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1

Zastosowanie języka programowania Python do modelowania rozprzestrzeniania się epidemii

Koziej Krzysztof, Kazimierska-Drobny Katarzyna

Studia i Materiały Informatyki Stosowanej

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2023

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T. 15, nr 2

10--17

PL

Niniejszy artykuł przedstawia numeryczne rozwiązania wybranych modeli rozprzestrzeniania się epidemii w języku programowania Python. Rozwiązania oparto na modelach epidemii SIS, SIR, SIRS oraz SEIR. Do rozwiązań numerycznych w języku Python wykorzystano biblioteki NumPy, SciPy oraz Matplotlib.

EN

This article presents numerical solutions of selected epidemic spread models in the Python programming language. The solutions were base on the SIS, SIR, SIRS and SEIR epidemic models. NumPy, SciPy and Matplotlib libraries were used for numerical solutions in Python.

2

Performance of four different nanoparticles in boundary layer flow over a stretching sheet in porous medium driven by buoyancy force

Ammani Kuttan B., Manjunatha S., Jayanthi S., Gireesha B. J.

International Journal of Applied Mechanics and Engineering

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2020

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Vol. 25, no. 2

1--10

EN

This contemporary work explores the theoretical analysis of energy transfer performance of distinct nanoparticles (silver, copper, aluminium oxide and titanium oxide) adjacent to a moving surface under the influence of a porous medium which is driven by the buoyancy force. A mathematical model is presented which is converted to similarity equations by employing similarity transformation. The condensed nonlinear equations were approximated by the iterative method called RKF 45th-order. The flow and energy transference characteristics are explained through graphs and tabulated values. The notable findings are: silver- water is an appropriate nanofluid for enhancing the thermal conductivity of the base fluid. Titanium oxide – water shows a lower fluid flow movement due to porosity.

3

On the evolution of solutions of Burgers equation on the positive quarter-plane

Hanaç Esen

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

237--248

EN

In this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; vt + vvx - vxx = 0, x>0, t>0, v(x,0) = u+, x>0, v(0,t) = ub, t>0, where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.

4

Numerical solutions of a steady 2-D incompressible flow in a rectangular domain with wall slip boundary conditions using the finite volume method

Ambethkar V., Srivastava M. K.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 2

5--16

EN

In this study, a finite volume method (FVM) is suitably used for solving the problem of a fully coupled fluid flow in a rectangular domain with slip boundary conditions. Numerical solutions for the flow variables, viz. velocity, and pressure have been computed. The FVM, with an upwind scheme, has been implemented to discretize the governing equations of the present problem. The well known SIMPLE algorithm is employed for pressure-velocity coupling. This was executed with the aid of a computer program developed and run in a C-compiler. Computations have been performed for unknown variables with Reynolds numbers (Re) = 50, 100, 250, 500, 750 and 1000. The behavior of steady-state solutions of velocity and pressure of the fluid along horizontal and vertical through geometric center of the rectangular domain have been illustrated. We observed that, with the increase of the Reynolds number, the absolute value of velocity components decreases whereas the pressure value increases.

5

MRS2016: rigid Moon Rotation Series in the relativistic approximation

Pashkevich V. V.

Artificial Satellites : Journal of Planetary Geodesy

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2017

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Vol. 52, No. 1

1--8

EN

The rigid Moon rotation problem is studied for the relativistic (kinematical) case, in which the geodetic perturbations in the Moon rotation are taken into account. As the result of this research the high-precision Moon Rotation Series MRS2016 in the relativistic approximation was constructed for the first time and the discrepancies between the highprecision numerical and the semi-analytical solutions of the rigid Moon rotation were investigated with respect to the fixed ecliptic of epoch J2000, by the numerical and analytical methods. The residuals between the numerical solution and MRS2016 in the perturbing terms of the physical librations do not exceed 80 mas and 10 arc seconds over 2000 and 6000 years, respectively.

6

A proposition of a torsional-bending vibrations modelling of combustion engines

Dąbrowski Z., Chiliński B., Pankiewicz J.

