Search results for: Stokes equation

Authors: A. Guezane-Lakoud, S. Bensebaa

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In this paper, we study a boundary value problem of nonlinear fractional differential equation. Existence and positivity results of solutions are obtained.

Keywords: positive solution, fractional caputo derivative, Banach contraction principle, Avery and Peterson fixed point theorem

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Authors: Farid Nouioua, Abdelouaheb Ardjouni

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In this article, we study the existence of positive periodic solutions of certain delay differential equations. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ Krasnoselskii's fixed point theorem to obtain sufficient conditions for the existence of a positive periodic solution of the differential equation. The obtained results improve and extend the results in the literature. Finally, an example is given to illustrate our results.

Keywords: delay differential equations, positive periodic solutions, integral equations, Krasnoselskii fixed point theorem

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Authors: Shubham Jaiswal

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In the present article, a drive is taken to compute the solution of spatial fractional order advection-dispersion equation having source/sink term with given initial and boundary conditions. The equation is converted to a system of ordinary differential equations using second-kind shifted Chebyshev polynomials, which have finally been solved using finite difference method. The striking feature of the article is the fast transportation of solute concentration as and when the system approaches fractional order from standard order for specified values of the parameters of the system.

Keywords: spatial fractional order advection-dispersion equation, second-kind shifted Chebyshev polynomial, collocation method, conservative system, non-conservative system

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Authors: W. Meron Mebrahtu, R. Absi

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Velocity distribution in turbulent open-channel flows is organized in a complex manner. This is due to the large spatial and temporal variability of fluid motion resulting from the free-surface turbulent flow condition. This phenomenon is complicated further due to the complex geometry of channels and the presence of solids transported. Thus, several efforts were made to understand the phenomenon and obtain accurate mathematical models that are suitable for engineering applications. However, predictions are inaccurate because oversimplified assumptions are involved in modeling this complex phenomenon. Therefore, the aim of this work is to study velocity distribution profiles and obtain simple, more accurate, and predictive mathematical models. Particular focus will be made on the acceptable simplification of the general transport equations and an accurate representation of eddy viscosity. Wide rectangular open-channel seems suitable to begin the study; other assumptions are smooth-wall, and sediment-free flow under steady and uniform flow conditions. These assumptions will allow examining the effect of the bottom wall and the free surface only, which is a necessary step before dealing with more complex flow scenarios. For this flow condition, two ordinary differential equations are obtained for velocity profiles; from the Reynolds-averaged Navier-Stokes (RANS) equation and equilibrium consideration between turbulent kinetic energy (TKE) production and dissipation. Then different analytic models for eddy viscosity, TKE, and mixing length were assessed. Computation results for velocity profiles were compared to experimental data for different flow conditions and the well-known linear, log, and log-wake laws. Results show that the model based on the RANS equation provides more accurate velocity profiles. In the viscous sublayer and buffer layer, the method based on Prandtl’s eddy viscosity model and Van Driest mixing length give a more precise result. For the log layer and outer region, a mixing length equation derived from Von Karman’s similarity hypothesis provides the best agreement with measured data except near the free surface where an additional correction based on a damping function for eddy viscosity is used. This method allows more accurate velocity profiles with the same value of the damping coefficient that is valid under different flow conditions. This work continues with investigating narrow channels, complex geometries, and the effect of solids transported in sewers.

Keywords: accuracy, eddy viscosity, sewers, velocity profile

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Authors: Kajal K. Patel, M. N. Mehta, T. R. Singh

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When water is injected in oil formatted area in secondary oil recovery process the instability occurs near common interface due to viscosity difference of injected water and native oil. The governing equation gives rise to the non-linear partial differential equation and its solution has been obtained by Homotopy analysis method with appropriate guess value of the solution together with some conditions and standard relations. The solution gives the average cross-sectional area occupied by the schematic fingers during the occurs of instability phenomenon. The numerical and graphical presentation has developed by using Maple software.

