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1
Content available The Sturm-Liouville eigenvalue problem - a numerical solution using the control volume method
Siedlecki J., Ciesielski M, Błaszczyk T.
Journal of Applied Mathematics and Computational Mechanics
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2016
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Vol. 15, nr 2
127--136
EN
The solution of the 1D Sturm-Liouville problem using the Control Volume Method is discussed. The second order linear differential equation with homogeneous boundary conditions is discretized and converted to the system of linear algebraic equations. The matrix associated with this system is tridiagonal and eigenvalues of this system are an approximation of the real eigenvalues of the boundary value problem. The numerical results of the eigenvalues for various cases and the experimental rate of convergence are presented.
2
Content available Simulations of thermal processes in tooth proceeding during cold pulp vitality testing
Ciesielski M., Mochnacki B., Siedlecki J.
Acta of Bioengineering and Biomechanics
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2016
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Vol. 18, no. 3
33--41
EN
Purpose: This paper deals with the mathematical modeling of the thermal processes occurring in the tooth, during a very brief contact (a few seconds) with a very cold liquid on a part of the tooth crown. In this way one can simulate a heat transfer in tooth proceeding during a dental diagnostic test - pulp vitality testing. The impact of rapid ambient thermal changes acting on the tooth can cause toothache. Methods: The mathematical model: a system of partial differential equations with initialboundary conditions (the axially-symmetrical problem) and their numerical solutions using the control volume method is discussed. Results: Simulation results of the kinetics of the temperature changes inside the tooth are presented. The example of the control volume mesh (using the Voronoi polygons) well describing the shape of a molar tooth is given. Conclusions: The simulation results (the temperature distribution in the tooth at any moment of the simulation time and the kinetics of temperature variation at the points of the considered tooth domain) can help dentists in the selection of an appropriate method of treatment.
3
Content available Transformation of the second order boundary value problem into integral form : different approaches and a numerical solution
Siedlecki J., Ciesielski M., Błaszczyk T.
Journal of Applied Mathematics and Computational Mechanics
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2015
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Vol. 14, nr 3
103--108
EN
In this paper we present different approaches to the transformation of the second order ordinary differential equation, with respect to adequate boundary conditions, into integral equations. The obtained equations are Fredholm integral equations of the second kind. Next, a numerical method based on quadrature methods has been proposed to get an approximate solution of these equations.
4
Content available Application of the control volume method in numerical modeling of cryosurgery treatment
Szopa R., Siedlecki J.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 4
131--136
EN
In the paper the problems of cryosurgery treatment are discussed. In particular the heat exchange between external cylindrical cryoprobe and human tissue is analyzed. The cryoprobe tip temperature changes from the initial temperature to the minimal one and back in a linear way (broken line). Thermophysical parameters determining the thermal properties of tissue can be treated as the temperature-dependent functions. The freezing model bases on the approach called ‘a one domain method’, and in the energy equation called a substitute thermal capacity appears. At the stage of computations the control volume method (CVM) is used. In the final part of paper the examples of computations are presented.
5
Content available An approximation of the fractional integrals using quadratic interpolation
Błaszczyk T., Siedlecki J.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 4
13--18
EN
In this paper we present a numerical scheme to calculations of the left fractional integral. To calculate it we use the fractional Simpson’s rule (FSR). The FSR is derived by applying quadratic interpolation. We calculate errors generated by the method for particular functions and compare the obtained results with the fractional trapezoidal rule (FTR).
6
Content available Simulations of thermal processes in a restored tooth
Siedlecki J., Ciesielski M.
Journal of Applied Mathematics and Computational Mechanics
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2013
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Vol. 12, nr 4
103--108
EN
In this paper the analysis of the temperature distribution in restored tooth is considered. The aim of the research is to evaluate the impact of temperature changes in the pulp layer of tooth under the influence of rapid temperature changes in the ambient fluid. The axially-symmetrical model of the molar tooth (in vitro) without filling and with fillings (amalgam and composite resin) is analyzed. The equations of mathematical model are solved numerically using the control volume method. Finally, some examples of simulations are presented.
