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At least 379 records · Page 21

Analysis of Smart Composite Structures Including Debonding

Smart composite structures with distributed sensors and actuators have the capability to actively respond to a changing environment while offering significant weight savings and additional passive controllability through ply tailoring. Piezoelectric sensing and actuation of composite laminates is the most promising concept due to the static and dynamic control capabilities. Essential to the implementation of these smart composites are the development of accurate and efficient modeling techniques and experimental validation. This research addresses each of these important topics. A refined higher order theory is developed to model composite structures with surface bonded or embedded piezoelectric transducers. These transducers are used as both sensors and actuators for closed loop control. The theory accurately captures the transverse shear deformation through the thickness of the smart composite laminate while satisfying stress free boundary conditions on the free surfaces. The theory is extended to include the effect of debonding at the actuator-laminate interface. The developed analytical model is implemented using the finite element method utilizing an induced strain approach for computational efficiency. This allows general laminate geometries and boundary conditions to be analyzed. The state space control equations are developed to allow flexibility in the design of the control system. Circuit concepts are also discussed. Static and dynamic results of smart composite structures, obtained using the higher order theory, are correlated with available analytical data. Comparisons, including debonded laminates, are also made with a general purpose finite element code and available experimental data. Overall, very good agreement is observed. Convergence of the finite element implementation of the higher order theory is shown with exact solutions. Additional results demonstrate the utility of the developed theory to study piezoelectric actuation of composite laminates with pre-existing debonding. Significant changes in the modes shapes and reductions in the control authority result due to partially debonded actuators. An experimental investigation addresses practical issues, such as circuit design and implementation, associated with piezoelectric sensing and actuation of composite laminates. Composite specimens with piezoelectric transducers were designed, constructed and tested to validate the higher order theory. These specimens were tested with various stacking sequences, debonding lengths and gains for both open and closed loop cases. Frequency changes of 15% and damping on the order of more than 20% of critical damping, via closed loop control, was achieved. Correlation with the higher order theory is very good. Debonding is shown to adversely affect the open and closed loop frequencies, damping ratios, settling time and control authority.

Chattopadhyay, Aditi↗

Full-scale wind-tunnel study of wing-fuselage interaction and comparison with paneling method

Full-scale wind-tunnel tests, conducted to determine the effects of three different airfoil sections on the aft-fuselage drag of a low-wing aircraft, are described. The measurements indicate a maximum difference in aft fuselage drag between the three airfoils of about 0.002. Measured changes in the locations of the fuselage pressure contours with airfoil section correlated well with the changes predicted by a three dimensional paneling code. A criterion, based on the inviscid computer code, is then proposed as an indicator for possible adverse viscous interactions at the wing-fuselage juncture.

Ross, J. C.↗

Development of an Improved RELAP5-3D Model for the High Temperature Test Facility

High-temperature gas-cooled reactors (HTGRs) are rapidly approaching deployment. Confidence in transient analysis of these systems requires modeling and simulation tools that have been validated against data relevant to HTGR conditions. The High Temperature Test Facility (HTTF) is an integral effects thermal hydraulics test facility for prismatic HTGRs. In spring and summer of 2019, HTTF was used for a series of experiments that now serve as the basis for the Organization of Economic Cooperation and Development / Nuclear Energy Agency Thermal Hydraulic Code Validation Benchmark for High Temperature Gas-Cooled Reactors using HTTF Data (HTGR T/H Benchmark). Previous analyses as part of the HTGR T/H benchmark used a RELAP5-3D model developed at Idaho National Laboratory (INL) and demonstrated an ability to reproduce trends in the measured data but difficulties reproducing experimental values within their uncertainty. These difficulties were largely attributed to assumptions made during the development of the initial RELAP5-3D model, which predated the HTTF experiments. A significant cause of difficulty reproducing the measured temperatures may be the radial nodalization of the previous RELAP5-3D model. In this paper, we present a new RELAP5-3D model of HTTF with finer radial nodalization built to assess the impact of radial heat transfer. We describe the new model and compare it against the old one at full-power steady state and for the pressurized conduction cooldown (PCC) transient. These analyses are based on the code-to-code comparison exercise for the PCC problem of the HTGR T/H benchmark. We compare maximum block temperature as the primary figure of merit and include discussion on intracore natural circulation.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Structural-Thermal-Optical-Performance (STOP) Model Development and Analysis of a Field-widened Michelson Interferometer

