Search results for: CFD numerical simulations

Authors: Paulo Mendonca, Nzar Faiq Naqeshbandi

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The objective of this study is to propose an interactive design environment that outlines the underlying computational framework to reach self-supporting overhangs. The research demonstrates the digital printability of overhangs taking into consideration factors related to the geometry design, the material used, the applied support, and the printing set-up of slicing and the extruder inclination. Parametric design tools can contribute to the design phase, form-finding, and stability optimization of self-supporting structures while printing in order to hold the components in place until they are sufficiently advanced to support themselves. The challenge is to ensure the stability of the printed parts in the critical inclinations during the whole fabrication process. Facilitating the identification of parameterization will allow to predict and optimize the process. Later, in the light of the previous findings, some guidelines of simulations and physical tests are given to be conducted for estimating the structural and functional performance.

Keywords: additive manufacturing, overhangs, self-support overhangs, printability, parametric tools

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Authors: A. Chellil, I. Gahlouz, S. Lecheb, A. Nour, S. Chellil, H. Mechakra, H. Kebir

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In this paper, the experimental study for the instability of a separator rotor is presented, under dynamic loading response in the harmonic analysis condition. The analysis of the stress which operates the rotor is done. Calculations of different energies and the virtual work of the aerodynamic loads from the rotor are developed. Numerical calculations on the model develop of three dimensions prove that the defects effect has a negative effect on the stability of the rotor. Experimentally, the study of the rotor in the transient system allowed to determine the vibratory responses due to the unbalances and various excitations.

Keywords: rotor, frequency, finite element, specter

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Authors: Alexandra K. Diem, Giles Richardson, Roxana O. Carare, Neil W. Bressloff

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Alzheimer's disease (AD) is the most common form of dementia and although it has been researched for over 100 years, there is still no cure or preventive medication. Its onset and progression is closely related to the accumulation of the neuronal metabolite Aβ. This raises the question of how metabolites and waste products are eliminated from the brain as the brain does not have a traditional lymphatic system. In recent years the rapid uptake of Aβ into cerebral artery walls and its clearance along those arteries towards the lymph nodes in the neck has been suggested and confirmed in mice studies, which has led to the hypothesis that interstitial fluid (ISF), in the basement membranes in the walls of cerebral arteries, provides the pathways for the lymphatic drainage of Aβ. This mechanism, however, requires a net reverse flow of ISF inside the blood vessel wall compared to the blood flow and the driving forces for such a mechanism remain unknown. While possible driving mechanisms have been studied using mathematical models in the past, a mechanism for net reverse flow has not been discovered yet. Here, we aim to address the question of the driving force of this reverse lymphatic drainage of Aβ (also called perivascular drainage) by using multi-scale numerical and analytical modelling. The numerical simulation software COMSOL Multiphysics 4.4 is used to develop a fluid-structure interaction model of a cerebral artery, which models blood flow and displacements in the artery wall due to blood pressure changes. An analytical model of a layer of basement membrane inside the wall governs the flow of ISF and, therefore, solute drainage based on the pressure changes and wall displacements obtained from the cerebral artery model. The findings suggest that an active role in facilitating a reverse flow is played by the components of the basement membrane and that stiffening of the artery wall during age is a major risk factor for the impairment of brain lymphatics. Additionally, our model supports the hypothesis of a close association between cerebrovascular diseases and the failure of perivascular drainage.

Keywords: Alzheimer's disease, artery wall mechanics, cerebral blood flow, cerebral lymphatics

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Authors: Nader Al-Rashidi, David Greenhalgh

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Mathematical modelling techniques are now being used by health organizations worldwide to help understand the likely impact that intervention strategies treatment options and combinations of these have on the prevalence and incidence of hepatitis C virus (HCV) in the people who inject drugs (PWID) population. In this poster, we develop a deterministic, compartmental mathematical model to approximate the spread of the HCV in a PWID population that has been divided into two groups by time since onset of injection. The model assumes that after injection needles adopt the most infectious state of their previous state or that of the PWID who last injected with them. Using analytical techniques, we find that the model behaviour is determined by the basic reproductive number R₀, where R₀ = 1 is a critical threshold separating two different outcomes. The disease-free equilibrium is globally stable if R₀ ≤ 1 and unstable if R₀ > 1. Additionally, we make some simulations where have confirmed that the model tends to this endemic equilibrium value with realistic parameter values giving an HCV prevalence.

