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At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

Evaluation of on-line pulse control for vibration suppression in flexible spacecraft

A numerical simulation was performed, by means of a large-scale finite element code capable of handling large deformations and/or nonlinear behavior, to investigate the suitability of the nonlinear pulse-control algorithm to suppress the vibrations induced in the Spacecraft Control Laboratory Experiment (SCOLE) components under realistic maneuvers. Among the topics investigated were the effects of various control parameters on the efficiency and robustness of the vibration control algorithm. Advanced nonlinear control techniques were applied to an idealized model of some of the SCOLE components to develop an efficient algorithm to determine the optimal locations of point actuators, considering the hardware on the SCOLE project as distributed in nature. The control was obtained from a quadratic optimization criterion, given in terms of the state variables of the distributed system. An experimental investigation was performed on a model flexible structure resembling the essential features of the SCOLE components, and electrodynamic and electrohydraulic actuators were used to investigate the applicability of the control algorithm with such devices in addition to mass-ejection pulse generators using compressed air.

Masri, Sami F.↗
NASA NTRS1990

Dual-shaped offset reflector antenna designs from solutions of the geometrical optics first-order partial differential equations

In obtaining solutions to the first-order nonlinear partial differential equations (PDEs) for synthesizing offset dual-shaped reflectors, it is found that previously observed computational problems can be avoided if the integration of the PDEs is started from an inner projected perimeter and integrated outward rather than starting from an outer projected perimeter and integrating inward. This procedure, however, introduces a new parameter, the main reflector inner perimeter radius p(o), when given a subreflector inner angle 0(o). Furthermore, a desired outer projected perimeter (e.g., a circle) is no longer guaranteed. Stability of the integration is maintained if some of the initial parameters are determined first from an approximate solution to the PDEs. A one-, two-, or three-parameter optimization algorithm can then be used to obtain a best set of parameters yielding a close fit to the desired projected outer rim. Good low cross-polarization mapping functions are also obtained. These methods are illustrated by synthesis of a high-gain offset-shaped Cassegrainian antenna and a low-noise offset-shaped Gregorian antenna.

Galindo-Israel, V.↗
DOE OSTI2020

Simulations of Laser Experiments to Study the Origin of Cosmic Magnetic Fields

Magnetic fields are ubiquitous in the universe; however, their origin is not fully understood. Cosmologists and astrophysicists have proposed a variety of ways in which small seed magnetic fields could be created. It is widely thought that the much larger values of the cosmic magnetic fields that we observe are a result of the amplification of these seed fields by the nonlinear turbulent dynamo mechanism. Such a mechanism had not yet been demonstrated in a controlled laboratory environment. We conceived experiments designed to demonstrate and study the nonlinear turbulent dynamo mechanism in the laboratory. These experiments characterize the distribution of turbulent energy among the velocity, magnetic field, and density fluctuations, providing a comprehensive picture of the energy cascade in a magnetized, turbulent plasma. The experiments build on the pathfinder experiments we have conducted on the Vulcan laser at the Rutherford-Appleton Laboratory in the UK and the OMEGA laser at Laboratory for Laser Energetics at Rochester. They utilize the high-intensity lasers at the Omega Laser Facility, the National Ignition Facility (NIF) at LLNL and the Laser Megajoule (LMJ) Facility in France – the largest laser facilities in the world. The goal of this project was to design and model the highly demanding experiments through simulation campaigns using FLASH, a highly capable radiation-MHD code we have developed, and large-scale 3D simulations on the Mira supercomputer at ANL. The simulations were vital to ensuring the experiments achieve the strong turbulence and large magnetic Reynolds numbers required for the nonlinear turbulent dynamo mechanism to operate. The simulations were also critical to determining when to fire the diagnostics, since the experiments last tens of nanoseconds but the signals of the strongly amplified magnetic fields last only a few nanoseconds. Finally, high-fidelity, validated FLASH simulations were crucial to interpreting the results of the experiments. By combining theory, experiments, and simulations, we were able to succeed in demonstrating and characterizing the nonlinear turbulent dynamo mechanism. The effort leveraged (1) the emergence of the Flash Center for Computational Science as a leader in numerical modeling of academic High Energy Density Physics (HEDP) experiments; (2) the FLASH code, which has been developed by the Flash Center and is a leadership-class, multi- physics, publicly available community code that has been applied successfully in many laser-driven experiments; (3) the expertise acquired through pathfinder experiments we performed on the Vulcan laser at the Rutherford-Appleton Laboratory in the UK and on the Omega laser facility; (4) the experience we have gained in designing and interpreting these experiments using validated simulations done with FLASH; and (5) the close collaborations we have established with outstanding experimental groups in the US and abroad. The broader impacts of the project are two-fold: (1) the work furthered the transformation of the academic community’s ability to design and analyze HEDP experiments at large laser facilities that is happening through the availability of FLASH and its widespread adoption by the academic community – a community that previously had had limited access to open and validated hydro/MHD simulation tools for HEDP experiments; and (2) the work trained young scientists to design and interpret HEDP experiments using validated simulations – a critical national need.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Development and Analysis of Optimal Multilevel Solvers on Advanced Computers. Final Report