An integrated Structural-Thermal-Optical-Performance (STOP) model was developed for a field-widened Michelson interferometer which is being built and tested for the High Spectral Resolution Lidar (HSRL) project at NASA Langley Research Center (LaRC). The performance of the interferometer is highly sensitive to thermal expansion, changes in refractive index with temperature, temperature gradients, and deformation due to mounting stresses. Hand calculations can only predict system performance for uniform temperature changes, under the assumption that coefficient of thermal expansion (CTE) mismatch effects are negligible. An integrated STOP model was developed to investigate the effects of design modifications on the performance of the interferometer in detail, including CTE mismatch, and other three- dimensional effects. The model will be used to improve the design for a future spaceflight version of the interferometer. The STOP model was developed using the Comet SimApp'TM' Authoring Workspace which performs automated integration between Pro-Engineer®, Thermal Desktop®, MSC Nastran'TM', SigFit'TM', Code V'TM', and MATLAB®. This is the first flight project for which LaRC has utilized Comet, and it allows a larger trade space to be studied in a shorter time than would be possible in a traditional STOP analysis. This paper describes the development of the STOP model, presents a comparison of STOP results for simple cases with hand calculations, and presents results of the correlation effort to bench-top testing of the interferometer. A trade study conducted with the STOP model which demonstrates a few simple design changes that can improve the performance seen in the lab is also presented.

Scola, Salvatore J.↗

Analysis and comparison of diurnal variations of cloud radiative forcing: Earth Radiation Budget Experiment and International Satellite Cloud Climatology Project results

Cloud radiative forcing (CRF) is the radiative impact of clouds on the Earth's radiation budget. This study examines the diurnal variations of CRF using the Earth Radiation Budget Experiment (ERBE) monthly hourly flux data and the flux data derived from the International Satellite Cloud Climatology Project (ISCCP) using the Goddard Institute for Space Studies general circulation model radiation code. The results for the months of April, July, and October 1985 and January 1986 are analyzed. We found that, in general, two data sets agreed. For longwave (LW) CRF the diurnal range over land is generally greater than that observed over oceans. For the 4-month averages the ERBE values are 15.8 W/sq m and 6.8 W/sq m for land and ocean, respectively, compared with the ISCCP calculated values of 18.4 W/sq m and 8.0 W/sq m, respectively. The land/ocean contrast is largely associated with changes in cloud amount and the temperature difference between surface and cloud top. It would be more important to note that the clear-sky flux (i.e., surface temperature) variabilities are shown to be a major contributor to the large variabilities over land. The maximum diurnal range is found to be in the summer hemisphere, and the minimum values in the winter hemisphere. It is also shown that the daytime maximum and the nighttime minimum are seen over large portions of land, whereas they occur at any local hour over most oceans. For shortwave (SW) CRF the daytime maximum values are about twice as large as monthly averages, and their highest frequency occurs at local noon, indicating that solar insolation is a primary factor for the diurnal variation of SW CRF. However, the comparison of the ERBE data with the ISCCP results demonstrated that the largest differences in the diurnal range and monthly mean of LW CRF were associated with tropical convergence zones, where clear-sky fluxes could be easily biased by persistent cloudiness and the inadequate treatment of the atmospheric water vapor.

Kim, Yongseung↗

A Comparison of EAST Shock-Tube Radiation Measurements with a New Air Radiation Model

