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Recent Advances in Soft Matter Characterization Capabilities Developed at NASA GRC for Lunar Exploration: Differential Dynamic Microscopy to Spectroscopy to Computer Vision

In 1991, famous French scientist Pierre-Gilles de Genes was awarded Nobel prize for his impactful research in soft matter, more specifically polymers. He is defined as the founding father of soft matter. In his Nobel lecture (https://www.nobelprize.org/uploads/2018/06/gennes-lecture.pdf ) he described soft matter aka complex fluids as materials with two primary features – (a) complexity and (b) flexibility. The sub-categories of soft matter (e.g.- granular materials, polymers, foams, colloids etc.) are defined on the basis of Pierre-Gilles de Gennes’ definition. At NASA GRC, we are pushing the boundaries for fundamental study of soft matter on Lunar Surface. With regard to Lunar surface science, we are focusing on developing capabilities pertaining to granular materials and bio-soft/active matter to facilitate future efforts in ISRU and bio-ISRU capabilities. In order to achieve fundamental goals of soft matter research within the limitations of Lunar environment, the scientific capabilities need to be small, flexible, modular, off the shelf and the focus needs to be more on developing an interdisciplinary capability that leverages the recent growth in AI/ML and Computer Vision to augment our understanding of fundamental science. This strategy would allow us to reduce our resource requirement during launch, installation, and occupied real estate footprint on Lunar surface In this talk, we will go over 3 different capabilities that we have developed in house and in close collaboration – (a) Differential Dynamic Microscopy (DDM), (b) Portable In-situ Chemical Spectroscopy (PICS) and (c) Computer Vision Enabled Observation. At very high level, Differential Dynamic Microscopy (DDM) allows us to study the structure-property-process relation (microrheology) of bio-soft/active matter using optical microscope and improved image analysis capabilities. PICS uses AI/ML-based advanced signal deconvolution and analysis technique that can work with existing portable spectroscopy tools to perform materials analysis (e.g.- granular materials and bio-soft/active matter) inspection on the go. Finally, computer vision enabled analysis allows us to use simple camera images for 3D reconstruction of experimental process and tracking of objects of interest in an experiment. We expect that this detailed process will allow us reach a thorough understanding of soft matter in Lunar environment. The capabilities developed by us will help to validate and establish fundamental understanding in Lunar environment. This will, in turn, allow us to guide future space exploration missions and expand the knowledge base of the scientific and engineering communities.

Suman Sinha Ray↗

Recent Advances in Soft Matter Characterization Capabilities Developed at NASA GRC for Lunar Exploration: Differential Dynamic Microscopy to Spectroscopy to Computer Vision

In 1991, famous French scientist Pierre-Gilles de Genes was awarded Nobel prize for his impactful research in soft matter, more specifically polymers. He is defined as the founding father of soft matter. In his Nobel lecture (https://www.nobelprize.org/uploads/2018/06/gennes-lecture.pdf ) he described soft matter aka complex fluids as materials with two primary features – (a) complexity and (b) flexibility. The sub-categories of soft matter (e.g.- granular materials, polymers, foams, colloids etc.) are defined on the basis of Pierre-Gilles de Gennes’ definition. At NASA GRC, we are pushing the boundaries for fundamental study of soft matter on Lunar Surface. With regard to Lunar surface science, we are focusing on developing capabilities pertaining to granular materials and bio-soft/active matter to facilitate future efforts in ISRU and bio-ISRU capabilities. In order to achieve fundamental goals of soft matter research within the limitations of Lunar environment, the scientific capabilities need to be small, flexible, modular, off the shelf and the focus needs to be more on developing an interdisciplinary capability that leverages the recent growth in AI/ML and Computer Vision to augment our understanding of fundamental science. This strategy would allow us to reduce our resource requirement during launch, installation, and occupied real estate footprint on Lunar surface. In this talk, we will go over 3 different capabilities that we have developed in house and in close collaboration – (a) Differential Dynamic Microscopy (DDM), (b) Portable In-situ Chemical Spectroscopy (PICS) and (c) Computer Vision Enabled Observation. At very high level, Differential Dynamic Microscopy (DDM) allows us to study the structure-property-process relation (microrheology) of bio-soft/active matter using optical microscope and improved image analysis capabilities. PICS uses AI/ML-based advanced signal deconvolution and analysis technique that can work with existing portable spectroscopy tools to perform materials analysis (e.g.- granular materials and bio-soft/active matter) inspection on the go. Finally, computer vision enabled analysis allows us to use simple camera images for 3D reconstruction of experimental process and tracking of objects of interest in an experiment. We expect that this detailed process will allow us reach a thorough understanding of soft matter in Lunar environment. The capabilities developed by us will help to validate and establish fundamental understanding in Lunar environment. This will, in turn, allow us to guide future space exploration missions and expand the knowledge base of the scientific and engineering communities.