Constrained optimization in the context of time dependent, partial differential equations (PDE) leads to a symmetric, block-tridiagonal system of nonlinear equations that must be solved repeatedly in an iterative solution strategy. The blocks represent spatial discretization, while the connection between the blocks represents a forward and backward integration in time. The focus of this project is to apply a parallel-in-time (PiT) solution technique to the large block-triangular system.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Domain Aware Deep-learning Algorithms Integrated with Scientific-computing Technologies (DADAIST)

This technical report summarized the contribution of the DADAIST project funded by the Data Model Convergence Initiative via the Laboratory Directed Research and Development (LDRD) investments at Pacific Northwest National Laboratory (PNNL). Specifically, we report the development of the NeuroMANCER (Neural Modules with Adaptive Nonlinear Constraints and Efficient Regularizations), a new open-source Scientific Machine Learning library for formulating and solving parametric constrained optimization problems, physics-informed system identification, and parametric optimal control problems. NeuroMANCER is using differentiable programming to combine modern data-driven models and optimization modeling language into a coherent algorithmic and software framework. NeuroMANCER is a Pytorch-based framework and adopts much of its philosophy focused on research and development, rapid prototyping, and streamlined deployment. Strong emphasis is given to extensibility, interoperability with the PyTorch ecosystem, and quick adaptability to custom domain problems. Neuromancer repository contains a comprehensive library of differentiable modules, including custom activation functions, matrix factorizations, deep learning architectures, neural differential equations, differential equation solvers, implicit layers such as iterative solvers, high-level API for symbolic expressions, API for modeling and control of dynamical systems, and extensive set of tutorial code examples in the form of python scripts and jupyter notebooks.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

CI-MOR Final Report: Analysis and Validation of Critical Infrastructure Models using Model Order Reduction

This report summarizes the research and capabilities developed as part of the project “Analysis and Validation of Critical Infrastructure Models using Model Order Reduction” (CI-MOR) LDRD project. CI-MOR research enables the solution of large, complex optimization models that naturally arise in national security challenges involving critical infrastructures. Specifically, CI-MOR researchers developed methods to (1) rigorously approximate complex, nonlinear optimization formulations, (2) identify alternative near-optimal solutions, (3) accelerate optimization workflows used for complex applications, and (4) rigorously integrate domain knowledge in stochastic-process models. This report provides an overview of the research done in CI-MOR, and we describe application exemplars used to illustrate CI-MOR capabilities. Furthermore, we describe the software developed by CI-MOR that researchers can leverage to analyze new applications.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

A multivariate variational objective analysis-assimilation method. Part 1: Development of the basic model

The variational method of undetermined multipliers is used to derive a multivariate model for objective analysis. The model is intended for the assimilation of 3-D fields of rawinsonde height, temperature and wind, and mean level temperature observed by satellite into a dynamically consistent data set. Relative measurement errors are taken into account. The dynamic equations are the two nonlinear horizontal momentum equations, the hydrostatic equation, and an integrated continuity equation. The model Euler-Lagrange equations are eleven linear and/or nonlinear partial differential and/or algebraic equations. A cyclical solution sequence is described. Other model features include a nonlinear terrain-following vertical coordinate that eliminates truncation error in the pressure gradient terms of the horizontal momentum equations and easily accommodates satellite observed mean layer temperatures in the middle and upper troposphere. A projection of the pressure gradient onto equivalent pressure surfaces removes most of the adverse impacts of the lower coordinate surface on the variational adjustment.

