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EN
In this study, the locally one dimensional (LOD) method is used to solve the two dimensional time fractional diffusion equation. The fractional derivative is the Caputo fractional derivative of order α. The rate of convergence of the finite difference method is presented. It is seen that this method is in agreement with the obtained numerical solutions with acceptable central processing unit time (CPU time). Error estimates, numerical and exact results are tabulated. The graphics of errors are given.
EN
The natural vibrations of thin (Kirchhoff-Love) plates with constant and variable thickness are considered in the paper. Isotropic and orthotropic rectangular plates with different boundary conditions are analysed. The Finite Element Method and the Finite Difference Method are used to describe structural deformation. The elements of stiffness matrix are derived numerically using author’s approaches of localization of integration points. The plate inertia forces are expressed by diagonal, lumped mass matrix or consistent mass matrix. The presence of the external medium, which can be a fluid, is described by the fluid velocity potential of double layer and the fundamental solution of Laplace equation which leads to the fully-populated mass matrix. The influence of external additional liquid mass on natural frequencies of plate is analysed, too.
PL
W artykule przedstawiono wpływ powszechnie stosowanych materiałów budowlanych na wartości natężenia pola. Analiza dotyczyła obszaru zawierającego ścianę jedno- lub dwuwarstwową wykonaną z: pełnej cegły, betonu, gazobetonu oraz dwóch rodzajów cegieł klinkierowych (z drążeniami). W badaniach uwzględniono grubość ściany, konduktywność materiałów oraz złożoność cegieł klinkierowych. Głównie analizowano wpływ zmiany parametru elektrycznego materiału ceramicznego, tj. konduktywności, na wartości pola elektrycznego. Do analizy zastosowano metodę różnic skończonych z bezpośrednim całkowaniem równań Maxwella w dziedzinie czasu (FDTD). Celem badań było lepsze zrozumienie zachodzących zjawisk polowych wewnątrz jednorodnych i złożonych materiałów budowlanych. Wyniki analizy mogą stanowić źródło wiedzy przy ocenie problemów związanych z zanikami sygnału i pozwolą na lepsze planowanie lokalizacji nadajników sieci bezprzewodowych stosowanych m.in. w sieciach komórkowych, Wi-Fi, WiMAX.
EN
The article presents the influence of commonly used building materials on field intensity values. The analysis concerned an area containing a single or double-layer wall made of: full brick, concrete, aerated concrete and two types of clinker bricks (with hollows). The research included wall thickness, material conductivity and complexity of clinker bricks. The impact of changing the electrical parameter of the ceramic material, i.e. conductivity, on the electric field values was mainly analysed. The Finite Difference Time Domain Method with direct integration of Maxwell’s equations in time domain (FDTD) was used for the analysis. The aim of the research was to better understand the occurring field phenomena inside homogeneous and complex building materials. The results of the analysis can be a source of knowledge when assessing problems related to signal loss and allow for better planning of the location of wireless network transmitters used, among others in cellular networks, Wi-Fi, WiMAX.
EN
In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.
EN
In this paper, an incompressible, two-dimensional (2D), time-dependent, Newtonian, laminar, and internal channel fluid flow over a skewed equilateral cavity is simulated using the finite difference method (FDM) and alternating direction implicit (ADI) technique. Navier-Stokes equations are solved numerically in stream function-vorticity formulation. The goal of tackling this problem depends on its academic significance by studying the difference between lid-driven and shear-driven cavity flows in terms of the formation of Moffatt eddies at the sharp corner, also to obtain the length and intensity ratios of these counter-rotating vortices. The value of velocity components along the centerlines of the skewed cavity was revealed at low and intermediate Reynolds numbers (Re), typically (Re = 200 and 2000) at two different skew angles of mainly 30° and 45°. Likewise, the blocked-off regions’ method is used to deal with the geometry of the skewed cavity especially the sharp corners. Furthermore, as Re increases, the main vortex approaches the skewed cavity center and the counter-rotating vortices get bigger in size and intensity, and their number increases.