This paper presents a comparison between the recent EAST shock tube radiation measurements (Grinstead et al., AIAA 2008-1244) and the HARA radiation model. The equilibrium and nonequilibrium radiation measurements are studied for conditions relevant to lunar-return shock-layers; specifically shock velocities ranging from 9 to 11 kilometers per second at initial pressures of 0.1 and 0.3 Torr. The simulated shock-tube flow is assumed one-dimensional and is calculated using the LAURA code, while a detailed nonequilibrium radiation prediction is obtained in an uncoupled manner from the HARA code. The measured and predicted intensities are separated into several spectral ranges to isolate significant spectral features, mainly strong atomic line multiplets. The equations and physical data required for the prediction of these strong atomic lines are reviewed and their uncertainties identified. The 700-1020 nm wavelength range, which accounts for roughly 30% of the radiative flux to a peak-heating lunar return shock-layer, is studied in detail and the measurements and predictions are shown to agree within 15% in equilibrium. The plus or minus 1.5% uncertainty on the measured shock velocity is shown to cause up to a plus or minus 30% difference in the predicted radiation. This band of predictions contains the measured values in almost all cases. For the highly nonequilibrium 0.1 Torr cases, the nonequilibrium radiation peaks are under-predicted by about half. This under-prediction is considered acceptable when compared to the order-of-magnitude over-prediction obtained using a Boltzmann population of electronic states. The reasonable comparison in the nonequilibrium regions provides validation for both the non-Boltzmann modeling in HARA and the thermochemical nonequilibrium modeling in LAURA. The N2 (+)(1-) and N2(2+) molecular band systems are studied in the 290 480 nm wavelength range for both equilibrium and nonequilibrium regimes. The non-Boltzmann rate models for these systems, which have significant uncertainties, are tuned to improve the comparison with measurements.

Johnston, Christopher O.↗

Numerical challenges for energy conservation in N -body simulations of collapsing self-interacting dark matter halos

Dark matter (DM) halos can be subject to gravothermal collapse if the DM is not collisionless, but engaged in strong self-interactions instead. When the scattering is able to efficiently transfer heat from the centre to the outskirts, the central region of the halo collapses and reaches densities much higher than those for collisionless DM. This phenomenon is potentially observable in studies of strong lensing. Current theoretical efforts are motivated by observations of surprisingly dense substructures. However, a comparison with observations requires accurate predictions. One method to obtain such predictions is to use N-body simulations. Collapsed halos are extreme systems that pose severe challenges when applying state-of-the-art codes to model self-interacting dark matter (SIDM). In this work, we investigate the root of such problems, with a focus on energy non-conservation. Moreover, we discuss possible strategies to avoid them. We ran N-body simulations, both with and without SIDM, of an isolated DM-only halo and we adjusted the numerical parameters to check the accuracy of the simulation. We find that not only the numerical scheme for SIDM can lead to energy non-conservation, but also the modelling of gravitational interaction and the time integration are problematic. The main issues we find are: (a) particles changing their time step in a non-time-reversible manner; (b) the asymmetry in the tree-based gravitational force evaluation; and (c) SIDM velocity kicks breaking the time symmetry. Tuning the parameters of the simulation to achieve a high level of accuracy allows us to conserve energy not only at early stages of the evolution, but also later on. However, the cost of the simulations becomes prohibitively large as a result. Some of the problems that make the simulations of the gravothermal collapse phase inaccurate can be overcome by choosing appropriate numerical schemes. However, other issues still pose a challenge. Our findings motivate further works on addressing the challenges in simulating strong DM self-interactions.

dark matter↗

Developing Information Power Grid Based Algorithms and Software

This exploratory study initiated our effort to understand performance modeling on parallel systems. The basic goal of performance modeling is to understand and predict the performance of a computer program or set of programs on a computer system. Performance modeling has numerous applications, including evaluation of algorithms, optimization of code implementations, parallel library development, comparison of system architectures, parallel system design, and procurement of new systems. Our work lays the basis for the construction of parallel libraries that allow for the reconstruction of application codes on several distinct architectures so as to assure performance portability. Following our strategy, once the requirements of applications are well understood, one can then construct a library in a layered fashion. The top level of this library will consist of architecture-independent geometric, numerical, and symbolic algorithms that are needed by the sample of applications. These routines should be written in a language that is portable across the targeted architectures.

Dongarra, Jack↗

The Comparison Of Dome And HMD Delivery Systems: A Case Study

For effective astronaut training applications, choosing the right display devices to present images is crucial. In order to assess what devices are appropriate, it is important to design a successful virtual environment for a comparison study of the display devices. We present a comprehensive system, a Virtual environment testbed (VET), for the comparison of Dome and Head Mounted Display (HMD) systems on an SGI Onyx workstation. By writing codelets, we allow a variety of virtual scenarios and subjects' information to be loaded without programming or changing the code. This is part of an ongoing research project conducted by the NASA / JSC.