Keywords: hepatitis C, people who inject drugs, HCV, PWID

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Authors: S. W. Lo, S.-H. Lu, Y. H. Guo, L. C. Hsu

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In this paper, a novel design of aerostatic thrust bearing is proposed and is analyzed numerically. The capillary restrictor and bearing disk are made of elastomer like silicone and PU. The viscoelasticity of elastomer helps the capillary expand for more air flux and at the same time, allows conicity of the bearing surface to form when the air pressure is enhanced. Therefore, the bearing has the better ability of passive compensation. In the present example, as compared with the typical model, the new designs can nearly double the load capability and offer four times static stiffness.

Keywords: aerostatic, bearing, elastomer, static stiffness

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Authors: Piotr Kacejko, Lukasz Grabowski, Zdzislaw Kaminski

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This paper discusses a model of fuel consumption and on-board electricity generation. Rapid changes in speed result in a constantly changing kinetic energy accumulated in a bus mass and an increased fuel consumption due to hardly recuperated kinetic energy. The model is based on the results achieved from chassis dynamometer, airport and city street researches. The verified model was applied to simulate the on-board electricity generation during the Japanese JE05 Emission Test Cycle. The simulations were performed for several values of vehicle mass and electrical load applied to on-board devices. The research results show that driving dynamics has an impact on a consumption of fuel to drive alternators.

Keywords: city bus, heavy duty vehicle, Japanese JE05 test cycle, power generation

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Authors: Abhisek Sarkar, Abhimanyu Gaur

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In this report we have discussed the theoretical aspects of the flow transformation, occurring through a series of bifurcations. The parameters and their continuous diversion, the intermittent bursts in the transition zone, variation of velocity and pressure with time, effect of roughness in turbulent zone, and changes in friction factor and head loss coefficient as a function of Reynolds number for a transverse flow across a cylinder have been discussed. An analysis of the variation in the wake length with Reynolds number was done in FORTRAN.

Keywords: bifurcation, attractor, intermittence, energy cascade, energy spectra, vortex stretching

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Authors: Muhammad Sameer Ahmed, Piotr Remlein, Tansal Gucluoglu

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The technique called as Generalized frequency division multiplexing (GFDM) used in the free space optical channel can be a good option for implementation free space optical communication systems. This technique has several strengths e.g. good spectral efficiency, low peak-to-average power ratio (PAPR), adaptability and low co-channel interference. In this paper, the impact of weather conditions such as haze, rain and fog on GFDM over the gamma-gamma channel model is discussed. A Trade off between link distance and system performance under intense weather conditions is also analysed. The symbol error probability (SEP) of GFDM over the gamma-gamma turbulence channel is derived and verified with the computer simulations.

Keywords: free space optics, generalized frequency division multiplexing, weather conditions, gamma gamma distribution

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Authors: A. Elnady

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This paper presents an improved Direct Power Control (DPC) scheme applied to the multilevel inverter that forms a Distributed Generation Unit (DGU). This paper demonstrates the performance of active and reactive power injected by the DGU to the smart grid. The DPC is traditionally operated by the hysteresis controller with the Space Vector Modulation (SVM) which is applied on the 2-level inverters or 3-level inverters. In this paper, the DPC is operated by the PI controller with the Phase-Disposition Pulse Width Modulation (PD-PWM) applied to the 5-level diode clamped inverter. The new combination of the DPC, PI controller, PD-PWM and multilevel inverter proves that its performance is much better than the conventional hysteresis-SVM based DPC. Simulations results have been presented to validate the performance of the suggested control scheme in the grid-connected mode.

Keywords: direct power control, PI controller, PD-PWM, and power control

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Authors: A. Kudrev

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The duplex Poly(A)-Poly(U) denaturation in an aqueous solutions in mixtures with the tetracationic MeTMPyP4 (Me = 2H, Zn(II); TMPyP4 is 5,10,15,20-tetrakis(N-methylpyridinium-4-yl)porphyrin), was investigated by monitoring the changes in the UV-Vis absorbance spectrum with increasing temperatures from 20°С to 70°С (рН 7.0, I=0.15M). The absorbance data matrices were analyzed with a versatile chemometric procedure that provides the melting profile (distribution of species) and the pure spectrum for each chemical species present along the heating experiment. As revealed by the increase of Tm, the duplex structure was stabilized by these porphyrins. The values of stabilization temperature ΔTm in the presence of these porphyrins are relatively large, 1.2-8.4 °C, indicating that the porphyrins contribute differently in stabilizing the duplex Poly(A)-Poly(U) structure. Remarkable is the fact that the porphyrin TMPyP4 was less effective in the stabilization of the duplex structure than the metalloporphyrin Zn(X)TMPyP4 which suggests that metallization play an important role in porphyrin-RNA binding. Molecular Dynamics Simulations has been used to illustrate melting of the duplex dsRNA bound with a porphyrin molecule.

