984 resultados para NUMERICAL-SOLUTION
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Петър Господинов, Добри Данков, Владимир Русинов, Стефан Стефанов - Изследвано е стационарно течение на Кует на разреден газ в случая на въртене на вътрешния цилиндър и неподвижен външен цилиндър чрез използване на DSMC метод и числено решение на уравненията на Навие–Стокс за относително малка (дозвукова) скорост на въртене. Изследвани са различни случаи при промяна на температурата на въртящият се цилиндър и числото на Кнудсен. Целта на изследването е да се установи влиянието на малки скорости на въртене върху макрохарактеристиките – плътността, скоростта и температурата на газа. Установено е добро съвпадение на резултатите получени по двата метода за Kn = 0.02. Получените резултати са важни при решаването на неравнинни, задачи от микрофлуидиката с отчитане на ефектите на кривината. Ключови думи: механика на флуидите, кинетична теория, разреден газ, DSMC.
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2000 Mathematics Subject Classification: 65H10.
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The theory and experimental applications of optical Airy beams are in active development recently. The Airy beams are characterised by very special properties: they are non-diffractive and propagate along parabolic trajectories. Among the striking applications of the optical Airy beams are optical micro-manipulation implemented as the transport of small particles along the parabolic trajectory, Airy-Bessel linear light bullets, electron acceleration by the Airy beams, plasmonic energy routing. The detailed analysis of the mathematical aspects as well as physical interpretation of the electromagnetic Airy beams was done by considering the wave as a function of spatial coordinates only, related by the parabolic dependence between the transverse and the longitudinal coordinates. Their time dependence is assumed to be harmonic. Only a few papers consider a more general temporal dependence where such a relationship exists between the temporal and the spatial variables. This relationship is derived mostly by applying the Fourier transform to the expressions obtained for the harmonic time dependence or by a Fourier synthesis using the specific modulated spectrum near some central frequency. Spatial-temporal Airy pulses in the form of contour integrals is analysed near the caustic and the numerical solution of the nonlinear paraxial equation in time domain shows soliton shedding from the Airy pulse in Kerr medium. In this paper the explicitly time dependent solutions of the electromagnetic problem in the form of time-spatial pulses are derived in paraxial approximation through the Green's function for the paraxial equation. It is shown that a Gaussian and an Airy pulse can be obtained by applying the Green's function to a proper source current. We emphasize that the processes in time domain are directional, which leads to unexpected conclusions especially for the paraxial approximation.
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Different industrial induction melting processes involve free surface and melt-solid interface of the liquid metal subject to dynamic change during the technological operation. Simulation of the liquid metal dynamics requires to solve the non-linear, coupled hydrodynamic-electromagnetic-heat transfer problem accounting for the time development of the liquid metal free boundary with a suitable turbulent viscosity model. The present paper describes a numerical solution method applicable for various axisymmetric induction melting processes, such as, crucible with free top surface, levitation, semi-levitation, cold crucible and similar melting techniques. The presented results in the cases of semi-levitation and crucible with free top surface meltings demonstrate oscillating transient behaviour of the free metal surface indicating the presence of gravity-inertial-electromagnetic waves which are coupled to the internal fluid flow generated by both the rotational and potential parts of the electromagnetic force.
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We consider a mechanical problem concerning a 2D axisymmetric body moving forward on the plane and making slow turns of fixed magnitude about its axis of symmetry. The body moves through a medium of non-interacting particles at rest, and collisions of particles with the body's boundary are perfectly elastic (billiard-like). The body has a blunt nose: a line segment orthogonal to the symmetry axis. It is required to make small cavities with special shape on the nose so as to minimize its aerodynamic resistance. This problem of optimizing the shape of the cavities amounts to a special case of the optimal mass transfer problem on the circle with the transportation cost being the squared Euclidean distance. We find the exact solution for this problem when the amplitude of rotation is smaller than a fixed critical value, and give a numerical solution otherwise. As a by-product, we get explicit description of the solution for a class of optimal transfer problems on the circle.
