVPFFT. Here, the origin of these differences is analyzed in terms of heterogeneity, GND density and stress evolutions during the compression stage.

36 MATERIALS SCIENCE↗

DOE OSTI2022

The Effect of Geometry, Spin, and Orbital Optimization in Achieving Accurate, Correlated Results for Iron–Sulfur Cubanes

Iron–sulfur clusters comprise an important functional motif in the catalytic centers of biological systems, capable of enabling important chemical transformations at ambient conditions. This remarkable capability derives from a notoriously complex electronic structure that is characterized by a high density of states that is sensitive to geometric changes. The spectral sensitivity to subtle geometric changes has received little attention from correlated, large active space calculations, owing partly to the exceptional computational complexity for treating these large and correlated systems accurately. Here, to provide insight into this aspect, we report the first Complete Active Space Self Consistent Field (CASSCF) calculations for different geometries of the [Fe(II/III) 4 S 4 (SMe) 4 ] -2 clusters using two complementary, correlated solvers: spin-pure Adaptive Sampling Configuration Interaction (ASCI) and Density Matrix Renormalization Group (DMRG). We find that the previously established picture of a double-exchange driven magnetic structure, with minute energy gaps (<1 mHa) between consecutive spin states, has a weak dependence on the underlying geometry. However, the spin gap between the singlet and the spin state 2S + 1 = 19, corresponding to a maximal number of Fe-d electrons being unpaired and of parallel spin, is strongly geometry dependent, changing by a factor of 3 upon slight deformations that are still within biologically relevant parameters. The CASSCF orbital optimization procedure, using active spaces as large as 86 electrons in 52 orbitals, was found to reduce this gap compared to typical mean-field orbital approaches. Our results show the need for performing large active space calculations to unveil the challenging electronic structure of these complex catalytic centers and should serve as accurate starting points for fully correlated treatments upon inclusion of dynamical correlation outside the active space.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2015

Correction of MRI-induced geometric distortions in whole-body small animal PET-MRI

Purpose: The fusion of positron emission tomography (PET) and magnetic resonance imaging (MRI) data can be a challenging task in whole-body PET-MRI. The quality of the registration between these two modalities in large field-of-views (FOV) is often degraded by geometric distortions of the MRI data. The distortions at the edges of large FOVs mainly originate from MRI gradient nonlinearities. This work describes a method to measure and correct for these kind of geometric distortions in small animal MRI scanners to improve the registration accuracy of PET and MRI data. Methods: The authors have developed a geometric phantom which allows the measurement of geometric distortions in all spatial axes via control points. These control points are detected semiautomatically in both PET and MRI data with a subpixel accuracy. The spatial transformation between PET and MRI data is determined with these control points via 3D thin-plate splines (3D TPS). The transformation derived from the 3D TPS is finally applied to real MRI mouse data, which were acquired with the same scan parameters used in the phantom data acquisitions. Additionally, the influence of the phantom material on the homogeneity of the magnetic field is determined via field mapping. Results: The spatial shift according to the magnetic field homogeneity caused by the phantom material was determined to a mean of 0.1 mm. The results of the correction show that distortion with a maximum error of 4 mm could be reduced to less than 1 mm with the proposed correction method. Furthermore, the control point-based registration of PET and MRI data showed improved congruence after correction. Conclusions: The developed phantom has been shown to have no considerable negative effect on the homogeneity of the magnetic field. The proposed method yields an appropriate correction of the measured MRI distortion and is able to improve the PET and MRI registration. Furthermore, the method is applicable to whole-body small animal imaging routines including different standard MRI sequences.