Keywords: melting, Poly(A)-Poly(U), TMPyP4, Zn(X)TMPyP4

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Authors: Vladimir S. Timofeev

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In this article the problem of distributional moments estimation is considered. The new approach of moments estimation based on usage of the characteristic function is proposed. By statistical simulation technique, author shows that new approach has some robust properties. For calculation of the derivatives of characteristic function there is used numerical differentiation. Obtained results confirmed that author’s idea has a certain working efficiency and it can be recommended for any statistical applications.

Keywords: characteristic function, distributional moments, robustness, outlier, statistical estimation problem, statistical simulation

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Authors: M. A. Tayfun, M. A. Alkhalidi

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The exact theoretical expression describing the probability distribution of nonlinear sea-surface elevations derived from the second-order narrowband model has a cumbersome form that requires numerical computations, not well-disposed to theoretical or practical applications. Here, the same narrowband model is re-examined to develop a simpler closed-form approximation suitable for theoretical and practical applications. The salient features of the approximate form are explored, and its relative validity is verified with comparisons to other readily available approximations, and oceanic data.

Keywords: ocean waves, probability distributions, second-order nonlinearities, skewness coefficient, wave steepness

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Authors: S. Gómez, G. Sánchez, J. Zarama, M. Castañeda Ramos, J. Escoto Alcántar, J. Torres, A. Núñez, S. Santana, F. Nájera, J. A. Lopez

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This paper shows in detail the mathematical model of direct and inverse kinematics for a robot manipulator (welding type) with four degrees of freedom. Using the D-H parameters, screw theory, numerical, geometric and interpolation methods, the theoretical and practical values of the position of robot were determined using an optimized algorithm for inverse kinematics obtaining the values of the particular joints in order to determine the virtual paths in a relatively short time.

Keywords: kinematics, degree of freedom, optimization, robot manipulator

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Authors: Jose Daniel Hoyos Giraldo, Jesus Hernan Jimenez Giraldo, Juan Pablo Alvarado Perilla

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Range and endurance are the main limitations of electric aircraft due to the nature of its source of power. The improvement of efficiency on this kind of systems is extremely meaningful to encourage the aircraft operation with less environmental impact. The propeller efficiency highly affects the overall efficiency of the propulsion system; hence its optimization can have an outstanding effect on the aircraft performance. An optimization method is applied to an aircraft propeller in order to maximize its range and endurance by estimating the best combination of geometrical parameters such as diameter and airfoil, chord and pitch distribution for a specific aircraft design at a certain cruise speed, then the rotational speed at which the propeller operates at minimum current consumption is estimated. The optimization is based on the Blade Element Momentum (BEM) method, additionally corrected to account for tip and hub losses, Mach number and rotational effects; furthermore an airfoil lift and drag coefficients approximation is implemented from Computational Fluid Dynamics (CFD) simulations supported by preliminary studies of grid independence and suitability of different turbulence models, to feed the BEM method, with the aim of achieve more reliable results. Additionally, Vortex Theory is employed to find the optimum pitch and chord distribution to achieve a minimum induced loss propeller design. Moreover, the optimization takes into account the well-known brushless motor model, thrust constraints for take-off runway limitations, maximum allowable propeller diameter due to aircraft height and maximum motor power. The BEM-CFD method is validated by comparing its predictions for a known APC propeller with both available experimental tests and APC reported performance curves which are based on Vortex Theory fed with the NASA Transonic Airfoil code, showing a adequate fitting with experimental data even more than reported APC data. Optimal propeller predictions are validated by wind tunnel tests, CFD propeller simulations and a study of how the propeller will perform if it replaces the one of on known aircraft. Some tendency charts relating a wide range of parameters such as diameter, voltage, pitch, rotational speed, current, propeller and electric efficiencies are obtained and discussed. The implementation of CFD tools shows an improvement in the accuracy of BEM predictions. Results also showed how a propeller has higher efficiency peaks when it operates at high rotational speed due to the higher Reynolds at which airfoils present lower drag. On the other hand, the behavior of the current consumption related to the propulsive efficiency shows counterintuitive results, the best range and endurance is not necessary achieved in an efficiency peak.