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Finance is one of the fastest growing areas in modern applied mathematics with real world applications. The interest of this branch of applied mathematics is best described by an example involving shares. Shareholders of a company receive dividends which come from the profit made by the company. The proceeds of the company, once it is taken over or wound up, will also be distributed to shareholders. Therefore shares have a value that reflects the views of investors about the likely dividend payments and capital growth of the company. Obviously such value will be quantified by the share price on stock exchanges. Therefore financial modelling serves to understand the correlations between asset and movements of buy/sell in order to reduce risk. Such activities depend on financial analysis tools being available to the trader with which he can make rapid and systematic evaluation of buy/sell contracts. There are other financial activities and it is not an intention of this paper to discuss all of these activities. The main concern of this paper is to propose a parallel algorithm for the numerical solution of an European option. This paper is organised as follows. First, a brief introduction is given of a simple mathematical model for European options and possible numerical schemes of solving such mathematical model. Second, Laplace transform is applied to the mathematical model which leads to a set of parametric equations where solutions of different parametric equations may be found concurrently. Numerical inverse Laplace transform is done by means of an inversion algorithm developed by Stehfast. The scalability of the algorithm in a distributed environment is demonstrated. Third, a performance analysis of the present algorithm is compared with a spatial domain decomposition developed particularly for time-dependent heat equation. Finally, a number of issues are discussed and future work suggested.
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This research is based on a numerical model for forecasting the three-dimensional behavior of (sea) water motion due to the effect of a variable wind velocity. The results obtained are then analyzed and compared with observation. This model is based on the equations that overcome the current and distribution of temperature by applying the method of finite difference with assuming Δx, Δy as constant and Δz, variable. The model is based on the momentum equation, continuity equation and thermodynamic energy equation and tension at the surface and middle layers and bottom stress. The horizontal and vertical eddy viscosity and thermal diffusivity coefficients we used in accordance with that of the Bennet on Outario Lake (1977). Considering the Caspian Sea dimension in numerical model the Coriolis parameter used with β effects and the approximation Boussines have been used. For the program controlling some simple experiment with boundary condition similar to that of the Caspian Sea have been done. For modeling the Caspian Sea the grid of the field was done as follows: At horizontal surface grid size is 10×10km extension and at vertical in 10 layers with varying thickness from surface to bed respectively as: 5, 10, 20, 3, 50, 100, 150, 200, 25, 500 and higher. The data of wind as velocity، direction and temperature of water related to 15th September 1995 at 6،12 and 18 o’clock were obtained from synoptic station at the Caspian Sea shore and the research marine of Haji Alief. The information concerning shore wind was measured and by the method of SPM (shore protection manual) was transferred to far shore winds through interpolation and by use of inverse square distance of position distribution of the wind velocity at the Caspian surface field was obtained. The model has been evaluated according to the reports and observations. Through studying the position of the current in different layers، the velocity in the cross section in the northern، southern and the middle layers، will be discussed. The results reveal the presence of the circulation cells in the three above mentioned areas. The circulation with depth is reduced too. The results obtained through the numerical solution of the temperature equation have been compared with the observation. The temperature change in different layers in cross section illustrates the relative accordance of the model mentioned.
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A significant part of the life of a mechanical component occurs, the crack propagation stage in fatigue. Currently, it is had several mathematical models to describe the crack growth behavior. These models are classified into two categories in terms of stress range amplitude: constant and variable. In general, these propagation models are formulated as an initial value problem, and from this, the evolution curve of the crack is obtained by applying a numerical method. This dissertation presented the application of the methodology "Fast Bounds Crack" for the establishment of upper and lower bounds functions for model evolution of crack size. The performance of this methodology was evaluated by the relative deviation and computational times, in relation to approximate numerical solutions obtained by the Runge-Kutta method of 4th explicit order (RK4). Has been reached a maximum relative deviation of 5.92% and the computational time was, for examples solved, 130,000 times more higher than achieved by the method RK4. Was performed yet an Engineering application in order to obtain an approximate numerical solution, from the arithmetic mean of the upper and lower bounds obtained in the methodology applied in this work, when you don’t know the law of evolution. The maximum relative error found in this application was 2.08% which proves the efficiency of the methodology "Fast Bounds Crack".
