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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 1,423 records · Page 79
All sourcesNASA NTRSDOE OSTI
NASA NTRS2008

Cryogenic Fluid Technologies for Long Duration In-Space Operations

Reliable knowledge of low-gravity cryogenic fluid management behavior is lacking and yet is critical in the areas of storage, distribution, and low-gravity propellant management. The Vision for Space Exploration mission objectives will require the use of high performance cryogenic propellants (hydrogen, oxygen, and methane). Additionally, lunar missions will require success in storing and transferring liquid and gas commodities on the surface. The fundamental challenges associated with the in-space use of cryogens are their susceptibility to environmental heat, their complex thermodynamic and fluid dynamic behavior in low gravity and the uncertainty of the position of the liquid-vapor interface if the propellants are not settled. The Cryogenic Fluid Management (CFM) project is addressing these issues through ground testing and analytical model development, and has crosscutting applications and benefits to virtually all missions requiring in-space operations with cryogens. Such knowledge can significantly reduce or even eliminate tank fluid boil-off losses for long term missions, reduce propellant launch mass and on-orbit margins, and simplify vehicle operations. The Cryogenic Fluid Management (CFM) Project is conducting testing and performing analytical evaluation of several areas to enable NASA s Exploration Vision. This paper discusses the content and progress of the technology focus areas within CFM.

Motil, Susan M.↗
NASA NTRS1994

Numerical and Experimental Study of Transport Phenomena in Directional Solidification of Succinonitrile

A numerical and experimental study of the growth of succinonitrile (SCN) using a horizontal Bridginan furnace and transparent glass ampoule was conducted. Two experiments were considered: one in which the temperature profile was fixed relative to the ampoule (no-growth case); and a second in which the thermal profile was translated at a constant rate (steady growth case). Measured temperature profiles on the outer surface of the ampoule were used as thermal boundary conditions for the modelling. The apparent heat capacity formulation combined with the variable viscositymeth~ was used to model the phase change in SeN. Both 2-D and 3-D models were studied and numerical solutions obtained using the commercial finite element code, FIDAP1. Comparison of the numerical results to experimental data showed excellent agreement. The complex 3-D shallow-cavity flow in the melt, differences between 2-D and 3-D models, effects of natural convection on the thermal gradient and shape of the solid/liquid interface, and the sensitivity of simulations to specific assumptions, are also discussed.

de Groh, Henry C., III↗
NASA NTRS1981

Satellite and surface geophysical expression of anomalous crustal structure in Kentucky and Tennessee

An equivalent layer magnetization model is discussed. Inversion of long wavelength satellite magnetic anomaly data indicates a very magnetic source region centered in south central Kentucky. Refraction profiles suggest that the source of the gravity anomaly is a large mass of rock occupying much of the crustal thickness. The outline of the source delineated by gravity contours is also discernible in aeromagnetic anomaly patterns. The mafic plutonic complex, and several lines of evidence are consistent with a rift association. The body is, however, clearly related to the inferred position of the Grenville Front. It is bounded on the north by the fault zones of the 38th Parallel Lineament. It is suggested that such magnetization levels are achieved with magnetic mineralogies produced by normal oxidation and metamorphic processes and enhanced by viscous build-up, especially in mafic rocks of alkaline character.

Mayhew, M. A.↗
NASA NTRS2017

Synoptic Traveling Weather Systems on Mars: Effects of Radiatively-Active Water Ice Clouds

