Search NASA⌕ Search

SEARCH · Search NASA

Results for “Engine Simulation”

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 415 records · Page 23

Enhancement of Nuclear Engineering Technology Degree with a Web-Based PWR Simulator

Reactor plant simulators are used in the nuclear industry for initial and continuous training of reactor operators and senior reactor operators. The use of the generic pressurized water reactor (GPWR) plant simulator is valuable for the nuclear industry as it will ensure graduates of the Bachelor of Science in Nuclear Engineering Technology (BSNET) program will be better trained employees for the nuclear industry and address forecasted workforce shortages in the nuclear industry. By adopting simulation for education, a proven teaching and training method applied extensively in the plant environment, work quality among entrants to the nuclear power workforce will be improved, elevating performance in the field. This includes a positive influence on safety which is paramount to retaining public support of this clean energy source. Much of the research on the use of simulations within courses has argued that simulations are good tools to help students learn key concepts, terminology, and theories within courses. The GPWR simulator provides an opportunity for nuclear engineering technology students to improve their understanding of theoretical concepts associated with nuclear power plants. This hands-on experience will ensure graduates of the BSNET program will be better trained employees for the nuclear industry and address forecasted workforce shortages in the nuclear industry by providing a real-world experience in plant operations. The GPWR simulator will be utilized in five nuclear courses to begin, while providing opportunities for future expansion of the curriculum. All students graduating with a BSNET degree will interact with the simulator, experience better preparing them for further simulator training provided throughout the industry for licensure of operators. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Verification and Validation of Spectral Element Code for Supercritical CO2 Flow in Vertical Heated Tubes

The investigation of heat transfer in supercritical CO2 (sCO2) has garnered considerable attention in recent decades, given sCO2's potential as a promising working fluid for advanced power conversion cycles. Despite previous research efforts, there are still gaps in our understanding of sCO2 heat transfer, particularly in conditions associated with heat transfer deterioration. To delve into sCO2 heat transfer more comprehensively, we propose employing the high-fidelity computational fluid dynamics code NekRS to simulate sCO2 flow using the large eddy simulation technique. Through graphics processing unit acceleration, NekRS achieves a higher computational speed than traditional CPU-based systems. However, before using NekRS in practical applications involving sCO2, it is imperative to perform verification and validation. Here, this paper presents our efforts to verify and validate the NekRS code's capability for simulating sCO2 using heated vertical tubes, where heat transfer deterioration usually happens. To accommodate the unique properties of sCO2, we have modified the NekRS code by integrating third-party property modules, such as REFPROP and PROPATH. Our simulations are compared with experimental and numerical data from the literature, instilling confidence in leveraging NekRS for future engineering applications. Our simulations also reveal that the accuracy of the property module significantly impacts the results, with REFPROP outperforming PROPATH for sCO2 properties. Additionally, we observed that, depending on the flow direction, buoyancy can either enhance or suppress turbulence in sCO2 flow. In upward flow, under certain conditions, the suppressed turbulence leads to heat transfer deterioration, resulting in elevated wall temperatures.

NekRS↗

Advanced Simulation and Computing (ASC) Software Quality Plan: ASC Software Quality Engineering Practices (V.4.0)

The purpose of the Sandia National Laboratories (SNL) Advanced Simulation and Computing (ASC) Software Quality Plan is to clearly identify the software quality engineering practices that are the basis for continually improving the quality of ASC software products. This plan defines the SNL ASC Program software quality engineering practices and provides a mapping of these practices to Laboratory Policy System IT008: Provide Quality Software Policy. This plan also identifies ASC Program Management and the software project teams' responsibilities in implementing quality software engineering practices and in assessing progress towards achieving their software quality goals.

