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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 1,333 records · Page 74
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

SAR backscatter from coniferous forest gaps

A study is in progress comparing Airborne Synthetic Aperture Radar (AIRSAR) backscatter from coniferous forest plots containing gaps to backscatter from adjacent gap-free plots. Issues discussed are how do gaps in the range of 400 to 1600 sq m (approximately 4-14 pixels at intermediate incidence angles) affect forest backscatter statistics and what incidence angles, wavelengths, and polarizations are most sensitive to forest gaps. In order to visualize the slant-range imaging of forest and gaps, a simple conceptual model is used. This strictly qualitative model has led us to hypothesize that forest radar returns at short wavelengths (eg., C-band) and large incidence angles (e.g., 50 deg) should be most affected by the presence of gaps, whereas returns at long wavelengths and small angles should be least affected. Preliminary analysis of 1989 AIRSAR data from forest near Mt. Shasta supports the hypothesis. Current forest backscatter models such as MIMICS and Santa Barbara Discontinuous Canopy Backscatter Model have in several cases correctly predicted backscatter from forest stands based on inputs of measured or estimated forest parameters. These models do not, however, predict within-stand SAR scene texture, or 'intrinsic scene variability' as Ulaby et al. has referred to it. For instance, the Santa Barbara model, which may be the most spatially coupled of the existing models, is not truly spatial. Tree locations within a simulated pixel are distributed according to a Poisson process, as they are in many natural forests, but tree size is unrelated to location, which is not the case in nature. Furthermore, since pixels of a simulated stand are generated independently in the Santa Barbara model, spatial processes larger than one pixel are not modeled. Using a different approach, Oliver modeled scene texture based on an hypothetical forest geometry. His simulated scenes do not agree well with SAR data, perhaps due to the simple geometric model used. Insofar as texture is the expression of biological forest processes, such as succession and disease, and physical ones, such as fire and wind-throw, it contains useful information about the forest, and has value in image interpretation and classification. Forest gaps are undoubtedly important contributors to scene variance. By studying the localized effects of gaps on forest backscatter, guided by our qualitative model, we hope to understand more clearly the manner in which spatial heterogeneities in forests produce variations in backscatter, which collectively give rise to scene texture.

Day, John L.↗
NASA NTRS1994

Luminous blue variables at quiescence: The zone of avoidance in the Hertzsprung-Russell diagram

Two phases of dynamical instability are theoretically predicted to exist during the evolution of supergiants of normal metallicity that are initially more massive than approximately 60 solar mass. One phase occurs briefly in the yellow or red region of the Hertzsprung-Russell diagram for stars in the early stages of core helium burning, and the other phase occurs for a longer time in the blue or blue-white region for stars exhausting their core helium. Probably only the second phase exists in the case of supergiants with initial masses between approximately 60 solar mass and approximately 30 solar mass or with low metallicities The cause of instability is the partial ionization of hydrogen and helium in a quasi-isolated outer region of the stellar envelope, above the layer where the iron opacity attains a large local maximum. Predicted luminosities, effective temperatures, ejected nebular masses, remnant masses, eruption recurrence times, and lifetimes, though very approximate, are generally consistent with available observational data for the important class of unstable supergiants known as luminous blue variables.

Stothers, Richard B.↗
NASA NTRS1995

A description of the solar wind-magnetosphere coupling based on nonlinear filters

A nonlinear filtering method is introduced for the study of the solar wind -- magnetosphere coupling and related to earlier linear techniques. The filters are derived from the magnetospheric state, a representation of the magnetospheric conditions in terms of a few global variables, here the auroral electrojet indices. The filters also couple to the input, a representation of the solar wind variables, here the rectified electric field. Filter-based iterative prediction of the indices has been obtained for up to 20 hours. The prediction is stable with respect to perturbations in the initial magnetospheric state; these decrease exponentially at the rate of 30/min. The performance of the method is examined for a wide range of parameters and is superior to that of other linear and nonlinear techniques. In the magnetospheric state representation the coupling is modeled as a small number of nonlinear equations under a time-dependent input.

