Search NASA⌕ Search

SEARCH · Search NASA

Results for “Simulation Use Case”

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 217 records · Page 12

Missing Data and Multiple Imputation: An Unbiased Approach

The default method of dealing with missing data in statistical analyses is to only use the complete observations (complete case analysis), which can lead to unexpected bias when data do not meet the assumption of missing completely at random (MCAR). For the assumption of MCAR to be met, missingness cannot be related to either the observed or unobserved variables. A less stringent assumption, missing at random (MAR), requires that missingness not be associated with the value of the missing variable itself, but can be associated with the other observed variables. When data are truly MAR as opposed to MCAR, the default complete case analysis method can lead to biased results. There are statistical options available to adjust for data that are MAR, including multiple imputation (MI) which is consistent and efficient at estimating effects. Multiple imputation uses informing variables to determine statistical distributions for each piece of missing data. Then multiple datasets are created by randomly drawing on the distributions for each piece of missing data. Since MI is efficient, only a limited number, usually less than 20, of imputed datasets are required to get stable estimates. Each imputed dataset is analyzed using standard statistical techniques, and then results are combined to get overall estimates of effect. A simulation study will be demonstrated to show the results of using the default complete case analysis, and MI in a linear regression of MCAR and MAR simulated data. Further, MI was successfully applied to the association study of CO2 levels and headaches when initial analysis showed there may be an underlying association between missing CO2 levels and reported headaches. Through MI, we were able to show that there is a strong association between average CO2 levels and the risk of headaches. Each unit increase in CO2 (mmHg) resulted in a doubling in the odds of reported headaches.

Foy, M.↗

Critical design load case fatigue and ultimate failure simulation for a 10-m H-type vertical-axis wind turbine

While previous studies investigating critical VAWT design load cases have focused on large and relatively flexible Darrieus designs, the bulk of current commercial products seeking certification fall in the relatively small, stiff, and H-type configuration, such as the XFlow Energy Corporation turbine that this study compares against. Understanding the critical design load case impacts for both fatigue and ultimate failure for this size and type of VAWT are imperative for certification. The abil

Brownstein, Ian↗

A simple lightning parameterization for calculating global lightning distributions

A simple parameterization has been developed to simulate global lightning distributions. Convective cloud top height is used as the variable in the parameterization, with different formulations for continental and marine thunderstorms. The parameterization has been validated using two lightning data sets: one global and one regional. In both cases the simulated lightning distributions and frequencies are in very good agreement with the observed lightning data. This parameterization could be used for global studies of lightning climatology; the earth's electric circuit; in general circulation models for modeling global lightning activity, atmospheric NO(x) concentrations, and perhaps forest fire distributions for both the present and future climate; and, possibly, even as a short-term forecasting aid.

Price, Colin↗

New Techniques to Evaluate the Incendiary Behavior of Insulators

New techniques for evaluating the incendiary behavior of insulators is presented. The onset of incendive brush discharges in air is evaluated using standard spark probe techniques for the case simulating approaches of an electrically grounded sphere to a charged insulator in the presence of a flammable atmosphere. However, this standard technique is unsuitable for the case of brush discharges that may occur during the charging-separation process for two insulator materials. We present experimental techniques to evaluate this hazard in the presence of a flammable atmosphere which is ideally suited to measure the incendiary nature of micro-discharges upon separation, a measurement never before performed. Other measurement techniques unique to this study include; surface potential measurements of insulators before, during and after contact and separation, as well as methods to verify fieldmeter calibrations using a charge insulator surface opposed to standard high voltage plates. Key words: Kapton polyimide film, incendiary discharges, brush discharges, contact and frictional electrification, ignition hazards, insulators, contact angle, surface potential measurements.

