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1
Content available Fourier Image Methods for Evolution Equations of Deep-Water Waves
Jankowski Jan, Leszczyński Henryk
TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk
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2019
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Vol. 23, No 3
261--268
EN
The velocity potential of the fluid satisfies the Laplace equation with nonlocal boundary conditions on a free surface. This differential problem is transformed to an evolution equation in Fourier variables. The Fourier transform images of boundary functions are approximated by Picard's iterations and the method of lines on meshes related to roots of Hermite polynomials. Due to convolutions of sine and cosine functions the integral terms of Picard's iterations reveal unexpected instabilities for wave numbers in a neighborhood of zero.
2
Content available Solving nonlinear thermal problems of friction by using method of lines
Och E.
Acta Mechanica et Automatica
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2015
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Vol. 9, no. 1
33--37
EN
One-dimensional heat conduction problem of friction for two bodies (half spaces) made of thermosensitive materials was considered. Solution to the nonlinear boundary-value heat conduction problem was obtained in three stages. At the first stage a partial linearization of the problem was performed by using Kirchhoff transform. Next, the obtained boundary-values problem by using the method of lines was brought to a system of nonlinear ordinary differential equations, relatively to Kirchhoff’s function values in the nodes of the grid on the spatial variable, where time is an independent variable. At the third stage, by using the Adams's method from DIFSUB package, a numerical solution was found to the above-mentioned differential equations. A comparative analysis was conducted (Och, 2014) using the results obtained with the proposed method and the method of successive approximations.
3
Content available Optimizing the Structure of Vector Bend and Strain Sensor on the Base of Three-Core Microstructured Fiber
Goncharenko I., Marciniak M., Konojko A., Reabtsev V.
Journal of Telecommunications and Information Technology
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2012
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nr 4
63-69
EN
In the paper the optical sensor allowing measuring a direction, values and localization of bends and stresses in building structures is described. The sensitive element of the sensor is the microstructured fiber with three cores. The use of three-core fiber makes it possible to define the direction of deformation. Distribution of mode fields in fiber cores depending on fiber structure and bend value and direction is analyzed. The optimization of the sensitive element parameters depending on the application is proposed.
4
Content available remote Solutions of incompressible Navier-Stokes equations with the artificial compressibility method for two- and three-dimensional shear-driven cavity flows
Kosma Z.
International Journal of Applied Mechanics and Engineering
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2012
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Vol. 17, no. 1
113-125
EN
An efficient method for simulating laminar flows in complex geometries is presented. The artificial compressibility method was applied to solve two- and three-dimensional Navier-Stokes equations in primitive variables on Cartesian grids. Two numerical approaches were proposed in this work, which are based on the method of lines process in conjunction with transfer of all the variables from the boundaries to the nearest uniform grid knots. Initial value problems for the systems of ordinary differential equations for pressure and velocity components were computed using the one-step backward-differentiation predictor-corrector method or the Galerkin-Runge-Kutta method of third order. Some test calculations for laminar flows in square, half-square, triangular, semicircular, cubic, half-cubic, half-cylinder and hemisphere cavities with one uniform moving wall were reported. The present results were compared with the available data in the literature and the Fluent solver numerical simulations.
5
Content available remote Finite difference schemes and the method of lines for solving incompressible viscous flow problems
Kosma Z., Motyl P.
International Journal of Applied Mechanics and Engineering
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2011
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Vol. 16, no 1
15-35
EN
For the determination of viscous incompressible flows a pure stream-function formulation for the fourth-order equation, the artificial compressibility method, and velocity correction method are employed. Test calculations are performed for various flows inside square, triangular, semicircular and cubic cavities with one uniform wall, the backward-facing step, double bent channels, the flow around an aerofoil at large angle of attack and for flows over models of buildings. Some complex geometrical configurations can be decomposed into a set of simpler subdomains. A practical methodology for the computation of the Navier-Stokes equations in arbitrarily complex geometries is also considered. The simplest approach for specifying boundary conditions near curved or irregular boundaries is to transfer all the variables from the boundaries to the nearest grid knots.
6
Content available Differential difference inequalities related to parabolic functional differential equations
Netka M.
Opuscula Mathematica
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2010
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Vol. 30, no. 1
95-115
EN
Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the method of lines is given. Nonlinear estimates of the Perron type for given operators with respect to functional variables are used. Results obtained in the paper can be applied to differential integral problems and to equations with deviated variables.
7
Content available Modelowanie wypływu filmu spomiędzy kropel podczas ich koalescencji w obecności surfaktantów
Grabias M., Podgórska W.
Inżynieria i Aparatura Chemiczna
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2010
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Nr 2
45-46
PL
W pracy zbadano wpływ surfaktantów na wypływ filmu spomiędzy kropel, stanowiący zwykle najwolniejszy etap koalescencji. Rozpatrywany model matematyczny rozwiązano metodą linii. Rozwiązanie modelu dla różnych parametrów pokazało wpływ stosunku lepkości faz i powierzchniowej liczby Pedela na przebieg wypływu filmu. Wykazano liniową zależność czasu wypływu filmu od stosunku lepkości faz oraz spadek, a następnie wzrost i ustalenie się czasu wypływu w miarę wzrostu powierzchniowej liczby Pecleta.