97 MATHEMATICS AND COMPUTING↗

Rethinking Sensitivity Analysis of Nuclear Simulations with Topology

In nuclear engineering, understanding the safety margins of the nuclear reactor via simulations is arguably of paramount importance in predicting and preventing nuclear accidents. It is therefore crucial to perform sensitivity analysis to understand how changes in the model inputs affect the outputs. Modern nuclear simulation tools rely on numerical representations of the sensitivity information -- inherently lacking in visual encodings -- offering limited effectiveness in communicating and exploring the generated data. In this paper, we design a framework for sensitivity analysis and visualization of multidimensional nuclear simulation data using partition-based, topology-inspired regression models and report on its efficacy. We rely on the established Morse-Smale regression technique, which allows us to partition the domain into monotonic regions where easily interpretable linear models can be used to assess the influence of inputs on the output variability. The underlying computation is augmented with an intuitive and interactive visual design to effectively communicate sensitivity information to the nuclear scientists. Our framework is being deployed into the multi-purpose probabilistic risk assessment and uncertainty quantification framework RAVEN (Reactor Analysis and Virtual Control Environment). We evaluate our framework using an simulation dataset studying nuclear fuel performance.

computational topology↗

Defense Waste Processing Facility Simulant Chemical Processing Cell Studies for Sludge Batch 9

The Savannah River National Laboratory (SRNL) received a technical task request from Defense Waste Processing Facility (DWPF) and Saltstone Engineering to perform simulant tests to support the qualification of Sludge Batch 9 (SB9) and to develop the flowsheet for SB9 in the DWPF. These efforts pertained to the DWPF Chemical Process Cell (CPC). CPC experiments were performed using SB9 simulant (SB9A) to qualify SB9 for sludge-only and coupled processing using the nitric-formic flowsheet in the DWPF. Two simulant batches were prepared, one representing SB8 Tank 40H and another representing SB9 Tank 51H. The simulant used for SB9 qualification testing was prepared by blending the SB8 Tank 40H and SB9 Tank 51H simulants. The blended simulant is referred to as SB9A. Eleven CPC experiments were run with an acid stoichiometry ranging between 105% and 145% of the Koopman minimum acid equation (KMA), which is equivalent to 109.7% and 151.5% of the Hsu minimum acid factor. Three runs were performed in the 1L laboratory scale setup, whereas the remainder were in the 4L laboratory scale setup. Sludge Receipt and Adjustment Tank (SRAT) and Slurry Mix Evaporator (SME) cycles were performed on nine of the eleven. The other two were SRAT cycles only. One coupled flowsheet and one extended run were performed for SRAT and SME processing. Samples of the condensate, sludge, and off-gas were taken to monitor the chemistry of the CPC experiments.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

MOSCATO Solver Development and Integration Plan

During FY21, we conducted ongoing development work for the MOSCATO (Molten Salt Chemistry and Transport) solver. The code development work primarily consisted of transitioning capabilities from the original version of the solver, which was written in OpenFOAM, into Nek5000. In doing so, a fast, highly parallelizable solver was created that is capable of complex chemistry and corrosion simulations for engineering-scale molten salt systems. The Nek5000 version of MOSCATO is now fully featured and capable of higher-fidelity simulations than were previously possible. Demonstration cases including a thermal convection loop have been simulated to test these new capabilities. Although capable of large-scale simulations, MOSCATO is not well-suited to parametric studies of complete reactor geometries. These types of simulations are instead better handled by reduced-order modeling codes such as ORNL’s Mole code. Reduced-order simulation tools like Mole, however, are dependent on high fidelity correlations to account for complex, coupled three-dimensional phenomena that they do not directly simulate. Tools such as MOSCATO must therefore be used to create these correlations, as suitable empirical relationships are not available for most molten salt systems. Toward that end, we used the Nek-derived version of MOSCATO to create new mass transfer correlations for three relevant cases including tubular, tube bank, and subchannel geometries. These new correlations are more accurate than any existing ones and can be readily integrated into any reduced-order modeling tools that are targeting full-scale MSR simulations.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

