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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 613 records · Page 34
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Flexible and Modular Simultaneous Modeling of Flow and Reactive Transport in Rivers and Hyporheic Zones

Investigations of coupled multiphysics processes in rivers and hyporheic zones have extensively used numerical models. Most existing models use a sequential, one-way coupling between the surface and subsurface domains. Such one-way coupling potentially introduces error. To overcome this, a fully coupled model, hyporheicFoam, was developed using the open-source computational platform OpenFOAM. It captures the coupled flow and multicomponent reactive transport processes within both surface and subsurface domains and across their interface. The coupling between two domains is implemented by mapping conservative flux boundary conditions at the interface through an iterative algorithm. Reactive transport is enabled by specifying a reaction network. To start, we have implemented reaction kinetics following the double Monod-type model with inhibition. The model capability is illustrated through modeling of both conservative and reactive hyporheic flow and transport through dune bedforms. With the novel coupled model, it is now possible to quantify reactions wherein the reactants and products are constantly exchanging between domains and have feedbacks. hyporheicFoam can simulate large, three-dimensional cases owing to the computational flexibility and power offered by the code structure and parallel design of OpenFOAM.

58 GEOSCIENCES↗
DOE OSTI2022

Modeling the small-scale deposition of snow onto structured Arctic sea ice during a MOSAiC storm using snowBedFoam 1.0.

The remoteness and extreme conditions of the Arctic make it a very difficult environment to investigate. In these polar regions covered by sea ice, the wind is relatively strong due to the absence of obstructions and redistributes a large part of the deposited snow mass, which complicates estimates for precipitation hardly distinguishable from blowing or drifting snow. Moreover, the snow mass balance in the sea ice system is still poorly understood, notably due to the complex structure of its surface. Quantitatively assessing the snow distribution on sea ice and its connection to the sea ice surface features is an important step to remove the snow mass balance uncertainties (i.e., snow transport contribution) in the Arctic environment. In this work we introduce snowBedFoam 1.0., a physics-based snow transport model implemented in the open-source fluid dynamics software OpenFOAM. We combine the numerical simulations with terrestrial laser scan observations of surface dynamics to simulate snow deposition in a MOSAiC (Multidisciplinary Drifting Observatory for the Study of Arctic Climate) sea ice domain with a complicated structure typical for pressure ridges. The results demonstrate that a large fraction of snow accumulates in their vicinity, which compares favorably against scanner measurements. However, the approximations imposed by the numerical framework, together with potential measurement errors (precipitation), give rise to quantitative inaccuracies, which should be addressed in future work. The modeling of snow distribution on sea ice should help to better constrain precipitation estimates and more generally assess and predict snow and ice dynamics in the Arctic.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

A machine learning approach to quantify degradation of nuclear fuels and the effects of fission products

Nuclear fuel performance is critically dependent on understanding the evolution of fuel properties under operational conditions, a complex challenge driven by chemical changes and substantial radiation damage during fission. Traditionally, property evolution has been determined via empirical data collected following irradiation. However, these empirical correlations are limited in their applicability beyond the specific conditions in which they were obtained. This study explores a novel approach to address this challenge by applying materials informatics to develop a machine learning random forest (ML-RF) model that captures the effects of fission products on fuel compounds. The model predicts formation enthalpy (ΔH f ) by leveraging extensive quantum materials property data and correlating it with material descriptors such as composition, atomic and site features, and crystal lattice properties. This ML-RF model enables rapid interpolation across the compositional and structural spaces covered by the training data, thus supporting high-throughput screening and energetic ranking of candidate phases. The model demonstrates the ability to predict ΔH f with a mean absolute error (MAE) of approximately 0.1 to 0.2 eV/atom across a wide range of compounds, including key nuclear fuel systems (U-O, U-N, U-C, U-Si, and U-Mo). For example, it was used to assess shifts in stoichiometry for UO 2 (O/M) and UN (N/M) fuels, revealing their distinct tendencies in chemical potential variation and enabling preliminary convex hull analyses. Furthermore, the model provides insights into how individual fission products affect fuel properties. Results indicate that larger fission products (e.g., Nd, Pu, Ce) have a more pronounced impact on UO 2 , while lighter ones (e.g., Zr) strongly influence UN. Here, the model developed in this work can be used to support the Accelerated Fuel Qualification approach by facilitating preliminary evaluations prior to extensive materials modeling and experimentation. To this end, the trained model has been made available to the fuel community to support ongoing fuel development efforts.

