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1

Nieliniowa analiza ram stalowych z węzłami podatnymi

Reder Patrycja, Litewka Przemysław

Nowoczesne Hale

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2024

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Nr 1

21--28

PL

Artykuł porusza problem nieliniowej analizy węzłów podatnych w ramach stalowych. Na podstawie wyników badań laboratoryjnych zostały określone zależności między względnym kątem obrotu węzła a wartością momentu zginającego węzeł, które wykorzystano do obliczenia przemieszczeń i sił wewnętrznych w przekrojach ram. Proces ten był wykonywany iteracyjnie przy pomocy różnych technik nieliniowego rozwiązywania równań. Bazę do obliczeń stanowiła macierzowa wersja metody przemieszczeń. Głównym wnioskiem wynikającym z przeprowadzonych analiz jest stwierdzenie bezpieczeństwa przy zakładaniu liniowej zależności między momentem zginającym a kątem obrotu w projektowaniu inżynierskim. Zauważono także zdecydowanie większą efektywność metody Newtona-Raphsona nad metodą kolejnych przybliżeń. Ponadto zaobserwowano, że w konstrukcjach inżynierskich szybciej wyczerpie się nośność elementu, niż uzyska się nieliniową zależność moment - kąt.

EN

The paper addresses the issue of the non-linear analysis of semi-rigid joints in steel frames. The relationships between the relative angle of the joint rotation and the value of the bending moment of the joint were determined based on laboratory tests. The determined relationships were used to calculate displacements and forces in the frame elements. This process was performed iteratively using various non-linear equation solving techniques. The basis for the calculations was the matrix version of the displacement method. The main conclusion of the analyses is that the assumption of the linear relationship between the bending moment and the angle of rotation in engineering design is safe. It was also noted that the efficiency of the Newton-Raphson method is significantly higher than that of the method of successive approximations. In addition, it was observed that in engineering structures, the loss of the load carrying capacity of an element occurs before the non-linear moment-angle relationship zone is reached.

2

Control of chaos in the burke-shaw system of fractal-fractional order in the sense of Caputo-Fabrizio

Saber Sayed

Journal of Applied Mathematics and Computational Mechanics

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2024

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Vol. 23, nr 1

83-96

EN

For the Burke-Shaw system, we propose a fractal-fractional order in the sense of the Caputo-Fabrizio derivative. The proposed system is solved by utilizing the fractal-fractional derivative operator with an exponential decay kernel. Time-fractional Caputo-Fabrizio fractal fractional derivatives are applied to the Burke-Shaw-type nonlinear chaotic systems.Based on fixed point theory, it has been demonstrated that a fractal-fractional-order model under the Caputo-Fabrizio operator exists and is unique. Using a numerical power series method, we solve the fractional Burke-Shaw model. Using Newton’s interpolation polynomial, we solve the equation numerically by implementing a novel numerical scheme based on an efficient polynomial.

3

Doświadczenia z projektowania i produkcji betonów architektonicznych na bazie produktów RECKLI

Wójcik Andrzej

Materiały Budowlane

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2022

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nr 9

21--22

4

On rational functions related to algorithms for a computation of roots. II

Baran Mirosław

Science, Technology and Innovation

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2019

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Vol. 7, no. 4

26--29

EN

We discuss a nice composition properties related to algorithms for computation of N-roots. Our approach gives direct proof that a version of Newton's iterative algorithm is of order 2. A basic tool used in this note are properties of rational function Φ(w; z) = z-w/(z+w), which was used earlier in [1] in the case of algorithms for computations of square roots.

5

On rational functions related to algorithms for a computation of roots. I

Baran Mirosław

Science, Technology and Innovation

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2019

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Vol. 7, no. 4

17--25

EN

We discuss a less known but surprising fact: a very old algorithm for computing square root known as the Bhaskara-Brouncker algorithm contains another and faster algorithms. A similar approach was obtained earlier by A.K. Yeyios [8] in 1992. By the way, we shall present a few useful facts as an essential completion of [8]. In particular, we present a direct proof that k – th Yeyios iterative algorithm is of order k. We also observe that Chebyshev polynomials Tn and Un are a special case of a more general construction. The most valuable idea followed this paper is contained in applications of a simple rational function Φ(w; z) = z-w/z+w.