Journal of KONES

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2016

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Vol. 23, No. 4

71--77

EN

The article presents the problem of modelling coupled bending-torsional vibrations in crank systems. Because during the design of modern drive systems a growing number of phenomena are taken into account, model description of such vibrations has a practical meaning. Commonly used models of dynamics of systems assume the independence of torsional and bending vibrations, which leads to simultaneous analysis of transverse and angular vibrations. Further analysis is carried out with the use of superposition principle. Such an approach is justified in the case of quite rigid drive shafts, where vibrations are relatively small. Current trends in the design of reducing weight, reduction of toxic emissions and reducing fuel consumption, lead to the situation where shafts in crank systems become less stiff. Therefore, phenomena neglected earlier may have significant meaning. Analysis of couplings of transverse and torsional vibrations is so important that the occurrence of these phenomena usually leads to new critical states, which may be especially dangerous for engine operation. Considerations on the reasons of the occurrence and kinds of vibration couplings were presented in the introduction of the article. Further part of the article proposes the linear-bending model of the crankshaft, where transverse and angular displacements are dependent. It was tantamount to the assumption of linear relation between the vector of generalized co-ordinates and generalized forces occurring in the system. The next chapter presents the system of equations describing the dynamics of the crankshaft together with a discussion of the co-ordinate system used in the considerations. In addition, there were presented the results of numerical simulations in frequency domain confirming the conclusions taken from the analysis. The whole paper is concluded with synthetic conclusions on the formulated system of equations, simulations and the influence of the coupling on the dynamics of the whole crank system.

7

Determination of critical conditions of spontaneous combustion of coal in longwall gob areas

Cygankiewicz J.

Archives of Mining Sciences

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2015

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Vol. 60, no. 3

761--776

EN

Decades of experience in the fight against endogenous fire hazard in coal mines indicate a major influence of certain conditions in a given area of the mine on the possibility of fire occurrence, such as: susceptibility of coals to spontaneous combustion, oxygen content in the air incoming to the self-heating coal, conditions of heat exchange between the self-heating coal mass and the environment This paper presents a numerical method for determining the critical conditions of spontaneous combustion of coal in longwall gob areas, i.e. conditions under which spontaneous combustion may occur. It has been assumed that crushed coal in the gob has a shape of a flat layer adjacent from the roof and floor side to the rocks. Our considerations have been limited to coals containing small amount of moisture. A simple model of oxidation kinetics on the coal surface expressed by the Arrhenius equation has been adopted. His model assumes that oxidation rate is independent of the amount of oxygen absorbed by coal. The rate of reaction depends only on temperature, with the parameters of the equation changing after the coal Has reached the critical temperature. The article presents also a mathematical model of spontaneous heating of the coal layer in the gob area. It describes the heat balance in the coal as well as the oxygen and heat balance in the flowing gases. The model consists of a system of differential equations which are solved using numerical techniques. The developed computer program enables to perform the relevant calculations. In this paper, on the example of coal from a seam 405, we present the method for determining the following critical parameters of the layer of crushed coal: thickness of the layer, oxygen content in a stream of gases flowing through the layer and thermal conductivity of surrounding rocks.

PL

Wieloletnie doświadczenie w zakresie zwalczania zagrożenia pożarem endogenicznym w kopalniach wskazują na zasadniczy wpływ na możliwość zaistnienia pożaru niektórych warunków panujących w danym miejscu kopalni, takich jak: skłonność węgla do samozapalania zawartość tlenu w powietrzu dopływającym do zagrzewającego się węgla. W pracy przedstawiono numeryczną metodę wyznaczania warunków krytycznych samozapalania węgla w zrobach ścian, czyli takich po spełnieniu których może dojść do samozapalenia. Przyjęto, że znajdujący się w zrobach rozkruszony węgiel ma kształt płaskiej warstwy, sąsiadującej od stropu i spągu ze skałami. Rozważania ograniczono do węgli charakteryzujących się niską zawartością wilgoci. Przyjęto prosty model kinetyki utleniania na powierzchni węgla wyrażony równaniem Arrheniusa. Model ten zakłada niezależność szybkości utleniania od ilości tlenu pochłoniętego przez węgiel. Szybkość reakcji zależy jedynie od temperatury , przy czym występujące w równaniu parametry zmieniają się po osiągnięciu przez węgiel temperatury krytycznej. W artykule przedstawiono matematyczny model samozagrzewania warstwy węgla w zrobach opisujący bilans ciepła w węglu oraz bilans tlenu i bilans ciepła w przepływających gazach. Tworzące model układy równań różniczkowych cząstkowych rozwiązywane są metodami numerycznymi. Opracowany program komputerowy umożliwia wykonywanie stosownych obliczeń. W pracy, na przykładzie węgla z pokładu 405 , przedstawiono sposób wyznaczania warunków krytycznych warstwy rozkruszonego węgla: grubości warstwy, zawartości tlenu w przepływających przez warstwę gazach oraz przewodności cieplnej otaczających skał.

8

Irregular, Non-linear Vibration in the System with an Elastomer Element

Dyk J., Osiński J.