Vassiliadis, D.↗
NASA NTRS1992

Computation of Turbulent Recirculating Flow in Channels, and for Impingement Cooling

Fully elliptic forms of the transport equations have been solved numerically for two flow configurations. The first is turbulent flow in a channel with transverse rectangular ribs, and the second is impingement cooling of a plane surface. Both flows are relevant to proposed designs for active cooling of hypersonic vehicles using supercritical hydrogen as the coolant. Flow downstream of an abrupt pipe expansion and of a backward-facing step were also solved with various near-wall turbulence models as benchmark problems. A simple form of periodicity boundary condition was used for the channel flow with transverse rectangular ribs. The effects of various parameters on heat transfer in channel flow with transverse ribs and in impingement cooling were investigated using the Yap modified Jones and Launder low Reynolds number k-epsilon turbulence model. For the channel flow, predictions were in adequate agreement with experiment for constant property flow, with the results for friction superior to those for heat transfer. For impingement cooling, the agreement with experiment was generally good, but the results suggest that improved modelling of the dissipation rate of turbulence kinetic energy is required in order to obtain improved heat transfer prediction, especially near the stagnation point. The k-epsilon turbulence model was used to predict the mean flow and heat transfer for constant and variable property flows. The effect of variable properties for channel flow was investigated using the same turbulence model, but comparison with experiment yielded no clear conclusions. Also, the wall function method was modified for use in the variable properties flow with a non-adiabatic surface, and an empirical model is suggested to correctly account for the behavior of the viscous sublayer with heating.

Chang, Byong Hoon↗
NASA NTRS1995

Accelerated Test Methods

Neural network systems were evaluated for use in predicting wear of mechanical systems. Three different neural network software simulation packages were utilized in order to create models of tribological wear tests. Representative simple, medium, and high complexity simulation packages were selected. Pin-on-disk, rub shoe, and four-ball tribological test data was used for training, testing, and verification of the neural network models. Results showed mixed success. The neural networks were able to predict results with some accuracy if the number of input variables was low or the amount of training data was high. Increased neural network complexity resulted in more accurate results, however there was a point of diminishing return. Medium complexity models were the best trade off between accuracy and computing time requirements. A NASA Technical Memorandum and a Society of Tribologists and Lubrication Engineers paper are being published which detail the work.

Jansen, Ralph↗
NASA NTRS2000

Monsoon-Enso Relationships: A New Paradigm

This article is partly a review and partly a new research paper on monsoon-ENSO relationship. The paper begins with a discussion of the basic relationship between the Indian monsoon and ENSO dating back to the work of Sir Gilbert Walker up to research results in more recent years. Various factors that may affect the monsoon-ENSO, relationship, including regional coupled ocean-atmosphere processes, Eurasian snow cover, land-atmosphere hydrologic feedback, intraseasonal oscillation, biennial variability and inter-decadal variations, are discussed. The extreme complex and highly nonlinear nature of the monsoon-ENSO relationship is stressed. We find that for regional impacts on the monsoon, El Nino and La Nina are far from simply mirror images of each other. These two polarities of ENSO can have strong or no impacts on monsoon anomalies depending on the strength of the intraseasonal oscillations and the phases of the inter-decadal variations. For the Asian-Australian monsoon (AAM) as a whole, the ENSO impact is effected through a east-west shift in the Walker Circulation. For rainfall anomalies over specific monsoon areas, regional processes play important roles in addition to the shift in the Walker Circulation. One of the key regional processes identified for the boreal summer monsoon is the anomalous West Pacific Anticyclone (WPA). This regional feature has similar signatures in interannual and intraseasonal time scales and appears to determine whether the monsoon-ENSO relationship is strong or weak in a given year. Another important regional feature includes a rainfall and SST dipole across the Indian Ocean, which may have strong impact on the austral summer monsoon. Results are shown indicating that monsoon surface wind forcings may induce a strong biennial signal in ENSO and that strong monsoon-ENSO coupling may translate into pronounced biennial variability in ENSO. Finally, a new paradigm is proposed for the study of monsoon variability. This paradigm provides a unified framework in which monsoon predictability, the role of regional vs. basin-scale processes, its relationship with different climate subsystems, and causes of secular changes in monsoon-ENSO relationship can be investigated.