6

Optimal modelling of steel multi-span beams using the gradient-iterative method

Mikulski Leszek

Civil and Environmental Engineering Reports

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2019

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Vol. 29, no. 4

112--127

EN

The article describes the gradient-iterative optimization method and outlines the method’s basic assumptions and illustrates its general use. The method’s implementation was illustrated based on a steel I-beam. The described calculation example concerns the optimization of the height of the web of a multi-span beam. The method enables finding an optimal solution with the use of simple and commonly available software. To illustrate the effectiveness of the optimization method, multiple calculations were performed for beams with various spans and various load conditions.

7

A unified algorithm for solving split generalized mixed equilibrium problem, and for finding fixed point of nonspreading mapping in Hilbert spaces

Jolaoso L. O., Oyewole K. O., Okeke C. C., Mewomo O. T.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

211--232

EN

The purpose of this paper is to study a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. We introduce a new iterative algorithm and prove its strong convergence for approximating a common solution of a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. Our algorithm is developed by combining a modified accelerated Mann algorithm and a viscosity approximation method to obtain a new faster iterative algorithm for finding a common solution of these problems in real Hilbert spaces. Also, our algorithm does not require any prior knowledge of the bounded linear operator norm. We further give a numerical example to show the efficiency and consistency of our algorithm. Our result improves and compliments many recent results previously obtained in this direction in the literature.

8

Quartic non-polynomial spline solution for solving two-point boundary value problems by using Conjugate Gradient iterative method

Justine H., Chew J. V. L., Sulaiman J.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 1

41--50

EN

Solving two-point boundary value problems has become a scope of interest among many researchers due to its significant contributions in the field of science, engineering, and economics which is evidently apparent in many previous literary publications. This present paper aims to discretize the two-point boundary value problems by using a quartic non-polynomial spline before finally solving them iteratively with Conjugate Gradient (CG) method. Then, the performances of the proposed approach in terms of iteration number, execution time and maximum absolute error are compared with Gauss-Seidel (GS) and Successive Over-Relaxation (SOR) iterative methods. Based on the performances analysis, the two-point boundary value problems are found to have the most favorable results when solved using CG compared to GS and SOR methods.

9

Algorytmy równoległe w języku programowania C#

Schubring T.

TTS Technika Transportu Szynowego

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2016

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R. 23, nr 12

196--199

PL

W artykule opisano możliwość łatwego implementowania równoległych algorytmów w języku programowania C# . Zwrócono szczególną uwagę na zastosowanie metody For klasy Parallel dostępnej na platformie Microsoft.NET.

EN

The article describes the ability to easily implement parallel algorithms in the C# programming language. Special attention was paid to the application of the method For class Parallel available on the Microsoft .NET platform.

10

Iteracyjna metoda wariacyjna w analizie zwichrzenia belki

Jarczewska K., Hołubowski R., Glabisz W.

Inżynieria i Budownictwo

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2015

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R. 71, nr 12

657--661

PL

Przedstawiono ideę iteracyjnej metody wariacyjnej VIM i pokazano jej skuteczność w analizie zwichrzenia pryzmatycznej belki poddanej obciążeniu równomiernie rozłożonemu. Wartości obciążeń krytycznych uzyskanych z rozwiązań analitycznych przy zastosowaniu metody VIM porównano z wartościami obliczonymi metodą elementów skończonych.

EN

The paper presents the idea of variational iteration method (VIM) and indicates its efficiency in stability analysis of prismatic beam under uniformly distributed load. The results obtained by using VIM were compared with the critical loads computed by finite element method.

11

An iterative method for symmetric positive semidefinite linear system of equations

Salkuyeh D. K.

Demonstratio Mathematica

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2014

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Vol. 47, nr 2

482--492

EN

In this paper, a new two-step iterative method for solving symmetric positive semidefinite linear system of equations is presented. A sufficient condition for the semiconvergence of the method is also given. Some numerical experiments are presented to show the efficiency of the proposed method.

12

Równoległa implementacja metod iteracyjnych dla układu równań z macierzami rzadkimi

Schubring T.

TTS Technika Transportu Szynowego

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2013

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R. 20, nr 10

499--505, CD

PL

W artykule podano równoległą implementację metod rozwiązywania układu równań liniowych z macierzą rzadką w języku programowania UPC (Unified Parallel C). Uwzględniono możliwości środowiska programistycznego Berkeley UPC oraz format spakowanych wierszy CSR (Compressed Sparse Row).