EN
The aim of the study was to analyze changes in tissue oxygen distribution resulting from temperature changes by the use of the Krogh cylinder model with Michaelis-Menten kinetics. A Hill model was also used to describe the oxyhemoglobin dissociation curve. In particular, variable values of parameters of dissociation curve and blood velocity in capillary were considered. Mathematical description was based on two separate equations for radial and axial directions. An additional task related to determination of the temperature, tissue thermal damage and perfusion was also solved. At the stage of numerical realization, the finite difference method was used.
EN
Heat transfer and fluid flow in the rectangular channel with an obstacle are considered. The problem is described by the Fourier-Kirchhoff equation, Navier-Stokes equations and continuity equation supplemented by appropriate boundary and initial conditions. To solve this system of equations the finite difference method with a staggered grid is used. The results of computations obtained using authorial computer program are compared with ANSYS Fluent simulation. Computations are carried out for obstacles of various sizes and positions, and on this basis the conclusions are formulated.
EN
The article is about the issue of the influence of ballast resistance on the stability of the Continuous Welded Rail. The ballast resistance affects both the longitudinal and transverse displacements. It depends on the quality of the ballast, the degree of its compaction and contamination. The article contains an analysis of the impact of ballast resistance on the track based on the Finite Difference Method. The calculations showed that the resistance value directly affects the allowable critical force and the maximum temperature rise in the rail that does not endanger the safety of railway traffic.
EN
The deformation modulus of the rock mass as a very important parameter in rock mechanic projects generally is determined by the plate load in-situ tests. While this test is very expensive and time-consuming, so in this study a new method is developed to combin artificial neural networks and numerical modeling for predicting deformation modulus of rock masses. For this aim, firstly, the plate load test was simulated using a Finite Difference numerical model that was verified with actual results of the plate load test in Pirtaghi dam galleries in Iran. Secondly, an artificial neural network is trained with a set of data resulted from numerical simulations to estimate the deformation modulus of the rock mass. The results showed that an ANN with five neurons in the input layer, three hidden layers with 4, 3 and 2 neurons, and one neuron in the output layer had the best accuracy for predicting the deformation modulus of the rock mass.
PL
W pracy przedstawiono algorytm rozwiązywania równań ruchu płynu metody „Wir w Komórce” z wykorzystaniem różnicowego schematu kompaktowego rzędu czwartego do rozwiązywania równania Poissona i równania dyfuzji. Opisano kolejne kroki algorytmu wraz z badaniami dokładności poszczególnych schematów różnicowych. Program obliczeniowy został sprawdzony na przykładzie popularnego zagadnienia przepływu we wnęce. Otrzymane wyniki porównano z wynikami opublikowanymi przez innych autorów.
EN
In this paper, an axially symmetrical biological tissue domain subjected to an external heat source is analyzed. The thermal processes occurring in the domain considered are described using the generalized dual-phase lag model supplemented by the Neumann boundary conditions and the appropriate initial conditions. The problem of tissue heating is solved using the implicit scheme of the finite difference method. The obtained solution allows one to determine the local and temporary values of the Arrhenius integral. Next, the inverse problem related to the identification of the boundary heat flux assuring the postulated destruction of the tissue target region is considered. The problem is solved using the gradient method. In the final part of the paper, the results of computations and the conclusions are presented.
EN
Evaluation of moisture absorption in foodstuffs such as black chickpea is an important stage for skinning and cropping practices. Water uptake process of black chickpea was discussed through normal soaking in four temperature levels of 20, 35, 50 and 65 °C for 18 hours, and then the hydration kinetics was predicted by Peleg’s model and finite difference strategy. Model results showed that with increasing soaking temperature from 20 to 65 °C, Peleg’s rate and Peleg’s capacity constant reduced from 13.368×10-2 to 5.664×10-2 and 9.231×10-3 to 9.138×10-3, respectively. Based on key results, a rise in the medium temperature caused an increase in the diffusion coefficient from 5.24×10-10 m2/s to 4.36×10-9 m2/s, as well. Modelling of moisture absorption of black chickpea was also performed employing finite difference strategy. Comparing the experimental results with those obtained from the analytical solution of the theoretical models revealed a good agreement between predicted and experimental data. Peleg’s model and finite difference technique revealed their predictive function the best at the temperature of 65 °C.
