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1

Comparison of heat transfer phenomena for two different cryopreservation methods : slow freezing and vitrification

Skorupa Anna, Piasecka-Belkhayat Alicja

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 1

57--69

EN

The purpose of the research is to prepare a mathematical and numerical model for the phenomenon of heat transfer during cryopreservation. In the paper, two popular methods, slow freezing and vitrification, are compared. Furthermore, the basic model of thermal processes is supplemented by the phenomenon of phase transitions. To determine the temperature distribution during cryopreservation processes, one uses the heat transfer equation proposed by Pennes. An integral part of the energy equation is the substitute thermal capacity (STC) performed according to the concept named one domain method (fixed domain method), The numerical model is developed using the finite difference method (FDM) connected with directed interval arithmetic. The final part of the article contains the results of numerical simulations.

2

Simulation of infeasible instruments in a sound synthesizer - implementation and control

Pluta Marek

Vibrations in Physical Systems

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2022

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Vol. 33, nr 2

art. no. 2022206

EN

Sound synthesis using mathematical modelling of musical instruments is a method particularly well suited for live performance using a physical controller. Depending on model complexity, it may be able to reproduce various subtle phenomena related to excitation and real time control of an instrument, providing an intuitive tool for a musician. A variant of physical modelling synthesis, referred to as the simulation of infeasible instruments, uses a model of an object that does not have a physical counterpart. Such model has some properties of a real object, which makes it still intuitive for a musician. However, other features, such as geometry, or material properties, are intentionally altered in such manner, that it could not function in reality. These infeasible features introduce new properties to the sound it produces. The study presents a few such models with a discussion regarding their implementation and control issues in a real-time sound synthesizer.

3

Numerical solution of MHD channel flow in a porous medium with uniform suction and injection in the presence of an inclined magnetic field

Chutia Muhim

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 2

5--13

EN

In this paper, the steady fully developed MHD flow of a viscous incompressible electrically conducting fluid through a channel filled with a porous medium and bounded by two infinite walls is investigated numerically for the cases (i) Poiseuille flow and (ii) Couette-Poiseuille flow; with uniform suction and injection at the walls in the presence of an inclined magnetic field. The Brinkman equation is used for the flow in the porous channel and solved numerically using the finite difference method. Numerical results are obtained for velocity. The effects of various dimensionless parameters such as Hartmann number (M), suction/injection parameter (S), permeability parameter (α) and angle of inclination (θ) on the flow are discussed and presented graphically.

4

Numerical analysis of thermal damage and oxygen distribution in laser irradiated tissue

Jasiński Marek

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 2

51--62

EN

A numerical analysis of the thermal damage process that proceeds 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 optical diffusion equation is taken into account. Changes in tissue oxygen distribution resulting from temperature changes are analyzed using the Krogh cylinder model with Michaelis-Menten kinetics. A Hill model was used to describe the oxyhemoglobin dissociation curve. At the stage of numerical realization, the boundary element method and the finite difference method have been applied.

5

Numerical analysis of biological tissue heating using the dual-phase lag equation with temperature – dependent parameters

Majchrzak Ewa, Stryczyński Mikołaj

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 3

85--98

EN

The dual-phase lag equation is formulated for the case when the thermophysical parameters occurring in this equation are temperature-dependent. The axial-symmetrical domain of biological tissue heated by an external heat source is considered. The problem is solved using the implicit scheme of the finite difference method. At the stage of numerical computations, the analytical relationships taken from the literature describing changes in parameters are taken into account.

6

Mathematical modeling of the phenomena that occur in a biological tissue containing photosensitizer

Jasiński Marek, Zadoń Maria

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 4

40--51

EN

The aim of the study is to analyze photothermal and photochemical phenomena that occur during photodynamic therapy (PDT). In this type of therapy, under the influence of the laser, reactions take place related to the transformation of triplet oxygen form into its singlet form which is cytotoxic to the tissue. The increases in temperature resulting from the laser-tissue interaction during PDT are not big; however, they can lead to changes in tissue perfusion, which can affect oxygen delivery to the tissue. The proposed model uses optical diffusion equation, Pennes bioheat transfer equation, and reactions equations for PDT. The main findings of the analysis show the impact of temperature on the value of the perfusion coefficient and triplet oxygen distributions at the end of the treatment procedure.

7

Finite difference method for the fractional order pseudo telegraph integro-differential equation

Modanli Mahmut, Ozbag Fatih, Akgül Ali

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 1

41--54

EN

The main goal of this paper is to investigate the numerical solution of the fractional order pseudo telegraph integro-differential equation. We establish the first order finite difference scheme. Then for the stability analysis of the constructed difference scheme, we give theoretical statements and prove them. We also support our theoretical statements by performing numerical experiments for some fractions of order α.

8

An extended finite difference method for singular perturbation problems on a non-uniform mesh

Swarnakar D., Kumar V. Ganesh, Soujanya G. B. S. L.

International Journal of Applied Mechanics and Engineering

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2022

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Vol. 27, no. 1

203--214

EN

An extended second order finite difference method on a variable mesh is proposed for the solution of a singularly perturbed boundary value problem. A discrete equation is achieved on the non uniform mesh by extending the first and second order derivatives to the higher order finite differences. This equation is solved efficiently using a tridiagonal solver. The proposed method is analysed for convergence, and second order convergence is derived. Model examples are solved by the proposed scheme and compared with available methods in the literature to uphold the method.

