Search results for: Numerical model

Authors: Somayeh Komeylian

Abstract:

The direction of arrival (DoA) estimation is the crucial aspect of the radar technologies for detecting and dividing several signal sources. In this scenario, the antenna array output modeling involves numerous parameters including noise samples, signal waveform, signal directions, signal number, and signal to noise ratio (SNR), and thereby the methods of the DoA estimation rely heavily on the generalization characteristic for establishing a large number of the training data sets. Hence, we have analogously represented the two different optimization models of the DoA estimation; (1) the implementation of the decision directed acyclic graph (DDAG) for the multiclass least-squares support vector machine (LS-SVM), and (2) the optimization method of the deep neural network (DNN) radial basis function (RBF). We have rigorously verified that the LS-SVM DDAG algorithm is capable of accurately classifying DoAs for the three classes. However, the accuracy and robustness of the DoA estimation are still highly sensitive to technological imperfections of the antenna arrays such as non-ideal array design and manufacture, array implementation, mutual coupling effect, and background radiation and thereby the method may fail in representing high precision for the DoA estimation. Therefore, this work has a further contribution on developing the DNN-RBF model for the DoA estimation for overcoming the limitations of the non-parametric and data-driven methods in terms of array imperfection and generalization. The numerical results of implementing the DNN-RBF model have confirmed the better performance of the DoA estimation compared with the LS-SVM algorithm. Consequently, we have analogously evaluated the performance of utilizing the two aforementioned optimization methods for the DoA estimation using the concept of the mean squared error (MSE).

Keywords: DoA estimation, adaptive antenna array, Deep Neural Network, LS-SVM optimization model, radial basis function, MSE.

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Authors: Benjamin Gorry

Abstract:

Safety Critical hard Real-Time Systems are ever present in the avionics industry. The Model Driven Architecture (MDA) offers different levels of model abstraction and generation. This paper discusses our concerns relating to model development and generation when using the MDA approach in the avionics industry. These concerns are based on our experience when looking into adopting the MDA as part of avionics systems development. We place emphasis on transformations between model types and discuss possible benefits of adopting an MDA approach as part of the software development life cycle.

Keywords: Model Driven Architecture, Real-Time AvionicsApplications.

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Authors: Héctor Rivera-Gómez, Eva Selene Hernández-Gress, Oscar Montaño-Arango, Jose Ramon Corona-Armenta

Abstract:

This research studies the joint production, maintenance and subcontracting control policy for an unreliable deteriorating manufacturing system. Production activities are controlled by a derivation of the Hedging Point Policy, and given that the system is subject to deterioration, it reduces progressively its capacity to satisfy product demand. Multiple deterioration effects are considered, reflected mainly in the quality of the parts produced and the reliability of the machine. Subcontracting is available as support to satisfy product demand; also, overhaul maintenance can be conducted to reduce the effects of deterioration. The main objective of the research is to determine simultaneously the production, maintenance and subcontracting rate, which minimize the total, incurred cost. A stochastic dynamic programming model is developed and solved through a simulation-based approach composed of statistical analysis and optimization with the response surface methodology. The obtained results highlight the strong interactions between production, deterioration and quality, which justify the development of an integrated model. A numerical example and a sensitivity analysis are presented to validate our results.

Keywords: Deterioration, simulation, subcontracting, production planning.

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Authors: B. T. Chew, S. N. Kazi, A. Amiri

Abstract:

This paper presents an adaptive thermal comfort model study in the tropical country of Malaysia. A number of researchers have been interested in applying the adaptive thermal comfort model to different climates throughout the world, but so far no study has been performed in Malaysia. For the use as a thermal comfort model, which better applies to hot and humid climates, the adaptive thermal comfort model was developed as part of this research by using the collected results from a large field study in six lecture halls with 178 students. The relationship between the operative temperature and behavioral adaptations was determined. In the developed adaptive model, the acceptable indoor neutral temperatures lay within the range of 23.9-26.0C, with outdoor temperatures ranging between 27.0-34.6C. The most comfortable temperature for students in lecture hall was 25.7C.

Keywords: Hot and humid, Lecture halls, Neutral temperature, Adaptive thermal comfort model.

