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At least 163 records · Page 9

Free vibration and dynamic response analysis of spinning structures

The proposed effort involved development of numerical procedures for efficient solution of free vibration problems of spinning structures. An eigenproblem solution procedure, based on a Lanczos method employing complex arithmetic, was successfully developed. This task involved formulation of the numerical procedure, FORTRAN coding of the algorithm, checking and debugging of software, and implementation of the routine in the STARS program. A graphics package for the E/S PS 300 as well as for the Tektronix terminals was successfully generated and consists of the following special capabilities: (1) a dynamic response plot for the stresses and displacements as functions of time; and (2) a menu driven command module enabling input of data on an interactive basis. Finally, the STARS analysis capability was further improved by implementing the dynamic response analysis package that provides information on nodal deformations and element stresses as a function of time. A number of test cases were run utilizing the currently developed algorithm implemented in the STARS program and such results indicate that the newly generated solution technique is significantly more efficient than other existing similar procedures.

Source record↗

Dynamic Scaling Analysis of Accelerated Irradiation Testing on Additive Manufacturing Materials by Positron Annihilation

The timely applications of Additive Manufacturing (AM) materials in nuclear environments require accelerated irradiation tests, mainly ion irradiation to enable rapid prototyping. Low dose ion irradiation would cause sub-nanostructure changes by generation of lattice defects, vacancies, vacancy clusters and voids and void swelling caused by cellular dislocations. Positron Annihilation Lifetime (PAL), a novel technology, sensitive towards sub-nanostructure morphology with high accuracy (about 10-7 vacancy per atom), supported by Transition Electron Microscope (TEM) would be applied to identify the type and total size of the defects. The subsequent PAL measurements and TEM surface studies would be followed by PAL analysis that includes sophisticated trapping model. The PAS results would become an input to dynamic scaling analysis (that predicts radiation effects from low dose studies for high dose effects), which incorporate mean-field theory model. The final effect is an in-depth understanding of the microstructure evolution of AM materials under ion irradiation which can be extrapolated to the studies of neutron irradiation, since ion-irradiation takes less time and do not cause the irradiation hazard. The working hypothesis is that PAL technology, that have excellent sensitivity to low-defect concentration would help to identify ion-induced material damage on the atomic and nano-scale level, which then could be extrapolated to understand the neutron damage better.

accelerated irradiation testing↗

Benefits Analysis of Multi-Center Dynamic Weather Routes

Dynamic weather routes are flight plan corrections that can provide airborne flights more than user-specified minutes of flying-time savings, compared to their current flight plan. These routes are computed from the aircraft's current location to a flight plan fix downstream (within a predefined limit region), while avoiding forecasted convective weather regions. The Dynamic Weather Routes automation has been continuously running with live air traffic data for a field evaluation at the American Airlines Integrated Operations Center in Fort Worth, TX since July 31, 2012, where flights within the Fort Worth Air Route Traffic Control Center are evaluated for time savings. This paper extends the methodology to all Centers in United States and presents benefits analysis of Dynamic Weather Routes automation, if it was implemented in multiple airspace Centers individually and concurrently. The current computation of dynamic weather routes requires a limit rectangle so that a downstream capture fix can be selected, preventing very large route changes spanning several Centers. In this paper, first, a method of computing a limit polygon (as opposed to a rectangle used for Fort Worth Center) is described for each of the 20 Centers in the National Airspace System. The Future ATM Concepts Evaluation Tool, a nationwide simulation and analysis tool, is used for this purpose. After a comparison of results with the Center-based Dynamic Weather Routes automation in Fort Worth Center, results are presented for 11 Centers in the contiguous United States. These Centers are generally most impacted by convective weather. A breakdown of individual Center and airline savings is presented and the results indicate an overall average savings of about 10 minutes of flying time are obtained per flight.