9

Free vibrations of Kirchhoff plates considering plate variable thickness and interaction with water including water boundaries

Lenartowicz Agnieszka, Guminiak Michał, Przychodzki Maciej

Vibrations in Physical Systems

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2021

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Vol. 32, nr 2

art. no. 2021208

EN

The natural vibrations of thin (Kirchhoff-Love) plates with constant and variable thickness and interaction with water are considered in the paper. The influence of the water free surface on natural frequencies of the coupled water-plate system is analysed too. The Finite Element Method (FEM) and the Finite Difference Method (FDM) are used to describe structural deformation and the Boundary Element Method (BEM) is applied to describe the dynamic interaction of water on a plate surface. The plate inertia forces are expressed by diagonal or consistent mass matrix. The water inertia forces are described by fully-populated mass matrix which is obtained directly from the theory of double layer potential.

10

A new numerical scheme for solving the two dimensional fractional diffusion equation

Altan Koç Dilara, Gülsu Mustafa

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 2

5--16

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.

11

Elastic shakedown limit of a steel lattice girder

Brzuzy Aneta

Inżynieria Bezpieczeństwa Obiektów Antropogenicznych

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2021

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Nr 4

12--18

EN

This paper presents a solution for the problem concerning the behaviour of a steel lattice girder subjected to dynamic load pulses. The theory of shakedown is used in the analysis. It is assumed that such loads cause a non-elastic response which includes dissipation of energy causing deformations and residual forces developed in the structural members of the girder. At a certain intensity of these forces, the girder can react to subsequent load pulses without further dissipation of energy, behaving in the elastic region after shakedown. This condition is referred to as adaptation of the structure to assumed cyclic loading. Elastic shakedown limit is determined through a direct analysis of the girder's dynamic behaviour, i.e. by checking if energy dissipation decreases with loading cycles. This gives the number of load applications after which no further increase of the energy dissipation is observed. The existing permanent deformations persist and residual forces remain in the same state. The analysis takes into account the possibility that compressed members can buckle which may result in non-elastic, longitudinal and transverse vibrations of these members. Non-linear geometry of members is taken into account. Then a perfectly elastic-viscoplastic model of the material is used. The main goal is to determine the state of the non-elastic movements of the girder joints and the residual internal forces developed in the girder members after each load application. The values obtained in this way serve as the basis for describing the next loading cycle. It is possible to use the approach presented in the paper to evaluate the effects of accidental loads. Then it is checked whether a small number of repetitions of accidental load would result in exceeding the serviceability limit state criteria of the maximum permanent deformation or displacement and/or strain amplitudes. If so, the magnitude of accidental load is greater than the elastic shakedown limit. Some examples are given to illustrate the application of the theory of shakedown.

12

Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations

Appadu Appanah Rao, Kelil Abey Sherif

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

377--409

EN

The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.

13

Investigation of fault zone induced site effect in the İzmit basin, Turkey

Firtana-Elcomert Karolin, Kocaoğlu Argun

Acta Geophysica

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2021

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Vol. 69, no. 6

2069--2083

EN

The seismic hazard in the İzmit basin, located in Marmara region of western Turkey, is high due to the northern branch of the North Anatolian Fault (NAF) and the potential ground motion amplification that may be caused by local site conditions, sedimentary basin effect as well as fault zone (FZ) induced site effect resulting from the generation of guided waves. In this study, we elaborate the relevance of the FZ-induced site effect in the İzmit basin along a 16.5-km-long N-S profile across the basin and perpendicular to the NAF by time and frequency domain analysis of waveforms obtained from two-dimensional (2D) simulations of viscoelastic wave propagation for a double-couple source at 14 km depth using a reference (basin-only) model and three basin-with-fault models: shallow (6 km), intermediate (12 km) and deep (19 km) FZ models. Our results show that the FZ-induced site effect within and near the northern branch of NAF in the İzmit basin can be very prominent with amplification of about 5–10 in the frequency range of 0.05–4 Hz and about 20 at frequencies above 2 Hz, respectively. We obtain the most dramatic results for the deep FZ model causing shear- and surface-wave amplification of about 15 at frequencies higher than 2.5 Hz for the distances between 6 and 13 km.

14

Free vibrations of iso- and orthotropic plates considering plate variable thickness and interaction with water

Lenartowicz Agnieszka, Guminiak Michał

Vibrations in Physical Systems

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2020

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Vol. 31, nr 2

art. no. 2020216

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.

15

Wpływ materiałów budowlanych oraz ich konduktywności na wartości natężenia pola elektrycznego

Choroszucho Agnieszka, Druć Gabriela, Orzechowski Damian

Napędy i Sterowanie

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2020

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R. 22, nr 9

78--85

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.

16

The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction diffusion boundary value problems

Duru Hakkı, Güneş Baransel

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 4

45--56

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.

17

Numerical simulation of channel flow over a skewed equilateral cavity

Kamel Abanoub G., Haraz Eman H., Hanna Sarwat N.

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 3

29--43

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.

18

Numerical analysis of the temperature impact to the oxygen distribution in the biological tissue

Jasiński Marek

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 3

17--28

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.

19

Modeling of heat transfer and fluid flow in a rectangular channel with an obstacle

Stryczyński Mikołaj, Majchrzak Ewa

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 2

121--132

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.

20

Influence of the ballast resistance on the stability of Continuous Welded Rail

Pokropski Dominik

Inżynieria Bezpieczeństwa Obiektów Antropogenicznych

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2020

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Nr 4

1--13

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.