Suman Sinha-Ray↗

Recent Advances in Soft Matter Characterization Capabilities Developed at NASA GRC for Lunar Exploration: Differential Dynamic Microscopy to Spectroscopy to Computer Vision

In 1991, famous French scientist Pierre-Gilles de Genes was awarded Nobel prize for his impactful research in soft matter, more specifically polymers. He is defined as the founding father of soft matter. In his Nobel lecture (https://www.nobelprize.org/uploads/2018/06/gennes-lecture.pdf ) he described soft matter aka complex fluids as materials with two primary features – (a) complexity and (b) flexibility. The sub-categories of soft matter (e.g.- granular materials, polymers, foams, colloids etc.) are defined on the basis of Pierre-Gilles de Gennes’ definition. At NASA GRC, we are pushing the boundaries for fundamental study of soft matter on Lunar Surface. With regard to Lunar surface science, we are focusing on developing capabilities pertaining to granular materials and bio-soft/active matter to facilitate future efforts in ISRU and bio-ISRU capabilities. In order to achieve fundamental goals of soft matter research within the limitations of Lunar environment, the scientific capabilities need to be small, flexible, modular, off the shelf and the focus needs to be more on developing an interdisciplinary capability that leverages the recent growth in AI/ML and Computer Vision to augment our understanding of fundamental science. This strategy would allow us to reduce our resource requirement during launch, installation, and occupied real estate footprint on Lunar surface In this talk, we will go over 3 different capabilities that we have developed in house and in close collaboration – (a) Differential Dynamic Microscopy (DDM), (b) Portable In-situ Chemical Spectroscopy (PICS) and (c) Computer Vision Enabled Observation. At very high level, Differential Dynamic Microscopy (DDM) allows us to study the structure-property-process relation (microrheology) of bio-soft/active matter using optical microscope and improved image analysis capabilities. PICS uses AI/ML-based advanced signal deconvolution and analysis technique that can work with existing portable spectroscopy tools to perform materials analysis (e.g.- granular materials and bio-soft/active matter) inspection on the go. Finally, computer vision enabled analysis allows us to use simple camera images for 3D reconstruction of experimental process and tracking of objects of interest in an experiment. We expect that this detailed process will allow us reach a thorough understanding of soft matter in Lunar environment. The capabilities developed by us will help to validate and establish fundamental understanding in Lunar environment. This will, in turn, allow us to guide future space exploration missions and expand the knowledge base of the scientific and engineering communities.

Suman Sinha Ray↗

Recent Advances in Soft Matter Characterization Capabilities Developed at NASA GRC

In 1991, famous French scientist Pierre-Gilles de Genes was awarded Nobel prize for his impactful research in soft matter, more specifically polymers. He is defined as the founding father of soft matter. In his Nobel lecture (https://www.nobelprize.org/uploads/2018/06/gennes-lecture.pdf ) he described soft matter aka complex fluids as materials with two primary features – (a) complexity and (b) flexibility. The sub-categories of soft matter (e.g.- granular materials, polymers, foams, colloids etc.) are defined on the basis of Pierre-Gilles de Gennes’ definition. At NASA GRC, we are pushing the boundaries for fundamental study of soft matter on Lunar Surface. With regard to Lunar surface science, we are focusing on developing capabilities pertaining to granular materials and bio-soft/active matter to facilitate future efforts in ISRU and bio-ISRU capabilities. In order to achieve fundamental goals of soft matter research within the limitations of Lunar environment, the scientific capabilities need to be small, flexible, modular, off the shelf and the focus needs to be more on developing an interdisciplinary capability that leverages the recent growth in AI/ML and Computer Vision to augment our understanding of fundamental science. This strategy would allow us to reduce our resource requirement during launch, installation, and occupied real estate footprint on Lunar surface In this talk, we will go over 3 different capabilities that we have developed in house and in close collaboration – (a) Differential Dynamic Microscopy (DDM), (b) Portable In-situ Chemical Spectroscopy (PICS) and (c) Computer Vision Enabled Observation. At very high level, Differential Dynamic Microscopy (DDM) allows us to study the structure-property-process relation (microrheology) of bio-soft/active matter using optical microscope and improved image analysis capabilities. PICS uses AI/ML-based advanced signal deconvolution and analysis technique that can work with existing portable spectroscopy tools to perform materials analysis (e.g.- granular materials and bio-soft/active matter) inspection on the go. Finally, computer vision enabled analysis allows us to use simple camera images for 3D reconstruction of experimental process and tracking of objects of interest in an experiment. We expect that this detailed process will allow us reach a thorough understanding of soft matter in Lunar environment. The capabilities developed by us will help to validate and establish fundamental understanding in Lunar environment. This will, in turn, allow us to guide future space exploration missions and expand the knowledge base of the scientific and engineering communities.