Simulação numérica da convecção mista em cavidade preenchida com meio poroso heterogêneo e homogêneo
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In this work is presented mixed convection heat transfer inside a lid-driven cavity heated from below and filled with heterogeneous and homogeneous porous medium. In the heterogeneous approach, the solid domain is represented by heat conductive equally spaced blocks; the fluid phase surrounds the blocks being limited by the cavity walls. The homogeneous or pore-continuum approach is characterized by the cavity porosity and permeability. Generalized mass, momentum and energy conservation equations are obtained in dimensionless form to represent both the continuum and the pore-continuum models. The numerical solution is obtained via the finite volume method. QUICK interpolation scheme is set for numerical treatment of the advection terms and SIMPLE algorithm is applied for pressure-velocity coupling. Aiming the laminar regime, the flow parameters are kept in the range of 102≤Re≤103 and 103≤Ra≤106 for both the heterogeneous and homogeneous approaches. In the tested configurations for the continuous model, 9, 16, 36, and 64 blocks are considered for each combination of Re and Ra being the microscopic porosity set as constant φ=0,64 . For the pore-continuum model the Darcy number (Da) is set according to the number of blocks in the heterogeneous cavity and the φ. Numerical results of the comparative study between the microscopic and macroscopic approaches are presented. As a result, average Nusselt number equations for the continuum and the pore continuum models as a function of Ra and Re are obtained.
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We shall consider the weak formulation of a linear elliptic model problem with discontinuous Dirichlet boundary conditions. Since such problems are typically not well-defined in the standard H^1-H^1 setting, we will introduce a suitable saddle point formulation in terms of weighted Sobolev spaces. Furthermore, we will discuss the numerical solution of such problems. Specifically, we employ an hp-discontinuous Galerkin method and derive an L^2-norm a posteriori error estimate. Numerical experiments demonstrate the effectiveness of the proposed error indicator in both the h- and hp-version setting. Indeed, in the latter case exponential convergence of the error is attained as the mesh is adaptively refined.
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The remediation of paracetamol (PA), an emerging contaminant frequently found in wastewater treatment plants, has been studied in the low concentration range (0.3–10 mg L−1) using as adsorbent a biomass-derived activated carbon. PA uptake of up to 100 mg g−1 over the activated carbon has been obtained, with the adsorption isotherms being fairly explained by the Langmuir model. The application of Reichemberg and the Vermeulen equations to the batch kinetics experiments allowed estimating homogeneous and heterogeneous diffusion coefficients, reflecting the dependence of diffusion with the surface coverage of PA. A series of rapid small-scale column tests were carried out to determine the breakthrough curves under different operational conditions (temperature, PA concentration, flow rate, bed length). The suitability of the proposed adsorbent for the remediation of PA in fixed-bed adsorption was proven by the high PA adsorption capacity along with the fast adsorption and the reduced height of the mass transfer zone of the columns. We have demonstrated that, thanks to the use of the heterogeneous diffusion coefficient, the proposed mathematical approach for the numerical solution to the mass balance of the column provides a reliable description of the breakthrough profiles and the design parameters, being much more accurate than models based in the classical linear driving force.
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This work considered the micro-mechanical behavior of a long fiber embedded in an infinite matrix. Using the theory of elasticity, the idea of boundary layer and some simplifying assumptions, an approximate analytical solution was obtained for the normal and shear stresses along the fiber. The analytical solution to the problem was found for the case when the length of the embedded fiber is much greater than its radius, and the Young's modulus of the matrix was much less than that of the fiber. The analytical solution was then compared with a numerical solution based on Finite Element Analysis (FEA) using ANSYS. The numerical results showed the same qualitative behavior of the analytical solution, serving as a validation tool against lack of experimental results. In general this work provides a simple method to determine the thermal stresses along the fiber embedded in a matrix, which is the foundation for a better understanding of the interaction between the fiber and matrix in the case of the classical problem of thermal-stresses.