EN

Paper described a parallel implementation of the iterative methods for solving linear equation systems with sparse matrix in the UPC programming language (Unified Parallel C). Possibilities of the Berkeley UPC development environment and the CSR packed rows format (Compressed Sparse Row) were included.

13

Performance evaluation of iterative methods to unconstrained single variable minimization problems

Horla D.

Studia z Automatyki i Informatyki

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2013

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T. 38

7--33

EN

The paper considers performance issues of a class of iterative minimization methods of unconstrained single variable problems. Problem structures that assure superior performance of a specific method have been stipulated with appropriate conclusions drawn.

PL

W artykule poruszono zagadnienie szybkości działania metod minimalizacji funkcji jednej zmiennej dla zadań programowania nieliniowego bez ograniczeń. Wskazano przypadki, dla których konkretna metoda działa szybciej niż pozostałe oraz wyciągnięto wnioski odnośnie takiego stanu rzeczy.

14

System matrix computation for iterative reconstruction algorithms in SPECT based on direct measurements

Borys D., Szczucka-Borys K., Gorczewski K.

International Journal of Applied Mathematics and Computer Science

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2011

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Vol. 21, no 1

193-202

EN

A method for system matrix calculation in the case of iterative reconstruction algorithms in SPECT was implemented and tested. Due to a complex mathematical description of the geometry of the detector set-up, we developed a method for system matrix computation that is based on direct measurements of the detector response. In this approach, the influence of the acquisition equipment on the image formation is measured directly. The objective was to obtain the best quality of reconstructed images with respect to specified measures. This is indispensable in order to be able to perform reliable quantitative analysis of SPECT images. It is also especially important in non-hybrid gamma cameras, where not all physical processes that disturb image acquisition can be easily corrected. Two experiments with an 131I point source placed at different distances from the detector plane were performed allowing the detector response to be acquired as a function of the point source distance. An analytical Gaussian function was fitted to the acquired data in both the one- and the two-dimensional case. A cylindrical phantom filled with a water solution of 131I containing a region of 'cold' spheres as well as a uniform solution (without any spheres) was used to perform algorithm evaluation. The reconstructed images obtained by using four different of methods system matrix computation were compared with those achieved using reconstruction software implemented in the gamma camera. The contrast of the spheres and uniformity were compared for each reconstruction result and also with the ranges of those values formulated by the AAPM (American Association of Physicists in Medicine). The results show that the implementation of the OSEM (Ordered Subsets Expectation Maximization) algorithm with a one-dimensional fit to the Gaussian CDR (Collimator-Detector Response) function provides the best results in terms of adopted measures. However, the fit of the two-dimensional function also gives satisfactory results. Furthermore, the CDR function has the potential to be applied to a fully 3D OSEM implementation. The lack of the CDR in system matrix calculation results in a very noisy image that cannot be used for diagnostic purposes.

15

Two implementations of the preconditioned conjugate gradient method on heterogeneous computing grids

Collignon T. P., Van Gijzen M. B.

International Journal of Applied Mathematics and Computer Science

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2010

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Vol. 20, no 1

109-121

EN

Efficient iterative solution of large linear systems on grid computers is a complex problem. The induced heterogeneity and volatile nature of the aggregated computational resources present numerous algorithmic challenges. This paper describes a case study regarding iterative solution of large sparse linear systems on grid computers within the software constraints of the grid middleware GridSolve and within the algorithmic constraints of preconditioned Conjugate Gradient (CG) type methods. We identify the various bottlenecks induced by the middleware and the iterative algorithm. We consider the standard CG algorithm of Hestenes and Stiefel, and as an alternative the Chronopoulos/Gear variant, a formulation that is potentially better suited for grid computing since it requires only one synchronisation point per iteration, instead of two for standard CG. In addition, we improve the computation-to-communication ratio by maximising the work in the preconditioner. In addition to these algorithmic improvements, we also try to minimise the communication overhead within the communication model currently used by the GridSolve middleware. We present numerical experiments on 3D bubbly flow problems using heterogeneous computing hardware that show lower computing times and better speed-up for the Chronopoulos/Gear variant of conjugate gradients. Finally, we suggest extensions to both the iterative algorithm and the middleware for improving granularity.

16

Solution of the solidification problem by using the variational iteration method

Hetmaniok E., Słota D., Zielonka A.