Atmospheric aerosols on Mars are critical in determining the nature of its thermal structure, its large-scale circulation, and hence the overall climate of the planet. We conduct multi-annual simulations with the latest version of the NASA Ames Mars global climate model (GCM), gcm2.3+, that includes a modernized radiative-transfer package and complex water-ice cloud microphysics package which permit radiative effects and interactions of suspended atmospheric aerosols (e.g., water ice clouds, water vapor, dust, and mutual interactions) to influence the net diabatic heating. Results indicate that radiatively active water ice clouds profoundly affect the seasonal and annual mean climate. The mean thermal structure and balanced circulation patterns are strongly modified near the surface and aloft. Warming of the subtropical atmosphere at altitude and cooling of the high latitude atmosphere at low levels takes place, which increases the mean pole-to-equator temperature contrast (i.e., "baroclinicity"). With radiatively active water ice clouds (RAC) compared to radiatively inert water ice clouds (nonRAC), significant changes in the intensity of the mean state and forced stationary Rossby modes occur, both of which affect the vigor and intensity of traveling, synoptic period weather systems.Such weather systems not only act as key agents in the transport of heat and momentum beyond the extent of the Hadley circulation, but also the transport of trace species such as water vapor, water ice-clouds, dust and others. The northern hemisphere (NH) forced Rossby waves and resultant wave train are augmented in the RAC case: the modes are more intense and the wave train is shifted equatorward. Significant changes also occur within the subtropics and tropics. The Rossby wave train sets up, combined with the traveling synoptic period weather systems (i.e., cyclones and anticyclones), the geographic extent of storm zones (or storm tracks) within the NH. A variety of circulation features will be presented which indicate contrasts between the RAC and nonRAC cases, and which highlight key effects radiatively-active clouds have on physical and dynamical processes active in the current climate of Mars.

GCM-global climate model↗
NASA NTRS2017

Synoptic Traveling Weather Systems on Mars: Effects of Radiatively-Active Water Ice Clouds

Atmospheric aerosols on Mars are critical in determining the nature of its thermal structure, its large-scale circulation, and hence the overall climate of the planet. We conduct multi-annual simulations with the latest version of the NASA Ames Mars global climate model (GCM), gcm2.3+, that includes a modernized radiative-transfer package and complex water-ice cloud microphysics package which permit radiative effects and interactions of suspended atmospheric aerosols (e.g., water ice clouds, water vapor, dust, and mutual interactions) to influence the net diabatic heating. Results indicate that radiatively active water ice clouds profoundly affect the seasonal and annual mean climate. The mean thermal structure and balanced circulation patterns are strongly modified near the surface and aloft. Warming of the subtropical atmosphere at altitude and cooling of the high latitude atmosphere at low levels takes place, which increases the mean pole-to-equator temperature contrast (i.e., "baroclinicity"). With radiatively active water ice clouds (RAC) compared to radiatively inert water ice clouds (nonRAC), significant changes in the intensity of the mean state and forced stationary Rossby modes occur, both of which affect the vigor and intensity of traveling, synoptic period weather systems. Such weather systems not only act as key agents in the transport of heat and momentum beyond the extent of the Hadley circulation, but also the transport of trace species such as water vapor, water ice-clouds, dust and others. The northern hemisphere (NH) forced Rossby waves and resultant wave train are augmented in the RAC case: the modes are more intense and the wave train is shifted equatorward. Significant changes also occur within the subtropics and tropics. The Rossby wave train sets up, combined with the traveling synoptic period weather systems (i.e., cyclones and anticyclones), the geographic extent of storm zones (or storm tracks) within the NH. A variety of circulation features will be presented which indicate contrasts between the RAC and nonRAC cases, and which highlight key effects radiatively-active clouds have on physical and dynamical processes active in the current climate of Mars.