Accelerated fuel qualification↗
DOE OSTI2025

Data for: Climatic Imprint on Interfacially-Controlled Platinum-Palladium Resources

Data package for manuscript "Climatic Imprint on Interfacially-Controlled Platinum-Palladium Resources" by Emily G. Wright, Ivey Wang, Yihang Fang, Elaine D. Flynn, and Jeffrey G. Catalano. This dataset contains adsorption results from experiments designed to investigate the effect of chloride on Pd(II) adsorption to goethite and Pt(II) adsorption to hematite and goethite, including lab experiments, X-ray absorption fine structure spectroscopy, and models of retention within a laterite. See the associated manuscript for full methods information. The file "Wright2025_PtAds_data.csv" contains the target starting Pt concentration (uM), final aqueous Pt and associated error (in uM), calculated adsorbed Pt and associated error (in umol/m2), target and measured aqueous chloride (mM), target aqueous nitrate (mM), final pH, and mineral concentration/loading (g/L). Associated mineral-free controls (mineral loading = 0 g/L) are included; the aqueous Pd error was not calculated and chloride was not measured in every sample. These data appear in Figures 1, S3, S4, S5, S20, and S22 in the associated manuscript. The file "Wright2025_PdAds_data.csv" contains the target starting Pd concentration (uM), final aqueous Pd and associated error (in uM), calculated adsorbed Pd and associated error (in umol/m2), target and measured aqueous chloride (mM), and mineral concentration/loading (g/L). Associated mineral-free controls (mineral loading = 0 g/L) are included; the aqueous Pd error was not calculated and chloride was not measured in every sample. These data appear in Figures 1, S3, S4, S5, and S20 in the associated manuscript. The file "Wright2025_MineralBatches_data.csv" contains the mineral identity and BET specific surface area (m2/g) for every mineral batch synthesized and used in experiments. The annealing time used is listed for hydrothermally annealed goethite. These data appear in Table S2 in the associated manuscript. The file "Wright2025_XRD_data.csv" contains the XRD patterns for every mineral batch synthesized as the counts as a function of two theta (in degrees). See "Wright2025_MineralBatches_data.csv" for more details on specific mineral batches. These data appear in Figure S2 in the associated manuscript. The file "Wright 2025_ZetaPotential_data.csv" contains the measured zeta potentials for samples of goethite (batch G2) at pH 4 the presence of varying amounts of sodium chloride. These data appear in Table S3 in the associated manuscript. The file "Wright2025_XAFSSamples_data.csv" contains the specific mineral batch, measured final aqueous Pd or Pt (uM), measured final aqueous chloride (mM), and estimated adsorbed Pd or Pt (umol/m2) of all XAFS samples. These data appear in Tables S4, S7, S8, and S10 in the associated manuscript. The files "Wright2025_PdXAFS_data.csv" and "Wright2025_PtXAFS_data.csv" contain the normalized spectra of Pd and Pt, respectively, adsorbed to minerals at varying chloride concentrations. See "Wright2025_XAFSSamples_data.csv" for a guide to sample names. Note that "05" in a sample name is equivalent to "0.5". These data appear in Figures 2, S6, S7, S8, S12, S13, and S14 in the associated manuscript. The file "Wright2025_LateriteProfileProfileModelParameters_data.csv" include the ratio of hematite to hematite and goethite in two synthetic, modeled profiles, as well as the modeled surface areas of goethite and hematite as a function of relative depth within the modeled weathering zone. These data were used, in conjunction with equations presented in the paper, to calculate the theoretical concentrations of Pd and Pt (and the resulting Pt/Pd ratio) within the profiles. These data appear in Figure 3 in the associated manuscript. The file "Wright2025_Imagery_data.zip" is a zipped folder containing the TEM and STEM images appear in Figures S18 and S19. Individual files are labeled as either STEM (Fig. S18) or TEM (Fig. S19) with a letter representing the part of the multipart figure.