7
Content available Sensitivity analysis of the crystallization model with respect to cooling conditions
Szopa R., Tuzikiewicz W., Siedlecki J.
Journal of Applied Mathematics and Computational Mechanics
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2013
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Vol. 12, nr 3
115--122
EN
In the paper the pure metal crystallization process is considered. The cooling conditions of the metal domain are determined by the mould dimensions and thermophysical parameters of the mould material (volumetric specific heat and thermal conductivity). The mathematical model of heat transfer processes proceeding in the system metal – mould is created by the system of heat diffusion partial differential equations corresponding to the sub-domains considered supplemented by the adequate geometrical, physical and boundary-initial conditions. The sensitivity model determining the perturbations of thermal processes due to the perturbations of cooling processes can be constructed using the direct approach (differentiation with respect to parameters analyzed) or using the differential quotients. Both approaches will be discussed in the paper presented.
8
Content available remote Substitute thermal capacity of binary alloys : review of hypotheses
Siedlecki J., Tuzikiewicz W.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2012
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Vol. 11, nr 4
121-129
EN
In the paper the problems connected with the mathematical modeling of alloys solidification (macro scale) are presented. The similar lecture was delivered by the authors during the IV Conference on Mathematical Modelling in Physics and Technique, Kleszczów, 28-30 June 2012. In particular the one domain method (fixed domain approach) in which the parameter called 'a substitute thermal capacity' appears will be discussed. The one domain method bases on the Fourier equation describing the thermal processes in the whole conventionally homogeneous casting domain. The substitute thermal capacity (STC) can be defined in the different ways. The first group of definitions results from the differentiation of the function determining the local volumetric solid state fraction in the neighborhood of the point considered, while the second one is based on the direct assumption of the STC form, at the same time this function should fulfill the condition resulting from the change of alloy physical enthalpy during the phase change. The details of these approaches will be presented in the Sections 2 and 3.
9
Content available remote Mathematical model for thermal processes in photopolymerization
Siedlecki J., Ciesielski M.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2012
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Vol. 11, nr 4
113-119
EN
In the paper the mathematical model for interaction of an intense light source on a photosensitive monomer/polymer mass with photoinitiator component is presented. Heat is released during the photopolymerization process. The thermal and photopolymerization processes occurring in the monomer/polymer mass which is located in the porcelain form are analysed. The equations of mathematical model in axially-symmetrical domain are solved numerically. In the work the effect of light dose on the processes is investigated. In the final part of this paper, the examples of computations are presented.
10
Content available remote Sensitivity of tissue freezing process on changes of time-dependent cryoprobe tip cooling rate
Mochnacki B., Siedlecki J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2011
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Vol. 10, nr 2
191-197
EN
In the paper, the problems of thermal interactions between skin tissue and a cylindrical cryoprobe tip are discussed. The cryoprobe tip temperature is time-dependent and its changes from the initial temperature to the final one and back are assumed in the form of a broken line (successive sectors correspond to cooling and heating processes or cooling and heating ones divided by a horizontal sector (a constant freezing temperature)). The aim of the considerations resolves itself in the analysis of dependences between the tip cooling (heating) rate and the course of the freezing process. To solve the problem, the direct approach of sensitivity analysis is used. The mathematical model of freezing is based on the energy equation corresponding to the 'fixed domain method' in which a parameter called 'the substitute thermal capacity' appears. The sensitivity model results from the differentiation of energy equation and boundary-initial conditions with respect to the cooling (heating) rate. At the stage of computations, the finite differences method (FDM) is used. In the final part of paper, examples of computations are presented.
11
Numerical model of solidification using the control volume method
Ciesielski M., Siedlecki J., Mochnacki B.
Zeszyty Naukowe. Mechanika / Politechnika Opolska
|
2010
|
z. 97
57-58
EN
The problems of control volume method (CVM) application in the numerical modeling of thermal processes proceeding in a system casting-mould are discussed (a 2D problem is taken into account).The shape of control volumes (Thiessen polygons) assures the good approximation of heat transfer processes proceeding in domain considered. In the final part of the paper the example of computation is shown.