Achtemeier, Gary L.↗
NASA NTRS1999

Near-Simultaneous Spectroscopic and Broadband Polarimetric Observations of Be Stars

Near simultaneous optical spectroscopic (on four nights) and broadband linear continuum (B, V, R, and I bands) polarimetric (on seven nights) observations of 29 Be stars were carried out during 1993 November-December. The program Be stars displayed wavelength dependence of intrinsic polarizations with no frequency dependence of polarimetric position angles. Some of the Be stars displayed long-term polarization variability. The Be and Be-shell stars could not be distinguished from one another solely on the basis of their polarization values. Full widths at half-maximum of the H.alpha profiles and the intrinsic linear continuum polarizations are closely correlated with the projected rotational velocities of the program stars. Photospheric-absorption-corrected equivalent widths of H.alpha profiles [W(alpha)] and the radii of H.alpha-emitting or -absorbing envelopes (R(sub e) or R(sub a)) are nonlinearly correlated with the intrinsic continuum polarizations of these stars. However, W(alpha) and R(sub e) are linearly correlated. With large uncertainties, there is a trend of spectral dependence of polarization. Detailed discussion of these results is presented in this paper.

Ghosh, K.↗
NASA NTRS2008

Assessment of Near-Field Sonic Boom Simulation Tools

A recent study for the Supersonics Project, within the National Aeronautics and Space Administration, has been conducted to assess current in-house capabilities for the prediction of near-field sonic boom. Such capabilities are required to simulate the highly nonlinear flow near an aircraft, wherein a sonic-boom signature is generated. There are many available computational fluid dynamics codes that could be used to provide the near-field flow for a sonic boom calculation. However, such codes have typically been developed for applications involving aerodynamic configuration, for which an efficiently generated computational mesh is usually not optimum for a sonic boom prediction. Preliminary guidelines are suggested to characterize a state-of-the-art sonic boom prediction methodology. The available simulation tools that are best suited to incorporate into that methodology are identified; preliminary test cases are presented in support of the selection. During this phase of process definition and tool selection, parallel research was conducted in an attempt to establish criteria that link the properties of a computational mesh to the accuracy of a sonic boom prediction. Such properties include sufficient grid density near shocks and within the zone of influence, which are achieved by adaptation and mesh refinement strategies. Prediction accuracy is validated by comparison with wind tunnel data.

Casper, J. H.↗
NASA NTRS2018

Finite Element Simulations of Two Vertical Drop Tests of F-28 Fuselage Sections

In March 2017, a vertical drop test of a forward fuselage section of a Fokker F-28 MK4000 aircraft was conducted as part of a joint NASA/FAA project to investigate the performance of transport aircraft under realistic crash conditions. In June 2017, a vertical drop test was conducted of a wing-box fuselage section of the same aircraft. Both sections were configured with two rows of aircraft seats, in a triple-double configuration. A total of ten Anthropomorphic Test Devices (ATDs) were secured in seats using standard lap belt restraints. The forward fuselage section was also configured with luggage in the cargo hold. Both sections were outfitted with two hat racks, each with added ballast mass. The drop tests were performed at the Landing and Impact Research facility located at NASA Langley Research Center in Hampton, Virginia. The measured impact velocity for the forward fuselage section was 346.8-in/s onto soil. The wing-box section was dropped with a downward facing pitch angle onto a sloping soil surface in order to create an induced forward acceleration in the airframe. The vertical impact velocity of the wing-box section was 349.2-in/s. A second objective of this project was to assess the capabilities of finite element simulations to predict the test responses. Finite element models of both fuselage sections were developed for execution in LS-DYNA(Registered Trademark), a commercial explicit nonlinear transient dynamic code. The models contained accurate representations of the airframe structure, the hat racks and hat rack masses, the floor and seat tracks, the luggage in the cargo hold for the forward section, and the detailed under-floor structure in the wing-box section. Initially, concentrated masses were used to represent the inertial properties of the seats, restraints, and ATD occupants. However, later simulations were performed that included finite element representations of the seats, restraints, and ATD occupants. These models were developed to more accurately replicate the seat loading of the floor and to enable prediction of occupant impact responses. Models were executed to generate analytical predictions of airframe responses, which were compared with test data to validate the model. Comparisons of predicted and experimental structural deformation and failures were made. Finally, predicted and experimental soil deformation and crater depths were also compared for both drop test configurations.