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Authors: Hafizullah A. Hashim, Aniza. A

Abstract:

The Spiral development model has been used successfully in many commercial systems and in a good number of defense systems. This is due to the fact that cost-effective incremental commitment of funds, via an analogy of the spiral model to stud poker and also can be used to develop hardware or integrate software, hardware, and systems. To support adaptive, semantic collaboration between domain experts and knowledge engineers, a new knowledge engineering process, called Spiral_OWL is proposed. This model is based on the idea of iterative refinement, annotation and structuring of knowledge base. The Spiral_OWL model is generated base on spiral model and knowledge engineering methodology. A central paradigm for Spiral_OWL model is the concentration on risk-driven determination of knowledge engineering process. The collaboration aspect comes into play during knowledge acquisition and knowledge validation phase. Design rationales for the Spiral_OWL model are to be easy-to-implement, well-organized, and iterative development cycle as an expanding spiral.

Keywords: Domain Expert, Knowledge Base, Ontology, Software Process.

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Authors: R. Abdulrahman, A. Eardley, A. Soliman

Abstract:

The proliferation in the development of mobile learning (m-learning) has played a vital role in the rapidly growing electronic learning market. This relatively new technology can help to encourage the development of in learning and to aid knowledge transfer a number of areas, by familiarizing students with innovative information and communications technologies (ICT). M-learning plays a substantial role in the deployment of learning methods for nursing students by using the Internet and portable devices to access learning resources ‘anytime and anywhere’. However, acceptance of m-learning by students is critical to the successful use of m-learning systems. Thus, there is a need to study the factors that influence student’s intention to use m-learning. This paper addresses this issue. It outlines the outcomes of a study that evaluates the unified theory of acceptance and use of technology (UTAUT) model as applied to the subject of user acceptance in relation to m-learning activity in nurse education. The model integrates the significant components across eight prominent user acceptance models. Therefore, a standard measure is introduced with core determinants of user behavioural intention. The research model extends the UTAUT in the context of m-learning acceptance by modifying and adding individual innovativeness (II) and quality of service (QoS) to the original structure of UTAUT. The paper goes on to add the factors of previous experience (of using mobile devices in similar applications) and the nursing students’ readiness (to use the technology) to influence their behavioural intentions to use m-learning. This study uses a technique called ‘convenience sampling’ which involves student volunteers as participants in order to collect numerical data. A quantitative method of data collection was selected and involves an online survey using a questionnaire form. This form contains 33 questions to measure the six constructs, using a 5-point Likert scale. A total of 42 respondents participated, all from the Nursing Institute at the Armed Forces Hospital in Saudi Arabia. The gathered data were then tested using a research model that employs the structural equation modelling (SEM), including confirmatory factor analysis (CFA). The results of the CFA show that the UTAUT model has the ability to predict student behavioural intention and to adapt m-learning activity to the specific learning activities. It also demonstrates satisfactory, dependable and valid scales of the model constructs. This suggests further analysis to confirm the model as a valuable instrument in order to evaluate the user acceptance of m-learning activity.

Keywords: Mobile learning, nursing institute, unified theory of acceptance and use of technology model.

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Authors: Irena Markovska, Dimitar Rusev, Nikolai Zaicev, Bogdan Bogdanov, Dimitar Georgiev, Yancho Hristov

Abstract:

A prototype model of an emulsion separator was designed and manufactured. Generally, it is a cylinder filled with different fractal modules. The emulsion was fed into the reactor by a peristaltic pump through an inlet placed at the boundary between the two phases. For hydrodynamic design and sizing of the reactor the assumptions of the theory of filtration were used and methods to describe the separation process were developed. Based on this methodology and using numerical methods and software of Autodesk the process is simulated in different operating modes. The basic hydrodynamic characteristics - speed and performance for different types of fractal systems and decisions to optimize the design of the reactor were also defined.