Buhler, Charles↗

Simulation of the Impact of New Aircraft and Satellite-Based Ocean Surface Wind Measurements on H*Wind Analyses

Accurate observations of surface ocean vector winds (OVW) with high spatial and temporal resolution are required for understanding and predicting tropical cyclones. As NASA's QuikSCAT and Navy's WindSat operate beyond their design life, many members of the weather and climate science communities recognize the importance of developing new observational technologies and strategies to meet the essential need for OVW information to improve hurricane intensity and location forecasts. The Hurricane Imaging Radiometer (HIRAD) is an innovative technology development which offers new and unique remotely sensed satellite observations of both extreme oceanic wind events and strong precipitation. It is based on the airborne Stepped Frequency Microwave Radiometer (SFMR), which is the only proven remote sensing technique for observing tropical cyclone (TC) ocean surface wind speeds and rain rates. The proposed HIRAD instrument advances beyond the current nadir viewing SFMR to an equivalent wide-swath SFMR imager using passive microwave synthetic thinned aperture radiometer (STAR) technology. This sensor will operate over 4-7 GHz (C-band frequencies) where the required TC remote sensing physics has been validated by both SFMR and WindSat radiometers. The instrument is described in more detail in a paper by Jones et al. presented to the Tropical Meteorology Special Symposium at this AMS Annual Meeting. Simulated HIRAD passes through a simulation of hurricane Frances are being developed to demonstrate HIRAD estimation of surface wind speed over a wide swath in the presence of heavy rain. These are currently being used in "quick" OSSEs (Observing System Simulation Experiments) with H'Wind analyses as the discriminating tool. The H'Wind analysis, a product of the Hurricane Research Division of NOAA's Atlantic , Oceanographic and Meteorological Laboratory, brings together wind measurements from a variety of observation platforms into an objective analysis of the distribution of wind speeds in a tropical cyclone. This product is designed to improve understanding of the extent and strength of the wind field, and to improve the assessment of hurricane intensity. See http://www.aoml.noaa._ov/hrd/data sub/wind.html. Observations have been simulated from both aircraft altitudes and space. The simulated flight patterns for the aircraft platform cases have been designed to duplicate the timing and flight patterns used in routine NOAA and USAF hurricane surveillance flights, and the spaceborne case simulates a TRMM orbit and altitude.

Miller, TImothy L.↗

Boundary Avoidance Tracking for Instigating Pilot Induced Oscillations

In order to advance research in the area of pilot induced oscillations, a reliable method to create PIOs in a simulated environment is necessary. Using a boundary avoidance tracking task, researchers performing an evaluation of control systems were able to create PIO events in 42% of cases using a nominal aircraft, and 91% of cases using an aircraft with reduced actuator rate limits. The simulator evaluation took place in the NASA Ames Vertical Motion Simulator, a high-fidelity motion-based simulation facility.

pilot induced oscillation↗

NASA's GMAO Atmospheric Motion Vectors Simulator: Description and Application to the MISTiC Winds Concept

An atmospheric wind vectors (AMVs) simulator was developed by NASA's GMAO to simulate observations from future satellite constellation concepts. The synthetic AMVs can then be used in OSSEs to estimate and quantify the potential added value of new observations to the present Earth observing system and, ultimately, the expected impact on the current weather forecasting skill. The GMAO AMV simulator is a tunable and flexible computer code that is able to simulate AMVs expected to be derived from different instruments and satellite orbit configurations. As a case study and example of the usefulness of this tool, the GMAO AMV simulator was used to simulate AMVs envisioned to be provided by the MISTiC Winds, a NASA mission concept consisting of a constellation of satellites equipped with infrared spectral midwave spectrometers, expected to provide high spatial and temporal resolution temperature and humidity soundings of the troposphere that can be used to derive AMVs from the tracking of clouds and water vapor features. The GMAO AMV simulator identifies trackable clouds and water vapor features in the G5NR and employs a probabilistic function to draw a subset of the identified trackable features. Before the simulator is applied to the MISTiC Winds concept, the simulator was calibrated to yield realistic observations counts and spatial distributions and validated considering as a proxy instrument to the MISTiC Winds the Himawari-8 Advanced Imager (AHI). The simulated AHI AMVs showed a close match with the real AHI AMVs in terms of observation counts and spatial distributions, showing that the GMAO AMVs simulator synthesizes AMVs observations with enough quality and realism to produce a response from the DAS equivalent to the one produced with real observations. When applied to the MISTiC Winds scanning points, it can be expected that the MISTiC Winds will be able to collect approximately 60,000 wind observations every 6 hours, if considering a constellation composed of 12 satellites (4 orbital planes). In addition, one of the main expected impacts of the MISTiC Winds concept is the ability to derive water vapor feature tracking AMVs below 500-400 hPa, an unique feature among the water vapor AMVs derived from the current Earth observing system.