NAS based Benefits Analysis↗

Ultrafast time-resolved x-ray absorption spectroscopy of ionized urea and its dimer through ab initio nonadiabatic dynamics

Investigating the early dynamics of chemical systems following ionization is essential for our understanding of radiation damage. However, experimental as well as theoretical investigations are very challenging due to the complex nature of these processes. Time-resolved x-ray absorption spectroscopy on a femtosecond timescale, in combination with appropriate simulations, is able to provide crucial insights into the ultrafast processes that occur upon ionization due to its element-specific probing nature. In this theoretical study, we investigate the ultrafast dynamics of valence-ionized states of urea and its dimer employing Tully's fewest switches surface hopping approach using Koopmans' theorem to describe the ionized system. We demonstrate that following valence ionization through a pump pulse, the time-resolved x-ray absorption spectra at the carbon, nitrogen, and oxygen K-edges reveal rich insights into the dynamics. Excited states of the ionized system give rise to time-delayed blueshifts in the x-ray absorption spectra as a result of electronic relaxation dynamics through nonadiabatic transitions. Moreover, our statistical analysis reveals specific structural dynamics in the molecule that induce time-dependent changes in the spectra. For the urea monomer, we elucidate the possibility to trace effects of specific molecular vibrations in the time-resolved x-ray absorption spectra. For the urea dimer, where ionization triggers a proton transfer reaction, we show how the x-ray absorption spectra can reveal specific details on the progress of proton transfer.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Recent Results in ICESat Geolocation Parameter Calibration from Range Residual Analysis

The Ice, Cloud and land Elevation Satellite (ICESat), launched on January 12, 2003, carries the Geoscience Laser Altimeter System (GLAS), which has a primary measurement of short-pulse laser-ranging to the Earths surface at 1064 nm wavelength at a rate of 40 pulses per second. The instrument has collected (at the time of this writing) over 540 million observations of the Earth's surface including precise elevation measurements of the ice sheets, sea ice roughness and thickness, ocean and land surface elevations and surface reflectivity. The accurate geolocation of GLASs surface returns, the spots from which the laser energy reflects on the Earths surface, is a critical issue in the scientific application of these data. Pointing, ranging, timing and orbit errors must be compensated to accurately geolocate the laser altimeter surface returns. Towards this end, the laser range observations can be fully exploited in an integrated residual analysis to accurately calibrate these geolocation/instrument parameters. The ICESat laser altimeter data have been simultaneously processed as direct altimetry from ocean sweeps and around-the-world scans along with dynamic crossovers in order to calibrate pointing ranging and timing. The analysis has resolved complex time varying signal resulting from thermal variations of instrument pointing and boresite shadowing. The calibration methodology and current calibration results are discussed along with future efforts and current geolocation accuracies.

Luthcke, Scott B.↗

A metabolic modeling platform for the computation of microbial ecosystems in time and space (COMETS)

Genome-scale stoichiometric modeling of metabolism has become a standard systems biology tool for modeling cellular physiology and growth. Extensions of this approach are emerging as a valuable avenue for predicting, understanding and designing microbial communities. Computation of microbial ecosystems in time and space (COMETS) extends dynamic flux balance analysis to generate simulations of multiple microbial species in molecularly complex and spatially structured environments. Here we describe how to best use and apply the most recent version of COMETS, which incorporates a more accurate biophysical model of microbial biomass expansion upon growth, evolutionary dynamics and extracellular enzyme activity modules. In addition to a command-line option, COMETS includes user-friendly Python and MATLAB interfaces compatible with the well-established COBRA models and methods, as well as comprehensive documentation and tutorials. Overall, this protocol provides a detailed guideline for installing, testing and applying COMETS to different scenarios, generating simulations that take from a few minutes to several days to run, with broad applicability to microbial communities across biomes and scales.

59 BASIC BIOLOGICAL SCIENCES↗

RAPID

Parallel computer code for the simulator for dynamics of power systems which has the capability to initiate the system and create different faults for the dynamic analysis. The code is based on time-parallel method (Parareal) with Adaptive Method Reduction (AMR). The coarse solvers for the Parareal algorithm include several Semi Analytical Solution methods. Also, Integrated simulation of coupled transmission and distribution systems can be studied.