Keywords: fractal systems, reactor, separation, emulsions

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Authors: E. Asadollahi-Yazdi, J. Gardan, P. Lafon

Abstract:

Nowadays, manufactures are encountered with production of different version of products due to quality, cost and time constraints. On the other hand, Additive Manufacturing (AM) as a production method based on CAD model disrupts the design and manufacturing cycle with new parameters. To consider these issues, the researchers utilized Design For Manufacturing (DFM) approach for AM but until now there is no integrated approach for design and manufacturing of product through the AM. So, this paper aims to provide a general methodology for managing the different production issues, as well as, support the interoperability with AM process and different Product Life Cycle Management tools. The problem is that the models of System Engineering which is used for managing complex systems cannot support the product evolution and its impact on the product life cycle. Therefore, it seems necessary to provide a general methodology for managing the product’s diversities which is created by using AM. This methodology must consider manufacture and assembly during product design as early as possible in the design stage. The latest approach of DFM, as a methodology to analyze the system comprehensively, integrates manufacturing constraints in the numerical model in upstream. So, DFM for AM is used to import the characteristics of AM into the design and manufacturing process of a hybrid product to manage the criteria coming from AM. Also, the research presents an integrated design method in order to take into account the knowledge of layers manufacturing technologies. For this purpose, the interface model based on the skin and skeleton concepts is provided, the usage and manufacturing skins are used to show the functional surface of the product. Also, the material flow and link between the skins are demonstrated by usage and manufacturing skeletons. Therefore, this integrated approach is a helpful methodology for designer and manufacturer in different decisions like material and process selection as well as, evaluation of product manufacturability.

Keywords: Additive manufacturing, 3D printing, design for manufacturing, integrated design, interoperability.

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Authors: Juraj Camaj, Martin Kendra, Jaroslav Masek

Abstract:

The paper is focused to the evaluation railway tracks in the Slovakia by using Multi-Criteria method. Evaluation of railway tracks has important impacts for the assessment of investment in technical equipment. Evaluation of railway tracks also has an important impact for the allocation of marshalling yards. Marshalling yards are in transport model as centers for the operation assigned catchment area. This model is one of the effective ways to meet the development strategy of the European Community's railways. By applying this model in practice, a transport company can guarantee a higher quality of service and then expect an increase in performance. The model is also applicable to other rail networks. This model supplements a theoretical problem of train formation problem of new ways of looking at evaluation of factors affecting the organization of wagon flows.

Keywords: Railway track, multi-criteria methods, evaluation, transportation model.

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Authors: Man Young Kim, Gyo Woo Lee

Abstract:

In this numerical study, we want to present the design of highly efficient extruded-type heat sink. The symmetrically arranged extruded-type heat sinks are used instead of a single extruded or swaged-type heat sink. In this parametric study, the maximum temperatures, the base temperatures between heaters, and the heat release rates were investigated with respect to the arrangements of heat sources, air flow rates, and amounts of heat input. Based on the results we believe that the use of both side of heat sink is to be much better for release the heat than the use of single side. Also from the results, it is believed that the symmetric arrangement of heat sources is recommended to achieve a higher heat transfer from the heat sink.

Keywords: Heat Sink, Forced Convection, Heat Transfer, Performance Evaluation, Symmetrically Arranged.

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Authors: Y. Lasbet, A. L. Boukhalkhal, K. Loubar

Abstract:

The study of mixed convection is, usually, focused on the straight channels in which the onset of the mixed convection is well defined as function of the ratio between Grashof number and Reynolds number, Gr/Re. This is not the case for a complex channel wherein the mixed convection is not sufficiently examined in the literature. Our paper focuses on the study of the mixed convection in a complex geometry in which our main contribution reveals that the critical value of the ratio Gr/Re for the onset of the mixed convection increases highly in the type of geometry contrary to the straight channel. Furthermore, the accentuated secondary flow in this geometry prevents the thermal stratification in the flow and consequently the buoyancy driven becomes negligible. To perform these objectives, a numerical study in complex geometry for several values of the ratio Gr/Re with prescribed wall heat flux (H2), was realized by using the CFD code.

Keywords: Complex geometry, heat transfer, laminar flow, mixed convection, Nusselt number.

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Authors: Mahmoud Zarrini, R. N. Pralhad

Abstract:

Avalanche velocity (from start to track zone) has been estimated in the present model for an avalanche which is triggered artificially by an explosive devise. The initial development of the model has been from the concept of micro-continuum theories [1], underwater explosions [2] and from fracture mechanics [3] with appropriate changes to the present model. The model has been computed for different slab depth R, slope angle θ, snow density ¤ü, viscosity μ, eddy viscosity η*and couple stress parameter η. The applicability of the present model in the avalanche forecasting has been highlighted.

Keywords: Snow avalanche velocity, avalanche zones, shockwave, couple stress fluids.