Keywords: capillary pressure, homotopy analysis method, instability phenomenon, viscosity

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Authors: Clarinda Vitorino Nhangumbe, Ercília Sousa

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Weather derivatives are financial products used to cover non catastrophic weather events with a weather index as the underlying asset. The rainfall weather derivative pricing model is modeled based in the assumption that the rainfall dynamics follows Ornstein-Uhlenbeck process, and the partial differential equation approach is used to derive the convection-diffusion two dimensional time dependent partial differential equation, where the spatial variables are the rainfall index and rainfall depth. To compute the approximation solutions of the partial differential equation, the appropriate boundary conditions are suggested, and an explicit numerical method is proposed in order to deal efficiently with the different choices of the coefficients involved in the equation. Being an explicit numerical method, it will be conditionally stable, then the stability region of the numerical method and the order of convergence are discussed. The model is tested for real precipitation data.

Keywords: finite differences method, ornstein-uhlenbeck process, partial differential equations approach, rainfall derivatives

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Authors: S. A. M. Yatim, Z. B. Ibrahim, K. I. Othman, M. Suleiman

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The method of solving second order stiff ordinary differential equation (ODEs) that is based on backward differentiation formula (BDF) is considered in this paper. We derived the method by increasing the order of the existing method using an improved strategy in choosing the step size. Numerical results are presented to compare the efficiency of the proposed method to the MATLAB’s suite of ODEs solvers namely ode15s and ode23s. The method was found to be efficient to solve second order ordinary differential equation.

Keywords: backward differentiation formulae, block backward differentiation formulae, stiff ordinary differential equation, variable step size

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Authors: Raphael Zanella

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This works deals with the finite element approximation of axisymmetric problems. The weak formulation of the heat equation under the axisymmetry assumption is established for continuous finite elements. The weak formulation is implemented in a C++ solver with implicit march-in-time. The code is verified by space and time convergence tests using a manufactured solution. The solving of an example problem with an axisymmetric formulation is compared to that with a full-3D formulation. Both formulations lead to the same result, but the code based on the axisymmetric formulation is much faster due to the lower number of degrees of freedom. This confirms the correctness of our approach and the interest in using an axisymmetric formulation when it is possible.

Keywords: axisymmetric problem, continuous finite elements, heat equation, weak formulation

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Authors: Medine Demir, Volker John

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Fluid equations in a rotating frame of reference have a broad class of important applications in meteorology and oceanography, especially in the large-scale flows considered in ocean and atmosphere, as well as many physical and industrial applications. The Coriolis and the centripetal forces, resulting from the rotation of the earth, play a crucial role in such systems. For such applications it may be required to solve the system in complex three-dimensional geometries. In recent years, the Navier--Stokes equations in a rotating frame have been investigated in a number of papers using the classical inf-sup stable mixed methods, like Taylor-Hood pairs, to contribute to the analysis and the accurate and efficient numerical simulation. Numerical analysis reveals that these classical methods introduce a pressure-dependent contribution in the velocity error bounds that is proportional to some inverse power of the viscosity. Hence, these methods are optimally convergent but small velocity errors might not be achieved for complicated pressures and small viscosity coefficients. Several approaches have been proposed for improving the pressure-robustness of pairs of finite element spaces. In this contribution, a pressure-robust space discretization of the incompressible Navier--Stokes equations in a rotating frame of reference is considered. The discretization employs divergence-free, $H^1$-conforming mixed finite element methods like Scott--Vogelius pairs. However, this approach might come with a modification of the meshes, like the use of barycentric-refined grids in case of Scott--Vogelius pairs. However, this strategy requires the finite element code to have control on the mesh generator which is not realistic in many engineering applications and might also be in conflict with the solver for the linear system. An error estimate for the velocity is derived that tracks the dependency of the error bound on the coefficients of the problem, in particular on the angular velocity. Numerical examples illustrate the theoretical results. The idea of pressure-robust method could be cast on different types of flow problems which would be considered as future studies. As another future research direction, to avoid a modification of the mesh, one may use a very simple parameter-dependent modification of the Scott-Vogelius element, the pressure-wired Stokes element, such that the inf-sup constant is independent of nearly-singular vertices.