60 APPLIED LIFE SCIENCES↗

NASA NTRS1973

A generalized technique for using cones and dihedral angles in attitude determination, revision 1

Analytic development is presented for a general least squares attitude determination subroutine applicable to spinning satellites. The method is founded on a geometric approach which is completely divorced from considerations relating to particular types and configurations of onboard attitude sensors. Any mix of sensor measurements which can be first transformed (outside the program) to cone or dihedral angle data can be processed. A cone angle is an angle between the spin axis and a known direction line in space; a dihedral angle is an angle between two planes formed by the spin axis and each of two known direction lines. Many different kinds of sensor data can be transformed to these angles, which in turn constitute the actual program inputs, so that the subroutine can be applied without change to a variety of satellite missions. Either a constant or dynamic spin axis model can be handled. The program is also capable of solving for fixed biases in the input angles, in addition to the spin axis attitude solution.

Werking, R. D.↗

NASA NTRS1970

The far field diffraction pattern for corner reflectors with complex reflection coefficients

The far field diffraction pattern of a geometrically perfect corner reflector is examined analytically for normally incident monochromatic light. The states of polarization and the complex amplitudes of the emerging light are expressed through transformation matrices in terms of those of the original incident light for each sextant of the face in a single coordinate system. The analytic expression of the total diffraction pattern is obtained for a circular face. This expression consists of three component functions in addition to the basic Airy function. The coefficient of each function is expressed in terms of complex coefficients of reflectance of the reflecting surface. Some numerical results for different reflecting surfaces, including total internal reflection, are presented. The iso-intensity contours of the diffraction pattern evaluated from the analytical expressions for an uncoated solid corner reflector are also presented along with the photographs of the pattern.

Chang, R. F.↗

NASA NTRS1982

Synthesis of offset dual reflector antennas transforming a given feed illumination pattern into a specified aperture distribution

The problem of transforming a given primary feed pattern into a desired aperture field distribution through two reflections by an offset dual reflector system is investigated using the concepts of geometrical optics. A numerically rigorous solution for the reflector surfaces is developed which realizes an exact aperture phase distribution and an aperture amplitude distribution that is accurate to within an arbitrarily small numerical tolerance. However, this procedure does not always yield a smooth solution, i.e., the reflector surfaces thus realized may not be continuous or their slopes may vary too rapidly. In the event of nonexistence of a numerically rigorous smooth solution, an approximate solution that enforces the smoothness of the reflector surfaces can be obtained. In the approximate solution, only the requirement for the aperture amplitude distribution is relaxed, and the condition on the aperture phase distribution is continued to be satisfied exactly.

Mittra, R.↗

NASA NTRS1990

Automatic generation of forward and inverse kinematics for a reconfigurable modular manipulator system

An algorithm is proposed for automatically generating both the forward and inverse kinematics of a serial-link N-degree-of-freedom reconfigurable manipulator (RM). Generation of the kinematic equations that govern a modular manipulator starts with geometric descriptions of the units, or modules, as well as their sequence in the manipulator. This geometric information is used to obtain the Denavit-Hartenberg (DH) parameters of an RM. The DH kinematic parameters are then used to obtain the forward kinematic transformation of the system. The problem of obtaining the inverse kinematics of RMs is addressed, and the idea of scaling an RM to automate the inverse kinematics and make the procedure as general as possible is proposed.

Kelmar, Laura↗

DOE OSTI2026

Dispersion Loss Counteracts Embedding Condensation and Improves Generalization in Small Language Models

Large language models (LLMs) achieve remarkable performance through ever-increasing parameter counts, but scaling incurs steep computational costs. To better understand LLM scaling, we study representational differences between LLMs and their smaller counterparts, with the goal of replicating the representational qualities of larger models in smaller models. We observe a geometric phenomenon which we term embedding condensation, where token embeddings collapse into a narrow cone-like subspace in some language models. Through systematic analyses across multiple Transformer families, we show that small models such as GPT2 and Qwen3-0.6B exhibit severe condensation, whereas larger models such as GPT2-x1 and Qwen3-32B are more resistant to this phenomenon. Additional observations show that embedding condensation is not reliably mitigated by knowledge distillation from larger models. To fight against it, we formulate a dispersion loss that explicitly encourages embedding dispersion during training. Experiments demonstrate that it mitigates condensation, recovers dispersion patterns seen in larger models, and yields performance gains across 10 benchmarks. We believe this work offers a principled path toward improving smaller Transformers without additional parameters.