12
Content available remote Integration on hyperspheres in RN
Madej T., Biernat G., Siedlecki J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2010
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Vol. 9, nr 2
135-144
EN
The subject of this paper are the hyperspheres in n-dimensional Euclidean space, which are the intersection of sphere and few planes. The paper concerns only the spheres with the center in the origin of coordinate system and the planes crossing through this point. Hypersphere parametrization and some integration formulas will be shown.
13
Content available remote Influence of cryoprobe cooling/heating rate on the course of freezing process
Mochnacki B., Siedlecki J., Tuzikiewicz W.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2010
|
Vol. 9, nr 2
197-202
EN
The methods of sensitivity analysis are used in order to observe the influences of changes of the cryoprobe tip cooling/heating rate on a course of biological tissue freezing process. The temperature history of tip temperature is assumed in a form of broken line. The direct variant of sensitivity analysis is here applied. At the stage of numerical modeling the control volume method is used. In the final part of the paper the examples of computations are shown.
14
Identification of parameters determining the linear model of crystallization
Szopa R., Siedlecki J.
Zeszyty Naukowe. Mechanika / Politechnika Opolska
|
2009
|
z. 95
89-90
15
Content available remote Integration on hypercircles in Rn
Biernat G., Ligus M., Siedlecki J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2009
|
Vol. 8, nr 2
13-19
EN
This paper presents the formulas parametrizing hypercircles (the intersections of hyperplanes with the sphere in Rn, n ≥ 3). A hypothesis concerning the integral on hypercircle Cn-2 of the n-2 - canonical form ω n-2 is proposed.
16
Content available remote Curvilinear finite difference method (CFD) approximation of differential operators
Lara-Dziembek S., Siedlecki J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2009
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Vol. 8, nr 1
97-103
EN
Curvilinear finite difference method is a one of variants of generalized finite difference method. Geometrical mesh can be created by the optional set of points for which the n-points stars are defined. In this paper the 9-points stars are considered (2D task) and the method of differential operators approximation is presented. In the final part of the paper the example of computations is shown.
17
Content available remote 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
|
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.
18
Distribution of internal heat sources in solidifying metal according to mould parameters
Szopa R., Siedlecki J., Wojciechowska W.
Zeszyty Naukowe. Mechanika / Politechnika Opolska
|
2007
|
Vol. 321, z. 89
65-66
EN
The mathematical micro/macro model of pure metal solidification is considered. On the basis of direct sensitivity analysis method the mutual connections between the function controlling the course of solidification (source function) and thermophysical parameters of mould are analyzed. In the final part of paper the example of numerical computations is shown.
19
Content available remote Numerical simulation of solidification process using the Scheil model
Szopa R., Siedlecki J., Wojciechowska W.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2007
|
Vol. 6, nr 1
253-259
EN
In the paper the macro model of volumetric solidification is discussed. The mathematical description of the process bases on a one domain approach, this means in order to take into account the evolution of latent heat, the substitute thermal capacity is introduced to the energy equation. This parameter is determined using the well known Scheil model of macrosegregation. In the first part of paper the theoretical considerations are presented, while in the second one the example of simulation are presented (solidification of Al-Si alloy is considered). The numerical algorithm bases on the finite differences method.
20
Content available remote Identification of solidification model parameters
Mochnacki B., Metelski A., Siedlecki J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2006
|
Vol. 5, nr 1
170-175
EN
In the paper the parametric inverse problem [1, 2] concerning the identification of latent heat is considered. This parameter appears in the energy equation in the source term if the one domain approach is taken into account [4]. The identification of latent heat is possible under the condition that one disposes the additional information concerning the temperature field in the casting domain (cooling curves at the selected set of points). In order to solve the problem the algorithm using the sensitivity coefficients and least squares criterion has been used. The numerical example is presented in the final part of the paper.
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