Keywords: BEM, blade design, CFD, electric aircraft, endurance, optimization, range

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Authors: Arnab Sarkar, Michael Huang, Chuang Ren, Jun Li

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The increasing importance of computing in modern society has brought substantial growth in the demand for more computational power. In some problem domains such as scientific simulations, available computational power still sets a limit on what can be practically explored in computation. For many types of code, there is non-uniformity in the utility of computation. That is not every piece of computation contributes equally to the quality of the result. If this non-uniformity is understood well and exploited effectively, we can much more effectively utilize available computing power. In this paper, we discuss a case study of exploring such non-uniformity in a particle-in-cell simulation platform. We find both the existence of significant non-uniformity and that it is generally straightforward to exploit it. We show the potential of order-of-magnitude effective performance gain while keeping the comparable quality of output. We also discuss some challenges in both the practical application of the idea and evaluation of its impact.

Keywords: approximate computing, landau damping, non uniform utility computing, particle-in-cell

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Authors: Chor Nejmeddine, Ines Ben Omrane, Mohamed Abdelwahed

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In this paper, we present a posteriori analysis of the discretization of the heat equation by spectral element method. We apply Euler's implicit scheme in time and spectral method in space. We propose two families of error indicators, both of which are built from the residual of the equation and we prove that they satisfy some optimal estimates. We present some numerical results which are coherent with the theoretical ones.

Keywords: heat equation, spectral elements discretization, error indicators, Euler

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Authors: Mahmoudi Noureddine, Kaou Abdellah

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Energy absorption is a major requirement for automotive structures. Although crashworthy structures of composite based glass fiber have exhibited energy absorption greater than similar at other composites structures, the crush process in many cases is accompanied by fracture, rather than by plastic deformation. The crash experiments show that the tubes are crushed in progressive manner start from one end of the tubes and delamination takes place between the layers. To better understand details of the crash process, ABAQUS finite element code is used.

Keywords: Energy absorption, crash, PEHD

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Authors: Alexey V. Klyuev, Valery P. Samarin, Viktor F. Klyuev, Andrey V. Klyuev

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A power measurement algorithm of the input mix components of the noise signal and pulse interference is considered. The algorithm efficiency analysis has been carried out for different interference to signal ratio. Algorithm performance features have been explored by numerical experiment results.

Keywords: noise signal, pulse interference, signal power, spectrum width, detection

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Authors: Shady Sayed, Samer Wagdy, Ahmed Badawy, Moutaz M. Hegaze

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A symmetrical single mass resonant gyroscope is discussed in this paper. The symmetrical design allows matched resonant frequencies for driving and sensing vibration modes, which leads to amplifying the sensitivity of the gyroscope by the mechanical quality factor of the sense mode. It also achieves decoupled vibration modes for getting a low zero-rate output shift and more stable operation environment. A new suspension beams design is developed to get a symmetrical gyroscope with matched and decoupled modes at the same time. Finite element simulations are performed using ANSYS software package to verify the theoretical calculations. The gyroscope is fabricated from aluminum alloy 2024 substrate, the measured drive and sense resonant frequencies of the fabricated model are matched and equal 81.4 Hz with 5.7% error from the simulation results.

Keywords: decoupled mode shapes, resonant sensor, symmetrical gyroscope, finite element simulation

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Authors: Tetsiki Taniguchi, Yoshio Karasawa

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This paper describes a subarray based low computational design method of multiuser massive multiple input multiple output (MIMO) system. In our previous works, use of large array is assumed only in transmitter, but this study considers the case both of transmitter and receiver sides are equipped with large array antennas. For this aim, receive arrays are also divided into several subarrays, and the former proposed method is modified for the synthesis of a large array from subarrays in both ends. Through computer simulations, it is verified that the performance of the proposed method is degraded compared with the original approach, but it can achieve the improvement in the aspect of complexity, namely, significant reduction of the computational load to the practical level.