Jackson, Karen E.↗
DOE OSTI2026

High-resolution mountain topography can inform global snow vulnerability estimates

Snow is changing globally. Computationally intensive snow reanalysis products and downscaled climate model projections allow for the estimation of historical and projected changes in snow over ∼4–10 km resolutions, but these resolutions are coarse relative to the scales needed for water supply and flood planning. Fine-scale digital elevation models (DEMs) are widely available but are underutilized to make first-order assessments of snow vulnerability. Here, we leverage DEMs at a 7.5 arc s (∼250 m) resolution, combining these with historical freezing level height estimates from ERA-5 to derive estimates of changes in the snow-receiving area (SRA) and its variability across global mountain ranges. Results show estimated SRA declines in 29% (1.9 million km2) of the global mountain area from 1982–2020; 66% of the mountainous areas had no change over the historical period. At +1.5 °C of warming relative to the pre-industrial control, global mountain SRA would decline by 9.5% (1.0 million km2) relative to recent conditions. This loss would be approximately doubled with +2 °C of warming. In a +4 °C warming scenario, an additional 34% (3.6 million km2) of SRA would be lost beyond the +2 °C case. Across individual mountain ranges, SRA losses can occur nonlinearly with warming, with some locations that have historically had relatively minor SRA losses at risk of substantially larger losses in warmer climates. Analysis using coarser-resolution DEMs can underestimate or overestimate SRA and its rate of loss, with the largest impacts in relatively warm, low-elevation mountain ranges. Results of this work provide estimates of projected loss in SRA at policy-relevant warming levels; inform the resolutions needed for process-based snow modeling; identify snow vulnerability hotspots; and provide a new integrated approach to snow vulnerability assessment that is achievable at global scales and highlights potential nonlinearities from recent trends to a variety of future warming scenarios.

climate, mountains↗
DOE OSTI2023

Statistical Learning for Nonlinear Model Reduction from Local Simulations of Stochastic and Particle- and Agent-Based Systems

Stochastic physical systems across the sciences that have very high-dimensional state spaces, with a large number of fast degrees of freedom that force direct simulators to proceed by integration steps that are orders of magnitude smaller than events of interests (e.g., particle collisions). Examples range from molecular motion to dynamics of large populations of cells. A grand challenge in the simulation and understanding of such systems is the systematic construction of accurate, interpretable, reduced models, enabling faster simulations, revealing fundamental properties of the dynamics, and predicting phenomena of interest that the original simulator could not reached with sufficient accuracy or within a given computational budget. In this projected we developed novel statistical estimation/machine learning techniques for analyzing and building empirical reduced models for important families of high-dimensional stochastic systems, in particular: - we developed techniques for estimating interaction kernels in interacting particle- and agent-based systems, which are ubiquitous in Physics, Biology and many other sciences, given observed trajectories of the system; - we developed techniques for nonlinear model reduction for high-dimensional stochastic systems that have a small number of unknown, nonlinear slow variables, and a large number of fast modes, that are possibly of large magnitude, given observed short trajectories of the system in the form of bursts of trajectories from different initial conditions; - we developed novel techniques for estimating linear dynamical systems on graphs when both the dynamics and the underlying graph are unknown, and we have a sparse set of space-time observations; - we considered the problem of estimating an unknown nonlinear observation function of a standard process (e.g. Brownian motion), so that we can recognized if an observed dynamics is "just" a nonlinear version of a known dynamics; we also developed benchmarks for learning algorithms aimed at learning and classifying diffusion processes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Identifying rolling resistance and air resistance simultaneously for an electric truck