Atmospheric aerosols↗
NASA NTRS2014

Calculation of Operations Efficiency Factors for Mars Surface Missions

For planning of Mars surface missions, to be operated on a sol-by-sol basis by a team on Earth (where a "sol" is a Martian day), activities are described in terms of "sol types" that are strung together to build a surface mission scenario. Some sol types require ground decisions based on a previous sol's results to feed into the activity planning ("ground in the loop"), while others do not. Due to the differences in duration between Earth days and Mars sols, for a given Mars local solar time, the corresponding Earth time "walks" relative to the corresponding times on the prior sol/day. In particular, even if a communication window has a fixed Mars local solar time, the Earth time for that window will be approximately 40 minutes later each succeeding day. Further complexity is added for non-Mars synchronous communication relay assets, and when there are multiple control centers in different Earth time zones. The solution is the development of "ops efficiency factors" that reflect the efficiency of a given operations configuration (how many and location of control centers, types of communication windows, synchronous or non-synchronous nature of relay assets, sol types, more-or-less sustainable operations schedule choices) against a theoretical "optimal" operations configuration for the mission being studied. These factors are then incorporated into scenario models in order to determine the surface duration (and therefore minimum spacecraft surface lifetime) required to fulfill scenario objectives. The resulting model is used to perform "what-if" analyses for variations in scenario objectives. The ops efficiency factor is the ratio of the figure of merit for a given operations factor to the figure of merit for the theoretical optimal configuration. The current implementation is a pair of models in Excel. The first represents a ground operations schedule for 500 sols in each operations configuration for the mission being studied (500 sols was chosen as being a long enough time to capture variations in relay asset interactions, Earth/Mars time phasing, and seasonal variations in holidays). This model is used to estimate the ops efficiency factor for each operations configuration. The second model in a separate Excel spreadsheet is a scenario model, which uses the sol types to rack up the total number of "scenario sols" for that scenario (in other words, the ideal number of sols it would take to perform the scenario objectives). Then, the number of sols requiring ground in the loop is calculated based on the soil types contained in the given scenario. Next, the scenario contains a description of what sequence of operations configurations is used, for how many days each, and this is used with the corresponding ops efficiency factors for each configuration to calculate the "ops duration" corresponding to that scenario. Finally, a margin is applied to determine the minimum surface lifetime required for that scenario. Typically, this level of analysis has not been performed until much later in the mission, and has not been able to influence mission design. Further, the notion of moving to sustainable operations during Prime Mission - and the effect that that move would have on surface mission productivity and mission objective choices - has not been encountered until the most recent rover missions (MSL and Mars 2018).

Layback, Sharon L.↗
NASA NTRS2016

Synoptic Traveling Weather Systems on Mars: Effects of Radiatively-Active Water Ice Clouds

Atmospheric aerosols on Mars are critical in determining the nature of its thermal structure, its large-scale circulation, and hence the overall climate of the planet. We conduct multi-annual simulations with the latest version of the NASA Ames Mars global climate model (GCM), gcm2.3+, that includes a modernized radiative-transfer package and complex water-ice cloud microphysics package which permit radiative effects and interactions of suspended atmospheric aerosols (e.g., water ice clouds, water vapor, dust, and mutual interactions) to influence the net diabatic heating. Results indicate that radiatively active water ice clouds profoundly affect the seasonal and annual mean climate. The mean thermal structure and balanced circulation patterns are strongly modified near the surface and aloft. Warming of the subtropical atmosphere at altitude and cooling of the high latitude atmosphere at low levels takes place, which increases the mean pole-to-equator temperature contrast (i.e., "baroclinicity"). With radiatively active water ice clouds (RAC) compared to radiatively inert water ice clouds (nonRAC), significant changes in the intensity of the mean state and forced stationary Rossby modes occur, both of which affect the vigor and intensity of traveling, synoptic period weather systems. Such weather systems not only act as key agents in the transport of heat and momentum beyond the extent of the Hadley circulation, but also the transport of trace species such as water vapor, water ice-clouds, dust and others. The northern hemisphere (NH) forced Rossby waves and resultant wave train are augmented in the RAC case: the modes are more intense and the wave train is shifted equatorward. Significant changes also occur within the subtropics and tropics. The Rossby wave train sets up, combined with the traveling synoptic-period weather systems (i.e., cyclones and anticyclones), the geographic extent of storm zones (or storm tracks) within the NH. A variety of circulation features will be presented which indicate contrasts between the RAC and nonRAC cases, and which highlight key effects radiatively-active clouds have on physical and dynamical processes active in the current climate of Mars.

NASA Ames Mars global climate model (GCM)↗
DOE OSTI2022

Decoding the dynamics of poleward shifting climate zones using aqua-planet model simulations

Growing evidence indicates that the atmospheric and oceanic circulation experiences a systematic poleward shift in a warming climate. However, the complexity of the climate system, including the coupling between the ocean and the atmosphere, natural climate variability and land-sea distribution, tends to obfuscate the causal mechanism underlying the circulation shift. Here, using an idealised coupled aqua-planet model, we explore the mechanism of the shifting circulation, by isolating the contributing factors from the direct CO 2 forcing, the indirect ocean surface warming, and the wind-stress feedback from the ocean dynamics. We find that, in contrast to the direct CO 2 forcing, ocean surface warming, in particular an enhanced subtropical ocean warming, plays an important role in driving the circulation shift. This enhanced subtropical ocean warming emerges from the background Ekman convergence of surface anomalous heat in the absence of the ocean dynamical change. Additionally, it expands the tropical warm water zone, causes a poleward shift of the mid-latitude temperature gradient, hence forces a corresponding shift in the atmospheric circulation and the associated wind pattern. The shift in wind, in turn drives a shift in the ocean circulation. Our simulations, despite being idealised, capture the main features of the observed climate changes, for example, the enhanced subtropical ocean warming, poleward shift of the patterns of near-surface wind, sea level pressure, storm tracks, precipitation and large-scale ocean circulation, implying that increase in greenhouse gas concentrations not only raises the temperature, but can also systematically shift the climate zones poleward.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Direct Mapping of Polyclonal Epitopes in Serum by HDX-MS