Carvalho, David↗

NASA's GMAO Atmospheric Motion Vectors Simulator: Description and Application to the MISTiC Winds Concept

An atmospheric wind vectors (AMVs) simulator was developed by NASA's GMAO to simulate observations from future satellite constellation concepts. The synthetic AMVs can then be used in OSSEs to estimate and quantify the potential added value of new observations to the present Earth observing system and, ultimately, the expected impact on the current weather forecasting skill. The GMAO AMV simulator is a tunable and flexible computer code that is able to simulate AMVs expected to be derived from different instruments and satellite orbit configurations. As a case study and example of the usefulness of this tool, the GMAO AMV simulator was used to simulate AMVs envisioned to be provided by the MISTiC Winds, a NASA mission concept consisting of a constellation of satellites equipped with infrared spectral midwave spectrometers, expected to provide high spatial and temporal resolution temperature and humidity soundings of the troposphere that can be used to derive AMVs from the tracking of clouds and water vapor features. The GMAO AMV simulator identifies trackable clouds and water vapor features in the G5NR and employs a probabilistic function to draw a subset of the identified trackable features. Before the simulator is applied to the MISTiC Winds concept, the simulator was calibrated to yield realistic observations counts and spatial distributions and validated considering as a proxy instrument to the MISTiC Winds the Himawari-8 Advanced Imager (AHI). The simulated AHI AMVs showed a close match with the real AHI AMVs in terms of observation counts and spatial distributions, showing that the GMAO AMVs simulator synthesizes AMVs observations with enough quality and realism to produce a response from the DAS equivalent to the one produced with real observations. When applied to the MISTiC Winds scanning points, it can be expected that the MISTiC Winds will be able to collect approximately 60,000 wind observations every 6 hours, if considering a constellation composed of 12 satellites (4 orbital planes). In addition, one of the main expected impacts of the MISTiC Winds concept is the ability to derive water vapor feature tracking AMVs below 500-400 hPa, an unique feature among the water vapor AMVs derived from the current Earth observing system.

Carvalho, David↗

Explicit block encodings of boundary value problems for many-body elliptic operators

Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for partial differential equations, including high-dimensional instances for many-body simulations. When restricted to rectangular domains with separable boundary conditions, we provide explicit circuits to block encode the many-body Laplacian with separable periodic, Dirichlet, Neumann, and Robin boundary conditions, using standard discretization techniques from low-order finite difference methods. To obtain high-precision, we introduce a scheme based on periodic extensions to solve Dirichlet and Neumann boundary value problems using a high-order finite difference method, with only a constant increase in total circuit depth and subnormalization factor. We then present a scheme to implement block encodings of differential operators acting on more arbitrary domains, inspired by Cartesian immersed boundary methods. We then block encode the many-body convective operator, which describes interacting particles experiencing a force generated by a pair-wise potential given as an inverse power law of the interparticle distance. This work provides concrete recipes that are readily translated into quantum circuits, with depth logarithmic in the total Hilbert space dimension, that block encode operators arising broadly in applications involving the quantum simulation of quantum and classical many-body mechanics.

Kharazi, Tyler [University of California, Berkeley↗

Analysis of Unsteady Tip and Endwall Heat Transfer in a Highly Loaded Transonic Turbine Stage

In a previous study, vane-rotor shock interactions and heat transfer on the rotor blade of a highly loaded transonic turbine stage were simulated. The geometry consists of a high pressure turbine vane and downstream rotor blade. This study focuses on the physics of flow and heat transfer in the rotor tip, casing and hub regions. The simulation was performed using the Unsteady Reynolds-Averaged Navier-Stokes (URANS) code MSU-TURBO. A low Reynolds number k-epsilon model was utilized to model turbulence. The rotor blade in question has a tip gap height of 2.1 percent of the blade height. The Reynolds number of the flow is approximately 3x10(exp 6) per meter. Unsteadiness was observed at the tip surface that results in intermittent "hot spots". It is demonstrated that unsteadiness in the tip gap is governed by inviscid effects due to high speed flow and is not strongly dependent on pressure ratio across the tip gap contrary to published observations that have primarily dealt with subsonic tip flows. The high relative Mach numbers in the tip gap lead to a choking of the leakage flow that translates to a relative attenuation of losses at higher loading. The efficacy of new tip geometry is discussed to minimize heat flux at the tip while maintaining choked conditions. In addition, an explanation is provided that shows the mechanism behind the rise in stagnation temperature on the casing to values above the absolute total temperature at the inlet. It is concluded that even in steady mode, work transfer to the near tip fluid occurs due to relative shearing by the casing. This is believed to be the first such explanation of the work transfer phenomenon in the open literature. The difference in pattern between steady and time-averaged heat flux at the hub is also explained.