Keywords: navier-stokes equations in a rotating frame of refence, coriolis force, pressure-robust error estimate, scott-vogelius pairs of finite element spaces

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Authors: Razieh Khalafi

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The brain is the information processing centre of the human body. Stimuli in the form of information are transferred to the brain and then brain makes the decision on how to respond to them. In this research, we propose a new partial differential equation which analyses the EEG signals and make a relationship between the incoming stimuli and the brain response to them. In order to test the proposed model, a set of external stimuli applied to the model and the model’s outputs were checked versus the real EEG data. The results show that this model can model the EEG signal well. The proposed model is useful not only for modelling of EEG signal in case external stimuli but it can be used for modelling of brain response in case of internal stimuli.

Keywords: brain, stimuli, partial differential equation, response, EEG signal

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Authors: J. J. Peña, J. Morales, J. García-Ravelo, L. Arcos-Díaz

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The Point canonical transformation method is applied for solving the Schrödinger equation with position-dependent mass. This class of problem has been solved for continuous mass distributions. In this work, a staggered mass distribution for the case of a free particle in an infinite square well potential has been proposed. The continuity conditions as well as normalization for the wave function are also considered. The proposal can be used for dealing with other kind of staggered mass distributions in the Schrödinger equation with different quantum potentials.

Keywords: free particle, point canonical transformation method, position-dependent mass, staggered mass distribution

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Authors: Helnaz Soltani, J. N. Reddy

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Presence of fluid medium interacting with a structure can lead to failure of the structure. Since developing efficient computational model for fluid-structure interaction (FSI) problems has broader impact to realistic problems encountered in aerospace industry, ship industry, oil and gas industry, and so on, one can find an increasing need to find a method in order to investigate the effect of fluid domain on structural response. A coupled finite element formulation of problems involving FSI issue is an accurate method to predict the response of structures in contact with a fluid medium. This study proposes a finite element approach in order to study the transient response of the structures interacting with a fluid medium. Since beam and plate are considered to be the fundamental elements of almost any structure, the developed method is applied to beams and plates benchmark problems in order to demonstrate its efficiency. The formulation is a combination of the various structure theories and the solid-fluid interface boundary condition, which is used to represent the interaction between the solid and fluid regimes. Here, three different beam theories as well as three different plate theories are considered to model the solid medium, and the Navier-Stokes equation is used as the theoretical equation governed the fluid domain. For each theory, a coupled set of equations is derived where the element matrices of both regimes are calculated by Gaussian quadrature integration. The main feature of the proposed methodology is to model the fluid domain as an added mass; the external distributed force due to the presence of the fluid. We validate the accuracy of such formulation by means of some numerical examples. Since the formulation presented in this study covers several theories in literature, the applicability of our proposed approach is independent of any structure geometry. The effect of varying parameters such as structure thickness ratio, fluid density and immersion depth, are studied using numerical simulations. The results indicate that maximum vertical deflection of the structure is affected considerably in the presence of a fluid medium.

Keywords: beam and plate, finite element analysis, fluid-structure interaction, transient response

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Authors: Ali Raza, Rizwan Latif, Syed Adnan Qasim, Imran Shafi

