Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “surrogate model”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 685 records · Page 38
All sourcesNASA NTRSDOE OSTI
NASA NTRS2014

NASA Numerical and Experimental Evaluation of UTRC Low Emissions Injector

Computational and experimental analyses of a PICS-Pilot-In-Can-Swirler technology injector, developed by United Technologies Research Center (UTRC) are presented. NASA has defined technology targets for near term (called "N+1", circa 2015), midterm ("N+2", circa 2020) and far term ("N+3", circa 2030) that specify realistic emissions and fuel efficiency goals for commercial aircraft. This injector has potential for application in an engine to meet the Pratt & Whitney N+3 supersonic cycle goals, or the subsonic N+2 engine cycle goals. Experimental methods were employed to investigate supersonic cruise points as well as select points of the subsonic cycle engine; cruise, approach, and idle with a slightly elevated inlet pressure. Experiments at NASA employed gas analysis and a suite of laser-based measurement techniques to characterize the combustor flow downstream from the PICS dump plane. Optical diagnostics employed for this work included Planar Laser-Induced Fluorescence of fuel for injector spray pattern and Spontaneous Raman Spectroscopy for relative species concentration of fuel and CO2. The work reported here used unheated (liquid) Jet-A fuel for all fuel circuits and cycle conditions. The initial tests performed by UTRC used vaporized Jet-A to simulate the expected supersonic cruise condition, which anticipated using fuel as a heat sink. Using the National Combustion Code a PICS-based combustor was modeled with liquid fuel at the supersonic cruise condition. All CFD models used a cubic non-linear k-epsilon turbulence wall functions model, and a semi-detailed Jet-A kinetic mechanism based on a surrogate fuel mixture. Two initial spray droplet size distribution and spray cone conditions were used: (1) an initial condition (Lefebvre) with an assumed Rosin-Rammler distribution, and 7 degree Solid Spray Cone; and (2) the Boundary Layer Stripping (BLS) primary atomization model giving the spray size distribution and directional properties. Contour and line plots are shown in comparison with experimental data (where this data is available) for flow velocities, fuel, and temperature distribution. The CFD results are consistent with experimental observations for fuel distribution and vaporization. Analysis of gas sample results, using a previously-developed NASA NOx correlation, indicates that for sea-level takeoff, the PICS configuration is predicted to deliver an EINOx value of about three for the targeted supersonic aircraft. Emissions results at supersonic cruise conditions show potential for meeting the NASA goals with liquid fuel.

Hicks, Yolanda R.↗
NASA NTRS2014

NASA Numerical and Experimental Evaluation of UTRC Low Emissions Injector

Computational and experimental analyses of a PICS-Pilot-In-Can-Swirler technology injector, developed by United Technologies Research Center (UTRC) are presented. NASA has defined technology targets for near term (called "N+1", circa 2015), midterm ("N+2", circa 2020) and far term ("N+3", circa 2030) that specify realistic emissions and fuel efficiency goals for commercial aircraft. This injector has potential for application in an engine to meet the Pratt & Whitney N+3 supersonic cycle goals, or the subsonic N+2 engine cycle goals. Experimental methods were employed to investigate supersonic cruise points as well as select points of the subsonic cycle engine; cruise, approach, and idle with a slightly elevated inlet pressure. Experiments at NASA employed gas analysis and a suite of laser-based measurement techniques to characterize the combustor flow downstream from the PICS dump plane. Optical diagnostics employed for this work included Planar Laser-Induced Fluorescence of fuel for injector spray pattern and Spontaneous Raman Spectroscopy for relative species concentration of fuel and CO2. The work reported here used unheated (liquid) Jet-A fuel for all fuel circuits and cycle conditions. The initial tests performed by UTRC used vaporized Jet-A to simulate the expected supersonic cruise condition, which anticipated using fuel as a heat sink. Using the National Combustion Code a PICS-based combustor was modeled with liquid fuel at the supersonic cruise condition. All CFD models used a cubic non-linear k-epsilon turbulence wall functions model, and a semi-detailed Jet-A kinetic mechanism based on a surrogate fuel mixture. Two initial spray droplet size distribution and spray cone conditions were used: 1) an initial condition (Lefebvre) with an assumed Rosin-Rammler distribution, and 7 degree Solid Spray Cone; and 2) the Boundary Layer Stripping (BLS) primary atomization model giving the spray size distribution and directional properties. Contour and line plots are shown in comparison with experimental data (where this data is available) for flow velocities, fuel, and temperature distribution. The CFD results are consistent with experimental observations for fuel distribution and vaporization. Analysis of gas sample results, using a previously-developed NASA NOx correlation, indicates that for sea-level takeoff, the PICS configuration is predicted to deliver an EINOx value of about 3 for the targeted supersonic aircraft. Emissions results at supersonic cruise conditions show potential for meeting the NASA goals with liquid fuel.