Simunovic, Srdjan [Oak Ridge National Lab. (ORNL),↗

Amplified Mesoscale and Submesoscale Variability and Increased Concentration of Precipitation under Global Warming over Western North America

Abstract Cold-season precipitation statistics in simulations from the storm-resolving WRF Model at 6-km and 1-h resolution over western North America are analyzed. Pseudo–global warming future simulations for the 2041–80 period, constrained by GCMs under the RCP8.5 scenario, are compared to the 1981–2020 historical simulation. The analysis focuses on the dynamical properties of precipitation time series at subdaily scales and on the morphology of storms. The statistical distribution of precipitation intensities in each pixel of the simulation domain is characterized through nonparametric statistical indicators: frequency of wet hours, mean wet-hour precipitation intensity, and Gini coefficient as a measure of the temporal concentration of the precipitation volume. Additionally, the temporal and spatial Fourier power spectra of precipitation time series and precipitation fields are analyzed. The half-power period (HPP) and half-power wavelength (HPW) are defined as spectral measures of the characteristic scales of precipitation’s temporal and spatial patterns. The results show statistically significant increases in the mean wet-hour precipitation intensity and in the Gini coefficient in 99% of the pixels, indicating that the seasonal precipitation volume becomes more concentrated within a smaller number of hours with higher precipitation intensity. The statistics of change in the frequency of wet hours are more contrasted across the simulation domain. The changes are also reflected in the power spectra, which show the spatial and temporal variability increasing proportionally more with finer spatial and temporal scales and the HPW and HPP decreasing. These projected changes are expected to have consequences, not only in terms of hydrologic impacts but also in terms of the predictability of precipitation patterns. Significance Statement The precipitation characteristics of winter storms over the western United States and southwestern Canada are analyzed in future climate simulations for the 2041–80 period. As compared to present-day climate, the most intense parts of the storms are projected to produce a higher rainfall volume, with increased concentration over smaller areas and shorter time intervals. The propensity of rainfall intensity to vary rapidly over time will be enhanced in the future according to the simulations. These model predictions imply an increased risk of rapid flooding in small basins. They also suggest that predicting several hours ahead the time and location at which a storm will produce maximum rainfall may become more challenging in the future.

Climate change↗

H2 Materials Compatibility of Low Cost, High Pressure, Polymer H2 Dispensing Hoses

NanoSonic was recently awarded a DOE Phase IIB SBIR program for the commercialization of an innovative metal-free polymer based H2 dispensing hose to make H2 an economically viable fuel alternative to gasoline. During the Phase I and Phase II base programs, NanoSonic's H2 hose demonstrated ultra-high hydrostatic burst strength values > 31,000 psi where failure occurred due to fitting slippage, rather than hose burst. Additionally, our hoses survived > 51,000 pressure impulse cycles at 12,000 psi over a thermal cycle of -40 °C to 85 °C. These hoses also failed due to fitting slippage. Thus, our Phase IIB program is centralized on the addition of a new fitting, polymer hose refinement, and validation of a complete hose and fitting system certified for use with H2. While NanoSonic has partnered with the National Renewable Energy Laboratory (NREL) to test our hose on their robotic H2 dispensing system, we are also seeking durability with the H2 environment at the molecular level. Here, NanoSonic proposes a CRADA with Pacific Northwest National Laboratory (PNNL), who has expertise in H2 polymer materials compatibility. Together, we will determine the lifetime of our H2 hose polymer and composite constituents via: 1) time-temperature superposition (TTS) studies via dynamic mechanical analysis (DMA) under H2, 2) friction and wear resistance under in situ H2 tribometry, and 3) multi-axis strain testing under cryogenic conditions. PNNL has the unique test equipment essential to gain this type of insight and lifetime prediction data. Importantly, the work conducted under the proposed CRADA will increase the safety and reliability of our H2 hoses while expanding the market for use of our low H2 permeation polymers and durable cryogenic composites to realize the H2@Scale objectives to reduce the cost of H2.