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Diesel engines are most efficient and reliable in terms of efficiency, reliability, and adaptability. Most of the research and development up till now have been directed towards High Speed Diesel Engine, for Commercial use. In these engines, objective is to optimize maximum acceleration by reducing exhaust emission to meet international standards. In high torque low speed engines, the requirement is altogether different. These types of engines are mostly used in Maritime Industry, Agriculture Industry, Static Engines Compressors Engines, etc. On the contrary, high torque low speed engines are neglected quite often and are eminent for low efficiency and high soot emissions. One of the most effective ways to overcome these issues is by efficient combustion in an engine cylinder. Fuel spray dynamics play a vital role in defining mixture formation, fuel consumption, combustion efficiency and soot emissions. Therefore, a comprehensive understanding of the fuel spray characteristics and atomization process in high torque low speed diesel engine is of great importance. Evaporation in the combustion chamber has a rigorous effect on the efficiency of the engine. In this paper, multiphase evaporation of fuel is modeled for high torque low speed engine using the CFD (computational fluid dynamics) codes. Two distinct phases of evaporation are modeled using modeling soft wares. The basic model equations are derived from the energy conservation equation and Naiver-Stokes equation. O’Rourke model is used to model the evaporation phases. The results obtained showed a generous effect on the efficiency of the engine. Evaporation rate of fuel droplet is increased with the increase in vapor pressure. An appreciable reduction in size of droplet is achieved by adding the convective heat effects in the combustion chamber. By and large, an overall increase in efficiency is observed by modeling distinct evaporation phases. This increase in efficiency is due to the fact that droplet size is reduced and vapor pressure is increased in the engine cylinder.

Keywords: diesel fuel, CFD, evaporation, multiphase

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Authors: Ya-Fen Lee, Yun-Yao Chi

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The study aims to explore the relationship between risk perceptions of rockfall and revisit intention using a Structural Equation Modelling (SEM) analysis. A total of 573 valid questionnaires are collected from travelers to Taroko National Park, Taiwan. The findings show the majority of travellers have the medium perception of rockfall risk, and are willing to revisit the Taroko National Park. The revisit intention to Taroko National Park is influenced by hazardous preferences, willingness-to-pay, obstruction and attraction. The risk perception has an indirect effect on revisit intention through influencing willingness-to-pay. The study results can be a reference for mitigation the rockfall disaster.

Keywords: risk perception, rockfall, revisit intention, structural equation modelling

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Authors: Carla E. O. de Moraes, Gladson O. Antunes, Mauro A. Rincon

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The aim of this paper is to study the internal stabilization of the Bernoulli-Euler equation numerically. For this, we consider a square plate subjected to a feedback/damping force distributed only in a subdomain. An algorithm for obtaining an approximate solution to this problem was proposed and implemented. The numerical method used was the Finite Difference Method. Numerical simulations were performed and showed the behavior of the solution, confirming the theoretical results that have already been proved in the literature. In addition, we studied the validation of the numerical scheme proposed, followed by an analysis of the numerical error; and we conducted a study on the decay of the energy associated.

Keywords: Bernoulli-Euler plate equation, numerical simulations, stability, energy decay, finite difference method

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Authors: Shigeyuki Haruyama, Ken Kaminishi, Junji Kaneko, Tadayuki Kyoutani, Siti Ruhana Omar, Oke Oktavianty

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This study is used as a definition method to the value and function in manufacturing sector. In concurrence of discussion about present condition of modeling method, until now definition of 1D-CAE is ambiguity and not conceptual. Across all the physics fields, those methods are defined with the formulation of differential algebraic equation which only applied time derivation and simulation. At the same time, we propose semi-acausal modeling concept and differential algebraic equation method as a newly modeling method which the efficiency has been verified through the comparison of numerical analysis result between the semi-acausal modeling calculation and FEM theory calculation.

Keywords: system model, physical models, empirical models, conservation law, differential algebraic equation, object-oriented

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Authors: M. K. Yusuf, W. O. Hamman, U. E. Umana, S. B. Oladele

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Background: Dilatation of the inferior vena cava (IVC) is used as the ultrasonic diagnostic feature in patients suspected of congestive heart failure. The IVC diameter has been reported to vary among the various body mass indexes (BMI) and body shape indexes (ABSI). Knowledge of these variations is useful in precision diagnoses of CHF by imaging scientists. Aim: The study aimed to establish an equation for predicting the ultrasonic mean diameter of the IVC among the various BMI/ABSI of inhabitants of Azare, Bauchi State-Nigeria. Methodology: Two hundred physically healthy adult subjects of both sexes were classified into under, normal, over, and obese weights using their BMIs after selection using a structured questionnaire following their informed consent for an abdominal ultrasound scan. The probe was placed on the midline of the body, halfway between the xiphoid process and the umbilicus, with the marker on the probe directed towards the patient's head to obtain a longitudinal view of the IVC. The maximum IVC diameter was measured from the subcostal view using the electronic caliper of the scan machine. The mean value of each group was obtained, and the results were analysed. Results: A novel equation {(IVC Diameter = 1.04 +0.01(X) where X= BMI} has been generated for determining the IVC diameter among the populace. Conclusion: An equation for predicting the IVC diameter from individual BMI values in apparently healthy subjects has been established.