Machine Dynamics Research

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2013

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Vol. 37, No. 3

8-25

EN

The considered nonlinear system with an elastomer element is subjected to the harmonic excitation and undamped practically. The performed analysis confirms that there is possibility of irregular motion in the system, similarly as in reality chaotic motion in case of conservative and Hamiltonians systems. Such kind of motion seems to be specific to the non-linearity of elastomers for heavy loaded systems that are forced at the frequencies close to ones from the resonant zones.

9

Fractional Euler-Lagrange equations : numerical solutions and applications of reflection operator

Błaszczyk T., Ciesielski M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2010

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Vol. 9, nr 2

17-24

EN

In this work numerical solutions of fractional Euler-Lagrange equations describing free motion are considered. This type of equations contains a composition of left and right fractional derivatives. A reflection operator is applied to obtain relations between the Euler-Lagrange equations. In addition we verify the dependence between the respective numerical schemes using the same operator. In the final part of paper the examples of the numerical solutions are shown.

10

Stefan and Kolmogoroff models of solidification. Comparison of numerical solutions

Szopa R., Pawlak E., Siedlecki J.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2008

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Vol. 7, nr 1

185-192

EN

Problems connected with the mathematical description of pure metals solidification (macro approach) are often called the Stefan ones. The second generation models (micro/ macro approach) discussed in this paper base on a theory presented by Kolmogoroff (Mehl-Johnson-Avrami-Kolmogoroff models). Both macro and micro/macro problems can be analyzed using the numerical methods. The aim of investigations presented here was a comparison of numerical solutions obtained by use of macro and micro/macro approach. On a stage of numerical modelling the finite difference method has been applied.

11

Dynamic stability of fixed-fixed beam excited by a moving support

Buśkiewicz J.

Vibrations in Physical Systems

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2006

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Vol. 22

89--94

12

Knot Point Method in Extrapolation of Boundary Measurement Results

Pisarek J.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2006

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Vol. 11

67--78

EN

The new measurement technique that relies on distinguishing the ranges in object's space in which the searched quantities are approximated by multiparameter continuous functions has been introduced. The parameters of these functions are calculated by minimizing the functional describing the distance between different functions determined at the same points and distance between numerical solution and experimental results and assumed’a priori’ theoretical guidelines. Functions’ spaces might be determined in any way and might be changed during the computation. This method is especially usefull in extrapolation of the boundary measurement results on the object interior and in joining the local and whole object (global) solutions.

13

Limitation of the single-domain numerical approach: Comparisons of analytical and numerical solutions for a forced convection heat transfer problem in a composite duct

Kuznetsov A. V.

Computer Assisted Mechanics and Engineering Sciences

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2003

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Vol. 10, No. 1

34-43

EN

The aim of this paper is to establish the bounds of applicability of the single-domain numerical approach for computations of convection in composite porous/fluid domains. The large number of papers that have utilized this numerical approach motivates this research. The popularity of this approach is due to the simplicity of its numerical formulation. Since the utilization of the single-domain numerical approach does not require the explicit imposing of any boundary conditions at the porous/fluid interface, the aim of the this research is to investigate whether this method always produces accurate numerical solutions.

14

Rozwiązania numeryczne rozkładów ciśnień dla turbulentnego przepływu ferrosmaru w szczelinie poprzecznego łożyska ślizgowego

Miszczak A.