MOSCATO Development and Integration in Fiscal Year 2022

During FY21, we conducted ongoing development work for the MOSCATO (Molten Salt Chemistry and Transport) solver. The code development work primarily consisted of transitioning capabilities from the original version of the solver, which was written in OpenFOAM, into Nek5000. In doing so, a fast, highly parallelizable solver was created that is capable of complex chemistry and corrosion simulations for engineering-scale molten salt systems. The Nek5000 version of MOSCATO is now fully featured and capable of higher-fidelity simulations than were previously possible. Demonstration cases including a thermal convection loop have been simulated to test these new capabilities. We built upon the work for FY22 and improved the code from several different perspectives. First, we improved the user interface by adding a new component to the official Nek5000 input file (.par). This new part contains documents parameters like, salt properties (density, viscosity, Cp, thermal conductivity), diffusion coefficients, standard potential, etc. Second, we built a conversion script to extract salt properties from the MSTDB-TP salt database and write to MOSCATO input file. Third, we migrated the code to NekRS, which is the GPU branch of Nek5000 and suitable for next generation supercomputers. Verification and Validation (V&V) work was also continued in FY22. Two tasks were performed. The first V&V task involved the validation of the Poisson-Nernst-Planck equation solver and Butler-Volmer electrode kinetics, by comparing with numerical and experimental data about thermoelectric cells. The second task involved the comparisons to corrosion results from a thermal convection loop run during the MSRE era. Satisfactory agreement was obtained from both tasks.

Yuan, Haomin↗

Turbulent nonpremixed cool flames: Experimental measurements, Direct Numerical Simulation, and manifold-based combustion modeling

Turbulence, low-temperature chemistry, and their interactions in the form of turbulent cool flames are critical to understanding and improving advanced engines. Design of such engines requires tractable simulations which in turn necessitate turbulent combustion models that can account for cool flames. While manifold-based turbulent combustion models are an attractive option for hot flames, their applicability to cool flames is not yet fully understood. This is partially due to the lack of turbulent cool flame experiments, which has made model validation difficult. This work addresses these points with a combined experimental and computational investigation of turbulent nonpremixed dimethyl ether cool flames. First, a Co-flow Axisymmetric Reactor-Assisted Turbulent (CARAT) burner is developed and tested. Turbulent cool flames are studied using the formaldehyde planar laser-induced fluorescence (PLIF), acetone PLIF, and planar Rayleigh scattering techniques. The acetone PLIF signals are converted into mixture fraction values, and measurements of time-averaged temperature are derived from the Rayleigh scattering signals by taking advantage of the similarities between cool flames and unburned mixtures. Second, a Direct Numerical Simulation (DNS) of the CARAT burner is conducted. The flame is shown to be sensitive to thermal boundary conditions such that the stabilization method is dependent on a 10 K change in inlet temperature. Comparisons of first- and second-order statistics of temperature, mixture fraction, and formaldehyde between the DNS and experiment show good agreement. The validity of manifold-based turbulent combustion models in turbulent cool flames is then explored by first studying the effective Lewis numbers in the flame using a local differential diffusion parameter analysis. Turbulent cool flames are shown to have effective unity Lewis number transport even at a much lower Reynolds number than typically required for hot flames. Good agreement is shown for temperature and formaldehyde between DNS conditional means and one-dimensional nonpremixed flame solutions. Additionally, the motion of the cool flame in mixture fraction space is shown to be well described by one-dimensional nonpremixed flame solutions. These points indicate that the fundamental physics of turbulent cool flames are analogous to turbulent hot flames, implying that the same reduced-order manifold-based modeling approaches utilized for turbulent hot flames can be utilized for turbulent cool flames.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Surface engineering of MXenes for energy and environmental applications

We report MXenes, a new family of two-dimensional (2D) transition-metal carbides, nitrides, and carbonitrides, have received considerable attention in energy and environmental applications due to their intriguing electronic, electrochemical, optical, and chemical properties. Applications of MXenes have been witnessed in photocatalysis, electrocatalysis, thermocatalysis, sensing and biosensing, electrochemical energy storage, energy conversion and storage, rechargeable batteries, supercapacitors, and biomedicine. Notably, the tunable surface chemistry of MXenes plays an important role in their success in those applications. The surface composition of MXenes is considered to have a significant influence on their physicochemical properties and functionalities. In order to achieve a comprehensive understanding and rational design of MXenes by surface engineering, in this review, we highlight the application of several kinds of surface engineering approaches for MXenes, including tuning the surface termination groups, surface functionalization, surface defects, surface doping, surface oxidation, and the theoretical simulation of surface engineering of MXenes. Moreover, the relationship between the surface engineering and the physicochemical properties of MXenes is discussed. Finally, the ongoing challenges and opportunities for the future development of MXene surface engineering are also highlighted.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Filter-based control of particulate matter from a lean gasoline direct injection engine