08 HYDROGEN↗

Automated Flight Dynamics Product Generation for the EOS AM-1 Spacecraft

As part of NASA's Earth Science Enterprise, the Earth Observing System (EOS) AM-1 spacecraft is designed to monitor long-term, global, environmental changes. Because of the complexity of the AM-1 spacecraft, the mission operations center requires more than 80 distinct flight dynamics products (reports). To create these products, the AM-1 Flight Dynamics Team (FDT) will use a combination of modified commercial software packages (e.g., Analytical Graphic's Satellite ToolKit) and NASA-developed software applications. While providing the most cost-effective solution to meeting the mission requirements, the integration of these software applications raises several operational concerns: (1) Routine product generation requires knowledge of multiple applications executing on variety of hardware platforms. (2) Generating products is a highly interactive process requiring a user to interact with each application multiple times to generate each product. (3) Routine product generation requires several hours to complete. (4) User interaction with each application introduces the potential for errors, since users are required to manually enter filenames and input parameters as well as run applications in the correct sequence. Generating products requires some level of flight dynamics expertise to determine the appropriate inputs and sequencing. To address these issues, the FDT developed an automation software tool called AutoProducts, which runs on a single hardware platform and provides all necessary coordination and communication among the various flight dynamics software applications. AutoProducts, autonomously retrieves necessary files, sequences and executes applications with correct input parameters, and deliver the final flight dynamics products to the appropriate customers. Although AutoProducts will normally generate pre-programmed sets of routine products, its graphical interface allows for easy configuration of customized and one-of-a-kind products. Additionally, AutoProducts has been designed as a mission-independent tool, and can be easily reconfigured to support other missions or incorporate new flight dynamics software packages. After the AM-1 launch, AutoProducts will run automatically at pre-determined time intervals . The AutoProducts tool reduces many of the concerns associated with the flight dynamics product generation. Although AutoProducts required a significant effort to develop because of the complexity of the interfaces involved, its use will provide significant cost savings through reduced operator time and maximum product reliability. In addition, user satisfaction is significantly improved and flight dynamics experts have more time to perform valuable analysis work. This paper will describe the evolution of the AutoProducts tool, highlighting the cost savings and customer satisfaction resulting from its development. It will also provide details about the tool including its graphical interface and operational capabilities.

Matusow, Carla↗

Filling data analysis gaps in time-resolved crystallography by machine learning

There is a growing understanding of the structural dynamics of biological molecules fueled by x-ray crystallography experiments. Time-resolved serial femtosecond crystallography (TR-SFX) with x-ray Free Electron Lasers allows the measurement of ultrafast structural changes in proteins. Nevertheless, this technique comes with some limitations. One major challenge is the quality of data from TR-SFX measurements, which often faces issues like data sparsity, partial recording of Bragg reflections, timing errors, and pixel noise. To overcome these difficulties, conventionally, large volumes of data are collected and grouped into a few temporal bins. The data in each bin are then averaged and paired with the mean of their corresponding jittered timestamps. This procedure provides one structure per bin, resulting in a limited number of averaged structures for the entire time interval spanned by the experiment. Therefore, the information on ultrafast structural dynamics at high temporal resolution is lost. This has initiated research for advanced methods of analyzing experimental TR-SFX data beyond the standard binning and averaging method. To address this problem, we use a machine learning algorithm called Nonlinear Laplacian Spectral Analysis (NLSA), which has emerged as a promising technique for studying the dynamics of complex systems. In this work, we demonstrate the power of this algorithm using synthetic x-ray diffraction snapshots from a protein with significant data incompleteness, timing uncertainties, and noise. Our study confirms that NLSA is a suitable approach that effectively mitigates the effects of these artifacts in TR-SFX data and recovers accurate structural dynamics information hidden in such data.

Trujillo, Justin (ORCID:0000000285505360)↗

What is (quantitative) system dynamics modeling? Defining characteristics and the opportunities they create

A clear definition of system dynamics modeling can provide shared understanding and clarify the impact of the field. We introduce a set of characteristics that define quantitative system dynamics, selected to capture core philosophy, describe theoretical and practical principles, and apply to historical work but be flexible enough to remain relevant as the field progresses. The defining characteristics are: (1) models are based on causal feedback structure, (2) accumulations and delays are foundational, (3) models are equation-based, (4) concept of time is continuous, and (5) analysis focuses on feedback dynamics. We discuss the implications of these principles and use them to identify research opportunities in which the system dynamics field can advance. These research opportunities include causality, disaggregation, data science and AI, and contributing to scientific advancement. Progress in these areas has the potential to improve both the science and practice of system dynamics.