Lau, K. M.↗
NASA NTRS2008

Silent Aircraft Initiative Concept Risk Assessment

A risk assessment of the Silent Aircraft Initiative's SAX-40 concept design for extremely low noise has been performed. A NASA team developed a list of 27 risk items, and evaluated the level of risk for each item in terms of the likelihood that the risk would occur and the consequences of the occurrence. The following risk items were identified as high risk, meaning that the combination of likelihood and consequence put them into the top one-fourth of the risk matrix: structures and weight prediction; boundary-layer ingestion (BLI) and inlet design; variable-area exhaust and thrust vectoring; displaced-threshold and continuous descent approach (CDA) operational concepts; cost; human factors; and overall noise performance. Several advanced-technology baseline concepts were created to serve as a basis for comparison to the SAX-40 concept. These comparisons indicate that the SAX-40 would have significantly greater research, development, test, and engineering (RDT&E) and production costs than a conventional aircraft with similar technology levels. Therefore, the cost of obtaining the extremely low noise capability that has been estimated for the SAX-40 is significant. The SAX-40 concept design proved successful in focusing attention toward low noise technologies and in raising public awareness of the issue.

Nickol, Craig L.↗
NASA NTRS2015

Comparison of Instantaneous Frequency Scaling from Rain Attenuation and Optical Disdrometer Measurements at K/Q bands

Rain attenuation is strongly dependent on the rain rate, but also on the rain drop size distribution (DSD). Typically, models utilize an average drop size distribution, such as those developed by Laws and Parsons, or Marshall and Palmer. However, individual rain events may possess drop size distributions which could be significantly different from the average and will impact, for example, fade mitigation techniques which utilize channel performance estimates from a signal at a different frequency. Therefore, a good understanding of the characteristics and variability of the raindrop size distribution is extremely important in predicting rain attenuation and instantaneous frequency scaling parameters on an event-toevent basis. Since June 2014, NASA Glenn Research Center (GRC) and the Politecnico di Milano (POLIMI) have measured the attenuation due to rain in Milan, Italy, on the 20/40 GHz beacon signal broadcast from the Alphasat TDP#5 Aldo Paraboni Q/V-band Payload. Concomitant with these measurements are the measurements of drop size distribution and rain rate utilizing a Thies Clima laser precipitation monitor (disdrometer). In this paper, we discuss the comparison of the predicted rain attenuation at 20 and 40 GHz derived from the drop size distribution data with the measured rain attenuation. The results are compared on statistical and real-time bases. We will investigate the performance of the rain attenuation model, instantaneous frequency scaling, and the distribution of the scaling factor. Further, seasonal rain characteristics will be analysed.

millimeter wave↗
NASA NTRS2017

Multi-Beam Laser Beacon Propagation Over Lunar Distance: Comparison of Predictions and Measurements

A multi-beam beacon was transmitted from the Optical Communication Telescope Laboratory (OCTL) located at Table Mountain, CA to the Lunar Laser Space Terminal (LLST), on-board the Lunar Atmospheric Dust and Environment Explorer (LADEE) spacecraft, during NASA’s recent Lunar Laser Communication Demonstration (LLCD). The laser beacon (1568?0.1 nm) was square wave modulated and sensed by a LLST quadrant sensor to enable link acquisition and tracking. LLST computed the mean received beacon power and returned 5 kHz sampled power time series over the downlink. Post-processing of the decoded downlink received at OCTL retrieved the uplink power recorded at LLST. The predicted mean irradiance delivered to LLST consistently agreed with the variable beam divergence transmitted from OCTL to within < 1 decibel (dB). Irradiance fluctuations detected at LLST were reconciled with an uplink wave-propagation model derived using independently monitored atmospheric parameters. We report good agreement of the scintillation index and atmospheric coherence time.

Biswas, A↗
NASA NTRS2019

Forecasting the Ambient Solar Wind with Numerical Models. I. On the Implementation of an Operational Framework