Shyam, Vikram↗

Sensitivity of a Cloud-Resolving Model to the Bulk and Explicit Bin Microphysical Schemes

A cloud-resolving model is used to study sensitivities of two different microphysical schemes, one is the traditional bulk type, and the other is an explicit bin scheme, in simulating a mid-latitude squall line case (PRE-STORM, June 10-1 1,1985). Simulations using different microphysical schemes are compared with each other and also with the observations. Both the bulk and bin models reproduce the general features during the developing and mature stage of the system. Furthermore, the observations and the well-proven bulk scheme simulation serve as validations for the newly incorporated bin scheme. However, it is also shown that the bulk and bin simulations have distinct differences, most notably in the stratiform region of the squall line system. Weak convective cells exist in the stratiform region in the bulk simulation, but not in the bin simulation. These weak convective cells in the stratiform region simulated in the bulk scheme model are remnants of the stronger convections previously at the leading edge of the system, sustained by horizontal vorticity generated by its own cool pool near the surface. The bin simulation, on the other hand, has a horizontally homogeneous stratiform cloud structure, which agrees better with the observations. Examinations of the downdraft core strength, the potential temperature perturbation, and the evaporative cooling rate show that the differences between the bulk and bin models are due mainly to the stronger low-level evaporative cooling in the convective zone simulated in the bulk microphysical scheme, which is unrealistic because of the assumptions made in raindrop size distribution. Further sensitivity tests that reduce the evaporation rate in bulk scheme artificially produce more upright convective core and less weak cores in stratiform region. However, they produce weaker upper level outflow and consequently less stratiform rain area. The addition of a more realistic raindrop breakup scheme in the bin scheme results more realistic radar reflectivity and stronger surface rainfall. Despite the increase of the rain evaporation and strengthening of the near surface cool pool, bin scheme with rain breakup shows homogeneous stratiform rain. These sensitivity tests prove the robustness of the bin microphysical scheme and the difficulty of tuning the limited parameters in the bulk microphysical scheme to realistically reproduce detail structures in a mid-latitude squall line case study.

Li, Xiao-Wen↗

Effect of Ice Number Concentration on the Evolution of Boundary Layer Clouds During Arctic Marine Cold‐Air Outbreaks

Abstract Marine cold‐air outbreaks (MCAOs) are crucial for Arctic Ocean heat loss, featuring convective cloud rolls that transition into convection cells downstream. Understanding factors controlling this transformation is the key for improving MCAO cloud representation in climate models. This study employs large‐eddy simulations to investigate how cloud ice number concentrations () affect cloud evolution using a case from the Cold‐Air Outbreaks in the Marine Boundary Layer Experiment (COMBLE) campaign. The simulations, performed in a Lagrangian framework following an air mass trajectory, are driven by ERA5 reanalysis data. Initially, all simulations produce similar cloud patterns, but higher leads to earlier breakup of cloud rolls. Between 4 and 10 hr, surface precipitation rates are similar across simulations, but precipitation initiates earlier, and the cloud‐base precipitation rates are higher when is higher. The stronger precipitation evaporation leads to increased stability of the boundary layer and reduced intensity of vertical mixing between the surface and cloud layer. An increased sink of cloud layer moisture via precipitation and decreased source through diminished vertical transport result in earlier cloud breakup in higher conditions. Simulations with different sea surface temperatures (SST) indicate that this cloud breakup mechanism remains valid for MCAOs of different strengths, although the cloud organization is more sensitive to SST changes in low environments. This work highlights the importance of accurate representations of ice processes in simulating MCAO clouds and suggests the need for observational constraints of ice nucleating particles and over the mixed‐phase cloud regimes.