EN
The temperature of annealed steel coils is a determining variable of the future steel sheets quality. This variable also determines the energy consumption in operation. Unfortunately, the monitoring of coil inner temperature is problematic due to the furnace environment with high temperature, coil structure, and annealing principle. Currently, there are no measuring principles that can measure the temperature inside the heat-treated product in a non-destructive manner. In this paper, the soft sensing of inner temperature based on the theory of non-stationary heat conduction and approach based on Support Vector Regression (SVR) was presented. The results showed that a black-box approach based on the SVR could replace an analytic approach, though with lesser performance. Several annealing experiments were performed to create a training data set and model performance improvement in the estimation of inner coil temperatures. The proposed software based on non-stationary heat conduction can calculate the behavior of inner coil temperature from the measured boundary temperatures that are measured by thermocouples. The soft-sensing principles presented in this paper were verified under laboratory conditions and on the data obtained from a real annealing plant.
PL
W pracy przedstawiono algorytm rozwiązywania równań ruchu płynu z wykorzystaniem kompaktowej, czwartorzędowej metody „Vortex in Cell“. W pierwszej części zawarto sposób budowania układów równań liniowych z wykorzystaniem wysokowydajnej biblioteki hypre. Przedstawiono kolejne kroki algorytmu wraz z badaniami dokładności i wydajności obliczeniowej poszczególnych schematów różnicowych. Ostatecznie przedstawiono badanie dokładności działania metody na przykładzie zagadnienia Taylora-Greena ze znanym rozwiązaniem dokładnym i porównano otrzymane wyniki z wartościami dokładnymi.
15
Content available remote On thermoelasticity in FGL - tolerance averaging technique
EN
In this paper the problem of linear thermoelasticity in a laminate with functional gradation of properties is considered. In micro level this laminate is made of two different materials, microlaminas, distributed non-periodically but also not randomly along one of directions, what in macro level results in aforementioned functionally gradation of laminate properties. In order to describe behavior of such structure, equations of two models are here presented - the tolerance and the tolerance-asymptotic model. Both are obtained by the tolerance averaging technique. The basic aim of this work is to analyse the influence of some terms from these averaged equations on the distribution and the values of the displacements and the temperature functions. To solve the equations of two proposed models the finite difference method is used.
16
Content available remote Selected applications of differential equations in Vanilla Options valuation
EN
The purpose of this paper is to present the applications of differential equations in Vanilla Options pricing. At the beginning we introduce main assumptions of the Black-Scholes Model with necessary comments, which as a norm can not be easily found in literature. In section 2, we show what the diffusion equation and the fair price of the European option have in common. Please note that these considerations were originally presented as proof of the Black-Scholes Formula. In sections 3 and 4 we explain why valuation of the American option can be coming down to Free Boundary Problem. Note: It is very interesting that the same mathematical model is well known as Stefan Problem describing temperature distribution in a homogeneous medium undergoing a phase change. At the end, we introduce the Finite Difference Method which will be used to solve problem numerically. We will describe the main features of the method and highlight situations in which the method might fail. At the end we make a comparison with other, widely used methods. The contribution of this paper is in an empirical analysis of effectiveness of the Finite Difference Method applied to the considered problem. Beyond this, the article has a review character.
PL
Celem niniejszego artykułu jest przedstawienie zastosowań równań różniczkowych w wycenie opcji waniliowych. Na początku przedstawiamy główne założenia modelu Blacka-Scholesa z niezbędnymi komentarzami, których jako normę nie można łatwo znaleźć w literaturze. Następnie pokazujemy, co wspólnego ma równanie dyfuzji ze sprawiedliwą ceną opcji europejskiej. Należy pamiętać, że te rozważania zostały pierwotnie przedstawione jako dowód wzoru Blacka-Scholesa. Kolejno wyjaśniamy, dlaczego wycena opcji amerykańskiej może zostać sprowadzona do problemu ze swobodnym brzegiem, znanym jako problem Stefana opisującym propagacje temperatury w niejednorodnym ośrodku ulegającym przemianom fazowym. W końcowej części pracy wprowadzamy metodę różnic skończonych, która posłuży do numerycznego rozwiązania problemu. Opisujemy główne cechy metody pokazując potencjalne zagrożenia, które mogą pojawić się w wyniku zastosowania tej metody bez dokładnego zrozumienia jej struktury. Dokonujemy również porównania z innymi, szeroko stosowanymi metodami.