Accurately estimating rolling and air resistance is essential for predicting the energy consumption of vehicles. This study presents a field-based approach using a rolldown test to simultaneously determine rolling and air resistance coefficients. Unlike prior methods that frequently used simulations or models, we employ a goal programming methodology to improve precision and evaluate the actual vehicle and environmental conditions. Our methodology was tested using a Class 8 Freightliner eCascadia on a surveyed road section, ensuring controlled conditions for data collection. By analyzing the time–velocity relationship across multiple test runs, we derived resistance coefficients for both loaded and unloaded conditions. The study confirms that rolling resistance is largely independent of velocity at low speeds but exhibits a nonlinear dependency at higher speeds. Additionally, road surface conditions, tire condition, axle configuration, aerodynamic properties, and weather conditions significantly impact resistance values, emphasizing the need for real-world testing rather than relying solely on standardized projections. Our results align with existing literature while demonstrating the efficacy of the goal programming approach in refining resistance estimates. This work contributes to improved vehicle energy modeling, offering practical insights for fleet operators and policymakers seeking accurate energy consumption predictions for electric trucks operating under varying environmental conditions.

47 OTHER INSTRUMENTATION↗
NASA NTRS1984

Optically thick ablation fronts

The physical characteristics of optically thick ablation fronts such as interstellar clouds are analyzed. Attention is given to cold clumps in both planar and spherical geometries and modifications caused by accelerations in a gravitational field or by evaporation of the clumps when encountered hot gas. The effects of ablation on the appearance of the Rayleigh-Taylor instability are examined in both linear and nonlinear regimes. The results of the calculations are applied to the astrophysical phenomena of cold clumps immersed in a supersonic flow, optically thick jets, and ablation in stellar envelopes. Evaporation in an optically thick front is projected to be orders of magnitude larger than evaporation in electron-conduction fronts in optically thin conditions. The optically thick processes could then be useful for modeling flows from, e.g., newly formed stars and active galactic nuclei.

Konigl, A.↗
NASA NTRS1992

Wave equations and computational models for sonic boom propagation through a turbulent atmosphere

The improved simulation of sonic boom propagation through the real atmosphere requires greater understanding of how the transient acoustic pulses popularly termed sonic booms are affected by atmospheric turbulence. A nonlinear partial differential equation that can be used to simulate the effects of smaller-scale atmospheric turbulence on sonic boom waveforms is described. The equation is first order in the time derivative and involves an extension of geometrical acoustics to include diffraction phenomena. Various terms in the equation are explained in physical terms. Such terms include those representing convection at the wave speed, diffraction, molecular relaxation, classical dissipation, and nonlinear steepening. The atmospheric turbulence enters through an effective sound speed, which varies with all three spatial coordinates, and which is the sum of the local sound speed and the component of the turbulent flow velocity projected along a central ray that connects the aircraft trajectory with the listener.

Pierce, Allan D.↗
NASA NTRS2014

Improving Sensorimotor Function and Adaptation using Stochastic Vestibular Stimulation

Astronauts experience sensorimotor changes during adaption to G-transitions that occur when entering and exiting microgravity. Post space flight, these sensorimotor disturbances can include postural and gait instability, visual performance changes, manual control disruptions, spatial disorientation, and motion sickness, all of which can hinder the operational capabilities of the astronauts. Crewmember safety would be significantly increased if sensorimotor changes brought on by gravitational changes could be mitigated and adaptation could be facilitated. The goal of this research is to investigate and develop the use of electrical stochastic vestibular stimulation (SVS) as a countermeasure to augment sensorimotor function and facilitate adaptation. For this project, SVS will be applied via electrodes on the mastoid processes at imperceptible amplitude levels. We hypothesize that SVS will improve sensorimotor performance through the phenomena of stochastic resonance, which occurs when the response of a nonlinear system to a weak input signal is optimized by the application of a particular nonzero level of noise. In line with the theory of stochastic resonance, a specific optimal level of SVS will be found and tested for each subject [1]. Three experiments are planned to investigate the use of SVS in sensory-dependent tasks and performance. The first experiment will aim to demonstrate stochastic resonance in the vestibular system through perception based motion recognition thresholds obtained using a 6-degree of freedom Stewart platform in the Jenks Vestibular Laboratory at Massachusetts Eye and Ear Infirmary. A range of SVS amplitudes will be applied to each subject and the subjectspecific optimal SVS level will be identified as that which results in the lowest motion recognition threshold, through previously established, well developed methods [2,3,4]. The second experiment will investigate the use of optimal SVS in facilitating sensorimotor adaptation to system disturbances. Subjects will adapt to wearing minifying glasses, resulting in decreased vestibular ocular reflex (VOR) gain. The VOR gain will then be intermittently measured while the subject readapts to normal vision, with and without optimal SVS. We expect that optimal SVS will cause a steepening of the adaptation curve. The third experiment will test the use of optimal SVS in an operationally relevant aerospace task, using the tilt translation sled at NASA Johnson Space Center, a test platform capable of recreating the tilt-gain and tilt-translation illusions associated with landing of a spacecraft post-space flight. In this experiment, a perception based manual control measure will be used to compare performance with and without optimal SVS. We expect performance to improve in this task when optimal SVS is applied. The ultimate goal of this work is to systematically investigate and further understand the potential benefits of stochastic vestibular stimulation in the context of human space flight so that it may be used in the future as a component of a comprehensive countermeasure plan for adaptation to G-transitions.