54 ENVIRONMENTAL SCIENCES↗

Techniques for improved statistical convergence in quantification of eddy diffusivity moments

While recent approaches, such as the macroscopic forcing method (MFM) or Green's function-based approaches, can be used to compute Reynolds-averaged Navier-Stokes closure operators using forced direct numerical simulations, MFM can also be used to directly compute moments of the effective nonlocal and anisotropic eddy diffusivities. The low-order spatial and temporal moments contain limited information about the eddy diffusivity but are often sufficient for quantification and modeling of nonlocal and anisotropic effects. However, when using MFM to compute eddy diffusivity moments, the statistical convergence can be slow for higher-order moments. In this work, we demonstrate that using the same direct numerical simulation (DNS) for all forced MFM simulations improves statistical convergence of the eddy diffusivity moments. We present its implementation in conjunction with a decomposition method that handles the MFM forcing semianalytically and allows for consistent boundary condition treatment, which we develop for both scalar and momentum transport. We demonstrate that for a two-dimensional Rayleigh-Taylor instability case study, using the same DNS for all forced MFM simulations results in convergence with 𝒪⁡(100) simulations rather than 𝒪⁡(1000) simulations. In conclusion, we then demonstrate the impacts of improved convergence on the quantification of the eddy diffusivity.

general physics↗

Spectrally Resolving High Redshift Dual AGNs with the Line Emission Mapper

The Line Emission Mapper (LEM) is an X-ray Probe-class Mission concept which combines 1-2 eV spectral resolution in the soft X-ray band (0.2-2 keV) with an effective area of ~2600 cm^2 at 1 keV, and 10’’ HPD over a large 30'x30' field of view. The mission will directly address the Astro2020 Decadal Report’s Priority Area of Unveiling the Drivers of Galaxy Growth through the study of merging galaxies, which offer one of the most dramatic channels for galaxy growth and evolution and can potentially trigger both star formation and supermassive black hole (SMBH) growth. Dual and binary AGNs in late-stage mergers are predicted to be a critical stage of merger-induced SMBH growth and coincide with the most transformative period for the host galaxies. Thus, dual and binary AGNs represent ideal laboratories for studying examples of SMBH-galaxy co-evolution. Recent discoveries of dual AGNs at redshifts of z~>2 have been made via optical observations, but such dual AGNs are inaccessible to current X-ray observatories due to a lack of instrumental capabilities, including sensitivity, spectral and spatial resolution. LEM will be uniquely suited to study AGNs at z > 3, where the intrinsic X-ray power law continuum and Iron K alpha emission line - a nearly ubiquitous signpost for AGN accretion – are redshifted into the LEM bandpass. In this poster, we present a simulation-based study on LEM’s ability to discern dual, binary, and/or clustered AGNs at high redshift. We demonstrate the unique capabilities of LEM to spectrally resolve Iron K-alpha lines emitted by distinct AGNs with sufficient line-of-sight velocity differences, even in cases of convolved point sources. Using a suite of simulations probing a range of velocity differences, line strengths, and column densities, we demonstrate LEM’s ability to constrain the spectral properties of dual AGNs at z>3 for a variety of physically realistic pairings. LEM will uniquely provide crucial constraints on dual AGN environments and activity as a function of redshift and will complement optical and infrared observations from upcoming Extremely Large Telescopes.

Ryan W Pfeifle↗

First Stage Solid Propellant Multiply Debris Thermal Analysis

Destruction of a solid rocket stage of a launch vehicle can create a thermal radiation hazard for an aborting crew module. This hazard was assessed for the Constellation Program (Cx) crew and launch vehicle concept. For this concept, if an abort was initiated in first stage flight, the Crew Module (CM) will separate and be pulled away from the malfunctioning launch vehicle via a Launch Abort System (LAS). Having aborted the mission, the launch vehicle will likely be destroyed via a Flight Termination System (FTS) in order to prevent it from errantly traversing back over land and posing a risk to the public. The resulting launch vehicle debris field, composed primarily of first stage solid propellant, poses a threat to the CM. The harsh radiative thermal environment, caused by surrounding burning propellant debris, may lead to CM parachute failure. A methodology, detailed herein, has been developed to address this concern and to quantify the risk of first stage propellant debris leading to the thermal demise of the CM parachutes. Utilizing basic thermal radiation principles, a software program was developed to calculate parachute temperature as a function of time for a given abort trajectory and debris piece trajectory set. Two test cases, considered worst case aborts with regard to launch vehicle debris environments, were analyzed using the simulation: an abort declared at Mach 1 and an abort declared at maximum dynamic pressure (Max Q). For both cases, the resulting temperature profiles indicated that thermal limits for the parachutes were not exceeded. However, short duration close encounters by single debris pieces did have a significant effect on parachute temperature. Therefore while these two test cases did not indicate exceedance of thermal limits, in order to quantify the risk of parachute failure due to radiative effects from the abort environment, a more thorough probability-based analysis using the methodology demonstrated herein must be performed.