Elucidating the interactions that drive antigen recognition is central to understanding antibody-mediated protection and is vital for the rational design of immunogens. Often, structural knowledge of epitopes targeted by antibodies is derived from isolated studies of monoclonal antibodies, for which numerous structural techniques exist. In contrast, there are very few approaches capable of mapping the full scope of antigen surfaces targeted by polyclonal sera through the course of a natural antibody response. Here, in this study, we develop an approach using immobilized antigen coupled to hydrogen/deuterium exchange with mass spectrometry (HDX-MS) to probe epitope targeting in the context of the fully native serum environment. Using the well-characterized Staphylococcal enterotoxin B (SEB) as a model system, we show that complex combinations of epitopes can be detected and subtle differences across different anti-sera can be discerned. This work reveals new insight into how neutralizing antibodies and antisera target SEB, and more importantly, establishes a novel method for directly mapping the epitope landscape of polyclonal sera.

antigens↗
DOE OSTI2025

Approach to Evaluating Reorganization Energies of Interfacial Electrochemical Reactions

Reaction rate coefficients for electron-transfer processes at the electrode–electrolyte interface are commonly estimated by using the Butler–Volmer equation, but their values are inaccurate beyond a few tenths of volts of overpotential. The Marcus–Hush–Chidsey (MHC) formalism yields correct asymptotic behavior of the rate coefficients vs applied overpotential but has complex dependencies on the redox system’s intrinsic parameters, which can be difficult to model or measure. In this work, we bridge the two kinetics formalisms to estimate the reorganization energy, one of the important parameters for the MHC formalism, and investigate its dependence on other intrinsic parameters such as activation barriers, electronic coupling strength, and the density of states of the electrode surface. We examine the sensitivity of the reorganization energy to these parameters, establish some general relationships for accurately predicting rate coefficients using the MHC formalism over a wide range of applied overpotentials, and compare this approach to calculating MHC rate constants with other empirical approaches for the mechanisms of CO 2 reduction on different metal electrode surfaces.

Butler−Volmer↗
DOE OSTI2022

Building an extensible cell wall

Abstract This article recounts, from my perspective of four decades in this field, evolving paradigms of primary cell wall structure and the mechanism of surface enlargement of growing cell walls. Updates of the structures, physical interactions, and roles of cellulose, xyloglucan, and pectins are presented. This leads to an example of how a conceptual depiction of wall structure can be translated into an explicit quantitative model based on molecular dynamics methods. Comparison of the model’s mechanical behavior with experimental results provides insights into the molecular basis of complex mechanical behaviors of primary cell wall and uncovers the dominant role of cellulose–cellulose interactions in forming a strong yet extensible network.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2025

exoALMA. V. Gaseous Emission Surfaces and Temperature Structures

An analysis of the gaseous component in protoplanetary disks can inform us about their thermal and physical structure, chemical composition, and kinematic properties, all of which are crucial for understanding various processes within the disks. By exploiting the asymmetry of the line emission, or via line profile analysis, we can locate the emitting surfaces. Here, we present the emission surfaces of the exoALMA sources in 12 CO J = 3–2, 13 CO J = 3–2, and CS J = 7–6. We find that 12 CO traces the upper disk atmosphere, with mean $\langle$ z/r $\rangle$ values of ≈0.28, while 13 CO and CS trace lower regions of the disk with mean $\langle$ z/r $\rangle$ values of ≈0.16 and ≈0.18, respectively. We find that 12 CO $\langle$ z/r $\rangle$ and the disk mass are positively correlated with each other; this relationship offers a straightforward way to infer the disk mass. We derive 2D r – z temperature distributions of the disks. Additionally, we search for substructure in the surfaces and radial intensity profiles; we find evidence of localized substructure in the emission surfaces and peak intensity profiles of nearly every disk, with this substructure often being coincident between molecular tracers, intensity profiles, and kinematic perturbations. Four disks display evidence of potential photodesorption, implying that this effect may be common even in low far-ultraviolet star-forming regions. For most disks, we find that the physical and thermal structure is more complex than analytical models can account for, highlighting a need for more theoretical work and a better understanding of the role of projection effects on our observations.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1991