Xiao, Xi [ORNL] (ORCID:0009000009316982)↗

DOE OSTI2024

Massive twistor worldline in electromagnetic fields

We study the (ambi-)twistor model for spinning particles interacting via electromagnetic field, as a toy model for studying classical dynamics of gravitating bodies including effects of both spins to all orders. We compute the momentum kick and spin kick up to one-loop order and show precisely how they are encoded in the classical eikonal. The all-orders-in-spin effects are encoded as a dynamical implementation of the Newman-Janis shift, and we find that the expansion in both spins can be resummed to simple expressions in special kinematic configurations, at least up to one-loop order. We confirm that the classical eikonal can be understood as the generator of canonical transformations that map the in-states of a scattering process to the out-states. We also remark that cut contributions for converting worldline propagators from time-symmetric to retarded amount to the iterated action of the leading eikonal at one-loop order.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Development of Multistep and Degenerate Variational Integrators for Applications in Plasma Physics

Geometric integrators yield high-fidelity numerical results by retaining conservation laws in the time advance. A particularly powerful class of geometric integrators is symplectic integrators, which are widely used in orbital mechanics and accelerator physics. An important application presently lacking symplectic integrators is the guiding center motion of magnetized particles represented by non-canonical coordinates. Because guiding center trajectories are foundational to many simulations of magnetically confined plasmas, geometric guiding center algorithms have high potential for impact. The motivation is compounded by the need to simulate long-pulse fusion devices, including ITER, and opportunities in high performance computing, including the use of petascale resources and beyond. This dissertation uses a systematic procedure for constructing geometric integrators - known as variational integration - to deliver new algorithms for guiding center trajectories and other plasma-relevant dynamical systems. These variational integrators are non-trivial because the Lagrangians of interest are degenerate - the Euler-Lagrange equations are first-order differential equations and the Legendre transform is not invertible. The first contribution of this dissertation is that variational integrators for degenerate Lagrangian systems are typically multistep methods. Multistep methods admit parasitic mode instabilities that can ruin the numerical results. These instabilities motivate the second major contribution: degenerate variational integrators. By replicating the degeneracy of the continuous system, degenerate variational integrators avoid parasitic mode instabilities. The new methods are therefore robust geometric integrators for degenerate Lagrangian systems. These developments in variational integration theory culminate in one-step degenerate variational integrators for non-canonical magnetic field line flow and guiding center dynamics. The guiding center integrator assumes coordinates such that one component of the magnetic field is zero; it is shown how to construct such coordinates for nested magnetic surface configurations. Additionally, collisional drag effects are incorporated in the variational guiding center algorithm for the first time, allowing simulation of energetic particle thermalization. Advantages relative to existing canonical-symplectic and non-geometric algorithms are numerically demonstrated. All algorithms have been implemented as part of a modern, parallel, ODE-solving library, suitable for use in high-performance simulations.

Guiding Center↗

DOE OSTI2022

Beam dynamics in independent phased cavities

Linear accelerators containing the sequence of independently phased cavities with constant geometrical velocity along each structure are widely used in practice. The chain of cavities with identical cell lengths is utilized within a certain beam velocity range, with subsequent transformation to the next chain with higher cavity velocity. Design and analysis of beam dynamics in this type of accelerator are usually performed using numerical simulations. A full theoretical description of particle acceleration in an array of independent phased cavities has not been developed. In the present paper, we provide an analytical treatment of beam dynamics in such linacs employing Hamiltonian formalism. We begin our analysis with an examination of beam dynamics in an equivalent traveling wave of a single cavity, propagating within accelerating section with constant phase velocity. We then consider beam dynamics in arrays of cavities, utilizing an effective traveling wave propagating along with the whole accelerator with the velocity of synchronous (reference) particle. The analysis concluded with the determination of the matched beam conditions. Finally, we present a beam dynamics study in 805 MHz Coupled Cavity Linac of the LANSCE accelerator facility.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2025