58 GEOSCIENCES↗
NASA NTRS1983

Application of identification techniques to remote manipulator system flight data

This paper addresses the application of identification techniques to flight data from the Space Shuttle Remote Manipulator System (RMS). A description of the remote manipulator, including structural and control system characteristics, sensors, and actuators is given. A brief overview of system identification procedures is presented, and the practical aspects of implementing system identification algorithms are discussed. In particular, the problems posed by desampling rate, numerical error, and system nonlinearities are considered. Simulation predictions of damping, frequency, and system order are compared with values identified from flight data to support an evaluation of RMS structural and control system models. Finally, conclusions are drawn regarding the application of identification techniques to flight data obtained from a flexible space structure.

Shepard, G. D.↗
NASA NTRS1986

Towards automatic Markov reliability modeling of computer architectures

The analysis and evaluation of reliability measures using time-varying Markov models is required for Processor-Memory-Switch (PMS) structures that have competing processes such as standby redundancy and repair, or renewal processes such as transient or intermittent faults. The task of generating these models is tedious and prone to human error due to the large number of states and transitions involved in any reasonable system. Therefore model formulation is a major analysis bottleneck, and model verification is a major validation problem. The general unfamiliarity of computer architects with Markov modeling techniques further increases the necessity of automating the model formulation. This paper presents an overview of the Automated Reliability Modeling (ARM) program, under development at NASA Langley Research Center. ARM will accept as input a description of the PMS interconnection graph, the behavior of the PMS components, the fault-tolerant strategies, and the operational requirements. The output of ARM will be the reliability of availability Markov model formulated for direct use by evaluation programs. The advantages of such an approach are (a) utility to a large class of users, not necessarily expert in reliability analysis, and (b) a lower probability of human error in the computation.

Liceaga, C. A.↗
NASA NTRS1985

Navigation systems

The elements of the measurement and communications network comprising the global deep space navigation system (DSN) for NASA missions are described. Among the measurement systems discussed are: VLBI, two-way Doppler and range measurements, and optical measurements carried out on board the spacecraft. Processing of navigation measurement is carried out using two modules: an N-body numerical integration of the trajectory (and state transition partial derivatives) based on pre-guessed initial conditions; and partial derivatives of simulated observables corresponding to each actual observation. Calculations of velocity correction parameters is performed by precise modelling of all physical phenomena influencing the observational measurements, including: planetary motions; tracking station locations, gravity field structure, and transmission media effects. Some of the contributions to earth-relative orbit estimate errors for the Doppler/range system on board Voyager are discussed in detail. A line drawing of the DSN navigation system is provided.

Jordan, J. F.↗
NASA NTRS1987

A modern control theory based algorithm for control of the NASA/JPL 70-meter antenna axis servos

A digital computer-based state variable controller was designed and applied to the 70-m antenna axis servos. The general equations and structure of the algorithm and provisions for alternate position error feedback modes to accommodate intertarget slew, encoder referenced tracking, and precision tracking modes are descibed. Development of the discrete time domain control model and computation of estimator and control gain parameters based on closed loop pole placement criteria are discussed. The new algorithm was successfully implemented and tested in the 70-m antenna at Deep Space Network station 63 in Spain.

Hill, R. E.↗
NASA NTRS1988

Maximum likelihood identification for large space structures

This paper examines the use of on-orbit identification based on Maximum Likelihood Estimation (MLE) to provide these high-order, high-accuracy control design models for large space structures (LSS's). First, it outlines a general MLE identification algorithm, together with a covariance-analysis procedure to assess algorithm performance in terms of systematic and stochastic errors. Next, it examines various simplifications appropriate for the LSS identification application. Simplified analytical performance results are presented, as are numerical results to support these analyses. Finally, a graphical interpretation of these results is given.