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In this thesis, we explore three methods for the geometrico-static modelling of continuum parallel robots. Inspired by biological trunks, tentacles and snakes, continuum robot designs can reach confined spaces, manipulate objects in complex environments and conform to curvilinear paths in space. In addition, parallel continuum manipulators have the potential to inherit some of the compactness and compliance of continuum robots while retaining some of the precision, stability and strength of rigid-links parallel robots. Subsequently, the foundation of our work is performed on slender beam by applying the Cosserat rod theory, appropriate to model continuum robots. After that, three different approaches are developed on a case study of a planar parallel continuum robot constituted of two connected flexible links. We solve the forward and inverse geometrico-static problem namely by using (a) shooting methods to obtain a numerical solution, (b) an elliptic method to find a quasi-analytical solution, and (c) the Corde model to perform further model analysis. The performances of each of the studied methods are evaluated and their limits are highlighted. This thesis is divided as follows. Chapter one gives the introduction on the field of the continuum robotics and introduce the parallel continuum robots that is studied in this work. Chapter two describe the geometrico-static problem and gives the mathematical description of this problem. Chapter three explains the numerical approach with the shooting method and chapter four introduce the quasi-analytical solution. Then, Chapter five introduce the analytic method inspired by the Corde model and chapter six gives the conclusions of this work.
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The study of ultra-cold atomic gases is one of the most active field in contemporary physics. The main motivation for the interest in this field consists in the possibility to use ultracold gases to simulate in a controlled way quantum many-body systems of relevance to other fields of physics, or to create novel quantum systems with unusual physical properties. An example of the latter are Bose-Fermi mixtures with a tunable pairing interaction between bosons and fermions. In this work, we study with many-body diagrammatic methods the properties of this kind of mixture in two spatial dimensions, extending previous work for three dimensional Bose-Fermi mixtures. At zero temperature, we focus specifically on the competition between boson condensation and the pairing of bosons and fermions into molecules. By a numerical solution of the main equations resulting by our many-body diagrammatic formalism, we calculate and present results for several thermodynamic quantities of interest. Differences and similarities between the two-dimensional and three-dimensional cases are pointed out. Finally, our new results are applied to discuss a recent proposal for creating a p-wave superfluid in Bose-Fermi mixtures with the fermionic molecules which form for sufficiently strong Bose-Fermi attraction.
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For a two layered long wave propagation, linearized governing equations, which were derived earlier from the Euler equations of mass and momentum assuming negligible friction and interfacial mixing are solved analytically using Fourier transform. For the solution, variations of upper layer water level is assumed to be sinosoidal having known amplitude and variations of interface level is solved. As the governing equations are too complex to solve it analytically, density of upper layer fluid is assumed as very close to the density of lower layer fluid to simplify the lower layer equation. A numerical model is developed using the staggered leap-forg scheme for computation of water level and discharge in one dimensional propagation having known amplitude for the variations of upper layer water level and interface level to be solved. For the numerical model, water levels (upper layer and interface) at both the boundaries are assumed to be known from analytical solution. Results of numerical model are verified by comparing with the analytical solutions for different time period. Good agreements between analytical solution and numerical model are found for the stated boundary condition. The reliability of the developed numerical model is discussed, using it for different a (ratio of density of fluid in the upper layer to that in the lower layer) and p (ratio of water depth in the lower layer to that in the upper layer) values. It is found that as ‘CX’ increases amplification of interface also increases for same upper layer amplitude. Again for a constant lower layer depth, as ‘p’ increases amplification of interface. also increases for same upper layer amplitude.