The ambient solar wind conditions in interplanetary space and in the near-Earth environment are determined by activity on the Sun. Steady solar wind streams modulate the propagation behavior of interplanetary coronal mass ejections and are themselves an important driver of recurrent geomagnetic storm activity. The knowledge of the ambient solar wind flows and fields is thus an essential component of successful space weather forecasting. Here, we present an implementation of an operational framework for operating, validating, and optimizing models of the ambient solar wind flow on the example of Carrington Rotation 2077. We reconstruct the global topology of the coronal magnetic field using the potential field source surface model (PFSS) and the Schatten current sheet model (SCS) and discuss three empirical relationships for specifying the solar wind conditions near the Sun, namely the Wang–Sheeley (WS) model, the distance from the coronal hole boundary model (DCHB), and the Wang-Sheeley–Arge (WSA) model. By adding uncertainty in the latitude about the sub-Earth point, we select an ensemble of initial conditions and map the solutions to Earth by the Heliospheric Upwind eXtrapolation (HUX) model. We assess the forecasting performance from a continuous variable validation and find that the WSA model most accurately predicts the solar wind speed time series (RMSE≈83 km/s). We note that the process of ensemble forecasting slightly improves the forecasting performance of all solar wind models investigated. We conclude that the implemented framework is well suited for studying the relationship between coronal magnetic fields and the properties of the ambient solar wind flow in the near-Earth environment.

Reiss, Martin A.↗
NASA NTRS2020

Subseasonal-to-Seasonal Hindcast Skill Assessment of Ridging Events Related to Drought Over the Western United States

Persistent atmospheric ridging events centered near the western United States are associated with widespread precipitation deficits and meteorological drought. Due to the relatively low skill of- dynamical models in forecasting precipitation on subseasonal‐to‐seasonal (S2S) time scales across this region, forecasts of ridging are explored in this study as a potential bridge for early warning drought prediction. To assess skill, we evaluate deterministic and probabilistic S2S hindcasts (out to 6 week lead time) of ridging events and geopotential height anomalies over the western United States in five ensemble hindcast systems. Prediction skill for ridging events is shown to be highly variable across models. For some models, longer‐time‐averaged patterns of geopotential height anomalies across the first 6 weeks can be skillfully simulated, when evaluated probabilistically against climatology. The most skillful models show modest skill in forecasting above normal ridging occurrences at lead times of Weeks 3–4 and Weeks 5–6,with some sensitivity to the specific ridge location and method for determining skill. Using the European Centre for Medium‐Range Weather Forecasts (ECMWF) model as a case study, longer lead time forecast busts are shown to often occur under extended periods of highly active La Nina‐like tropical convection. Model errors at simulating these tropical convection features may have a disproportionately large impact and degrade downstream forecasts over the western United States. Despite the documented model shortcomings, our results highlight an opportunity to target skillful ridging forecasts from dynamical models for improving early warning drought forecasting at S2S lead times.

S2S↗
DOE OSTI2020

On the Formalization of Development and Assessment Process for Digital Twins in the Nearly Autonomous Management and Control System

In recent years, the autonomous control system has been encouraged in advanced reactors for restoring economic viability, simplifying the operation and maintenance, and enabling remote-site power generations [1]. Since the reactor is expected to be operated for a long period of time with a limited number of individuals onsite, it is recommended that the autonomous control system should have access to very realistic models of the state of processes in the whole lifecycle, together with these process behaviors in interaction with their environment in the real world. As a result, digital twin (DT) technology is suggested in autonomous control systems. DT is defined as a digital representation of a physical object or system, which contains a record for the histories of loads, operation and maintenance status, predictions for the near-term transient of important state variables, and decision-making process [2]. Since machine learning (ML) can recognize patterns within a complex system in real-time applications, it has been used to build DTs in the autonomous control systems for advanced reactors. Meanwhile, due to the rareness of operation data in accident scenarios, the development and assessment of DTs is expected to be mainly driven by simulations. Although the capability and feasibility of ML-based DTs are recognized in improving the safety and efficiency of reactor control, a major concern from the regulatory commission and the nuclear industry is whether the information from a DT is developed and assessed in accordance with expectation and requirements by the target decision. Such concerns not only affect the acceptance criteria for DTs, but also values that can be extracted from DTs and autonomous control system during operations. Inspired by the success of formal methods in improving the reliability and robustness of computer programming and software development, it is suggested that the development and assessment process (DAP) for both separate DTs and integral control system should be formalized in a transparent, consistent, and improvable manner. In this study, a digital-twin development and assessment process (DT-DAP) is proposed by adapting the evaluation model development and assessment process (EMDAP) [3] to requirements by the autonomous control system, ML algorithms, and DT technology. To demonstrate the framework, a baseline nearly autonomous management and control (NAMAC) system with ML-based DTs for diagnosis and prognosis is developed and assessed based on the framework. It is found that with selected testing methods and techniques, the DT-DAP can help identify errors in DTs and NAMAC which would otherwise be left unverified. Meanwhile, it is found that the DT-DAP can improve the DTs and NAMAC by continuously learning and iterating through different elements.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2024