Barrett, Michael F.↗
NASA NTRS2008

Direct Adaptive Control of Systems with Actuator Failures: State of the Art and Continuing Challenges

In this paper, the problem of controlling systems with failures and faults is introduced, and an overview of recent work on direct adaptive control for compensation of uncertain actuator failures is presented. Actuator failures may be characterized by some unknown system inputs being stuck at some unknown (fixed or varying) values at unknown time instants, that cannot be influenced by the control signals. The key task of adaptive compensation is to design the control signals in such a manner that the remaining actuators can automatically and seamlessly take over for the failed ones, and achieve desired stability and asymptotic tracking. A certain degree of redundancy is necessary to accomplish failure compensation. The objective of adaptive control design is to effectively use the available actuation redundancy to handle failures without the knowledge of the failure patterns, parameters, and time of occurrence. This is a challenging problem because failures introduce large uncertainties in the dynamic structure of the system, in addition to parametric uncertainties and unknown disturbances. The paper addresses some theoretical issues in adaptive actuator failure compensation: actuator failure modeling, redundant actuation requirements, plant-model matching, error system dynamics, adaptation laws, and stability, tracking, and performance analysis. Adaptive control designs can be shown to effectively handle uncertain actuator failures without explicit failure detection. Some open technical challenges and research problems in this important research area are discussed.

Tao, Gang↗
NASA NTRS2014

Adaptive Augmenting Control Flight Characterization Experiment on an F/A-18

This paper summarizes the Adaptive Augmenting Control (AAC) flight characterization experiments performed using an F/A-18 (TN 853). AAC was designed and developed specifically for launch vehicles, and is currently part of the baseline autopilot design for NASA's Space Launch System (SLS). The scope covered here includes a brief overview of the algorithm (covered in more detail elsewhere), motivation and benefits of flight testing, top-level SLS flight test objectives, applicability of the F/A-18 as a platform for testing a launch vehicle control design, test cases designed to fully vet the AAC algorithm, flight test results, and conclusions regarding the functionality of AAC. The AAC algorithm developed at Marshall Space Flight Center is a forward loop gain multiplicative adaptive algorithm that modifies the total attitude control system gain in response to sensed model errors or undesirable parasitic mode resonances. The AAC algorithm provides the capability to improve or decrease performance by balancing attitude tracking with the mitigation of parasitic dynamics, such as control-structure interaction or servo-actuator limit cycles. In the case of the latter, if unmodeled or mismodeled parasitic dynamics are present that would otherwise result in a closed-loop instability or near instability, the adaptive controller decreases the total loop gain to reduce the interaction between these dynamics and the controller. This is in contrast to traditional adaptive control logic, which focuses on improving performance by increasing gain. The computationally simple AAC attitude control algorithm has stability properties that are reconcilable in the context of classical frequency-domain criteria (i.e., gain and phase margin). The algorithm assumes that the baseline attitude control design is well-tuned for a nominal trajectory and is designed to adapt only when necessary. Furthermore, the adaptation is attracted to the nominal design and adapts only on an as-needed basis (see Figure 1). The MSFC algorithm design was formulated during the Constellation Program and reached a high maturity level during SLS through simulation-based development and internal and external analytical review. The AAC algorithm design has three summary-level objectives: (1) "Do no harm;" return to baseline control design when not needed, (2) Increase performance; respond to error in ability of vehicle to track command, and (3) Regain stability; respond to undesirable control-structure interaction or other parasitic dynamics. AAC has been successfully implemented as part of the Space Launch System baseline design, including extensive testing in high-fidelity 6-DOF simulations the details of which are described in [1]. The Dryden Flight Research Center's F/A-18 Full-Scale Advanced Systems Testbed (FAST) platform is used to conduct an algorithm flight characterization experiment intended to fully vet the aforementioned design objectives. FAST was specifically designed with this type of test program in mind. The onboard flight control system has full-authority experiment control of ten aerodynamic effectors and two throttles. It has production and research sensor inputs and pilot engage/disengage and real-time configuration of up to eight different experiments on a single flight. It has failure detection and automatic reversion to fail-safe mode. The F/A-18 aircraft has an experiment envelope cleared for full-authority control and maneuvering and exhibits characteristics for robust recovery from unusual attitudes and configurations aided by the presence of a qualified test pilot. The F/A-18 aircraft has relatively high mass and inertia with exceptional performance; the F/A-18 also has a large thrust-to-weight ratio, owing to its military heritage. This enables the simulation of a portion of the ascent trajectory with a high degree of dynamic similarity to a launch vehicle, and the research flight control system can simulate unstable longitudinal dynamics. Parasitic dynamics such as slosh and bending modes, as well as atmospheric disturbances, are being produced by the airframe via modification of bending filters and the use of secondary control surfaces, including leading and trailing edge flaps, symmetric ailerons, and symmetric rudders. The platform also has the ability to inject signals in flight to simulate structural mode resonances or other challenging dynamics. This platform also offers more test maneuvers and longer maneuver times than a single rocket or missile test, which provides ample opportunity to fully and repeatedly exercise all aspects of the algorithm. Prior to testing on an F/A-18, AAC was the only component of the SLS autopilot design that had not been flight tested. The testing described in this paper raises the Technology Readiness Level (TRL) early in the SLS Program and is able to demonstrate its capabilities and robustness in a flight environment.