Chen, Jian↗

Assessment of Reduced-Kinetics Mechanisms for Combustion of Jet Fuel in CFD Applications

A computational effort was undertaken to analyze the details of fluid flow in Lean-Direct Injection (LDI) combustors for next-generation LDI design. The National Combustor Code (NCC) was used to perform reacting flow computations on single-element LDI injector configurations. The feasibility of using a reduced chemical-kinetics approach, which optimizes the reaction rates and species to model the emissions characteristics typical of lean-burning gas-turbine combustors, was assessed. The assessments were performed with Reynolds- Averaged Navier-Stokes (RANS) and Time-Filtered Navier Stokes (TFNS) time-integration, with a Lagrangian spray model with the NCC code. The NCC predictions for EINOx and combustor exit temperature were compared with experimental data for two different single-element LDI injector configurations, with 60deg and 45deg axially swept swirler vanes. The effects of turbulence-chemistry interaction on the predicted flow in a typical LDI combustor were studied with detailed comparisons of NCC TFNS with experimental data.

Ajmani, Kumud↗

Numerical Predictions of Mode Reflections in an Open Circular Duct: Comparison with Theory

The NASA Broadband Aeroacoustic Stator Simulation code was used to compute the acoustic field for higher-order modes in a circular duct geometry. To test the accuracy of the results computed by the code, the duct was terminated by an open end with an infinite flange or no flange. Both open end conditions have a theoretical solution that was used to compare with the computed results. Excellent comparison for reflection matrix values was achieved after suitable refinement of the grid at the open end. The study also revealed issues with the level of the mode amplitude introduced into the acoustic held from the source boundary and the amount of reflection that occurred at the source boundary when a general nonreflecting boundary condition was applied.

Duct Modes↗

Shock Tube Measurements of Radiative Heating for Titan and Nitrogen

Detailed spectrally and spatially resolved radiance has been measured in the Electric Arc Shock Tube at NASA Ames Research Center for conditions relevant to Titan entry, with varying atmospheric composition, free-stream density (equivalently, altitude) and shock velocity. The test campaign measured radiation at velocities from 4.7 km/s to 8 km/s and free-stream pressures of 0.1, 0.28 and 0.47 Torr with a variety of compositions. Radiances measured in this work are substantially larger compared to that reported both in past EAST test campaigns and in other shock tube facilities. Depending on the metric used for comparison, the discrepancy can be as high as an order of magnitude. Due to the difference with previously reported data, a substantial effort was undertaken to provide confidence in the new results. The present work provides a new benchmark set of data to replace those published in previous studies. The effect of gas impurities identified in previous shock tube studies was also examined by testing in pure N2 and deliberate addition of air to the CH4/N2 mixtures. Furthermore, a test campaign in pure N2 was also conducted with the aim of providing data for improving fundamental understanding of high enthalpy flows containing N2, such as high-speed entries into Earth or Titan. These experiments cover conditions from approximately 6 km/s to 11 km/s at an initial pressure of 0.2 Torr. It is the intention of this paper to motivate code comparisons benchmarked against this data set.

Radiation↗

Extended magnetohydrodynamics simulations of thin-foil Z-pinch implosions with comparison to experiments

Cylindrical foil liners, with foil thicknesses on the order of 400 nm, are often used in university-scale Z-pinch experiments (~1 MA in 100 ns) to study physics relevant to inertial confinement fusion efforts on larger-scale facilities (e.g., the MagLIF effort on the 25-MA Z facility at Sandia National Laboratories). The use of ultrathin foil liners typically requires a central support rod to maintain the structural integrity of the liner target assembly prior to implosion. The radius of this support rod sets a limit on the maximum convergence ratio achievable for the implosion. In recent experiments with a support rod and a preimposed axial magnetic field, helical instability structures in the imploding foil plasma were found to persist as the foil plasma stagnated on the rod and subsequently expanded away from the rod. We have now used the 3D extended magnetohydrodynamics simulation code PERSEUS (which includes Hall physics) to study these experiments. The results suggest that it is the support rod which is responsible for the helical structures persisting beyond stagnation. Furthermore, we find that as the radius of the support rod decreases (i.e., as the convergence ratio increases), the integrity and persistence of the helical modes diminish. In the limit with no support rod, we find that the structure of the final stagnation column is governed by the structure of the central precursor plasma column. Furthermore, these simulation results and their comparisons to experiment are presented.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Comparison between MAST-U conventional and Super-X configurations through SOLPS-ITER modelling