Shape-anisotropy inverses the behavior of emergent vortices in active chiral fluids

Active colloidal fluids exhibit spontaneous emergence of correlated states, characterized by complex collective dynamics and self-organization. In geometrically confined systems, activity modulations trigger robust polar state reversals of a macroscopic vortex formed by colloidal rollers. Here, we reveal that the shape anisotropy of dielectric rollers transforms the system into a chiral active fluid comprised of rollers of arbitrary handedness. The chiral rollers self-assemble into multiple freestanding vortices with a spontaneously selected sense of rotation. We demonstrate that upon reactivation of the system after a complete cessation of activity beyond all relevant timescales, the vortices simultaneously restore their previous chiral states in striking contrast to the chiral state reversals demonstrated by spherical rollers. The analysis reveals that shape-anisotropy modifies the collective state memory of the ensemble. The findings provide insights into the emergence of complex collective behavior in chiral colloidal fluids governed by an interplay between shape anisotropy, chiral motion, and activity modulations.

Colloids↗

DOE OSTI2018

A Problem of Axisymmetric Torsion of a Multilayer Plate with Elastic Links between the Layers

By using the method of compliance functions and the Hankel transformation of the first order, we propose a procedure for the solution of the problem of torsion of a multilayer plate with elastic links between the layers. Two auxiliary functions connected with the transforms of stresses and displacements at the points of the top boundary of the layer are introduced for each layer. The recurrence relations for the auxiliary functions and the compliance functions of neighboring layers of the plate are constructed. For a two-layer plate, we analyze the influence of the coefficients of elastic links and the geometric and mechanical parameters of layers on the distribution of stresses at points of the common boundary of the layers.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

Adopting Biophysics Methods in Pursuit of Biogeophysical Research: Advancing the measurement and modeling of electrical signatures of microbe-mineral transformations impacting contaminant transport

This exploratory project involved laboratory experiments to investigate three hypotheses: (H1) Physics-based modeling of low-frequency dispersions (henceforth referred to as alpha) measured in broadband dielectric spectroscopy (DS) data can quantify pore-scale geometric changes impacting contaminant transport resulting from biomineralization; (H2) Physics-based modeling of high-frequency dispersions (henceforth referred to as beta) measured in broadband dielectric spectroscopy data can quantify rates of mineral growth in/on the cell wall; (H3) Application of this measurement and modeling approach can enhance geophysical interpretation of bioremediation experiments conducted at the RIFLE IFC by providing constraints on bioremediation efficiency (biomass concentration, mineral uptake within the cell wall, biomineralization rate). We tested H1 by performing DS measurements (alpha and beta range) on iron (Fe) particles of dimensions similar to microbial cells, dispersed within agar gels over a range of Fe concentrations. We have tested the ability of the physics-based modeling to predict volume concentrations of the Fe particles by assuming that the Fe particles are polarizable inclusions within an otherwise nonpolarizable medium. We evaluated the smallest volume concentration that can be detected with the DS method. Similar experiments and modeling have been performed on the sulfate-reducing bacteria D. vulgaris. Synchrotron x-ray absorption measurements were conducted to determine the local structure of biominerals coatings on D. vulgaris which were grown in the presence of different Fe concentrations. We imaged the mineral growth on cell wall using SEM. We used dielectric spectroscopy to differentiate between iron sulfide precipitates of biotic and abiotic nature. Biotic measurements were made on D. vulgaris bacteria grown in the presence of different concentrations of iron to form different thicknesses of iron sulfide precipitates around themselves and abiotic measurements were made on different concentrations of pyrrhotite particles suspended in agar. Results show a decrease in dielectric permittivity as a function of frequency for biotic minerals and an opposite trend is observed for abiotic minerals. Our results suggest that dielectric spectroscopy offers a noninvasive and fast approach for distinguishing between abiotic and biotic mineral precipitates.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Gaussian-preserved, non-volatile shape morphing in three-dimensional microstructures for dual-functional electronic devices