Galvan, R. C.↗
NASA NTRS2023

Fully Coupled Aeroelastic Stability Analysis of Adaptive Shape Memory Alloy Structural Technologies for Airframe Noise Reduction

This final report documents work performed by ATA Engineering, Inc., (ATA) to develop computational models and analyze the coupled fluid-structure response of two types of noise treatments applied to the leading-edge-slat component of a high-lift system typical of modern transport aircraft. The first treatment is a slat-gap filler (SGF), which closes the gap between the suction surfaces of a deployed slat and an aircraft main wing, and the second treatment is a slat-cove filler (SCF), which replaces the recirculating flow on the slat cove with a surface that promotes flow attachment. The representative airframe chosen for this work was NASA’s High-Lift Common Research Model (CRM-HL) in a baseline high-lift configuration. Superelastic shape memory alloys (SMAs) have been identified as enabling materials for these structural treatments. Since the technology elements rely upon having a highly reconfigurable structure, designs were assessed for their static aeroelastic deflection as well as their dynamic aeroelastic stability using coupled computational fluid dynamics (CFD) and nonlinear computational structural dynamics (NL-CSD) tools. Specifically, fluid-structure interaction (FSI) problems were solved computationally using the CFD solver Loci/CHEM and the NL-CSD solver Abaqus. As a part of the overall project, a similar capability was implemented using the CFD solver FUN3D coupled to Abaqus, although that work is documented in a separate report and that FSI framework was not used to analyze any of NASA’s SGF and SCF configurations. The technical approach consisted of solving for the flow field around the entire vehicle using a global CFD model, followed by extraction of relevant local subdomain data for CFD and NL-CSD co-simulations. The SGF design was analyzed using both 2D and 3D co-simulations to predict quasi-static aeroelastic deformations and to assess dynamic aeroelastic stability, whereas the SCF was analyzed in 2D only. SGF static aeroelastic response predictions focused on characterizing the deformed shape, with maximum displacements predicted to be on the order of magnitude of the technology element panel thickness. SGF dynamic aeroelastic response predictions used Partial Floquet analysis of the temporal evolution of selected nodal displacements to quantify the sign and magnitude of aeroelastic damping. Results suggest that the CRM-HL operating conditions would result in a dynamically stable response. The simulated dynamic pressure was also increased up to a factor of about four, and resulting responses suggest that predicted dynamic stability would be achieved with some margin.

Fluid Structure Interaction↗
DOE OSTI2026

Multiresolution Quantum Chemistry: Nonlinear Response Properties at the Basis Set Limit

We benchmark the accuracy of Dunning correlation-consistent Gaussian basis sets for computing frequencydependent second-order hyperpolarizabilities relevant to second-harmonic generation (SHG), using multiresolution analysis (MRA) as a reference. Basis set errors are analyzed using a unit-sphere representation of the effective hyperpolarizability vector, enabling direct assessment of directional error structure. We introduce a relative RMS total error metric that integrates directional deviations over the unit sphere and complement it with signed projection errors that distinguish over- and underestimation. Unsupervised clustering based on these signed directional metrics reveals four distinct convergence behaviors across a set of 68 molecules. Unitsphere visualizations of representative systems show that basis set errors are often highly anisotropic and localized along specific bond directions, even when global error measures appear small. Doubly augmented basis sets consistently outperform singly augmented ones, and core-polarization functions are required for uniform convergence in second-row systems. Overall, this work demonstrates that directional analysis combined with clustering provides a robust framework for understanding basis set convergence in nonlinear optical response properties.

Basis sets↗