VanZwieten, Tannen S.↗
NASA NTRS2017

Aeroelastic Modeling of X-56A Stiff-Wing Configuration Flight Test Data

Aeroelastic stability and control derivatives for the X-56A Multi-Utility Technology Testbed (MUTT), in the stiff-wing configuration, were estimated from flight test data using the output-error method. Practical aspects of the analysis are discussed. The orthogonal phase-optimized multisine inputs provided excellent data information for aeroelastic modeling. Consistent parameter estimates were determined using output error in both the frequency and time domains. The frequency domain analysis converged faster and was less sensitive to starting values for the model parameters, which was useful for determining the aeroelastic model structure and obtaining starting values for the time domain analysis. Including a modal description of the structure from a finite element model reduced the complexity of the estimation problem and improved the modeling results. Effects of reducing the model order on the short period stability and control derivatives were investigated.

Grauer, Jared A.↗
DOE OSTI2021

Using Neural Network Force Fields to Ascertain the Quality of Ab Initio Simulations of Liquid Water

Accurately simulating the properties of bulk water, despite the apparent simplicity of the molecule, is still a challenge. In order to fully understand and reproduce its complex phase diagram, it is necessary to perform simulations at the ab initio level, including quantum mechanical effects both for electrons and nuclei. This comes at a high computational cost, given that the structural and dynamical properties tend to require long timescales and large simulation cells. In this work, we evaluate the errors that density functional theory (DFT)-based simulations routinely incur into due time- and size-scale limitations. These errors are evaluated using neural-network-trained force fields that are accurate at the level of DFT methods. We compare different exchange and correlation potentials for properties of bulk water that require large timescales. Here we show that structural properties are less dependent on the system size and that dynamical properties such as the diffusion coefficient have a strong dependence on the simulation size and timescale. Our results facilitate comparisons of DFT-based simulation results with experiments and offer a path to discriminate between model and convergence errors in these simulations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Transformer Leakage Inductance Design Methodology

The leakage inductance exhibited by a transformer depends on its winding geometry, which generally involves the selection of several key design parameters in addition to the winding structure and the interleaving configuration. With few resources explaining the effects of these design choices on the observed leakage inductance, numerous trial-and-error iterations become necessary to realize the desired leakage inductance. This paper explores more than a hundred winding geometries feasible in a 2-winding transformer comprising the same magnetic core, number of turns, and wire gauge, and finds the leakage inductance for each unique design using 2-D finite element method (FEM) simulations in association with the semi-analytical double- 2-D model. These leakage inductances are plotted and further analyzed to understand the effects of different design parameters on the effective leakage inductance. The results presented herein and the conclusions drawn from this research can serve as a valuable resource for future design practitioners from both industry and academia.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