Hicks, Yolanda R.↗
NASA NTRS2006

Beam Heating of Samples: Modeling and Verification

Energy absorbed from the X-ray beam by the sample requires cooling by forced convection (i.e. cryostream) to minimize temperature increase and the damage caused to the sample by the X-ray heating. In this presentation we will first review the current theoretical models and recent studies in the literature, which predict the sample temperature rise for a given set of beam parameters. It should be noted that a common weakness of these previous studies is that none of them provide actual experimental confirmation. This situation is now remedied in our investigation where the problem of x-ray sample heating is taken up once more. We have theoretically investigated, and at the same time, in addition to the numerical computations, performed experiments to validate the predictions. We have modeled, analyzed and experimentally tested the temperature rise of a 1 mm diameter glass sphere (sample surrogate) exposed to an intense synchrotron X-ray beam, while it is being cooled in a uniform flow of nitrogen gas. The heat transfer, including external convection and internal heat conduction was theoretically modeled using CFD to predict the temperature variation in the sphere during cooling and while it was subjected to an undulator (ID sector 19) X-ray beam at the APS. The surface temperature of the sphere during the X-ray beam heating was measured using the infrared camera measurement technique described in a previous talk. The temperatures from the numerical predictions and experimental measurements are compared and discussed. Additional results are reported for the two different sphere sizes and for two different supporting pin orientations.

Kazmierczak, Michael↗
DOE OSTI2025

Material Fracturing and Failure Simulation Datasets

Fracturing is a fundamental physics phenomena with broad relevance across multiple domains, ranging from infrastructure integrity, aerospace durability, reservoir production, and seismic events. We present a diverse dataset of simulated fracture evolution and material failure generated from two numerical solvers: the phase-field method and the combined finite-discrete element method (FDEM). These solvers differ in formulation, physical fidelity, and computational efficiency. The dataset includes five materials: PBX, anisotropic shale, tungsten, aluminum, and steel. For each, phase-field simulations span 400,000 cases: 200,000 under uniaxial tension and 200,000 under biaxial tension. The computationally expensive FDEM simulations include 90,000 split evenly among PBX, shale, and tungsten under uniaxial loading. All simulations begin with randomized initial fracture patterns. Each entry includes temporal data capturing fracture propagation dynamics. This comprehensive dataset is designed to support the development of foundational or surrogate machine learning approaches for predicting material failure. While no such models are introduced here, the dataset lays a robust foundation for advancing future research and innovation in these areas.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Accelerating discoveries at DIII-D with the Integrated Research Infrastructure

DIII-D research is being accelerated by leveraging high performance computing (HPC) and data resources available through the National Energy Research Scientific Computing Center (NERSC) Superfacility initiative. As part of this initiative, a high-resolution, fully automated, whole discharge kinetic equilibrium reconstruction workflow was developed that runs at the NERSC for most DIII-D shots in under 20 min. This has eliminated a long-standing research barrier and opened the door to more sophisticated analyses, including plasma transport and stability. These capabilities would benefit from being automated and executed within the larger Department of Energy Advanced Scientific Computing Research program’s Integrated Research Infrastructure (IRI) framework. The goal of IRI is to empower researchers to meld DOE’s world-class research tools, infrastructure, and user facilities seamlessly and securely in novel ways to radically accelerate discovery and innovation. For transport, we are looking at producing flux matched profiles and also using particle tracing to predict fast ion heat deposition from neutral beam injection before a shot takes place. Our starting point for evaluating plasma stability focuses on the pedestal limits that must be navigated to achieve better confinement. This information is meant to help operators run more effective experiments, so it needs to be available rapidly inside the DIII-D control room. So far this has been achieved by ensuring the data is available with existing tools, but as more novel results are produced new visualization tools must be developed. In addition, all of the high-quality data we have generated has been collected into databases that can unlock even deeper insights. This has already been leveraged for model and code validation studies as well as for developing AI/ML surrogates. The workflows developed for this project are intended to serve as prototypes that can be replicated on other experiments and can be run to provide timely and essential information for ITER, as well as next stage fusion power plants.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1999