New regulations requiring increases in vehicle fuel economy are challenging automotive manufacturers to identify fuel-efficient engines for future vehicles. Lean gasoline direct injection (GDI) engines offer significant increases in fuel efficiency over the more common stoichiometric GDI engines already in the marketplace. However, particulate matter (PM) emissions from lean GDI engines, particularly during stratified combustion modes, are problematic for lean GDI technology to meet U.S. Environmental Protection Agency Tier 3 and other future emission regulations. As such, the control of lean GDI PM with wall-flow filters, referred to as gasoline particulate filter (GPF) technology, is of interest. Since lean GDI PM chemistry and morphology differ from diesel PM (where more filtration experience exists), the functionality of GPFs needs to be studied to determine the operating conditions suitable for efficient PM removal. In addition, lean GDI engine exhaust temperatures are generally higher than diesel engines which results in more continuous regeneration of the GPF and less presence of the soot cake layer common to diesel particulate filters. Since the soot layer improves filtration efficiency, this distinction is important to consider. Research on the emission control of PM from a lean GDI engine with a GPF was conducted on an engine dynamometer. PM, after dilution, was characterized with membrane filters, organic vs. elemental carbon characterization, and size distribution techniques at various steady state engine speed and load points. The engine was operated in three primary combustion modes: stoichiometric, lean homogeneous, and lean stratified. In addition, rich combustion was utilized to simulate PM from engine operation during active regeneration of lean NOx control technologies. High (>95%) PM filtration efficiencies were observed over a wide range of conditions; however, some PM was observed to slip through the GPF at high speed and load conditions. The PM characterization at various engine speeds and loads will help enable optimized GPF design and control to achieve more fuel efficient lean GDI vehicles with low PM emissions.

Lewis, Sr, Samuel Arthur↗

Hyperbolic Reformulation Approach to Enable Efficient Simulation of Groundwater Flow and Reactive Transport

We apply Cattaneo's relaxation approach to the one-dimensional coupled Boussinesq groundwater flow and advection-diffusion-reaction equations, commonly used in engineering applications to simulate contaminant transport in the subsurface. The diffusion-type governing equations are reformulated as a hyperbolic system, augmented by an equation that can be interpreted as a momentum balance. The hyperbolization enables an efficient unified computation of the primary variable and its gradients, for example piezometric head and unit discharge in the Boussinesq equation. An augmented Roe scheme is used to solve the hyperbolic system. The hyperbolized system of equations is studied in a set of steady state and transient test cases with idealized geometry. These test cases confirm the equivalence of the hyperbolic system to its original formulation. The larger time step size of the hyperbolic equation is verified theoretically by means of a stability analysis and numerically in the test cases. Finally, a reach-scale application of flow and transport across a river meander is considered. This application case shows that the performance of the hyperbolic relaxation approach holds for more realistic groundwater flow and transport problems, relevant to water resources management.

54 ENVIRONMENTAL SCIENCES↗

Enabling Low-Temperature (LTP) Ignition Technologies for Multi-Mode Engines through the Development of a Validated High-Fidelity LTP Model for Predicative Simulations Tools