Abstract MAST-U has recently started operating with a Super-X divertor, designed to increase total flux expansion and neutral trapping, both predicted through simple analytic models and SOLPS calculations to reduce the plasma and impurity density detachment thresholds. In this study, utilising the SOLPS-ITER code, we are quantifying the possible gain allowed by the MAST-U Super-X and neutral baffling geometry, in terms of access to detachment. We show that a significant reduction of the upstream density detachment threshold (up to a factor 1.6) could be achieved in MAST-U, for the Super-X, as opposed to conventional divertor geometry, mainly through an increased total flux expansion, neutral trapping being found very similar between the different configurations. We also show that variations of the strike-point angle are complex to interpret in such a tightly baffled geometry, and that a case in which the target normal points more towards the separatrix does not necessarily imply a lower detachment threshold. As in previous calculations for TCV, we quantify the role of neutral effects through developing and applying a quantitative definition of neutral trapping.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Experimental Space Shuttle Orbiter Studies to Acquire Data for Code and Flight Heating Model Validation

In an experimental study to obtain detailed heating data over the Space Shuttle Orbiter, CUBRC has completed an extensive matrix of experiments using three distinct models and two unique hypervelocity wind tunnel facilities. This detailed data will be employed to assess heating augmentation due to boundary layer transition on the Orbiter wing leading edge and wind side acreage with comparisons to computational methods and flight data obtained during the Orbiter Entry Boundary Layer Flight Experiment and HYTHIRM during STS-119 reentry. These comparisons will facilitate critical updates to be made to the engineering tools employed to make assessments about natural and tripped boundary layer transition during Orbiter reentry. To achieve the goals of this study data was obtained over a range of Mach numbers from 10 to 18, with flight scaled Reynolds numbers and model attitudes representing key points on the Orbiter reentry trajectory. The first of these studies were performed as an integral part of Return to Flight activities following the accident that occurred during the reentry of the Space Shuttle Columbia (STS-107) in February of 2003. This accident was caused by debris, which originated from the foam covering the external tank bipod fitting ramps, striking and damaging critical wing leading edge heating tiles that reside in the Orbiter bow shock/wing interaction region. During investigation of the accident aeroheating team members discovered that only a limited amount of experimental wing leading edge data existed in this critical peak heating area and a need arose to acquire a detailed dataset of heating in this region. This new dataset was acquired in three phases consisting of a risk mitigation phase employing a 1.8% scale Orbiter model with special temperature sensitive paint covering the wing leading edge, a 0.9% scale Orbiter model with high resolution thin-film instrumentation in the span direction, and the primary 1.8% scale Orbiter model with detailed thin-film resolution in both the span and chord direction in the area of peak heating. Additional objectives of this first study included: obtaining natural or tripped turbulent wing leading edge heating levels, assessing the effectiveness of protuberances and cavities placed at specified locations on the orbiter over a range of Mach numbers and Reynolds numbers to evaluate and compare to existing engineering and computational tools, obtaining cavity floor heating to aid in the verification of cavity heating correlations, acquiring control surface deflection heating data on both the main body flap and elevons, and obtain high speed schlieren videos of the interaction of the orbiter nose bow shock with the wing leading edge. To support these objectives, the stainless steel 1.8% scale orbiter model in addition to the sensors on the wing leading edge was instrumented down the windward centerline, over the wing acreage on the port side, and painted with temperature sensitive paint on the starboard side wing acreage. In all, the stainless steel 1.8% scale Orbiter model was instrumented with over three-hundred highly sensitive thin-film heating sensors, two-hundred of which were located in the wing leading edge shock interaction region. Further experimental studies will also be performed following the successful acquisition of flight data during the Orbiter Entry Boundary Layer Flight Experiment and HYTHIRM on STS-119 at specific data points simulating flight conditions and geometries. Additional instrumentation and a protuberance matching the layout present during the STS-119 boundary layer transition flight experiment were added with testing performed at Mach number and Reynolds number conditions simulating conditions experienced in flight. In addition to the experimental studies, CUBRC also performed a large amount of CFD analysis to confirm and validate not only the tunnel freestream conditions, but also 3D flows over the orbiter acreage, wing leading edge, and controlurfaces to assess data quality, shock interaction locations, and control surface separation regions. This analysis is a standard part of any experimental program at CUBRC, and this information was of key importance for post-test data quality analysis and understanding particular phenomena seen in the data. All work during this effort was sponsored and paid for by the NASA Space Shuttle Program Office at the Johnson Space Center in Houston, Texas.