Measuring Directional Wave Spectra and Wind Speed with a Scanning Radar Altimeter

The geometry for the NASA Scanning Radar Altimeter (SRA) is shown. It transmits a 8-ns duration pulse at Ka-band (8.3 mm) and measures time of flight as it scans a 1 degree (two-way) beam from left to right across the aircraft ground track. The most recent configuration determines the surface elevation at 64 points spaced at uniform angular intervals of about 0.7 across a swath whose width is about 0.8 times the aircraft altitude. The system generates these raster lines of the surface topography beneath the aircraft at about a 10 Hz rate. In postflight processing the SRA wave topographic data are transformed with a two-dimensional Fast Fourier Transformation (FFT) and Doppler corrected to produce directional wave spectra. The SRA is not absolutely calibrated in power, but by measuring the relative fall-off of backscatter with increasing incidence angle, the SRA can also determine the mean square slope (mss) of the sea surface, a surrogate for wind speed. For the slope-dependent specular point model of radar sea surface scattering, an expression approximated by a geometric optics form, for the relative variation with incidence angle of the normalized backscatter radar cross section would be sigma (sup 0) (sub rel) = sec (exp 4) theta exp (-tan squared theta/mss) where theta is the off-nadir incidence angle.

Walsh, E. J.↗
NASA NTRSDate not supplied

Structured Covariance Gaussian Networks for Orion Crew Module Aerodynamic Uncertainty Quantification

In this paper we propose a new approach for nonlinear regression and uncertainty quantification. The method is based on a pair of neural networks which parameterize mean and dense covariance functions of a multivariate Gaussian process, trained together to maximize the log-likelihood of observing the given data. The covariance matrix is made positive definite at every input by construction. We also propose a sampling approach that produces viable surrogate function realizations from the Gaussian process. We call the proposed model a Structured Covariance Gaussian Network (SCGN). We illustrate the use of SCGNs for learning an aerodynamic response surface with built-in uncertainty for the Orion crew module. We find that SCGN provides an efficient and systematic way to learn nonlinear functional relationships and dense covariances. We compare results to a baseline Gaussian process regressor and observe that the SCGN provides comparable uncertainty descriptions with improved scalability to dataset size. The sample functions generated by SCGN are fast to evaluate online and are therefore convenient for use in trajectory simulations. These results suggest that SCGN may be a viable computational method for aerodynamic uncertainty quantification.

machine learning↗
NASA NTRSDate not supplied

Structured Covariance Gaussian Networks for Orion Crew Module Aerodynamic Uncertainty Quantification

In this paper we propose a new approach for nonlinear regression and uncertainty quantification. The method is based on a pair of neural networks which parameterize mean and dense covariance functions of a multivariate Gaussian process, trained together to maximize the log-likelihood of observing the given data. The covariance matrix is made positive definite at every input by construction. We also propose a sampling approach that produces viable surrogate function realizations from the Gaussian process. We call the proposed model a Structured Covariance Gaussian Network (SCGN). We illustrate the use of SCGNs for learning an aerodynamic response surface with built-in uncertainty for the Orion crew module. We find that SCGN provides an efficient and systematic way to learn nonlinear functional relationships and dense covariances. We compare results to a baseline Gaussian process regressor and observe that the SCGN provides comparable uncertainty descriptions with improved scalability to dataset size. The sample functions generated by SCGN are fast to evaluate online and are therefore convenient for use in trajectory simulations. These results suggest that SCGN may be a viable computational method for aerodynamic uncertainty quantification.