The goal of multi-mode engine architectures is to extend current lean-burn dilution limits with renewable fuels, which requires spark plugs to deposit high energies (hundreds of mJ) in order to initiate ignition and complete combustion. At elevated energy deposition rates, spark plugs experience increased electrode erosion and thermal losses, which ultimately shortens the spark-plug lifetime and lowers ignition efficiency. As such, in order to safeguard the efficiency gains of multi-mode concepts, new and improved ignition technologies are required. Recently, non-equilibrium low-temperature plasmas (LTP) have been shown to promote energy-efficient ignition via quenching and transport of electronically excited atoms and molecules, selective radical production and fast heating of hydrocarbon/air mixtures [1-2]. Thus, LTP is seen as a technology that can potentially improve the energy extraction efficiency of fuels, while enabling kinetically controlled combustion modes towards fuel leaner conditions to realize current DOE VTO goals of improving the sustainability of future mobility [3]. Although many previous studies have demonstrated the efficacy of plasma-assisted ignition to enhance combustion, the detailed enhancement mechanisms remain largely unknown, especially for oxygenated fuels and at elevated pressures that are most relevant to practical engine conditions. These barriers hinder the development of accurate and comprehensive numerical models that seek to describe LTP-based ignition in existing engine design software tools and methods. Current state-of-the-art simulation capabilities for LTP ignition systems are in need of improvements since they deliver qualitative results only due to important limitations of existing approaches. Firstly, validated kinetic models with elementary steps for plasma discharges in oxygenated fuel/air mixtures of relevance to the transportation sector are required. Such kinetic models do not exist at present and will be developed and validated within this project. Secondly, plasma discharges and reactive mixture ignition are multi-scale, unsteady processes requiring high-performance numerical methods and software that execute efficiently on DOE supercomputers. Such software does not exist at present and will be developed and applied to practical LTP ignition scenarios as part of this project. Thirdly, experimental databases that are tailored to serve as benchmark in support of the development of predictive computational models of LTP ignition do not exist and will be part of this project.

33 ADVANCED PROPULSION SYSTEMS↗

Large-eddy simulation of a multi-injection flame in a diesel engine environment using an unsteady flamelet/progress variable approach

In this work, large-eddy simulations (LESs) are conducted for a multiple-injection flame in a diesel engine environment using an unsteady flamelet/progress variable (UFPV) approach in which differential diffusion is considered. The suitability of the UFPV tabulation approach is first evaluated through a priori analyses using the state-of-the-art direct numerical simulation (DNS) dataset. Both the instantaneous data and the conditional values for the major and minor species' mass fractions are compared between the UFPV and the DNS. The comparisons show that the proposed UFPV tabulation approach can give good predictions for the multiple-injection flame at different injection phases. While the gas temperature and major species mass fractions can be accurately predicted with or without differential diffusion being considered in the UFPV flamelet library, the prediction accuracy for the highly diffusive species (e.g., hydrogen) in the main injection phase can be noticeably improved when differential diffusion is taken into account. The fully coupled LES/UFPV simulations show that the overall structure of the multiple-injection flame can be predicted, and the conditional thermo-chemical values are close to the filtered DNS dataset. The reasons for the remaining discrepancies found in the a priori analyses and the a posteriori simulations using the UFPV approach are analyzed.

Wen, Xu (ORCID:0000000331334292)↗

Convergence Criteria for Multiphysics Simulations

The behavior of engineered systems is often influenced by multiple physical phenomena, such as mechanical deformation, heat transfer, and chemical species transport and reactions. There are often strong interactions between these phenomena, and there is increasing interest in applying coupled-physics models to improve understanding of physical behavior under complex environmental conditions. Multiple simulation frameworks that facilitate coupled-physics simulations are in widespread use, and these employ a variety of techniques to account for interactions between those physics. Many frameworks solve the physics models independently and transfer results between them. Alternatively, a single monolithic system of equations for every physics model can be formed and solved. Each of these approaches has its benefits and drawbacks, and the optimal approach varies depending on the nature of the problem. The open-source MOOSE framework was developed targeting solution of large-scale multiphysics problems. Although it provides options for all these coupling approaches, its standard approach for multiphysics solutions is to form and solve a single monolithic system of equations containing the unknowns for all physics models. MOOSE provides a streamlined approach for users to define the solution variables, the terms in the partial differential equations pertaining to each variable, and interactions between solution variables. One aspect of the monolithic solution approach that can be problematic, however, is defining appropriate convergence criteria for the nonlinear system. A standard approach is to determine convergence is to simply take a norm of the residual vector corresponding to the full vector of unknowns. However, if the residual vector contains variables for multiple physics models, the magnitudes of those variables can differ significantly, and the variables can converge at significantly different rates from each other. It is important to ensure that the variables for each of the physics are converged, and also ensure that the convergence criteria are not excessively stringent in cases when there is little change in the solution. This talk presents representative multiphysics problems to highlight these issues, and shows strategies for convergence criteria in MOOSE that are robust for multiphysics models under a variety of conditions.