Keywords: equation, ultrasonic, IVC diameter, body adiposities

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Authors: Jonny, T. Yuri M. Zagloel

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This paper is made to present an Indonesian Healthcare model. Currently, there are nine TQM (Total Quality Management) practices in healthcare industry. However, these practices are not integrated yet. Therefore, this paper aims to integrate these practices as a model by using Structural Equation Modeling (SEM). After administering about 210 questionnaires to various stakeholders of this industry, a LISREL program was used to evaluate the model's fitness. The result confirmed that the model is fit because the p-value was about 0.45 or above required 0.05. This has signified that previously mentioned of nine TQM practices are able to be integrated as an Indonesian healthcare model.

Keywords: healthcare, total quality management (TQM), structural equation modeling (SEM), linear structural relations (LISREL)

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Authors: Lim Yi Wei

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Structural equation modelling (SEM) is well-known in statistics due to its flexibility and accessibility. It plays an increasingly important role in the development of the education field. The number of research publications using SEM in education has increased in recent decades. However, there is a lack of scientific review conducted on SEM in education. The purpose of this study is to investigate research trends related to SEM in education. The researcher will use Vosviewer, Datawrapper, and SciMAT to do bibliometric analysis on 5549 papers that have been published in the Scopus database in the last five years. The result will show the publication trends of the most cited documents, the top contributing authors, countries, institutions, and journals in the research field. It will also look at how they relate to each other in terms of co-citation, collaboration, and co-occurrence of keywords. This study will benefit researchers and practitioners by identifying research trends and the current state of SEM in education.

Keywords: structural equation modeling, education, bibliometric analysis, Vosviewer

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Authors: J. R. Bhalla, Gautam, Gurcharan Singh, Sanjeev Naval

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Mathematics is everywhere behind the every things on the earth as well as in the universe. Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. Now a day’s we can made and apply an equation on a complex geometry through applied mathematics. Here we work and focus on to create a formula which apply in the field of civil engineering in new concrete technology. In this paper our target is to make a formula which is applied on “BAUT” Metal Aggregate. In this paper our approach is to make formulation and equation on special “BAUT” Metal Aggregate by Applied Mathematical Study Case 1. BASIC PHYSICAL FORMULATION 2. ADVANCE EQUATION which shows the mechanical performance of special metal aggregates for concrete technology. In case 1. Basic physical formulation shows the surface area and volume manually and in case 2. Advance equation shows the mechanical performance has been discussed, the metal aggregates which had outstandingly qualities to resist shear, tension and compression forces. In this paper coarse metal aggregates is 20 mm which used for making high performance concrete (H.P.C).

Keywords: applied mathematical study case, special metal aggregates, concrete technology, basic physical formulation, advance equation

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Authors: Sherry Ann Ganase, Sandra Sookram

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This study investigates the level of awareness to climate change and major factors that influence such awareness in Grande Riviere, Trinidad. Through the development of an Awareness Index and application of a Structural Equation Model to survey data, the findings suggest an Awareness index value of 0.459 in Grande Riviere. These results suggest that households have climate smart attitudes and behaviors but climate knowledge is lacking. This is supported by the structural equation model which shows a negative relationship between awareness and causes of climate change. The study concludes by highlighting the need for immediate action on increasing knowledge.