Handling Iterative Solvers in an Algorithmic Differentiation Framework Using Implicit Methods

Differentiable programming is a powerful concept as it enables the seemly propagation of gradients through functions, algorithms, and/or whole physics simulations. These gradients are useful for a wide variety of applications, including sensitivity studies and machine learning, but one of particular interest is optimization. Gradient-based optimization, enabled through automatic/algorithmic differentiation (AD), can be used on predictive physical models to efficiently optimize a set of design variables. AD methods are a particularly promising approach to complex physics simulations because they can be shown to scale well with an increasing number of design variables; however, care must be taken when coupling between different models or different states of a single model.

algorithmic differentiation↗
NASA NTRS2019

Integrating Remote Sensing Observations with NASA's GEOS-5 Modeling Framework in Support of Retrospective Analyses and Seasonal Prediction of Biosphere-Atmosphere CO2 Flux

The absorption of nearly half of all anthropogenic carbon dioxide (CO2) emissions by terrestrial and marine ecosystems has played a critical role in mitigating climate change. However, a persistent lack of understanding of these uptake processes impedes attribution, and thereby introduces large uncertainties into projections of the trajectory of carbon and climate in coming decades. A better understanding and quantification of the seasonal and interannual variability of carbon flux between the atmosphere and ocean and terrestrial biospheres is critically needed to improve predictions from Earth system models. We propose to bring together a diverse suite of remote sensing observations, mechanistic models, and NASA's Goddard Earth Observing System, version 5 (GEOS-5) model to address two objective 1) quantify the processes controlling the temporal variability of biosphere-atmosphere CO2 flux on seasonal and interannual timescales from 1989 to 2018 and 2) evaluating the predictability of the components of carbon flux on seasonal timescales. Because GEOS-5 has been developed to incorporate a variety of Earth system observations and includes a complete and physically consistent depiction of the atmosphere, land, and ocean carbon cycles, it represents a unique and ideal synthesis framework for this effort. By expanding the realism of processes that could be included in GEOS-5 seasonal forecasts, this effort supports NASA's goals toward an integrated Earth system modeling and prediction framework.We propose to integrate a number of observations into the GEOS-5 modeling system to refine retrospective estimates of both terrestrial and marine carbon flux and to test hypotheses regarding the effects of climate drivers on terrestrial carbon flux. This will include assimilating several satellite ocean color products in the NASA Ocean Biogeochemistry Model (NOBM) to improve the representation of ocean biology. We will also utilize remote sensing datasets to increase the temporal resolution of global land-use change estimates and fire emissions to better constrain the temporal variability of the terrestrial carbon cycle. This information will be incorporated into two NASA terrestrial biosphere models Catchment-CN, a dynamic vegetation model that is integrated within GEOS-5 and the Ecosystem Demography (ED) model, a core modeling resource supporting the Global Ecosystem Dynamics Investigation (GEDI) mission to map biomass. Fluxes will be evaluated through 1) comparison to top-down flux estimates inferred from surface CO2 observations and 2) comparison against available aircraft and satellite observations.By combining predictions of net ecosystem exchange and air-sea CO2 flux from land and ocean models with statistical predictions of land use change, fire, and fossil fuel emissions, we propose to create the first dynamical CO2 forecast on seasonal timescales. Because of the experimental nature of this endeavor, an important project objective will be to evaluate the limits to the predictability of different carbon cycle processes over lead times of 1-12 months using different satellite and surface data streams.

Ott, Lesley↗
DOE OSTI2024

Mapping the Binary Covalent Alloy Space to Pursue Superior Nitrogen Reduction Reaction Catalysts

The electrochemical nitrogen reduction reaction (NRR) has the potential to decarbonize industrial ammonia production. However, NRR has poor activity and selectivity versus the competing hydrogen evolution reaction for catalysts that adhere to scaling relations. Overcoming the limitations imposed by scaling relations requires more complex catalyst materials, however, evaluating materials beyond simple metal systems is a large combinatorial problem that requires an improved understanding of the electrocatalyst surface to rationally guide the discovery of superior catalysts. The study uses grand canonical density functional theory to uncover NRR trends on a large and disparate set of binary covalent alloys (BCA) with variable compositions and active-site geometries. The studied BCAs generally follow scaling relations, albeit with larger variance and several systems that significantly break scaling. BCAs with early- to mid-transition metals tend to lie near the volcano peak and activate the N 2 triple bond via a side-on binding configuration. Trends in the BCA space cannot be readily predicted using simple electronic descriptors, which is ascribed to the large geometric variability of the BCA surfaces. Furthermore, it is anticipated that these findings will provide a foundation for the rational design of superior NRR electrocatalysts with increasing material complexity.