Suman Sinha Ray↗

Is Radioactive Tank Waste a Bingham Fluid?

Slurries and sludges across the United States Department of Energy (DOE) complex rank among the most rheologically interesting. Their composition is heterogeneous, spanning a very broad range of particle sizes, densities, and interparticle forces. All exhibit shear thinning, some have yield stresses, and many are thixotropic. Despite the variety, these complex fluids are often represented using the historic Bingham fluid model, which fits higher shear rate data to a simple straight line. The intercept provides a yield stress, which has been a key design parameter in construction of large-scale waste processing facilities. However, many radioactive wastes are simply not Bingham fluids, which extrapolate but poorly across low to intermediate shear rates that are characteristic of typical processing conditions. Indeed, processing shear rates as high as 200 1/s, which has been a typical minimum shear rate used in fitting, are seldom encountered in nuclear waste processing. Therefore, more realistic rheological models are necessary to accurately predict waste processing performance. PNNL recently re-evaluated the rheology of reconstituted Hanford REDOX sludge waste against a wide variety of rheological models including the Bingham, Cross, Cross with yield stress, Carreau, biviscous, Herschel-Bulkley (which includes a power law dependence), Casson, and Gay models. They found that all of the models provided a closer fit than the Bingham model and that the biviscous model and Cross with yield stress model were convincing. However, reconstituted Hanford REDOX sludge waste is but one type of DOE waste and a direct contrast and comparison of these three models against undiluted, unmixed tank waste (actual not simulant) has not been performed previously. Therefore, the purpose of this paper is to evaluate the rheology of actual tank waste with these more accurate rheological models. In this paper, we evaluate select rheological data for slurry samples from Hanford’s AZ-101, AZ-102, and SY-101 waste tanks. In each of these cases, we find that Cross’ model with yield stress and the biviscous model significantly outperform the Bingham fluid model. Furthermore, the AZ-101 data also shows that the shear stress peak that startup significantly exceeds the Bingham yield stress, which is commonly observed in the incipient moments of rheological measurements on simulants. In contrast, Cross’ model may empirically accommodate an initial spike in shear stress at modest shear rates. These are important observations because computational and analytical fluid dynamics simulations rely on rheological constitutive models for accurately and conservatively predict waste processing performance. These findings suggest the need for better rheological modeling of and validation against radioactive waste.

Rheology, Radioactive Waste↗

Active Mixing of Reactive Materials for 3D Printing

The mixing of materials during additive manufacturing is a major benefit which allows one to compositionally and spatially tailor material properties, for example to locally control the reactivity in fuel–oxidizer systems known as thermites. This work characterizes an active mixing printhead used in conjunction with a 3D printing process known as Direct Ink Writing. Besides compositional control, a major benefit of this approach is that it offers a safe method for working with these materials, which can otherwise be hazardous once mixed. Custom fuel and oxidizer inks are fed at fixed volumetric rates into an active mixing printhead, and both the rotational speed of the mixing impeller as well as the fuel–oxidizer ratio are varied. Upon ignition, the propagation speed increases with the rotational speed of the mixer and plateaus above a critical value of ≈750 rpm. The critical mixing speed is corroborated by computational fluid simulations and an analytical expression that considers the inks’ complex fluid behavior. Additionally, varying the composition results in a wide range of propagation speeds with peak reactivity corresponding to a fuel–rich formulation ( Φ = 1.5). As a result, a test article incorporating a fast– and slow–burning region demonstrates how spatial composition can manipulate the reactivity.