Motile plant structures such as Mimosa pudica leaves, Impatiens glandulifera seedpods, and Dionaea muscipula leaves exhibit fast nastic movements in a few seconds or less. This motion is stimuli-independent mechanical movement following theorema egregium rules. Artificial analogs of tropistic motion in plants are exemplified by shape-morphing systems, which are characterized by high functional robustness and resilience for creating 3D structures. However, all shape-morphing systems developed so far rely exclusively on continuous external stimuli and result in slow response. Here, we report a Gaussian-preserved shape-morphing system to realize ultrafast shape morphing and non-volatile reconfiguration. Relying on the Gaussian-preserved rules, the transformation can be triggered by mechanical or thermal stimuli within a microsecond. Moreover, as localized energy minima are encountered during shape morphing, non-volatile configuration is preserved by geometrically enhanced rigidity. Using this system, we demonstrate a suite of electronic devices that are reconfigurable, and therefore, expand functional diversification.

36 MATERIALS SCIENCE↗

DOE OSTI2024

On the Topotactic Phase Transition Achieving Superconducting Infinite‐Layer Nickelates

Abstract Topotactic reduction is critical to a wealth of phase transitions of current interest, including synthesis of the superconducting nickelate Nd 0.8 Sr 0.2 NiO 2 , reduced from the initial Nd 0.8 Sr 0.2 NiO 3 /SrTiO 3 heterostructure. Due to the highly sensitive and often damaging nature of the topotactic reduction, however, only a handful of research groups have been able to reproduce the superconductivity results. A series of in situ synchrotron‐based investigations reveal that this is due to the necessary formation of an initial, ultrathin layer at the Nd 0.8 Sr 0.2 NiO 3 surface that helps to mediate the introduction of hydrogen into the film such that apical oxygens are first removed from the Nd 0.8 Sr 0.2 NiO 3 / SrTiO 3 (001) interface and delivered into the reducing environment. This allows the square‐planar / perovskite interface to stabilize and propagate from the bottom to the top of the film without the formation of interphase defects. Importantly, neither geometric rotations in the square planar structure nor significant incorporation of hydrogen within the films is detected, obviating its need for superconductivity. These findings unveil the structural basis underlying the transformation pathway and provide important guidance on achieving the superconducting phase in reduced nickelate systems.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Dynamics of McMillan mappings II. axially symmetric map

Here, in this article, we investigate the transverse dynamics of a single particle in a model integrable accelerator lattice, based on a McMillan axially-symmetric electron lens. Although the McMillan e-lens has been considered as a device potentially capable of mitigating collective space charge forces, some of its fundamental properties have not been described yet. The main goal of our work is to close this gap and understand the limitations and potentials of this device. It is worth mentioning that the McMillan axially symmetric map provides the first-order approximations of dynamics for a general linear lattice plus an arbitrary thin lens with motion separable in polar coordinates. Therefore, advancements in its understanding should give us a better picture of more generic and not necessarily integrable round beams. In the first part of the article, we classify all possible regimes with stable trajectories and find the canonical action-angle variables. This provides an evaluation of the dynamical aperture, Poincaré rotation numbers as functions of amplitudes, and thus determines the spread in nonlinear tunes. Also, we provide a parameterization of invariant curves, allowing for the immediate determination of the map image forward and backward in time. The second part investigates the particle dynamics as a function of system parameters. We show that there are three fundamentally different configurations of the accelerator optics causing different regimes of nonlinear oscillations. Each regime is considered in great detail, including the limiting cases of large and small amplitudes. In addition, we analyze the dynamics in Cartesian coordinates and provide a description of observable variables and corresponding spectra.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