97 MATHEMATICS AND COMPUTING↗

Development of High Speed Imaging and Analysis Techniques Compressible Dynamics Stall

Dynamic stall has limited the flight envelope of helicopters for many years. The problem has been studied in the laboratory as well as in flight, but most research, even in the laboratory, has been restricted to surface measurement techniques such as pressure transducers or skin friction gauges, except at low speed. From this research, it became apparent that flow visualization tests performed at Mach numbers representing actual flight conditions were needed if the complex physics associated with dynamic stall was to be properly understood. However, visualization of the flow field during compressible conditions required carefully aligned and meticulously reconstructed holographic interferometry. As part of a long-range effort focused on exposing of the physics of compressible dynamic stall, a research wind tunnel was developed at NASA Ames Research Center which permits visual access to the full flow field surrounding an oscillating airfoil during compressible dynamic stall. Initially, a stroboscopic schlieren technique was used for visualization of the stall process, but the primary research tool has been point diffraction interferometry(PDI), a technique carefully optimized for use in th is project. A review of the process of development of PDI will be presented in the full paper. One of the most valuable aspects of PDI is the fact that interferograms are produced in real time on a continuous basis. The use of a rapidly-pulsed laser makes this practical; a discussion of this approach will be presented in the full paper. This rapid pulsing(up to 40,000 pulses/sec) produces interferograms of the rapidly developing dynamic stall field in sufficient resolution(both in space and time) that the fluid physics of the compressible dynamic stall flowfield can be quantitatively determined, including the gradients of pressure in space and time. This permits analysis of the influence of the effect of pitch rate, Mach number, Reynolds number, amplitude of oscillation, and other parameters on the dynamic stall process. When interferograms can be captured in real time, the potential for real-time mapping of a developing unsteady flow such as dynamic stall becomes a possibility. This has been achieved in the present case through the use of a high-speed drum camera combined with electronic circuitry which has resulted in a series of interferograms obtained during a single cycle of dynamic stall; images obtained at the rate of 20 KHz will be presented as a part of the formal presentation. Interferometry has been available for a long time; however, most of its use has been limited to visualization. The present research has focused on use of interferograms for quantitative mapping of the flow over oscillating airfoils. Instantaneous pressure distributions can now be obtained semi-automatically, making practical the analysis of the thousands of interferograms that are produced in this research. A review of the techniques that have been developed as part of this research effort will be presented in the final paper.

Chandrasekhara, M. S.↗

Detailed biomass fast pyrolysis kinetics integrated to computational fluid dynamic (CFD) and discrete element modeling framework: Predicting product yields at the bench-scale

Fast pyrolysis is an intricate process due to the variability and anisotropy of lignocellulosic biomass and the complicated chemistry and physics during conversion in a bubbling fluidized bed reactor (BFBR). The complexity of biomass fast pyrolysis lends itself well to computational fluid dynamics (CFD) and discrete element (DEM) analysis, which promises to reduce experimental time and its associated cost. This work investigated switchgrass fast pyrolysis simulated by computational fluid dynamics coupled with a discrete element method to track individual reacting biomass particles throughout a bench-scale BFBR reactor. We accounted for the fast pyrolysis chemistry through a comprehensive reaction scheme with secondary cracking reactions. We performed a three-step reduction for secondary cracking reactions to convert the full cracking scheme into a reduced scheme easily incorporated into our model. We assessed the impact of operational conditions on the steady-state yields of liquid bio-oil, non-condensable gases (NCG), at 550 °C over a range of fluidization numbers (2 – 6 Umf), reported as a ratio to the minimum fluidization velocity (Umf). At steady-state, the volatile bio-oil yield had a range of 49.3–50.4 wt%. Levoglucosan was the primary volatile component present with 21 wt% of the bio-oil while water was the second largest with 20 wt%. The reduction of the secondary reaction schemes did not appreciably affect the overall yields of switchgrass pyrolysis compared to the full secondary scheme.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Nonlinear Dynamic Models in Advanced Life Support

To facilitate analysis, ALS systems are often assumed to be linear and time invariant, but they usually have important nonlinear and dynamic aspects. Nonlinear dynamic behavior can be caused by time varying inputs, changes in system parameters, nonlinear system functions, closed loop feedback delays, and limits on buffer storage or processing rates. Dynamic models are usually cataloged according to the number of state variables. The simplest dynamic models are linear, using only integration, multiplication, addition, and subtraction of the state variables. A general linear model with only two state variables can produce all the possible dynamic behavior of linear systems with many state variables, including stability, oscillation, or exponential growth and decay. Linear systems can be described using mathematical analysis. Nonlinear dynamics can be fully explored only by computer simulations of models. Unexpected behavior is produced by simple models having only two or three state variables with simple mathematical relations between them. Closed loop feedback delays are a major source of system instability. Exceeding limits on buffer storage or processing rates forces systems to change operating mode. Different equilibrium points may be reached from different initial conditions. Instead of one stable equilibrium point, the system may have several equilibrium points, oscillate at different frequencies, or even behave chaotically, depending on the system inputs and initial conditions. The frequency spectrum of an output oscillation may contain harmonics and the sums and differences of input frequencies, but it may also contain a stable limit cycle oscillation not related to input frequencies. We must investigate the nonlinear dynamic aspects of advanced life support systems to understand and counter undesirable behavior.