3D printing↗

An Integrated Solution for Performing Thermo-fluid Conjugate Analysis

A method has been developed which integrates a fluid flow analyzer and a thermal analyzer to produce both steady state and transient results of 1-D, 2-D, and 3-D analysis models. The Generalized Fluid System Simulation Program (GFSSP) is a one dimensional, general purpose fluid analysis code which computes pressures and flow distributions in complex fluid networks. The MSC Systems Improved Numerical Differencing Analyzer (MSC.SINDA) is a one dimensional general purpose thermal analyzer that solves network representations of thermal systems. Both GFSSP and MSC.SINDA have graphical user interfaces which are used to build the respective model and prepare it for analysis. The SINDA/GFSSP Conjugate Integrator (SGCI) is a formbase graphical integration program used to set input parameters for the conjugate analyses and run the models. The contents of this paper describes SGCI and its thermo-fluids conjugate analysis techniques and capabilities by presenting results from some example models including the cryogenic chill down of a copper pipe, a bar between two walls in a fluid stream, and a solid plate creating a phase change in a flowing fluid.

Kornberg, Oren↗

Evaluating Chemical Kinetics Predictions for Propane Using 3-D and 0-D Models in a Boosted Spark-Ignited Engine

Propane has been shown to be a promising alternative fuel to reduce emissions while simultaneously achieving high efficiencies in medium- and heavy-duty engines. These high-power density applications require boosted engines which, combined with high compression ratio, can lead to auto-ignition and knock. While three-dimensional (3-D) computational fluid dynamics (CFD) models are often used for resolving the complex fluid flow in engines, these models can become computationally expensive when simulating detailed chemical kinetics. Likewise, zero-dimensional (0-D) models are computationally concise enough for kinetics development, but lack any flow-field information which governs the flame propagation processes in spark ignition (SI) engines. This work presents a comprehensive comparison between 3-D and 0-D closed cycle simulations at knocking conditions in a high compression ratio high stroke-to-bore ratio propane engine. In order to initialize the flow-field for the 3-D closed cycle (intake valve closing, (IVC) to exhaust valve opening, (EVO)) simulation, a motored multi-cycle 3-D model was run using Converge to create a map at IVC, reducing the computational time. The map allowed a non-homogeneous 3-D closed cycle simulation to be satisfactorily validated against experiments, while a homogeneous case using only the turbulence field mapping was also simulated, mimicking 0-D modeling. The 3-D simulations were used to prescribe the initial conditions (e.g., IVC thermodynamics, speciation, burn-rate profile) for a 2-zone 0-D SI engine model in Chemkin Pro for both cases. It was found that 2-zone 0-D modeling underpredicted the knock onset timing, likely due to the lack of thermal stratification in the unburned gas region. Future work will carry multi-zone 0-D modeling to capture the fuel auto-ignition in the unburned region.

Douvry-Rabjeau, Julien [Oakland University, Roches↗

Method and Apparatus for Predicting Unsteady Pressure and Flow Rate Distribution in a Fluid Network

A method and apparatus for analyzing steady state and transient flow in a complex fluid network, modeling phase changes, compressibility, mixture thermodynamics, external body forces such as gravity and centrifugal force and conjugate heat transfer. In some embodiments, a graphical user interface provides for the interactive development of a fluid network simulation having nodes and branches. In some embodiments, mass, energy, and specific conservation equations are solved at the nodes, and momentum conservation equations are solved in the branches. In some embodiments, contained herein are data objects for computing thermodynamic and thermophysical properties for fluids. In some embodiments, the systems of equations describing the fluid network are solved by a hybrid numerical method that is a combination of the Newton-Raphson and successive substitution methods.