Keywords: large array, massive multiple input multiple output (MIMO), multiuser, singular value decomposition, subarray, zero forcing

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Authors: Tamou Nasser, Ahmed Essadki, Ali Boukhriss

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This paper presents the control of doubly fed induction generator (DFIG) used in the wind energy conversion systems. Maximum power point tracking (MPPT) strategy is used to extract the maximum of power during the conversion and taking care that the system does not exceed the operating limits. This is done by acting on the pitch angle to control the orientation of the turbine's blades. Having regard to its robustness and performance, active disturbance rejection control (ADRC) based on the extended state observer (ESO) is employed to achieve the control of both rotor and grid side converters. Simulations are carried out using matlab simulink.

Keywords: active disturbance rejection control, extended state observer, doubly fed induction generator, maximum power point tracking

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Authors: Tamou Nasser, Ahmed Essadki, Ali Boukhriss

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This paper presents the control of doubly fed induction generator (DFIG) used in the wind energy conversion systems. Maximum power point tracking (MPPT) strategy is used to extract the maximum of power during the conversion and taking care that the system does not exceed the operating limits. This is done by acting on the pitch angle to control the orientation of the turbine's blades. Having regard to its robustness and performance, active disturbance rejection control (ADRC) based on the extended state observer (ESO) is employed to achieve the control of both rotor and grid side converters. Simulations are carried out using matlab simulink.

Keywords: active disturbance rejection control, extended state observer, doubly fed induction generator, maximum power point tracking

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Authors: Yeongyu Choi, Ju H. Park, S. M. Lee, Ho-Youl Jung

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In this paper, an improved method for estimating fundamental matrix is proposed. The method is applied effectively to monocular camera based moving object detection. The method consists of corner points detection, moving object’s motion estimation and fundamental matrix calculation. The corner points are obtained by using Harris corner detector, motions of moving objects is calculated from pyramidal Lucas-Kanade optical flow algorithm. Through epipolar geometry analysis using RANSAC, the fundamental matrix is calculated. In this method, we have improved the performances of moving object detection by using two threshold values that determine inlier or outlier. Through the simulations, we compare the performances with varying the two threshold values.

Keywords: corner detection, optical flow, epipolar geometry, RANSAC

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Authors: M. Zareh, H. Mohammadi, B. Naser

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Atherosclerotic plaque rupture is the main trigger of heart attack and brain stroke which are the leading cause of death in developed countries. Better understanding of rupture-prone plaque can help clinicians detect vulnerable plaques- rupture prone or instable plaques- and apply immediate medical treatment to prevent these life-threatening cardiovascular events. Therefore, there are plenty of studies addressing disclosure of vulnerable plaques properties. Necrotic core and fibrous tissue are two major tissues constituting atherosclerotic plaque; using histopathological and numerical approaches, many studies have demonstrated that plaque rupture is strongly associated with a large necrotic core and a thin fibrous cap, two morphological characteristic which can be acquired by two-dimensional imaging of atherosclerotic plaque present in coronary and carotid arteries. Plaque rupture is widely considered as a mechanical failure inside plaque tissue; this failure occurs when the stress within plaque excesses the strength of tissue material; hence, finite element method, a strong numerical approach, has been extensively applied to estimate stress distribution within plaques with different compositions which is then used for assessment of various vulnerability characteristics including plaque morphology, material properties and blood pressure. This study aims to evaluate significance of three-dimensional morphology on vulnerability degree of atherosclerotic plaque. To reach this end, different two-dimensional geometrical models of atherosclerotic plaques are considered based on available data and named Main 2D Models (M2M). Then, for each of these M2Ms, two three-dimensional idealistic models are created. These two 3D models represent two possible three-dimensional morphologies which might exist for a plaque with similar 2D morphology to one of M2Ms. Finite element method is employed to estimate stress, von-Mises stress, within each 3D models. Results indicate that for each M2Ms stress can significantly varies due to possible 3D morphological changes in that plaque. Also, our results show that an atherosclerotic plaque with thick cap may experience rupture if it has a critical 3D morphology. This study highlights the effect of 3D geometry of plaque on its instability degree and suggests that 3D morphology of plaque might be necessary to more effectively and accurately assess atherosclerotic plaque vulnerability.