Wadhams, T. P.↗

Studying visual search without an eye tracker: an assessment of artificial foveation

Abstract Eye tracking is a useful tool for studying human cognition, both in the laboratory and in real-world applications. However, there are cases in which eye tracking is not possible, such as in high-security environments where recording devices cannot be introduced. After facing this challenge in our own work, we sought to test the effectiveness of using artificial foveation as an alternative to eye tracking for studying visual search performance. Two groups of participants completed the same list comparison task, which was a computer-based task designed to mimic an inventory verification process that is commonly performed by international nuclear safeguards inspectors. We manipulated the way in which the items on the inventory list were ordered and color coded. For the eye tracking group, an eye tracker was used to assess the order in which participants viewed the items and the number of fixations per trial in each list condition. For the artificial foveation group, the items were covered with a blurry mask except when participants moused over them. We tracked the order in which participants viewed the items by moving their mouse and the number of items viewed per trial in each list condition. We observed the same overall pattern of performance for the various list display conditions, regardless of the method. However, participants were much slower to complete the task when using artificial foveation and had more variability in their accuracy. Our results indicate that the artificial foveation method can reveal the same pattern of differences across conditions as eye tracking, but it can also impact participants’ task performance.

47 OTHER INSTRUMENTATION↗

Verification and Validation Activities for the Multi-Fidelity Toolkit

The Multi-Fidelity Toolkit (MFTK) is a simulation tool being developed at Sandia National Laboratories for aerodynamic predictions of compressible flows over a range of physics fidelities and computational speeds. These models include the Reynolds-Averaged-Navier-Stokes (RANS) equations, the Euler equations, and modified Newtonian aerodynamics (MNA) equations, and they can be invoked independently or coupled with hierarchical Kriging to interpolate between high-fidelity simulations using lower-fidelity data. However, as with any new simulation capability, verification and validation are necessary to gather credibility evidence. This work describes formal code- and solution-verification activities as well as model validation with uncertainty considerations. Code verification is performed on the MNA model by comparing with an analytical solution for flat-plate and inclined-plate geometries. Solution-verification activities include grid-refinement studies of HIFiRE-1 wind tunnel measurements, which are used for validation, for all model fidelities. A thorough treatment of the validation comparison with prediction error and validation uncertainty is also presented.

42 ENGINEERING↗

Using Likwid and Byfl to Benchmark Hardware Performance

This paper outlines a benchmarking study conducted during my internship at LANL, focusing on CPU (Computer Processing Unit) and program performance assessment. The primary goal was to gather memory access data using three methods across five polybench kernels The data gathered would then be used to compare and contrast to one another and calculate operational intensity for performance comparisons. Benchmarking tools like Byfl and Likwid were employed, with Byfl offering hardware-independent data through LLVM compiler communication and Likwid directly interacting with computer hardware. The study considered various benchmarking factors, including optimization levels, Big O notation ((n)), CPU diversity and specific kernel equations. Big O notation was utilized to simplify code complexity, with detailed breakdwons of operations and memory components for each polybench application. Specific O(n) equations enabled nuanced kernel compariosns, facilitating the identification of performance variations. CPU efficiency assessments were conducted using Likwid tests on two CPUs. The central focus on code optimization aimed at achieving higher speeds and reduced memory usage through streamlined code. Future work propsoes creating a roofline model, synthesizing benchmarking data into a comprehensive data graph to assist in optimizing code and improving hardware performance. The potential impact on the laboratory or national mission was underscored, emphasizing the importance of optimizing applications and hardware to conserve resources and accelerate program execution. The specific relevance to LANL’s operations in math-intensive fields such as Nuclear Fission, Space Exploration, and Nanotechnology highlights the necessity of efficient benchmarking for resource conservation and proram speed. Overall, this study contributes to the understanding of CPU and program performance, providing insights for future optimization efforts in a laboratory setting

97 MATHEMATICS AND COMPUTING↗