Majumdar, Alok K.↗

Experimental and Numerical Investigation of Reduced Gravity Fluid Slosh Dynamics for the Characterization of Cryogenic Launch and Space Vehicle Propellants

As space programs increasingly investigate various options for long duration space missions the accurate prediction of propellant behavior over long periods of time in microgravity environment has become increasingly imperative. This has driven the development of a detailed, physics-based understanding of slosh behavior of cryogenic propellants over a range of conditions and environments that are relevant for rocket and space storage applications. Recent advancements in computational fluid dynamics (CFD) models and hardware capabilities have enabled the modeling of complex fluid behavior in microgravity environment. Historically, launch vehicles with moderate duration upper stage coast periods have contained very limited instrumentation to quantify propellant stratification and boil-off in these environments, thus the ability to benchmark these complex computational models is of great consequence. To benchmark enhanced CFD models, recent work focuses on establishing an extensive experimental database of liquid slosh under a wide range of relevant conditions. In addition, a mass gauging system specifically designed to provide high fidelity measurements for both liquid stratification and liquid/ullage position in a micro-gravity environment has been developed. This pUblication will summarize the various experimental programs established to produce this comprehensive database and unique flight measurement techniques.

Walls, Laurie K.↗

Recent advances in computational mathematics and applications

We are honored to bring you this special issue dedicated to recent advances in computational mathematics and applications in science and engineering. The papers in this special issue were presented in ”Conference on Computational Mathematics and Applications (CCMA)” held at the University of Nevada Las Vegas (UNLV) during October 25–27, 2019. The conference has attracted about 90 attendees from six countries. The 15 papers in the special issue comprise a diverse collection of theoretical numerical analysis as well as applications in various subjects. Topics cover optimal control of PDEs, discontinuous Galerkin methods, fluid flow in porous media, turbulence flow, static and moving interface problems, complex fluids composed by the mixture of Newtonian fluid and nematic (liquid crystal) flows, modeling of ocean–atmosphere system, image segmentation algorithm, uncertainty quantification problems for turbulence model and Maxwell’s equations, iterative solver for systems resulting from mixed methods, and novel methods for solving systems of nonlinear equations. As the Guest Editors we would like to thank Dr. Leland Jameson at NSF (National Science Foundation) for kindly supporting our conference proposal which made this conference possible. We would also like to thank Darren Sugrue from Elsevier who provided partial support for our conference, and Thennarasu Gunasekaran and his production team from Elsevier who produced this nice special issue. We are grateful to many people (Dr. Zhijian Wu, Lori Ornelas, Elsa Juarez, and many Ph.D. students) at the Department of Mathematical Sciences of UNLV who provided tremendous support for the conference. Finally, we very much appreciate all authors who contributed their precious results to this special issue.

97 MATHEMATICS AND COMPUTING↗

Progress and future directions in computational fluid dynamics

Computational fluid dynamics (CFD) has made great strides in the detailed simulation of complex fluid flows, including the fluid physics of flows heretofore not understood. It is now being routinely applied to some rather complicated problems, and starting to impact the design cycle of aerospace vehicles and their components. In addition, it is being used to complement and is being complemented by experimental studies. In this paper some major elements of contemporary CFD research, such as code validation, turbulence physics, and hypersonic flows are discussed, along with a review of the principal pacing items that currently govern CFD. Several examples are presented to illustrate the current state of the art. Finally, prospects for the future of the development and application of CFD are suggested.

Kutler, Paul↗

Computational fluid dynamics at NASA Ames Research Center

Computational fluid dynamics (CFD) has made great strides in the detailed simulation of complex fluid flows, including the fluid physics of flows heretofore not understood. It is now being routinely applied to some rather complicated problems, and starting to impact the design cycle of aerospace flight vehicles and their components. In addition, it is being used to complement, and is being complemented by, experimental studies. In the present paper, some major elements of contemporary CFD research, such as code validation, turbulence physics, and hypersonic flows are discussed, along with a review of the principal pacing items that currently govern CFD. Several examples of pioneering CFD research are presented to illustrate the current state of the art. Finally, prospects for the future development and application of CFD are suggested.

Kutler, Paul↗

Computational fluid dynamics - Current capabilities and directions for the future

Computational fluid dynamics (CFD) has made great strides in the detailed simulation of complex fluid flows, including some of those not before understood. It is now being routinely applied to some rather complicated problems and starting to affect the design cycle of aerospace flight vehicles and their components. It is being used to complement, and is being complemented by, experimental studies. Several examples are presented in the paper to illustrate the current state of the art. Included is a discussion of the barriers to accomplishing the basic objective of numerical simulation. In addition, the directions for the future in the discipline of computational fluid dynamics are addressed.