Keywords: atherosclerotic plaque, plaque rupture, finite element method, 3D model

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Authors: Brando Vagenende, Marie-Anne Guerry

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For stochastic matrices of any order, the geometric description of the convex set of eigenvalues is completely known. The purpose of this study is to investigate the subset of the monotone matrices. This type of matrix appears in contexts such as intergenerational occupational mobility, equal-input modeling, and credit ratings-based systems. Monotone matrices are stochastic matrices in which each row stochastically dominates the previous row. The monotonicity property of a stochastic matrix can be expressed by a nonnegative lower-order matrix with the same eigenvalues as the original monotone matrix (except for the eigenvalue 1). Specifically, the aim of this research is to focus on the properties of eigenvalues of monotone matrices. For those matrices up to order 3, there already exists a complete description of the convex set of eigenvalues. For monotone matrices of order at least 4, this study gives, through simulations, more insight into the geometric description of their eigenvalues. Furthermore, this research treats in a geometric and algebraic way the properties of eigenvalues of monotone matrices of order at least 4.

Keywords: eigenvalues of matrices, finite Markov chains, monotone matrices, nonnegative matrices, stochastic matrices

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Authors: Mohammadhosein Hasanbeig, Lacra Pavel

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In this paper, we study a distributed control algorithm for the problem of unknown area coverage by a network of robots. The coverage objective is to locate a set of targets in the area and to minimize the robots’ energy consumption. The robots have no prior knowledge about the location and also about the number of the targets in the area. One efficient approach that can be used to relax the robots’ lack of knowledge is to incorporate an auxiliary learning algorithm into the control scheme. A learning algorithm actually allows the robots to explore and study the unknown environment and to eventually overcome their lack of knowledge. The control algorithm itself is modeled based on game theory where the network of the robots use their collective information to play a non-cooperative potential game. The algorithm is tested via simulations to verify its performance and adaptability.

Keywords: distributed control, game theory, multi-agent learning, reinforcement learning

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Authors: Gürkan Şakar, Akgün Alsaran, Emrah E. Özbaldan

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In this study, lateral buckling analysis of composite beams with particle additive was carried out experimentally and numerically. The effects of particle type, particle addition ratio on buckling loads of composite beams were determined. The numerical studies were performed with ANSYS package. In the analyses, clamped-free boundary condition was assumed. The load carrying capabilities of composite beams were influenced by different particle types and particle addition ratios.

Keywords: lateral buckling, nanoparticle, composite beam, numeric analysis

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Authors: H. Pasternak, A. Chwastek

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The paper describes the results from a research project about repair of welds. The repair was carried out by grinding the flawed seams and re-welding them. The main task was to determine the FAT classes of original state and after repair of seams according to the assessment procedures, such as nominal, structural and effective notch stress approach. The first part shows the results of the tests, the second part encloses numerical analysis and evaluation of results to determine the fatigue strength classes according to three assessment procedures.

Keywords: cyclic loading, fatigue crack, post-weld treatment, seams’ repair

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Authors: Umesh Kumar, Arun Kumar Shukla, B. V. V. Ravindra Babu

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The paper presents a compact tri band Curved U-Slot patch antenna with improved bandwidth and isolation characteristics. The proposed antenna excited by coaxial feed resonates at tri band of 2.8 GHz, 4.1 GHz and 5.7 GHz for VSWR ≤ 1.5 with an improved bandwidth of 99.7% and also for getting high gain antenna of 11.31 dB. A 2×2 MIMO is developed using the proposed antenna giving an excellent isolation of 28 dB between the two antennas. The simulation results of return loss, Mutual Coupling, Gain, VSWR, Surface Current Distribution and Electrical Distribution are presented. By keeping the substrate thickness constant over various dielectric constants, simulations were carried out using MATLAB® and HFSS (High Frequency Structure Simulator) software.

Keywords: performance analysis, curved U-slot patch, antenna with enhanced bandwidth, isolation for mimo systems

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Authors: Anuar Ishak

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The characteristics of fluid flow and heat transfer over a permeable shrinking sheet is studied. The governing partial differential equations are transformed into a set of ordinary differential equations, which are then solved numerically using MATLAB routine boundary value problem solver bvp4c. Numerical results show that dual solutions are possible for a certain range of the suction parameter. A stability analysis is performed to determine which solution is linearly stable and physically realizable.

Keywords: dual solutions, heat transfer, shrinking sheet, stability analysis

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