Archives of Foundry Engineering

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2009

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Vol. 9, iss. 4

63-68

EN

The paper presents the approximated solution of the solidification problem, modelled with the aid of the one-phase Stefan problem with the boundary condition of the second kind, by using the variational iteration method. For solving this problem one needs to determine the distribution of temperature in the given domain and the position of the moving interface. The proposed method of solution consists of describing the considered problem with a system of differential equations in a domain with known boundary, and solving the received system with the aid of VIM method. The accuracy of the obtained approximated solution is verified by comparing it with the analytical solution.

17

Computation of current harmonics from voltage waveforms: optimization approach

Kwiczak S., Mikołajuk K.

Przegląd Elektrotechniczny

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2005

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R. 81, nr 2

70-73

EN

The group harmonic compensation in electrical systems, especially compensation of scattered nonlinear loads, needs a proper approach to acquisition of reference signals controlling compensators. Such signal cannot be worked out from the current of individual load. The reference signal should be obtained from voltage waveform. Presented method of compensation current estimation can be classified as an invasive method. The change of voltage waveform caused by compensator current is the source of information for proposed iterative algorithm. This algorithm allows to compute the current reducing voltage distortion. It has been shown that optimization algorithms available in MATLAB can be used for considered purpose. Numerical tests have been made for the example nonlinear circuit. The quasi-Newton algorithm was used in these tests.

PL

Kompensacja równoległa harmonicznych dla grup odbiorników, szczególnie rozproszonych, stwarza specjalne wymagania w odniesieniu do sygnału sterującego kompensator. Problemem jest jaki prąd powinien generować taki kompensator. Prąd ten nie może być bezpośrednio uzyskany na podstawie prądów odbiorników, które mogą być nawet niedostępne dla pomiarów. Sygnał sterujący może być obliczany w oparciu o przebieg napięcia. Taka metoda jest prezentowana w pracy. Metoda ta może być sklasyfikowana jako inwazyjna i iteracyjna. Źródłem informacji jest zmiana przebiegu napięcia spowodowana zmianą przebiegu prądu kompensatora. Pokazano, że do tego celu może być użyty algorytm optymalizacyjny dostępny w programie MATLAB. Przedstawiono testy uzyskane na przykładzie kilku obwodów nieliniowych.

18

On The Two-Step Iterative Method of Solving Frictional Contact Problems in Elasticity

Angelov T. A., Liolios A. A.

International Journal of Applied Mathematics and Computer Science

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2005

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Vol. 15, no 2

197-203

EN

A class of contact problems with friction in elastostatics is considered. Under a certain restriction on the friction coefficient, the convergence of the two-step iterative method proposed by P.D. Panagiotopoulos is proved. Its applicability is discussed and compared with two other iterative methods, and the computed results are presented.

19

Numerical stability of the Richardson second order method

Smoktunowicz A., Wróbel I.

Demonstratio Mathematica

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2005

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Vol. 38, nr 1

255--263

EN

In this paper we study numerical properties of the Richardson second order method (RS) for solving a linear system Ax = b, where A € Rnxn is infinitysymmetric and positive definite. We consider the standard model of floating point arithmetic (cf. [6], [7], [11]). We prove that the RS-algorithm is numerically stable. This means that the algorithm computes approximations xk to the exact solution x* = A-1b such that the error limfk||xk - x*ll2 ls of order eMcond(A), where eM is the machine precision and cond(A) = ||A || 2 ||A-1|| denotes the condition number of the matrix A.

20

Sensitivity analysis of electromagnetic quantities by means of FEM

Gawrylczyk K. M.

Journal of Technical Physics

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2002

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Vol. 43, no 3

323-334

EN

Sensitivity analysis determines the dependence of global or local electromagnetic quantities on geometrical or physical parameters expressed in the form of an objective function. The final aim of the field calculation methods is generally the design of an electromagnetic device. Solution of inverse problem on the basis of the finite elements method (FEM) makes the optimal shape design possible, as well as the identification of material cracks and flaws inside the conducting materials with the help of the eddy-current method. These tasks can be defined similarly for recognition of the spatial distribution of material parameters. The recognition takes place in iterative manner, basing on gradient information derived from the sensitivity analysis. The work deals with sensitivity evaluation in conjunction with finite element algorithms. It will be shown that the sensitivity analysis necessary to solve an inverse problem, requires only a low additional calculation cost.