machine learning↗
DOE OSTI2025

Ensemble cure kinetics network (ECK-Net): A method to derive cure kinetics of thermosetting resin

This paper introduces an Ensemble Cure Kinetics Network (ECK-Net), a neural network (NN)–based framework for modeling the cure kinetics of thermosetting resins within a phenomenological context. ECK-Net replaces traditional analytic models, which require extensive chemical insight and multiple isothermal/non-isothermal experiments, with a data-driven surrogate that maps nonlinear relationships between temperature, degree of cure, and reaction rate from differential scanning calorimetry data. The proposed approach predicts input-dependent kinetic coefficients of a generalized nth-order reaction equation rather than reaction rates directly, enabling a single unified model to represent various epoxy systems without relying on iso-conversional analysis or predefined functional forms. To ensure robustness, multiple independently trained networks under different random initializations are blended through an ensemble strategy, effectively mitigating the stochastic variability inherent to neural networks. The framework is validated using experimental datasets from multiple resin systems, including aerospace-grade materials (Toray 3900-2, Cycom 5320-1, and Hexcel 8552) and a windmill-grade resin (RIMR 035c). The model accurately reproduces the temporal evolution of the degree of cure under manufacturers’ recommended cure cycles across all tested resins systems, yielding Pearson’s correlation coefficients of 0.992, 0.994, 0.993, 0.997, respectively. To demonstrate process-level applicability, the trained network was implemented within the Abaqus environment to simulate out-of-autoclave (OOA) curing process of the CFRP panel composed of Toray T830H-6K/3900-2D prepreg. The simulation results showed excellent agreement with experimental temperature response (maximum peak temperature, simulation: 189.6 °C, experiment: 188.5 °C) and the final degree of cure (simulation: 0.948, experiment: 0.960 ± 0.013), confirming ECK-Net’s capability as a reliable alternative to conventional cure kinetics modeling methods.

Composite curing↗
DOE OSTI2026

A fractional calculus framework for open quantum dynamics: From Liouville to Lindblad to memory kernels

Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini–Kossakowski–Sudarshan–Lindblad equation uniquely characterizes Markovian completely positive and trace-preserving (CPTP) evolution, many physical platforms display non-Markovian features such as algebraic relaxation and coherence backflow. Fractional calculus provides a natural way to model such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here, we develop a unified framework that embeds fractional master equations within the broader hierarchy of open-system formalisms. The fractional equation forms a structured subclass of memory-kernel models, reduces to the Lindblad form at unit order, and, through Bochner–Phillips subordination, admits a CPTP representation as an average over Lindblad semigroups. Its resolvent structure further connects fractional dynamics to established non-Markovian approaches, including Nakajima–Zwanzig kernels and hierarchical equations of motion, providing a compact surrogate for long-memory effects. This formulation positions fractional calculus as a rigorous and practical language for modeling non-Markovian quantum dynamics in chemical physics and physical chemistry, providing a CPTP-preserving, computationally efficient surrogate for structured condensed-phase environments where long-time memory and dissipation play a central role.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

MULTI-LEADER: MULTI-source LEarning-Accelerated Design of high-Efficiency multi-stage compRessor (Final Technical Report)

The objective of MULTI-LEADER is to cut design costs by 80% while generating more energy-efficient designs of multi-stage compressors by developing and implementing novel machine learning (ML) techniques, which enable faster and fewer design iterations, improved solver performance, and concurrent multi-disciplinary design. Current industrial practices for the design of multi-stage compressors involve simulation-based design optimization with successive levels of model fidelity, iteratively evaluated between distinct disciplines, one stage at a time to tackle the high dimensional design variations. This project addresses these key design challenges: (1) concurrent optimization of multiple stages under many non-linear constraints; (2) multitude of evaluation of high-fidelity and expensive solvers and their gradients during optimization convergence in high-dimensional design; (3) multi-disciplinary design to maximize aerodynamic performance while guaranteeing structural integrity and additive manufacturability; (4) utilization of multiple fidelity of solvers with disparate parameterization and modeling assumptions. MULTI-LEADER achieved more than 5x speed up in detailed design of more energy-efficient compressors via these machine learning (ML) innovations: (i) rapid design surrogates by multi-source learning from diverse fidelities across multiple disciplines, (ii) physics-constrained data-augmented modeling for improved empiricism, (iii) generative manifold embedding for high dimensional concurrent design without gradient information; (iv) budget-constrained fidelity-adaptive sampling towards fewer design iterations.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2026