Keywords: awareness, climate change, climate education, index structural equation model

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Authors: Ikram Slamani, Hicheme Ferdjani

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This study focuses on analyzing a Griffith crack situated at the center of an infinite strip. The problem is reformulated as a hyper-singular integral equation and solved numerically using second-order Chebyshev polynomials. The primary objective is to calculate the stress intensity factor in mode 1, denoted as K1. The obtained results reveal the influence of the strip width and crack length on the stress intensity factor, assuming stress-free edges. Additionally, a comparison is made with relevant literature to validate the findings.

Keywords: center crack, Chebyshev polynomial, hyper singular integral equation, Griffith, infinite strip, stress intensity factor

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Authors: Varvara Roubtsova, Mohamed Chekired

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Recent studies on the numerical simulation of geotechnical problems show the importance of considering the soil micro-structure. At this scale, soil is a discrete particle medium where the particles can interact with each other and with water flow under external forces, structure loads or natural events. This paper presents research conducted in a virtual laboratory named SiGran, developed at IREQ (Institut de recherche d’Hydro-Quebec) for the purpose of investigating a broad range of problems encountered in geotechnics. Using Discrete Element Method (DEM), SiGran simulated granular materials directly by applying Newton’s laws to each particle. The water flow was simulated by using Marker and Cell method (MAC) to solve the full form of Navier-Stokes’s equation for non-compressible viscous liquid. In this paper, examples of numerical simulation and their comparisons with real experiments have been selected to show the complexity of geotechnical research at the micro level. These examples describe transient flows into a porous medium, interaction of particles in a viscous flow, compacting of saturated and unsaturated soils and the phenomenon of liquefaction under seismic load. They also provide an opportunity to present SiGran’s capacity to compute the distribution and evolution of energy by type (particle kinetic energy, particle internal elastic energy, energy dissipated by friction or as a result of viscous interaction into flow, and so on). This work also includes the first attempts to apply micro discrete results on a macro continuum level where the Smoothed Particle Hydrodynamics (SPH) method was used to resolve the system of governing equations. The material behavior equation is based on the results of simulations carried out at a micro level. The possibility of combining three methods (DEM, MAC and SPH) is discussed.

Keywords: discrete element method, marker and cell method, numerical simulation, multi-scale simulations, smoothed particle hydrodynamics

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Authors: Damir Latypov

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A closed-form solution for 4-D double surface integrals arising in boundary integrals equations of a potential theory is obtained for arbitrary coplanar polygonal surfaces. The solution method is based on the construction of exact differential forms followed by the application of Stokes' theorem for each surface integral. As a result, the 4-D double surface integral is reduced to a 2-D double line integral. By an appropriate change of variables, the integrand is transformed into a separable function of integration variables. The closed-form solutions to the corresponding 1-D integrals are readily available in the integration tables. Previously closed-form solutions were known only for the case of coincident triangle surfaces and coplanar rectangles. Solutions for these cases were obtained by surface-specific ad-hoc methods, while the present method is general. The method also works for non-polygonal surfaces. As an example, we compute in closed form the 4-D integral for the case of coincident surfaces in the shape of a circular disk. For an arbitrarily shaped surface, the proposed method provides an efficient quadrature rule. Extensions of the method for non-coplanar surfaces and other than 1/R integral kernels are also discussed.

Keywords: boundary integral equations, differential forms, integration, stokes' theorem

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Authors: Kobamelo Mashaba

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The molten slag has a high substantial temperatures range between 1723-1923, carrying a huge amount of useful energy for reducing energy consumption and CO₂ emissions under the heat recovery process. Therefore in this study, we investigated the performance of the modified crank Nicolson method for a delayed partial differential equation on the heat recovery of molten slag in the metallurgical mining environment. It was proved that the proposed method converges quickly compared to the classic method with the existence of a unique solution. It was inferred from numerical result that the proposed methodology is more viable and profitable for the mining industry.