Toleman, Benjamin M.↗

Hadoop for High-Performance Climate Analytics: Use Cases and Lessons Learned

Scientific data services are a critical aspect of the NASA Center for Climate Simulations mission (NCCS). Hadoop, via MapReduce, provides an approach to high-performance analytics that is proving to be useful to data intensive problems in climate research. It offers an analysis paradigm that uses clusters of computers and combines distributed storage of large data sets with parallel computation. The NCCS is particularly interested in the potential of Hadoop to speed up basic operations common to a wide range of analyses. In order to evaluate this potential, we prototyped a series of canonical MapReduce operations over a test suite of observational and climate simulation datasets. The initial focus was on averaging operations over arbitrary spatial and temporal extents within Modern Era Retrospective- Analysis for Research and Applications (MERRA) data. After preliminary results suggested that this approach improves efficiencies within data intensive analytic workflows, we invested in building a cyber infrastructure resource for developing a new generation of climate data analysis capabilities using Hadoop. This resource is focused on reducing the time spent in the preparation of reanalysis data used in data-model inter-comparison, a long sought goal of the climate community. This paper summarizes the related use cases and lessons learned.

analytics↗

First Stage Solid Propellant Multi Debris Thermal Analysis

The crew launch vehicle considered for the Constellation (Cx) Program utilizes a first stage solid rocket motor. If an abort is initiated in first stage flight the Crew Module (CM) will separate and be pulled away from the launch vehicle via a Launch Abort System (LAS) in order to safely and quickly carry the crew away from the malfunction launch vehicle. Having aborted the mission, the launch vehicle will likely be destroyed via a Flight Termination System (FTS) in order to prevent it from errantly traversing back over land and posing a risk to the public. The resulting launch vehicle debris field, composed primarily of first stage solid propellant, poses a threat to the CM. The harsh radiative thermal environment induced by surrounding burning propellant debris may lead to CM parachute failure. A methodology, detailed herein, has been developed to address this concern and quantify the risk of first stage propellant debris leading to radiative thermal demise of the CM parachutes. Utilizing basic thermal radiation principles, a software program was developed to calculate parachute temperature as a function of time for a given abort trajectory and debris piece trajectory set. Two test cases, considered worst-case aborts with regard to launch vehicle debris environments, were analyzed using the simulation: an abort declared at Mach 1 and an abort declared at maximum dynamic pressure (Max Q). For both cases, the resulting temperature profiles indicated that thermal limits for the parachutes were not exceeded. However, short duration close encounters by single debris pieces did have a significant effect on parachute temperature, with magnitudes on the order of 10 s of degrees Fahrenheit. Therefore while these two test cases did not indicate exceedance of thermal limits, in order to quantify the risk of parachute failure due to radiative effects from the abort environment, a more thorough probability-based analysis using the methodology demonstrated herein must be performed.

Toleman, Benjamin M.↗

Direct simulation of hypersonic rarefied flow about a delta wing

Three-dimensional hypersonic rarefied flow about a delta wing at angle of incidence has been studied using the direct simulation Monte Carlo technique. Results are obtained for a transitional flow case which has been investigated in a nitrogen wind-tunnel experiment. The simulations are performed using a nonreacting, single-species gas model that accounts for rotational and vibrational internal energies. The computations yield an attached leeside flow associated with supersonic expansion over the leeside of the wing. Results are presented for the computed flowfield and surface quantities and overall aerodynamic coefficients.

Celenligil, M. Cevdet↗