Evaluation of a three-dimensional Navier-Stokes solver for transonic flow over a low-aspect-ratio wing mounted in a wind tunnel

A three dimensional Navier-Stokes solver is evaluated for transonic flow over a thin, swept, low-aspect ratio wing. The computational study was undertaken in support of a wind tunnel experimental program. The computational results are compared to experimental surface pressure data obtained in a cryogenic wind tunnel with an adaptive wall test section. The results show favorable agreement over a wide range of conditions, further the numerical results provide additional data of the complex three-dimensional flow field. Differences in the predictions and experiment suggest a need to conduct further experiments to evaluate the adaptive wall testing technique, and to model the tunnel sidewall boundary layer in the computations.

Milholen, William E., II↗
NASA NTRS1993

Investigation of a hypersonic crossing shock wave/turbulent boundary layer interaction

A combined theoretical and experimental study is presented for the interaction between crossing shock waves generated by (10 deg, 10 deg) sharp fins and a flat plate turbulent boundary layer at Mach 8.3. The theoretical model is the full 3D mean compressible Reynolds-averaged Navier-Stokes (RANS) equations incorporating the algebraic turbulent eddy viscosity model of Baldwin and Lomax (1978). A grid refinement study indicated that adequate resolution of the flow field has been achieved. Computed results agree well with experiment for surface pressure and surface flow patterns and for pitot pressure and yaw angle profiles in the flow field. The computations, however, significantly overpredict surface heat transfer. Analysis of the computed flow field results indicates the formation of complex streamline and wave structures within the interaction region.

Narayanswami, N↗
NASA NTRS2026

Petermann Glacier on the Brink: Progress, Challenges and Insights

Petermann Glacier, the largest marine-terminating glacier in northern Greenland based on catchment area and ice discharge, plays a key role in regulating ice discharge from the Greenland Ice Sheet into the Arctic Ocean. With an upstream catchment connected to the ice sheet interior via a deep subglacial canyon, its future stability has major implications for sea level rise. In this review, we synthesize recent advances in understanding Petermann’s dynamics across three critical interfaces: the ice–ocean, ice–atmosphere, and ice–bed boundaries. At the surface, observations show that reanalysis products underestimate air temperatures and melt, while regional climate models diverge significantly in their estimates of surface mass balance, underscoring the need for improved in situ data and models. At the ocean boundary, enhanced basal melting driven by both subglacial runoff and Atlantic water intrusions is identified as the dominant driver of recent mass loss of Petermann Glacier, with continued warming posing a serious threat to the stability of the floating tongue. At the bed, new geophysical synthesis reveals complex geology, likely spatial variability in geothermal heat flux, and the influence of the megacanyon on seasonal hydrology and velocity fluctuations. Petermann’s mass balance has been negative in recent decades without corresponding flow acceleration. However, the glacier has undergone significant calving events, and the current rifting that began in September 2025 highlights its vulnerability. This upcoming calving event underscores the timeliness of this review, as it will put Petermann Glacier’s terminus at its most retreated position since records began in 1923. The anticipated retreat of the ice tongue also reduces buttressing and brings the terminus closer to the grounding zone, and modeling studies suggest that calving within 12 km of the grounding zone could potentially trigger dynamic retreat, accelerating ice discharge, and a doubling of flow speeds. We conclude that improved observations, sustained monitoring of oceanographic and atmospheric properties, and high-resolution modeling are critical to constraining projections of Petermann Glacier’s future and its role in the stability of the Greenland Ice Sheet.