Explainable physics-based constraints on reinforcement learning for accelerator optimization

We present a reinforcement learning (RL) framework for optimizing particle accelerator experiments that builds explainable physics-based constraints on agent behavior. The goal is to increase transparency and trust by letting users verify that the agent’s decision-making process incorporates suitable physics. Our algorithm uses a learnable surrogate function for physical observables, such as energy, and uses them to fine-tune how actions are chosen. This surrogate can be represented by a neural network or by an interpretable sparse dictionary model. We test our algorithm on a range of particle accelerator optimization environments designed to emulate the Continuous Electron Beam Accelerator Facility at Jefferson Lab. By examining the mathematical form of the learned constraint function, we are able to confirm the agent has learned to use the established physics of each environment. In addition, we find that the introduction of a physics-based surrogate enables our RL algorithms to reliably converge for difficult high-dimensional accelerator optimization environments.

explainability↗
NASA NTRS2026

Benchmarking Bayesian Optimization Frameworks and Acquisition Strategies for Materials Discovery and Autonomous Laboratories

Bayesian optimization (BO) can accelerate materials discovery by guiding expensive experiments toward the most promising processing conditions. We systematically compare five BO surrogate and framework combinations (Gaussian processes in Ax, Gaussian processes and Monte-Carlo neural networks in BayBE, random forests in Lolopy, and tree-structured Parzen (TPE) estimators in Hyperopt) on three benchmarks that mimic common materials design tasks (a discrete solid-electrolyte composition space, a hybrid discrete/continuous laminate-composite design problem solved with micromechanics modeling, and the continuous Ishigami analytic function which is a standard optimization benchmark). Each BO surrogate is paired with posterior mean, probability of improvement, and expected improvement acquisition functions and run for 100 trials from randomized initial samples with uniform random search providing a control. Across five random seeds per setting, BayBE’s Gaussian-process surrogate with expected improvement consistently reached ≥95 % of the known optimum in the fewest evaluations, while Lolopy’s random forest matched or exceeded GP performance on purely categorical or mixed spaces at a higher computational cost. Posterior mean alone often stagnated at local optima, underscoring the need for exploration, whereas probability and expected improvement balanced exploration and exploitation leading to better optimization in fewer trials. Execution times ranged from milliseconds for TPE to minutes for neural-network and random-forest surrogates. These results establish baseline expectations for BO in automated materials laboratories and highlight expected improvement with Gaussian processes as a reliable first choice, with random forests offering a strong alternative when categorical variables dominate. The benchmark suite and code are released to facilitate future surrogate, acquisition, and constraint-handling research in data-driven materials optimization.

Bayesian optimization↗
DOE OSTI2026

Attention-based explainability for structure–property relationships

Machine learning methods are emerging as a universal paradigm for constructing correlative structure–property relationships in materials science based on multimodal characterization. However, this necessitates the development of methods for the physical interpretability of the resulting correlative models. Here, we demonstrate the potential of attention-based neural networks for revealing structure–property relationships and the underlying physical mechanisms, using the ferroelectric properties of PbTiO3 thin films as a case study. Through the analysis of attention scores, we disentangle the influence of distinct domain patterns on the polarization switching process. The attention-based Transformer model is explored both as a direct interpretability tool and as a surrogate for explaining representations learned via unsupervised machine learning, enabling the identification of physically grounded correlations. We compare attention-derived interpretability scores with classical SHapley Additive exPlanations analysis and show that, in contrast to applications in natural language processing, attention mechanisms in materials science exhibit high efficiency in highlighting meaningful structural features.