Keywords: delayed partial differential equation, modified Crank-Nicolson Method, molten slag, heat recovery, parabolic equation

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Authors: Mezghiche Kamel

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We present solitary wave solutions for the perturbed nonlinear Schrodinger (PNLS) equation describing propagation of femtosecond light pulses through the fiber Bragg grating structure where the pulse dynamics is governed by the nonlinear-coupled mode (NLCM) equations. Using the multiple scale analysis, we reduce the NLCM equations into the perturbed nonlinear Schrodinger (PNLS) type equation. Unlike the reported solitary wave solutions of the PNLS equation, the novel ones can describe W shaped of solitons and their properties.

Keywords: fiber bragg grating, nonlinear-coupled mode equations, w shaped of solitons, PNLS

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Authors: Keke Gao, Tao Lin, Yonghui Xie, Di Zhang

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Complicated unsteady flow in axial turbines produces high-frequency unsteady aerodynamic exciting force, which threatens the safe operation of turbines. This paper illustrates how negative-bowed stator reduces the rotor unsteady aerodynamic exciting force by unsteady flow field. With the support of three-dimensional viscous compressible Navier-Stokes equation, the single axial turbines with 0, -10 and -20 degree bowed stator are comparably investigated, aiming to identify the flow field structure difference caused by various negative-bowed degrees. The results show that negative-bowed stator strengthens the turbulence kinetic energy, which is further strengthened with the increase of negative-bowed degree. Meanwhile, the flow phenomenon including stator wakes and passage vortex is shown. In addition, the interaction of upstream negative-bowed wakes contributes to the reduction of unsteady blade load fluctuation. Furthermore, the aerodynamic exciting force decreases with the increasing negative bowed degree, while the efficiency is correspondingly reduced. This paper provides the reference for the alleviation of the harmful impact caused by unsteady interaction with the method of wake control.

Keywords: unsteady flow, axial turbine, wake, aerodynamic force, loss

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Authors: Norhashidah Hj. Mohd Ali, Teng Wai Ping

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In this paper, we present a high order group explicit method in solving the two dimensional Helmholtz equation. The presented method is derived from a nine-point fourth order finite difference approximation formula obtained from a 45-degree rotation of the standard grid which makes it possible for the construction of iterative procedure with reduced complexity. The developed method will be compared with the existing group iterative schemes available in literature in terms of computational time, iteration counts, and computational complexity. The comparative performances of the methods will be discussed and reported.

Keywords: explicit group method, finite difference, helmholtz equation, rotated grid, standard grid

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Authors: Suleiman Bashir Adamu, Lawan Sani Taura

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We study the role of boundary conditions via -symmetric quantum mechanics, where denotes parity operator and denotes time reversal operator. Using the one-dimensional Schrödinger Hamiltonian for a free particle in an infinite square well, we introduce symmetric boundary conditions. We find solutions of the 1D Klein-Gordon equation for a free particle in an infinite square well with Hermitian boundary and symmetry boundary conditions, where in both cases the energy eigenvalues and eigenfunction, respectively, are obtained.

Keywords: Eigenvalues, Eigenfunction, Hamiltonian, Klein- Gordon equation, PT-symmetric quantum mechanics

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Authors: Nurul Izzati Abd Karim, Samira Albati Kamaruddin, Rozaimi Che Hasan

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The appropriate petrophysical relationship is needed for Soil Water Content (SWC) estimation especially when using Ground Penetrating Radar (GPR). Ground penetrating radar is a geophysical tool that provides indirectly the parameter of SWC. This paper examines the performance of few published petrophysical relationships to obtain SWC estimates from in-situ GPR common- offset survey measurements with gravimetric measurements at peat soil area. Gravimetric measurements were conducted to support of GPR measurements for the accuracy assessment. Further, GPR with dual frequencies (250MHhz and 700MHz) were used in the survey measurements to obtain the dielectric permittivity. Three empirical equations (i.e., Roth’s equation, Schaap’s equation and Idi’s equation) were selected for the study, used to compute the soil water content from dielectric permittivity of the GPR profile. The results indicate that Schaap’s equation provides strong correlation with SWC as measured by GPR data sets and gravimetric measurements.

Keywords: common-offset measurements, ground penetrating radar, petrophysical relationship, soil water content

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