Dominik Fahrner↗
DOE OSTI2022

Automatic Waveform Quality Control for Surface Waves Using Machine Learning

Surface-wave seismograms are widely used by researchers to study Earth’s interior and earthquakes. To extract information reliably and robustly from a suite of surface waveforms, the signals require quality control screening to reduce artifacts from signal complexity and noise. This process has usually been completed by human experts labeling each waveform visually, which is time consuming and tedious for large data sets. We explore automated approaches to improve the efficiency of waveform quality control processing by investigating logistic regression, support vector machines, K-nearest neighbors, random forests (RF), and artificial neural networks (ANN) algorithms. To speed up signal quality assessment, we trained these five machine learning (ML) methods using nearly 400,000 human-labeled waveforms. The ANN and RF models outperformed other algorithms and achieved a test accuracy of 92%. We evaluated these two best-performing models using seismic events from geographic regions not used for training. The results show that the two trained models agree with labels from human analysts but required only 0.4% of the time. Although the original (human) quality assignments assessed general waveform signal-to-noise, the ANN or RF labels can help facilitate detailed waveform analysis. Our investigations demonstrate the capability of the automated processing using these two ML models to reduce outliers in surface-wave-related measurements without human quality control screening.

58 GEOSCIENCES↗
DOE OSTI2020

Microscale mechanical modeling of deformable geomaterials with dynamic contacts based on the numerical manifold method

Abstract Micromechanical modeling of geomaterials is challenging because of the complex geometry of discontinuities and potentially large number of deformable material bodies that contact each other dynamically. In this study, we have developed a numerical approach for micromechanical analysis of deformable geomaterials with dynamic contacts. In our approach, we detect contacts among multiple blocks with arbitrary shapes, enforce different contact constraints for three different contact states of separated, bonded, and sliding, and iterate within each time step to ensure convergence of contact states. With these features, we are able to simulate the dynamic contact evolution at the microscale for realistic geomaterials having arbitrary shapes of grains and interfaces. We demonstrate the capability with several examples, including a rough fracture with different geometric surface asperity characteristics, settling of clay aggregates, compaction of a loosely packed sand, and failure of an intact marble sample. With our model, we are able to accurately analyze (1) large displacements and/or deformation, (2) the process of high stress accumulated at contact areas, (3) the failure of a mineral cemented rock samples under high stress, and (4) post-failure fragmentation. The analysis highlights the importance of accurately capturing (1) the sequential evolution of geomaterials responding to stress as motion, deformation, and high stress; (2) large geometric features outside the norms (such as large asperities and sharp corners) as such features can dominate the micromechanical behavior; and (3) different mechanical behavior between loosely packed and tightly packed granular systems.

58 GEOSCIENCES↗
NASA NTRS2003

Flutter and Divergence Analysis using the Generalized Aeroelastic Analysis Method

The Generalized Aeroelastic Analysis Method (GAAM) is applied to the analysis of three well-studied checkcases: restrained and unrestrained airfoil models, and a wing model. An eigenvalue iteration procedure is used for converging upon roots of the complex stability matrix. For the airfoil models, exact root loci are given which clearly illustrate the nature of the flutter and divergence instabilities. The singularities involved are enumerated, including an additional pole at the origin for the unrestrained airfoil case and the emergence of an additional pole on the positive real axis at the divergence speed for the restrained airfoil case. Inconsistencies and differences among published aeroelastic root loci and the new, exact results are discussed and resolved. The generalization of a Doublet Lattice Method computer code is described and the code is applied to the calculation of root loci for the wing model for incompressible and for subsonic flow conditions. The error introduced in the reduction of the singular integral equation underlying the unsteady lifting surface theory to a linear algebraic equation is discussed. Acknowledging this inherent error, the solutions of the algebraic equation by GAAM are termed 'exact.' The singularities of the problem are discussed and exponential series approximations used in the evaluation of the kernel function shown to introduce a dense collection of poles and zeroes on the negative real axis. Again, inconsistencies and differences among published aeroelastic root loci and the new 'exact' results are discussed and resolved. In all cases, aeroelastic flutter and divergence speeds and frequencies are in good agreement with published results. The GAAM solution procedure allows complete control over Mach number, velocity, density, and complex frequency. Thus all points on the computed root loci can be matched-point, consistent solutions without recourse to complex mode tracking logic or dataset interpolation, as in the k and p-k solution methods.

Edwards, John W.↗