Tribologia

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2002

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nr 2

461-474

PL

Zjawisko przepływów turbulentnych w szczelinie poprzecznych łożysk ślizgowych zachodzi wówczas, gdy liczba Taylora Ty = Re ź (pierwiastek kwadratowy z psi) > lub = 41,2 lub też zmodyfikowana liczba Reynoldsa Re* = Re ź psi > lub = 2. Ma to miejsce w szybkoobrotowych łożyskach ślizgowych, np. łożyskach napędów dysków HDD, w łożyskach w elektrowniach atomowych czy pojazdach kosmicznych, gdzie dodatkowo może występować pole magnetyczne a czynnikiem smarującym może być ferrosmar. Czynnik smarujący o własnościach magnetycznych, tzw. ferrosmar, jest to układ koloidalny cząstek magnetycznych najczęściej Fe304 lub Gd2O2 z substancją rozpraszającą taką jak: olej mineralny lub syntetyczny, węglowodory, fluoropochodne węglowodorów, woda itp. Wyniki numeryczne przedstawione w niniejszej pracy wykazują, że istotne są również rozkłady ciśnienia dla łożysk o skończonej długości przy założeniu turbulentnego przepływu dla izotermicznej ferrocieczy w polu magnetycznym o wartości indukcji magnetycznej bliskiej punktowi nasycenia magnetycznego ferrocieczy. W przypadku uwzględniania nieizotermicznego turbulentnego przepływu autor przewiduje dalsze istotne zmiany wartości ciśnienia. Analiza nieustalonego przepływu ferrosmaru [L. l, 3] dla niezależnej od czasu gęstości rho jest przeprowadzana z użyciem równania ciągłości i równań zachowania pędu. Prędkość cieczy w stanie turbulentnym charakteryzuje się pulsacją wokół średnich wartości. Pulsacja ta jest wywołana przez wymianę energii kinetycznej pomiędzy małymi masami płynu. Celem niniejszej pracy i wkładem autora do rozpatrywanego problemu jest: - numeryczno-analityczne modelowanie pola magnetycznego w szczelinie cylindrycznego łożyska ślizgowego pod kątem przeprowadzenia obliczeń wartości ciśnienia hydrodynamicznego, - numeryczne wyznaczenie rozkładów ciśnień dla różnych przypadków łożyska ślizgowego o skończonej długości dla przepływów turbulentnych ferrosmaru w polu magnetycznym, - przeprowadzenie analizy porównawczej uzyskanych wyników dla różnych wariantów rozwiązań (Tipei, Wierzcholski) [L. 4, 5].

EN

The phenomenon of turbulent flows within the gap of journal slide bearings has a place when Taylor number Ty = Re ź (square root of psi) > or = 41.2 or modified Reynolds number Re* = Re ź psi > or = 2. It occurs in high-speed slide bearings e.g. HDD (hard disk drive), nuclear power plants or spacecrafts, where additionally magnetic field may act and ferrofluids are lubricant agent. An lubricant agent having magnetic properties, so called ferrolubricant, is the colloidal system of magnetic particles, most often Fe304 or Gd202, with dispersing substance e.g mineral or synthetic oil, hydrocarbons, fluorocarbons, water etc. The author presents the results of numerical calculations of pressure distributions for journals of finite length at assumption of isothermal flow of ferrofluid in the magnetic field of the value of magnetic induction approaching the point of magnetic saturation of the ferrofluid. The pressure distributions are comparable to each other both in the case of presence of magnetic field and lack of it, for various values of relative eccentricity and dimensionless length of the journal. The analysis of transient flow of ferrolubricant [L. 1, 3] for time independent density rho is realised by means of the continuity equation and the equations of conservation of momentum. The velocity of fluid in the turbulent state characterises by oscillation of its value in the vicinity of the mean value. That oscillation is caused by interchange of kinetic energy between small volumes of the fluid. The aim of this work and the author's own contribution to the considered problem is: - Numerically-analytical modelling of magnetic field within the gap of the cylindrical slide bearing in the aspect of calculating the value of hydrodynamic pressure, - Numerical determining of pressure distribution for different cases of slide bearings of finite length and turbulent flows of ferrolubricant, - Carrying out the comparable analysis of results obtained for different solutions (Tipei, Wierzcholski) [L. 4, 5].

15

Layered orthotropic plate bending element of discrete Kirchhof theory

Belevicius R., Michnevic E., Rusakevicius D.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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1998

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z. 7

29-36

EN

The new finite element of multilayered built up with an arbitrary series of layers plate for plate bending problem is formulated on a base f widely used, effective finite element of Discrete Kirchhof Theory (DKT). The material of each layer is supposed to be different and orthotropic . Triangular element has 6 d.o.f.'s a t eac h o f 3 nodal points : 3 displacements and 2 rotations about coordinate axes. The 6th, fictitious rotation about axis perpendicular to the element is also introduced due to numerical requirements. The element takes into account all th e in-plane/out-of-plate effects except the shear . The element could find an application in the slab bending problems or in the plate, where the shear influence could be neglected , bending problems . The numerical examples are presented . Present solutions are compared with available analytical and numerical solutions.

PL

W prac y przedstawiono nowy płytowy wielowarstwowy element skończony dla zagadnień zginania zbudowany na bazie dyskretnej teorii Kirchhoffa. Materiał każdej warstwy jest inny i ortotropowy . Element trójkątny posiada 6 stopni swobody każdego węzła: 3 przemieszczenia, 2 obroty wokół osi układu. Szósty , fikcyjny , dookoła osi prostopadłej do elementu wprowadzono ze względu na wymogi obliczeń numerycznych . Element uwzględnia wszystkie efekty wewnętrzne i zewnętrzne oprócz ścinania . Zaprezentowano wynik i obliczeń numerycznych , które podano z rozwiązaniam i analitycznym i i numerycznymi .