Kutler, Paul↗

A Computational Fluid Dynamics Algorithm on a Massively Parallel Computer

The discipline of computational fluid dynamics is demanding ever-increasing computational power to deal with complex fluid flow problems. We investigate the performance of a finite-difference computational fluid dynamics algorithm on a massively parallel computer, the Connection Machine. Of special interest is an implicit time-stepping algorithm; to obtain maximum performance from the Connection Machine, it is necessary to use a nonstandard algorithm to solve the linear systems that arise in the implicit algorithm. We find that the Connection Machine ran achieve very high computation rates on both explicit and implicit algorithms. The performance of the Connection Machine puts it in the same class as today's most powerful conventional supercomputers.

Jespersen, Dennis C.↗

Numerical Modeling of Propellant Boil-Off in a Cryogenic Storage Tank

A numerical model to predict boil-off of stored propellant in large spherical cryogenic tanks has been developed. Accurate prediction of tank boil-off rates for different thermal insulation systems was the goal of this collaboration effort. The Generalized Fluid System Simulation Program, integrating flow analysis and conjugate heat transfer for solving complex fluid system problems, was used to create the model. Calculation of tank boil-off rate requires simultaneous simulation of heat transfer processes among liquid propellant, vapor ullage space, and tank structure. The reference tank for the boil-off model was the 850,000 gallon liquid hydrogen tank at Launch Complex 39B (LC- 39B) at Kennedy Space Center, which is under study for future infrastructure improvements to support the Constellation program. The methodology employed in the numerical model was validated using a sub-scale model and tank. Experimental test data from a 1/15th scale version of the LC-39B tank using both liquid hydrogen and liquid nitrogen were used to anchor the analytical predictions of the sub-scale model. Favorable correlations between sub-scale model and experimental test data have provided confidence in full-scale tank boil-off predictions. These methods are now being used in the preliminary design for other cases including future launch vehicles

Majumdar, A. K.↗

Corresponding-states behavior of an ionic model fluid with variable dispersion interactions

Guggenheim’s corresponding-states approach for simple fluids leads to a remarkably universal representation of their thermophysical properties. For more complex fluids, such as polar or ionic ones, deviations from this type of behavior are to be expected, thereby supplying us with valuable information about the thermodynamic consequences of the interaction details in fluids. Here, the gradual transition of a simple fluid to an ionic one is studied by varying the relative strength of the dispersion interactions compared to the electrostatic interactions among the charged particles. In addition to the effects on the reduced surface tension that were reported earlier [F. Leroy and V. C. Weiss, J. Chem. Phys. 134, 094703 (2011)], we address the shape of the coexistence curve and focus on properties that are related to and derived from the vapor pressure. These quantities include the enthalpy and entropy of vaporization, the boiling point, and the critical compressibility factor Z{sub c}. For all of these properties, the crossover from simple to characteristically ionic fluid is seen once the dispersive attraction drops below 20%–40% of the electrostatic attraction (as measured for two particles at contact). Below this threshold, ionic fluids display characteristically low values of Z{sub c} as well as large Guggenheim and Guldberg ratios for the reduced enthalpy of vaporization and the reduced boiling point, respectively. The coexistence curves are wider and more skewed than those for simple fluids. The results for the ionic model fluid with variable dispersion interactions improve our understanding of the behavior of real ionic fluids, such as inorganic molten salts and room temperature ionic liquids, by gauging the importance of different types of interactions for thermodynamic properties.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Generalized Fluid System Simulation Program

A general purpose program implemented on a computer analyzes steady state and transient flow in a complex fluid network, modeling phase changes, compressibility, mixture thermodynamics and external body forces such as gravity and centrifugal force. A preprocessor provides for the inter- active development of a fluid network simulation having nodes and branches. Mass, energy, and specie conservation equations are solved at the nodes, and momentum conservation equations are solved in the branches. Contained herein are subroutines for computing "real fluid" thermodynamic and thermophysical properties for 12 fluids, and a number of different source options are provided for model- ing momentum sources or sinks in the branches. The system of equations describing the fluid network is solved by a hybrid numerical method that is a combination of the Newton-Raphson and successive substitution methods. Application and verification of this invention are provided through an example problem, which demonstrates that the predictions of the present invention compare most reasonably with test data.

Alok Kumar Majumdar↗