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1

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.

2

Stationary heat conduction in a solid with functionally graded thermal properties

Bagdasaryan Vazgen, Szołucha Jan

Przegląd Naukowy Inżynieria i Kształtowanie Środowiska

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2019

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Vol. 28, No. 4

539--546

EN

In the paper the solutions for stationary heat conduction in a two dimensional composite with functionally graded heat properties were obtained. Numerical solutions for the taken boundary conditions are shown for several types of changes of composite’s thermal conductivity. The solutions were obtained with the use of the finite-difference method.

3

On diffusion of chemically reactive species in a convective flow past an inclined plate with variable surface temperature and variable mass diffusion

Kirubavathi J. D., Palani G.

International Journal of Applied Mechanics and Engineering

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2016

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Vol. 21, no. 4

905--922

EN

A numerical solution of a transient natural convection flow past a semi-infinite inclined plate under the combined buoyancy effects of heat and mass transfer along with chemical reaction is presented herewith. The governing boundary layer equations for the above flow problem for a first order homogeneous chemical reaction are set up and non-dimensionalised. An implicit finite difference method is employed to solve the unsteady, nonlinear, integro and coupled partial differential equation. Numerical results are presented for various parameters occurring in the problem. The unsteady velocity, temperature and concentration profiles, local and average skin friction, Nusselt number and Sherwood number are studied for both a generative and destructive reaction.

4

Application of analytical solution for extended surfaces on curved and squared ribs

Brestovič T., Jasminská N., Lázár M.

Acta Mechanica et Automatica

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2015

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Vol. 9, no. 2

75--83

EN

The presented article discusses how to increase heat transfer through ribbed surfaces and it is oriented to the mathematical representation of temperature fields and the total thermal flow. The complexity of solving for some types of ribs with variable cross-section requires the application of numerical methods, which are applied consequently to the planar rib as well. In this case there was chosen the finite-difference method (FDM). During solution of the cylindrical ribs the FDM method is preferably used directly with regard to the complexity of solving for infinite sums and improper integrals in Bessel functions. In conclusion is assessed the application suitability of the calculation procedure applied to curved ribs. This procedure is usually used to planar ribs. At the same time it is pointed out the possibility of using this method for calculation of the total thermal flow through cylindrical ribs, which have got the squared form.

5

Development of adaptive methods for reaction-diffusion and other transport problems arising in electrochemistry

Bieniasz L.

Computer Methods in Materials Science

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2009

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Vol. 9, No. 2

296-301

EN

Development of adaptive methods for reaction-diffusion and other transport problems arising in electrochemistry Lesław K. Bieniasz Institute of Physical Chemistry of the Polish Academy of Sciences, Department of Complex Systems and Chemical Processing of Information, ul. Niezapominajek 8, 30-239 Cracow, Poland. Tel. (+48 12) 639 52 12, Fax. (+48 12) 425 19 23, E-mail: nbbienia@cyf-kr.edu.pl, URL: http://www.cyf-kr.edu.pl/~nbbienia, and Institute of Teleinformatics, Faculty of Electrical and Computer Engineering, Cracow University of Technology, ul. Warszawska 24, 31-155 Cracow, Poland. Computational modelling of reaction-diffusion and other reactive transport phenomena presents a challenging task in many areas of science and technology related to chemistry and biology, materials science not being excluded. Problems of this kind become particularly difficult to solve when the governing equations (for example partial or ordinary differential equations) are singularly perturbed, so that their solutions possess local layers or moving fronts. In such cases, adaptive methods that detect such difficult local solution structures, and appropriately concentrate the computational effort on resolving them, are necessary. Another reason for developing the adaptive methods is the modern trend towards automation of computational procedures: users of the simulation software want to obtain solutions having a guaranteed prescribed accuracy, independently of the location, extension and duration of the local solution structures, which may well not be known a priori. For the past 15 years, the present author has been developing finite-difference adaptive approaches to the numerical solution of reaction-diffusion and other reactive transport equations occurring in electrochemistry. The work concentrated on initial boundary value problems for partial differential equations in one-dimensional geometry [1-17], boundary value problems for ordinary differential equations [18-19], and recently on integral equations [20]. In the present communication the results of this work will be briefly summarized. Experiments with the patch-adaptive grid strategy [1-17] and with the local grid node insertion/deletion [18-19] will be used to demonstrate the advantages and disadvantages of the various methods. Some conclusions of potential interest to modellers in other areas will be attempted. In particular, it will be argued that much more work still has to be done to design satisfactory methods, even for spatially one-dimensional equations, despite the fact that for such problems the adaptive methodology is currently regarded to be mature.