EN
The numerical analysis of thermal damage process proceeding in biological tissue during laser irradiation is presented. Heat transfer in the tissue is assumed to be transient and two-dimensional. The internal heat source resulting from the laser irradiation based on the solution of the diffusion equation is taken into account. The tissue is regarded as a homogeneous domain with perfusion coefficient and effective scattering coefficient treated as dependent on tissue injury. At the stage of numerical realization, the boundary element method and the finite difference method have been used. In the final part of the paper the results of computations are shown.
EN
The aim of this paper is to reduce the necessary CPU time to solve the three-dimensional heat diffusion equation using Dirichlet boundary conditions. The finite difference method (FDM) is used to discretize the differential equations with a second-order accuracy central difference scheme (CDS). The algebraic equations systems are solved using the lexicographical and red-black Gauss-Seidel methods, associated with the geometric multigrid method with a correction scheme (CS) and V-cycle. Comparisons are made between two types of restriction: injection and full weighting. The used prolongation process is the trilinear interpolation. This work is concerned with the study of the influence of the smoothing value (v), number of mesh levels (L) and number of unknowns (N) on the CPU time, as well as the analysis of algorithm complexity.
PL
Artykuł dotyczy sposobu wskazania rodzaju ruchu przy pomocy przetwarzania i analizy obrazu cyfrowego na przykładzie sekwencji przemieszczania się pyłu węglowego. Prezentowane podejście ma poszerzyć perspektywę zastosowania systemów wizyjnych w kontroli procesów spalania, podnosząc tym samym ich efektywność. Wyniki pokazały właściwe odzwierciedlenie rzeczywistego ruchu paliwa na dyskretnej powierzchni pól wektorowych przy zastosowaniu metody przepływu optycznego. Dołączenie metody numerycznej, do analizy pól wektorowych, pozwoliło wyznaczyć sześć rodzajów ruchu. Obserwacje, obejmujące fragmentu badanego zjawiska, wyczerpują wszystkie przypadki oceny przemieszczenia cząstek paliwa.
EN
The article refers to the method of indicating the type of movement by means the digital image processing and analysis on the example of a coal dust movement sequence. Presented approach is broaden the perspective of use the vision systems in the combustion processes control, thus increasing their effectiveness. The results showed a correct reflection of the real fuel movement in the discrete surface of the vector fields using the optical flow method. Including a numerical method for vector field analysis allowed to indicate six types of movement. Research, covering a fragment of the studied phenomenon, exhaust all cases of assessing the fuel particles displacement.
20
Content available Analysis of an anti-parallel memristor circuit
EN
The basic purpose of the present paper is to propose an extended investigation and computer analysis of an anti-parallel memristor circuit with two equivalent memristor elements with different initial values of the state variables using a modified Boundary Condition Memristor (BCM) Model and the finite differences method. The memristor circuit is investigated for sinusoidal supply current at different magnitudes – for soft-switching and hardswitching modes, respectively. The influence of the initial values of the state variables on the circuit’s behaviour is presented as well. The equivalent i-v and memristance-flux and the other important relationshipsof the memristor circuit are also analyzed.
PL
Podstawowym celem niniejszego artykułu jest zaproponowanie rozszerzonego badania i komputerowej analizy przeciwrównoległego układu memrystorowego z dwoma równoważnymi elementami memrystorowymi o różnych wartościach początkowych zmiennych stanu z wykorzystaniem zmodyfikowanego modelu Boundary Condition Memristor (BCM) i metody różnic skończonych. Obwód memrystorowy jest badany dla sinusoidalnego prądu zasilania o różnych wielkościach – odpowiednio dla trybów miękkiego przełączania i twardego przełączania. Przedstawiono również wpływ wartości początkowych zmiennych stanu na zachowanie obwodu. Analizowane są również równoważne charakterystyki prądowo-napięciowe zależność między memrystancją i strumieniem magnetycznym oraz inne ważne cechy obwodu memrystora.
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