Slautin, Boris [Independent Researcher]↗
DOE OSTI2025

An Experiment with LLMs as Database Design Tutors: Persistent Equity and Fairness Challenges in Online Learning

As large language models (LLMs) continue to evolve, their capacity to replace humans as their surrogates is also improving. As increasing numbers of intelligent tutoring systems (ITSs) are embracing the integration of LLMs for digital tutoring, questions are arising as to how effective they are and if their hallucinatory behaviors diminish their perceived advantages. One critical question that is seldom asked if the availability, plurality, and relative weaknesses in the reasoning process of LLMs are contributing to the much discussed digital divide and equity and fairness in online learning. In this paper, we present an experiment with database design theory assignments and demonstrate that while their capacity to reason logically is improving, LLMs are still prone to serious errors. We demonstrate that in online learning and in the absence of a human instructor, LLMs could introduce inequity in the form of “wrongful” tutoring that could be devastatingly harmful for learners, which we call ignorant bias, in increasingly popular digital learning. We also show that significant challenges remain for STEM subjects, especially for subjects for which sound and free online tutoring systems exist. Based on the set of use cases, we formulate a possible direction for an effective ITS for online database learning classes of the future.

Jamil, Hasan M. (ORCID:0000000231243780)↗
DOE OSTI2025

"Hidden" hydrothermal technical potential & technoeconomics: Revealing permeability & fluids with more data

Historical hydrothermal estimates have largely relied on temperature or heat flow estimates ignoring the need for natural flowing fluids. More accurate hydrothermal estimates require some indication of permeability and fluids that naturally exist in the subsurface. This paper describes a novel approach that includes proxies of permeability and fluids in hydrothermal estimates by leveraging the relatively data-rich Great Basin. Specifically, nameplate capacities (megawatts) of operating geothermal plants, negative (0 megawatt) locations and 48 geophysical and geologic features are used to used in eXtreme Gradient Boosting (XGBoost) regression to make hydrothermal capacity predictions. Additionally, this work inputs the XGBoost-based hydrothermal predictions into the Renewable Energy Potential (reV) model to quantify technical capacity, its uncertainty and techno-economics. Compared to historical hydrothermal estimates, these predictions adhere to the 37 operating geothermal plants and negative locations. We present a method for subsampling the negative sites to bring the labels into balance that uses the geologic domain knowledge to proportionally represent negatives. Overall, the distributions of the hydrothermal technical capacity and the site levelized cost of energy are respectively much tighter, lower and more accurate than the previous estimates for the Great Basin, as they include geological and geophysical surrogates for permeability and fluids. Percentile (50th and 90th, median and high estimate, respectively) models provide bookends for these metrics.

13 HYDRO ENERGY↗
DOE OSTI2024

Mini-Canister Radiolysis Testing of ASNF Materials and Surrogates

An experimental irradiation campaign to investigate radiolysis behavior of ASNF was conducted using in situ gas monitoring of small, sealed stainless-steel vessels (mini-canisters) containing aluminum samples with adherent (oxy)hydroxide films under helium backfill. The samples were irradiated with gamma radiation from a Co-60 irradiator. The samples tested included aluminum plate assemblies with lab-grown (oxy)hydroxides as surrogates for fuel as well as an end cropping from an actual ASNF assembly retrieved from long-term wet storage. These experiments enabled investigation of the impacts of various fuel drying approaches on the radiolytic generation rate and measurement of the H 2 yield associated with a reactor exposed sample with reactor-formed (oxy)hydroxide. The resulting data can be incorporated into model development for ASNF in dry storage. This report presents the cumulative results from four surrogate assemblies tested after application of different preparation (drying) conditions as well as the ASNF cropping; some post-irradiation testing was included. The mini-canister results are compared to data from related experimental campaigns that also tested the impact of drying conditions using samples irradiated in glass ampoules and discusses implications of the combined data

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
NASA NTRS1983

A mathematical model for crop spectral-temporal trajectories based on a plant growth model

The Kubelka-Munk radiative transfer model is combined with an approximation of Kauth-Thomas greeness and brightness transforms to derive approximate closed form expressions for crop greeness and brightness surrogates in terms of canopy biomass. The greeness relation derived resembles an existing empirical relation between leaf area index and greeness. A simple growth model based on interception and utilization of photosynthetically active radiation is developed and used to describe the time evolution of greeness and brightness. The model developed does not yet yield definitive profile calculations but suggests a conceptual framework which may be found useful for further profile analysis.

Woolford, T. L.↗