PL

W ciągu ponad 15 lat pracy autora nad rozwojem adaptacyjnych metod różnic skończonych dla zagadnień reakcji-dyfuzji oraz innych zjawisk transportu reakcyjnego występujących w elektrochemii nagromadziło się wiele doświadczeń, które mogą być interesujące dla badaczy zajmujących się modelowaniem w innych dziedzinach, łącznie z nauką o materiałach. Wyniki tych badań zostaną krótko przedstawione, ze wskazaniem na wady i zalety różnych metod. Przedstawione zostaną argumenty na rzecz tezy, że potrzeba znacznie więcej pracy aby zaprojektować zadowalające metody adaptacyjne, nawet dla równań w przestrzeni jednowymiarowej, mimo iż obecnie uważa się, że metodologia adaptacyjna dla takich problemów osiągnęła stan dojrzały.

6

Ograniczenia metody różnic skończonych w dokumentowaniu zasobów oraz dróg przepływu w ośrodku szczelinowo-krasowym

Staśko Stanisław, Wcisło Marek

Geologos

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2006

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Vol. 10

241--252

EN

In regional modeling approach vide application of finite-difference method was established due to well documentation and standard of the modeling code (MODFLOW). The authors illustrated limitation of the program on example from modeling prevailing flow zone (tectonic and karstic) in Tarnów Opolski region. Experience gained during modelling fractured-karstic groundwater Triassic system (MGWB 333) indicates that interpolation of parameters provide good results in regional scale when in local calibration high discrepancy was noticed in comparison with measured values. Deficiency of precise simulation of such issues results for example in case of protection zone delineation (advection mass transport) as well as detailed head calculation. Moreover negative feature is production of zones characterised by values far away from observed by field methods.

7

2nd and 3rd-order finite difference methods with counter-error formulations for solving the convection-diffusion equation

Wang X., Guo B., Ghalambor A.

Archives of Mining Sciences

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2006

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Vol. 51, no 1

101-122

EN

Numerical simulation is becoming a common means of predicting performance of oil and gas reservoirs in the petroleum industry. It is also a time-consuming task due to the large dimension of the simulation grids and computing time required to complete a simulation job. Commercial software used in the petroleum reservoir simulation employs the first-order-accuracy finite difference method to solve the convection-diffusion equation. This method introduces numerical dispersion because of truncation error caused by neglecting higher-order terms in Taylor's expansion. This study focused on providing solutions to the above problems. We developed and tested two new algorithms to speed up computation and minimize numerical dispersion. In this research, we have derived the second- and third-order accuracy finite difference formulations to solve the convection-diffusion equation and applied a counter-error mechanism to reduce numerical dispersion. The results indicated that the use of the second- and third-order accuracy finite difference formulations can speed up numerical simulations and retain a sharp displacing slope controlled by the physical diffusion coefficient.

PL

Symulacja numeryczna jest powszechnie stosowanym narzędziem w projektowaniu procesów eksploatacji złóż ropy naftowej i gazu ziemnego. Obliczenia numeryczne są niezwykle czasochłonne, ze względu na wielkowymiarowe z dużą ilością oczek siatki, a także na czas potrzebny do przeprowadzenia pełnej symulacji. Handlowe oprogramowanie używane w inżynierii złożowej do rozwiązywania równań konwekcyjno-dyfuzyjnych wykorzystuje metody różnic skończonych o dokładności pierwszego rzędu. Te metody wprowadzają tzw. numeryczną dyspersję, której powodem jest zaniedbanie wyrazów wyższego rzędu w rozwinięciach w szereg Taylora. Numeryczna dyspersja zniekształca obraz rozwiązań. Artykuł dotyczy właśnie tego problemu. Autorzy rozwinęli i sprawdzili dwa nowe algorytmy przyspieszające proces obliczeniowy i minimalizujące dyspersję numeryczną. Zbudowany formalizm obliczeń, to metody różnic skończonych o 2-gim i 3-cim rzędzie dokładności dla równania typu kon wekcyjno-dyfuzyjnego, z oszacowaniem błędu wyrównującego. Efektem jest zredukowanie numerycznej dyspersji. Rezultaty obliczeń pokazały, że użycie tych metod może przyspieszyć przebieg symulacji komputerowych oraz otrzymać nachylenie konturu płynów wypierającego i wypieranego, wywołanego tylko przez współczynnik dyfuzji fizycznej.