Jones, Harry↗

Application of Reduced Order Transonic Aerodynamic Influence Coefficient Matrix for Design Optimization

Supporting the Aeronautics Research Mission Directorate guidelines, the National Aeronautics and Space Administration [NASA] Dryden Flight Research Center is developing a multidisciplinary design, analysis, and optimization [MDAO] tool. This tool will leverage existing tools and practices, and allow the easy integration and adoption of new state-of-the-art software. Today s modern aircraft designs in transonic speed are a challenging task due to the computation time required for the unsteady aeroelastic analysis using a Computational Fluid Dynamics [CFD] code. Design approaches in this speed regime are mainly based on the manual trial and error. Because of the time required for unsteady CFD computations in time-domain, this will considerably slow down the whole design process. These analyses are usually performed repeatedly to optimize the final design. As a result, there is considerable motivation to be able to perform aeroelastic calculations more quickly and inexpensively. This paper will describe the development of unsteady transonic aeroelastic design methodology for design optimization using reduced modeling method and unsteady aerodynamic approximation. The method requires the unsteady transonic aerodynamics be represented in the frequency or Laplace domain. Dynamically linear assumption is used for creating Aerodynamic Influence Coefficient [AIC] matrices in transonic speed regime. Unsteady CFD computations are needed for the important columns of an AIC matrix which corresponded to the primary modes for the flutter. Order reduction techniques, such as Guyan reduction and improved reduction system, are used to reduce the size of problem transonic flutter can be found by the classic methods, such as Rational function approximation, p-k, p, root-locus etc. Such a methodology could be incorporated into MDAO tool for design optimization at a reasonable computational cost. The proposed technique is verified using the Aerostructures Test Wing 2 actually designed, built, and tested at NASA Dryden Flight Research Center. The results from the full order model and the approximate reduced order model are analyzed and compared.

Pak, Chan-gi↗

Dynamic mode decomposition for gyrokinetic eigenmode analysis

Dynamic mode decomposition (DMD) is a post-processing approach to decompose a complex time series into a set of modes via spectral analysis. DMD provides a new and powerful method to recover gyrokinetic drift-wave eigenfrequencies and eigenfunctions based only on the solution of the gyrokinetic-Maxwell initial value problem with almost no added cost to the initial value solver. In the present paper, DMD is applied to the CGYRO gyrokinetic code using a newly-developed CGYRO-DMD post-processor. CGYRO-DMD is numerically efficient, even on a single CPU. It does not set any restrictions on the plasma shape, beta (ratio of the plasma pressure to the magnetic field pressure), collisionality or number of species, and allows one to resolve numerous eigenmodes, even of comparable growth rates. In addition, DMD is not limited to unstable modes, but rather can capture stable and unstable branches simultaneously. In this work, we illustrate the accuracy of DMD through gyrokinetic analysis of mode transition for electromagnetic drift wave instabilities.

drift-wave eigenmodes↗

Engineering micromotion in Floquet prethermalization via space-time symmetries

We present a systematic framework for Floquet prethermalization under strong resonant driving, emphasizing the pivotal role of dynamical space-time symmetries. Our approach demonstrates how dynamical space-time symmetries map onto the projective static symmetry group of the prethermal Hamiltonian governing the prethermal regime. We introduce techniques for detecting dynamical symmetries through the time evolution of local observables, facilitating a detailed analysis of micromotion within each period and surpassing the limitations of conventional stroboscopic Floquet prethermal dynamics. To implement this framework, we present a prethermal protocol that preserves order-2 dynamical symmetry in a spin-ladder model, confirming the predicted relationships between the expectation values of local observables at distinct temporal points in the Floquet cycle, linked by this symmetry.

floquet systems↗