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Authors: Susovan Chakrabortty, Madhumangal Pal, Prasun Kumar Nayak

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A manufacturing inventory model with shortages with carrying cost, shortage cost, setup cost and demand quantity as imprecise numbers, instead of real numbers, namely interval number is considered here. First, a brief survey of the existing works on comparing and ranking any two interval numbers on the real line is presented. A common algorithm for the optimum production quantity (Economic lot-size) per cycle of a single product (so as to minimize the total average cost) is developed which works well on interval number optimization under consideration. Finally, the designed algorithm is illustrated with numerical example.

Keywords: EOQ, Inventory, Interval Number, Demand, Production, Simulation

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Authors: Aymen Laadhari, Gábor Székely

Abstract:

In this work, the hemodynamics in the sinuses of Valsalva after Transcatheter Aortic Valve Implantation is numerically examined. We focus on the physical results in the two-dimensional case. We use a finite element methodology based on a Lagrange multiplier technique that enables to couple the dynamics of blood flow and the leaflets’ movement. A massively parallel implementation of a monolithic and fully implicit solver allows more accuracy and significant computational savings. The elastic properties of the aortic valve are disregarded, and the numerical computations are performed under physiologically correct pressure loads. Computational results depict that blood flow may be subject to stagnation in the lower domain of the sinuses of Valsalva after Transcatheter Aortic Valve Implantation.

Keywords: Hemodynamics, Transcatheter Aortic Valve Implantation, blood flow stagnation, numerical simulations.

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Authors: M. Yousaf, S. Usman

Abstract:

The present study focused on the investigation of the effects of roughness elements on heat transfer during natural convection in a rectangular cavity using numerical technique. Roughness elements were introduced on the bottom hot wall with a normalized amplitude (A*/H) of 0.1. Thermal and hydrodynamic behaviors were studied using computational method based on Lattice Boltzmann method (LBM). Numerical studies were performed for a laminar flow in the range of Rayleigh number (Ra) from 103 to 106 for a rectangular cavity of aspect ratio (L/H) 2.0 with a fluid of Prandtl number (Pr) 1.0. The presence of the sinusoidal roughness elements caused a minimum to maximum decrease in the heat transfer as 7% to 17% respectively compared to smooth enclosure. The results are presented for mean Nusselt number (Nu), isotherms and streamlines.

Keywords: Natural convection, Rayleigh number, surface roughness, Nusselt number, Lattice Boltzmann Method.

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Authors: Tapas Kumar Sinha, Joseph Mathew

Abstract:

Based on the Lagrangian for the Gross –Pitaevskii equation as derived by H. Sakaguchi and B.A Malomed [5] we have derived a double well model for the nonlinear optical lattice. This model explains the various features of nonlinear optical lattices. Further, from this model we obtain and simulate the probability for tunneling from one well to another which agrees with experimental results [4].

Keywords: Double well model, nonlinear optical lattice, Solitons, tunneling.

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Authors: Mahvash Zandy Navgaran, Maryam Momeni Feili

Abstract:

The bag radius of the nucleon can be determined by MIT bag model based on electric and magnetic form factors of the nucleon. Also we determined the masses and magnetic moment of the nucleon with MIT bag model, using bag radius and compared with other results, suggests a suitable compatibility.

Keywords: MIT bag model, masses and magnetic moment of thenucleon, bag radius of the nucleon.

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Authors: Abdur Rosyid, Mohamed El-Madany, Mohanad Alata

Abstract:

Due to simplicity and low cost, rotordynamic system is often modeled by using lumped parameters. Recently, finite elements have been used to model rotordynamic system as it offers higher accuracy. However, it involves high degrees of freedom. In some applications such as control design, this requires higher cost. For this reason, various model reduction methods have been proposed. This work demonstrates the quality of model reduction of rotor-bearing-support system through substructuring. The quality of the model reduction is evaluated by comparing some first natural frequencies, modal damping ratio, critical speeds, and response of both the full system and the reduced system. The simulation shows that the substructuring is proven adequate to reduce finite element rotor model in the frequency range of interest as long as the number and the location of master nodes are determined appropriately. However, the reduction is less accurate in an unstable or nearly-unstable system.

Keywords: Finite element model, rotordynamic system, model reduction, substructuring.