A discussion on various experimental methods of impact ionization coefficient measurement in GaN

Impact ionization coefficients play a critical role in semiconductors. In addition to silicon, silicon carbide and gallium nitride are important semiconductors that are being seen more as mainstream semiconductor technologies. As a reflection of the maturity of these semiconductors, predictive modeling has become essential to device and circuit designers, and impact ionization coefficients play a key role here. Recently, several studies have measured impact ionization coefficients. We dedicated the first part of our study to comparing three experimental methods to estimate impact ionization coefficients in GaN, which are all based on photomultiplication but feature characteristic differences. The first method inserts an InGaN hole-injection layer, the accuracy of which is challenged by the dominance of ionization in InGaN, leading to possible overestimation of the coefficients. The second method utilizes the Franz–Keldysh effect for hole injection but not for electrons, where the mixed injection of induced carriers would require a margin of error. The third method uses complementary p–n and n–p structures that have been at the basis of this estimation in Si and SiC and leans on the assumption of a constant electric field, and any deviation would require a margin of error. In the second part of our study, we evaluated the models using recent experimental data from diodes demonstrating avalanche breakdown.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2005

Estimating Effects of Multipath Propagation on GPS Signals

Multipath Simulator Taking into Account Reflection and Diffraction (MUSTARD) is a computer program that simulates effects of multipath propagation on received Global Positioning System (GPS) signals. MUSTARD is a very efficient means of estimating multipath-induced position and phase errors as functions of time, given the positions and orientations of GPS satellites, the GPS receiver, and any structures near the receiver as functions of time. MUSTARD traces each signal from a GPS satellite to the receiver, accounting for all possible paths the signal can take, including all paths that include reflection and/or diffraction from surfaces of structures near the receiver and on the satellite. Reflection and diffraction are modeled by use of the geometrical theory of diffraction. The multipath signals are added to the direct signal after accounting for the gain of the receiving antenna. Then, in a simulation of a delay-lock tracking loop in the receiver, the multipath-induced range and phase errors as measured by the receiver are estimated. All of these computations are performed for both right circular polarization and left circular polarization of both the L1 (1.57542-GHz) and L2 (1.2276-GHz) GPS signals.

Byun, Sung↗
DOE OSTI2024

Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning

Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Finally, our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.

Auto-encoder↗
DOE OSTI2020

General Protocol for the Accurate Prediction of Molecular 13 C/ 1 H NMR Chemical Shifts via Machine Learning Augmented DFT

An accurate prediction of NMR chemical shifts at affordable computational cost is very important for different types of structural assignments in experimental studies. Density functional theory (DFT) and gauge-including atomic orbital (GIAO) are two of the most popular computational methods for NMR calculation, yet they often fail to resolve ambiguities in structural assignments. In this work, we present a new method that uses machine learning (ML) techniques (DFT + ML) that significantly increases the accuracy of 13 C/ 1 H NMR chemical shift prediction for a variety of organic molecules. The input of the generalizable DFT + ML model contains two critical parts: one is a vector providing insights into chemical environments, which can be evaluated without knowing the exact geometry of the molecule; the other one is the DFT-calculated isotropic shielding constant. The DFT + ML model was trained with a data set containing 476 13 C and 270 1 H experimental chemical shifts. For the DFT methods used here, the root mean square deviations (RMSDs) for the errors between predicted and experimental 13 C/ 1 H chemical shifts can be as small as 2.10/0.18 ppm, which is much lower than those from simple DFT (5.54/0.25 ppm), or DFT + linear regression (LR) (4.77/0.23 ppm) approaches. It also has a smaller maximum absolute error than two previously proposed NMR-predicting ML models. The robustness of the DFT + ML model is tested on two classes of organic molecules (TIC10 and hyacinthacines), where the correct isomers were unambiguously assigned to the experimental ones. Overall, the DFT + ML model shows promise for structural assignments in a variety of systems, including stereoisomers, that are often challenging to determine experimentally.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