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In this paper we consider a pentadiagonal matrix which consists of only three non-zero bands. We prove that the determinant of such a matrix can be represented by a product of two determinants of corresponding tridiagonal matrices. It is shown that such an approach gives greatly shorter time of computer calculations.
This paper presents the numerical model of the measuring stand - the wind tunnel, in which there is fixed a cylinder with a turbulent inlet oscillating stream. The aim of this work is the juxtaposition and comparison of characteristic values concerning the oscillating turbulent flow around the cylinder, obtained from the experiment conducted in the wind tunnel at the Institute of Thermal Machinery...
A non-homogeneous system being the composition of burn wound and healthy tissue is considered. The heat exchange between sub-domains and environment is described by the system of partial differential equations (the Pennes equations) supplemented by the assumed boundary conditions. Additional problems associated with sensitivity analysis with respect to thermal parameters occurring in the mathematical...
This paper presents a solution to the free vibration problem of a system of two cantilever nanobeams. The nanobeams of the system are axially loaded and coupled by discrete translational springs. The solution of the problem by using the Green’s function method has been obtained. A numerical example shows the effect of the small scale on free vibration frequencies of the nanobeam system considered.
The paper presents a numerical model, which allows one to determine the wrinkling stress in sandwich panels. The wrinkling of the compressed face of sandwich panel can be caused by a bending moment or axial force. The developed model enables one to take into account various boundary conditions and micro-profiling of sandwich faces. Damage initiation and propagation in the interface between the face...
In the article the combined algorithm for finding conservation laws and implectic operators has been proposed. Using the Novikov-Bogoyavlensky method the finite dimensional reductions have been found. The structure of invariant submanifolds has been examined. Having analyzed phase portraits of Hamiltonian systems, partial periodical solutions have been found.
In the paper, the solution of second order differential equations with various coefficients is presented. The concerning equations are written as first order matrix differential equations and solved with the use of the power series method. Examples of application of the proposed method to the equations occurring in the technical problems are presented.
The paper presents a method of solving two-dimensional wave equations which describe vibrations of the membrane with variable thickness and with damping. The differential operator is decomposed into two parts. The first one describes vibrations of the membrane with constant thickness without damping. The second contains the rest of the original operator and is treated as inhomogeneity for the first...
We use (quasi-) Young measures associated with strictly monotonic functions with a differentiable inverse to prove an L1 ([0,1],ℝ) weak convergence of the monotonic sequence of such functions. The result is well known, but the method seems to be new.
This article discusses an application of an algorithm for modeling the chain-like structures’ motion. The basic and the most important assumption of the adopted approach is sequentialization of the moves. In the presented version of the algorithm there is a possibility of modeling the tension propagation mechanism. The main purpose of the paper is to investigate the impact of different algorithm parameters...
In the continuum mechanics there is a class of problems that cannot be solved directly or the solutions of these problems are affected by large errors when the classical equations of thermoviscoelasticity are considered. The paper discusses a special case of such problems - the cyclic bending of a composite plate with a circular hole subjected to the stationary self-heating, which was solved within...
We study mechanical-damage avalanches occurring in axially loaded nanopillars located in the nodes of the supporting square lattice. Nanopillars are treated as fibres in the framework of the stochastic Fibre Bundle Model and they are characterised by random strength thresholds. Once an element crashes, its load is transferred to the other intact elements according to a given load transfer rule. In...
In this paper, an exact analytical solution of the radial heat conduction problem in a hollow multilayered sphere is presented. The boundary conditions of the third kind and perfect contact at the interfaces are assumed. The temperatures are obtained by using the Green’s function method. The Green’s functions are expressed in the form of the series of appropriate eigenfunctions.
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).
The work is a continuation of the method of calculating the determinant of the block matrix in the two-dimensional case. In this paper we use the Finite Differences Method and the symmetric polynomials.
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...
The report presents a new model of linear elastodynamics of periodic composites. The study deals with a certain extension of the ideas yield tolerance modelling. In the proposed model the displacement field is determined by an infinite number of unknowns. This disappointment seems to be apparent in many cases of periodicity since the sequence of tolerance shape functions used in the modelling procedure...
In this paper we show that the determinant of heptadiagonal symmetric Toeplitz matrix can be represented by a particular solution of the system of three homogeneous linear recurrence equations. The general considerations are illustrated by certain numerical example implementated in the Maple system.
Exploiting the mapping between a binary mixture and the Ising model we have analyzed the critical fluctuations by means of the density-matrix renormalization group technique. The calculations have been carried out for a two-dimensional Ising strip subject to equally strong surface fields. It was found that the critical Casimir force has significantly different behavior on opposite sides of the capillary...
In the paper, the topological derivative for the Laplace equation is taken into account. The governing equation is solved by means of the Boundary Element Method. The topological-shape sensitivity method is used to determine the points showing the lowest sensitivities. On the selected points, material is eliminated by opening a hole, using the appropriate iterative process. This one is halted when...
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