EN
The work investigated influence of surfactants on film drainage, usually the slowest stage of coalescence. The considered mathematical model was solved using the method of lines. Solution of the model with various parameters showed influence of phase viscosity ratio and the surface Peclet number on film drainage. Drainage time tended to drop, then increased, finally leading to a constant value with increasing the surface Peclet number, and increased linearly with increasing the phase viscosity ratio.
8
Content available remote Fast Algorithms for Calculations of Viscous Incompressible Flows Using the Artificial Compressibility Method
Kosma Z.
TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk
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2008
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Vol. 12, No 3-4
273-287
EN
An artificial compressibility method is designed to simulate stationary two-and threedimensional motions of a viscous incompressible fluid. A standard method of lines approach is applied in this contribution. A partial differential equation system is discretized in space by secondorder finite-difference schemes on uniform computational grids, and the time-variable is preserved as continuous. Initial value problems for systems of ordinary differential equations for pressure and velocity components are computed using the Galerkin-Runge-Kutta method of third order. Some test calculations for laminar flows in square, cubic, triangular and semicircular cavities with one uniform moving wall and double bent channels are reported.
9
Content available remote Fast Algorithms For Calculations of Viscous Incompressible Flows Using the Velocity Correction Method
Kosma Z., Motyl P.
TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk
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2008
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Vol. 12, No 3-4
289-301
EN
The velocity correction method is designed to simulate stationary and non-stationary, two- and three-dimensional motions of a viscous incompressible fluid. The basic assumption of this method consists in splitting the velocity and the pressure fields and calculations are performed in two steps. In the first step, a tentative velocity field is determined by simplified equations for momentum conservation. In the second step the Neumann problem for the Poisson equations is solved to obtain the computational pressure, and the velocity components are corrected. A standard method of lines approach and the two grids method are applied in this contribution. Some test calculations for laminar and transitional flows in square and cubic cavities with one moving wall as well as in a backward-facing step are reported.
10
Content available Analysis of propagation of orthogonally polarized supermode in straight and curved multicore microstructured fibres
Goncharenko I. A., Marciniak M.
Journal of Telecommunications and Information Technology
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2007
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nr 4
63-71
11
Content available remote Method of lines for the incompressible Navier-Stokes equations in the stream-function formulation
Kosma Z.
TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk
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2005
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Vol. 9, No 1
17-35
EN
The aim of this paper is to simulate the laminar motion of viscous incompressible fluid and the transition between the laminar and the turbulent state in simply connected domains. The developed numerical algorithms are based on the solution of an initial-boundary value problem for the full incompressible Navier-Stokes equations, written in the form of a fourth-order equation for the stream function. The spatial derivatives and the boundary conditions are discretized on uniform grids by means of sixth-order compact schemes together with fourth-order finite-difference formulas, while the continuity of the time variable is preserved. The resulting system of ordinary differential equations has been integrated using the backward-differentiation predictor-corrector method. The efficiency of the numerical algorithms is demonstrated by solving two problems of viscous liquid plane flows in a square driven cavity and a backward-facing step. Calculations for the cavity flow configuration have been obtained for Reynolds numbers ranging from Re=100 to Re=30000 on uniform 50×50 and 100×100 grids. Calculations for the backward-facing step have been made for Re ≤ 3000 with channel lengths, L, within the range 10-30, on 30L×30 uniform grids. The computed stream-function contours and velocity fields have been compared with numerical results reported in the literature.
12
Content available Photonic crystal fibre characterisation with the method of lines
Goncharenko I.A., Marciniak M.
Journal of Telecommunications and Information Technology
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2004
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nr 1
106-111
EN
Photonic crystal fibres are longitudinally uniform fibres in which in lateral directions periodic refractive index changes occur. Two basically different light guiding mechanisms occur in crystal fibres: index guiding and bandgap guiding. In the paper different modelling methods have been evaluated when applied to photonic crystal fibres. In particular, the method of lines has been shown to be effective and reliable for both classes of photonic crystal fibres. High accuracy results for optical field distribution and dispersion characteristics in a photonic crystal fibre have been achieved with the method of lines.
13
Content available remote Differential difference inequalities generated by infinite systems of parabolic functional differential equations
Kozieł S.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2004
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Vol. 44, nr 1
99--126
EN
The paper deals with the initial boundary value problem for infinite systems of parabolic functional differential equations. A comparison theorem concerning infinite systems of differential difference inequalities generated by the original problem is proved. The comparison result is used in an existence theorem and in investigating the stability of the numerical method of lines. A theorem on the error estimate of the metod is given.
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