97 - MATHEMATICS AND COMPUTING↗

Determination of Ice Characteristics for Marine Hydroengineering Structures

Problems and potential approaches to determining ice characteristics for sea hydroengineering structures design are considered. A system for numerical modeling of ice dynamics is presented. The system may be used to define ice characteristics on approaches to structures with due regard for local hydrometeorological conditions and ice loads on structures. System application examples are presented for determining computational scenarios for ice loads at structures of the Pevek floating nuclear power plant (FNPP), as well as for the breakwater pier under reconstruction in Vanino. A scenario approach is used to determined ice loads.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Analysis of Space-Conditioning Loads in Commercial Buildings

Space conditioning end-uses, which include heating, cooling, and ventilation, represent a significant fraction of commercial building energy use, with a wide variety of heating and cooling technology options available in the market. In the interest of improving the overall efficiency of heating, ventilation and air-conditioning (HVAC) technologies, governments, utilities and private sector entities have implemented a variety of market transformation policies that aim to influence consumer purchase decisions. To evaluate the costs and benefits of such programs, analysts typically postulate a hypothetical default equipment choice, and compare it to one that provides comparable service with lower energy and/or power use. The corresponding reduced operating cost provides a benefit that offsets the potential higher cost of improved efficiency. Typically, life-cycle cost or cash-flow analyses are used to quantify the net economic benefit. These analyses require the capability to assess how a given equipment design would perform across a broad range of characteristics, both of the building and of the local weather. While these assessments can be performed using customized building simulations, it is generally not practical to develop and validate detailed building simulation code to cover all the potential variations of equipment design and installation. An alternative, and somewhat simpler, approach is to solely use detailed building simulations to generate time series of heating and cooling loads in commercial buildings. These loads can then be used as input to more detailed, stand-alone engineering models that simulate HVAC system performance under different equipment designs. This approach was used to evaluate a range of high-efficiency commercial packaged air conditioner design options for the Department of Energy’s Appliance and Equipment Standards Program (DOE-EERE 2015). While there may be some loss of precision relative to full simulation, the accuracy of this approach is sufficient for practical applications of cost-benefit analysis. This report describes the development of a database of commercial building heating and cooling loads, generated using the EnergyPlus software package, a whole building energy use model supported by the Department of Energy (DOE-EERE 2020a). EnergyPlus takes as input a set of configuration files that describe the building itself (size, zoning, envelope characteristics, etc.) and the various systems within it (HVAC, lighting, water heating, etc.). This analysis uses a publicly available collection of commercial reference buildings (CRB), comprised of sixteen building types and three vintages (DOE-EERE 2020b; Deru et al 2011). Each building is simulated in eighteen different locations, covering a wide range of climatic conditions. The prototype building description files assign the type of HVAC equipment used, and capacities across climate zones.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

Predicting Failure Using Deep Learning SAND Report

Accurate prediction of ductile failure is critical to Sandia’s NW mission, but the models are computationally heavy. The costs of including high-fidelity physics and mechanics that are germane to the failure mechanisms are often too burdensome for analysts either because of the person-hours it requires to input them or because of the additional computational time, or both. In an effort to deliver analysts a tool for representing these phenomena with minimal impact to their existing workflow, our project sought to develop modern data-driven methods that would add microstructural information to business-as-usual calculations and expedite failure predictions. The goal is a tool that receives as input a structural model with stress and strain fields, as well as a machine-learned model, and output predictions of structural response in time, including failure. As such, our project spent substantial time performing high-fidelity, three-dimensional experiments to elucidate materials mechanisms of void nucleation and evolution. We developed crystal-plasticity finite-element models from the experimental observations to enrich the findings with fields not readily measured. We developed engineering length-scale simulations of replicated test specimens to understand how the engineering fields evolve in the presence of fine-scale defects. Finally, we developed deep learning convolutional neural networks, and graph-based neural networks to encode the findings of the experiments and simulations and make forward predictions in time for structural performance. This project demonstrated the power of data-driven methods for model development, which have the potential to vastly increase both the accuracy and speed of failure predictions. These benefits and the methods necessary to develop them are highlighted in this report. However, many challenges remain to implementing these in real applications, and these are discussed along with potential methods for overcoming them.

97 MATHEMATICS AND COMPUTING↗