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Authors: Dimitra Alexiou

Abstract:

In an urban area the location allocation of emergency services mobile units, such as ambulances, police patrol cars must be designed so as to achieve a prompt response to demand locations. In this paper the partition of a given urban network into distinct sub-networks is performed such that the vertices in each component are close and simultaneously the sums of the corresponding population in the sub-networks are almost uniform. The objective here is to position appropriately in each sub-network a mobile emergency unit in order to reduce the response time to the demands. A mathematical model in framework of graph theory is developed. In order to clarify the corresponding method a relevant numerical example is presented on a small network.

Keywords: Distances, Emergency Service, Graph Partition, location.

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Authors: A. K. Sharma, N. P. Patidar, G. Agnihotri, D. K. Palwalia

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This paper presents the performance state analysis of Self-Excited Induction Generator (SEIG) using Artificial Bee Colony (ABC) optimization technique. The total admittance of the induction machine is minimized to calculate the frequency and magnetizing reactance corresponding to any rotor speed, load impedance and excitation capacitance. The performance of SEIG is calculated using the optimized parameter found. The results obtained by ABC algorithm are compared with results from numerical method. The results obtained coincide with the numerical method results. This technique proves to be efficient in solving nonlinear constrained optimization problems and analyzing the performance of SEIG.

Keywords: Artificial bee colony, Steady state analysis, Selfexcited induction generator, Nonlinear constrained optimization.

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Authors: M. K. Pradhan, C. K. Biswas,

Abstract:

In the current research, neuro-fuzzy model and regression model was developed to predict Material Removal Rate in Electrical Discharge Machining process for AISI D2 tool steel with copper electrode. Extensive experiments were conducted with various levels of discharge current, pulse duration and duty cycle. The experimental data are split into two sets, one for training and the other for validation of the model. The training data were used to develop the above models and the test data, which was not used earlier to develop these models were used for validation the models. Subsequently, the models are compared. It was found that the predicted and experimental results were in good agreement and the coefficients of correlation were found to be 0.999 and 0.974 for neuro fuzzy and regression model respectively

Keywords: Electrical discharge machining, material removal rate, neuro-fuzzy model, regression model, mountain clustering.

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Authors: T. H. Lee, J. H. Park, S. M. Lee, S. C. Lee

Abstract:

In this paper, we consider Wiener nonlinear model for solid oxide fuel cell (SOFC). The Wiener model of the SOFC consists of a linear dynamic block and a static output non-linearity followed by the block, in which linear part is approximated by state-space model and the nonlinear part is identified by a polynomial form. To control the SOFC system, we have to consider various view points such as operating conditions, another constraint conditions, change of load current and so on. A change of load current is the significant one of these for good performance of the SOFC system. In order to keep the constant stack terminal voltage by changing load current, the nonlinear model predictive control (MPC) is proposed in this paper. After primary control method is designed to guarantee the fuel utilization as a proper constant, a nonlinear model predictive control based on the Wiener model is developed to control the stack terminal voltage of the SOFC system. Simulation results verify the possibility of the proposed Wiener model and MPC method to control of SOFC system.

Keywords: SOFC, model predictive control, Wiener model.

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Authors: Balgaisha Mukanova, Tolkyn Mirgalikyzy

Abstract:

Theory of interpretation of electromagnetic fields studied in the electrical prospecting with direct current is mainly developed for the case of a horizontal surface observation. However in practice we often have to work in difficult terrain surface. Conducting interpretation without the influence of topography can cause non-existent anomalies on sections. This raises the problem of studying the impact of different shapes of ground surface relief on the results of electrical prospecting's research. This research examines the numerical solutions of the direct problem of electrical prospecting for two-dimensional and three-dimensional media, taking into account the terrain. The problem is solved using the method of integral equations. The density of secondary currents on the relief surface is obtained.

Keywords: Ground surface relief, method of integral equations, numerical method.

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Authors: Susan J. Simmons, Fang Fang, Qijun Fang, Karl Ricanek

Abstract:

Quantitative trait loci (QTL) experiments have yielded important biological and biochemical information necessary for understanding the relationship between genetic markers and quantitative traits. For many years, most QTL algorithms only allowed one observation per genotype. Recently, there has been an increasing demand for QTL algorithms that can accommodate more than one observation per genotypic distribution. The Bayesian hierarchical model is very flexible and can easily incorporate this information into the model. Herein a methodology is presented that uses a Bayesian hierarchical model to capture the complexity of the data. Furthermore, the Markov chain Monte Carlo model composition (MC3) algorithm is used to search and identify important markers. An extensive simulation study illustrates that the method captures the true QTL, even under nonnormal noise and up to 6 QTL.