36 MATERIALS SCIENCE↗
NASA NTRS2011

Can Functional Cardiac Age be Predicted from ECG in a Normal Healthy Population

In a normal healthy population, we desired to determine the most age-dependent conventional and advanced ECG parameters. We hypothesized that changes in several ECG parameters might correlate with age and together reliably characterize the functional age of the heart. Methods: An initial study population of 313 apparently healthy subjects was ultimately reduced to 148 subjects (74 men, 84 women, in the range from 10 to 75 years of age) after exclusion criteria. In all subjects, ECG recordings (resting 5-minute 12-lead high frequency ECG) were evaluated via custom software programs to calculate up to 85 different conventional and advanced ECG parameters including beat-to-beat QT and RR variability, waveform complexity, and signal-averaged, high-frequency and spatial/spatiotemporal ECG parameters. The prediction of functional age was evaluated by multiple linear regression analysis using the best 5 univariate predictors. Results: Ignoring what were ultimately small differences between males and females, the functional age was found to be predicted (R2= 0.69, P < 0.001) from a linear combination of 5 independent variables: QRS elevation in the frontal plane (p<0.001), a new repolarization parameter QTcorr (p<0.001), mean high frequency QRS amplitude (p=0.009), the variability parameter % VLF of RRV (p=0.021) and the P-wave width (p=0.10). Here, QTcorr represents the correlation between the calculated QT and the measured QT signal. Conclusions: In apparently healthy subjects with normal conventional ECGs, functional cardiac age can be estimated by multiple linear regression analysis of mostly advanced ECG results. Because some parameters in the regression formula, such as QTcorr, high frequency QRS amplitude and P-wave width also change with disease in the same direction as with increased age, increased functional age of the heart may reflect subtle age-related pathologies in cardiac electrical function that are usually hidden on conventional ECG.

Schlegel, Todd↗
DOE OSTI2022

ASP (Adaptive Splines for Prediction) [SWR-22-66]

This software package provides functionality to fit piecewise cubic splines to model the relationship between a univariate X (independent variable) and a univariate Y (dependent variable). The output is akin to linear regression but has higher "capacity" in that it can learn more complicated relationships despite leveraging only a single predictor. It thus can learn very reasonable relationships between X and Y with minimal effort. The canonical intended use case is for predicting electric demand (Y) from temperature (X). There can be complicated, non-linear relationships between these two variables, but these can be reasonably approximated by the sum of: 1) an "aggregate" relationship between daily average temperature and daily average demand, and 2) intra-day patterns represented as hourly offsets from the daily average load. The spline functionality is used for learning a robust "aggregate" relationship without the inconvenience of manual feature engineering while still ensuring robust results. The model fitting workhorse is R's built-in smooth.spline. This package provides a suite of options for controlling how smooth.spline fits a spline to the data. In particular, it is designed to return a model with a single critical point (i.e., a single location along the domain/support where the first derivative is zero). This is to ensure that predicted changes in electric demand are always positive as temperatures become more extreme. This helps to prevent overfitting, particularly when the input data has few observations or has significant uncertainty from other sources.

Murphy, Sinnott↗
NASA NTRS1984

Determination of equivalent model geometry for tunnel wall interference assessment/correction

A formula for the determination of equivalent model geometry with two variables measured at the interface is derived, based on two dimensional subsonic flow. This predicted model profile is a reasonable initial estimate for transonic flow as long as the sonic region does not reach the interface. A general formula is given in two forms. One is in terms of complex variable functions and the other is an integral equation. The complex-function formula has the advantage of using analytic expressions. The integral equation form requires a numerical solution after assuming the model geometry as a polynomial function. Examples are given to illustrate the application of the formulas.

Lo, C. F.↗