Keywords: Bayesian hierarchical model, Markov chain MonteCarlo model composition, quantitative trait loci.

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Authors: Aichun Feng, Zhi-Min Chen, Jing Tang Xing

Abstract:

A two-dimensional linear wave-body interaction problem can be solved using a desingularized integral method by placing free surface Rankine sources over calm water surface and satisfying boundary conditions at prescribed collocation points on the calm water surface. A new free-surface Rankine source distribution scheme, determined by the intersection points of free surface and body surface, is developed to reduce numerical computation cost. Associated with this, a new treatment is given to the intersection point. The present scheme results are in good agreement with traditional numerical results and measurements.

Keywords: Source point distribution, panel method, Rankine source, desingularized algorithm.

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Authors: Dmitry V. Fomichev, Vladimir I. Solonin

Abstract:

This paper presents the findings from a numerical simulation of the flow in 37-rod fuel assembly models spaced by a double-wire trapezoidal wrapping as applied to the BREST-OD-300 experimental nuclear reactor. Data on a high static pressure distribution within the models, and equations for determining the fuel bundle flow friction factors have been obtained. Recommendations are provided on using the closing turbulence models available in the ANSYS Fluent. A comparative analysis has been performed against the existing empirical equations for determining the flow friction factors. The calculated and experimental data fit has been shown.

An analysis into the experimental data and results of the numerical simulation of the BREST-OD-300 fuel rod assembly hydrodynamic performance are presented.

Keywords: BREST-OD-300, ware-spaces, fuel assembly, computation fluid dynamics.

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Authors: Alireza Bozorgian, Navid Majdi Nasab, Abdolreza Memari

Abstract:

A continuum model is presented to study vdW interaction on buckling analysis of multi-walled walled carbon nanotube. In previous studies, only the vdW interaction between adjacent two layers was considered and the vdW interaction between the other two layers was neglected. The results show that the vdW interaction cofficients are dependent on the change of interlayer spacing and the radii of tubes. With increase of radii the vdW coefficients approach a constant value. The numerical results show that the effect of vdW interaction on the critical strain for a doublewalled CNT is negligible when the radius is large enough for the both the cases of before and after buckling.

Keywords: Buckling, Carbon nanotube, van der Waals interaction, Multi-walled Carbon nanotube, Critical Strain, Prebuckling Pressure

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Authors: K. M. Alsager, N. O. Alshehri

Abstract:

In this paper, we proposed the notion of single valued neutrosophic hesitant fuzzy rough set, by combining single valued neutrosophic hesitant fuzzy set and rough set. The combination of single valued neutrosophic hesitant fuzzy set and rough set is a powerful tool for dealing with uncertainty, granularity and incompleteness of knowledge in information systems. We presented both definition and some basic properties of the proposed model. Finally, we gave a general approach which is applied to a decision making problem in disease diagnoses, and demonstrated the effectiveness of the approach by a numerical example.

Keywords: Single valued neutrosophic hesitant set, single valued neutrosophic hesitant relation, single valued neutrosophic hesitant fuzzy rough set, decision making method.

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Authors: Hui Hsin Huang

Abstract:

The downstream manufactures will order their materials from different upstream suppliers to maintain a certain level of the demand. This paper proposes a bivariate model to portray this phenomenon of material demand. We use empirical data to estimate the parameters of model and evaluate the RMSD of model calibration. The results show that the model has better fitness.

Keywords: Farlie-Gumbel-Morgenstern family of bivariate distributions, multi-source ordering, materials demand quantity, recency, ordering time.

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Authors: Kholod M. Abu-Alnaja

Abstract:

In this work, we derive two numerical schemes for solving a class of nonlinear partial differential equations. The first method is of second order accuracy in space and time directions, the scheme is unconditionally stable using Von Neumann stability analysis, the scheme produced a nonlinear block system where Newton-s method is used to solve it. The second method is of fourth order accuracy in space and second order in time. The method is unconditionally stable and Newton's method is used to solve the nonlinear block system obtained. The exact single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitary waves for different parameters are also discussed.

Keywords: Crank-Nicolson Scheme, Douglas Scheme, Partial Differential Equations

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