Informacja

Drogi użytkowniku, aplikacja do prawidłowego działania wymaga obsługi JavaScript. Proszę włącz obsługę JavaScript w Twojej przeglądarce.

English (EN)·Polski (PL)·Yкраїнський (UK)
Przejdź do strony domowej biblioteki Centralna Biblioteka Wojskowa Link otwiera się w nowym oknie Logo biblioteki Prolib Integro - strona głównaCentralna Biblioteka Wojskowa

Wyszukujesz frazę "Lagrangian method" wg kryterium: Temat

  Lista filtrów
Wyrównaj
Wyświetlanie 1-15 z 18
Tytuł:
Simulation of Solitary Wave Mechanics by a Corrected Smoothed Particle Hydrodymamics Method
Autorzy:
Staroszczyk, R.
Powiązania:
https://bibliotekanauki.pl/articles/973863.pdf
Data publikacji:
2011
Wydawca:
Polska Akademia Nauk. Instytut Budownictwa Wodnego PAN
Tematy:
solitary wave
non-linear wave interaction
meshfree Lagrangian method
smoothed particle hydrodynamics
Opis:
The paper is devoted to numerical modelling of solitary wave propagation phenomena in shallow water of uniform depth. The problem governing equations are solved by applying a corrected smoothed particle hydrodynamics (SPH) method in which standard smoothing kernel functions are modified in such a way that so-called linear reproducing conditions for kernel approximations and their first-order spatial derivatives are satisfied. Numerical performance of the proposed SPH model has been verified by comparing its predictions with analytical results for a solitary wave travelling over the horizontal bottom. Also, the results obtained by applying the corrected SPH method and those given by the standard SPH method, with no kernel correction, are compared. Further, an impact of the solitary wave on a vertical rigid wall is investigated, and finally an interaction of two colliding solitary waves is considered.
Źródło:
Archives of Hydro-Engineering and Environmental Mechanics; 2011, 58, 1-4; 23-45
1231-3726
Pojawia się w:
Archives of Hydro-Engineering and Environmental Mechanics
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Simulation of Dam-Break Flow by a Corrected Smoothed Particle Hydrodynamics Method
Autorzy:
Staroszczyk, R.
Powiązania:
https://bibliotekanauki.pl/articles/241454.pdf
Data publikacji:
2010
Wydawca:
Polska Akademia Nauk. Instytut Budownictwa Wodnego PAN
Tematy:
dam-break problem
free surface flow
meshfree Lagrangian method
smoothed particle hydrodynamics
Opis:
The paper deals with numerical modelling of water flow which is generated by the break of a dam. The problem is solved by applying a smoothed particle hydrodynamics (SPH) method in which standard smoothing kernel functions are corrected in such a way that so-called linear reproducing conditions for kernel approximations and their gradients are satisfied. The proposed SPH model has been used to simulate a two-dimensional problem of the collapse of a water column inside a rectangular tank. The simulations illustrate the formation and subsequent propagation of a wave over the horizontal plane. It is shown that the model predictions of the changes of the advancing wave-front position, and of the changes of the free surface elevation of water, compare well with experimental data. Also, the results obtained with the corrected SPH method are compared with those given by the standard SPH method with no kernel correction. In addition, an impact of the surging wave against a vertical rigid wall is illustrated.
Źródło:
Archives of Hydro-Engineering and Environmental Mechanics; 2010, 57, 1; 61-79
1231-3726
Pojawia się w:
Archives of Hydro-Engineering and Environmental Mechanics
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Hermite spline interpolation on patches for parallelly solving the Vlasov-Poisson equation
Autorzy:
Crouseilles, N.
Latu, G.
Sonnendrücker, E.
Powiązania:
https://bibliotekanauki.pl/articles/929688.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Oficyna Wydawnicza
Tematy:
równanie Vlasova-Poissona
metoda Lagrangiana
równoległość
Vlasov-Poisson equation
semi-Lagrangian method
parallelism
Opis:
This work is devoted to the numerical simulation of the Vlasov equation using a phase space grid. In contrast to Particle- In-Cell (PIC) methods, which are known to be noisy, we propose a semi-Lagrangian-type method to discretize the Vlasov equation in the two-dimensional phase space. As this kind of method requires a huge computational effort, one has to carry out the simulations on parallel machines. For this purpose, we present a method using patches decomposing the phase domain, each patch being devoted to a processor. Some Hermite boundary conditions allow for the reconstruction of a good approximation of the global solution. Several numerical results demonstrate the accuracy and the good scalability of the method with up to 64 processors. This work is a part of the CALVI project.
Źródło:
International Journal of Applied Mathematics and Computer Science; 2007, 17, 3; 335-349
1641-876X
2083-8492
Pojawia się w:
International Journal of Applied Mathematics and Computer Science
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Integrals of the one-dimensional continuity equation
Autorzy:
Dowkontt, S.
Dowkontt, G.
Powiązania:
https://bibliotekanauki.pl/articles/252056.pdf
Data publikacji:
2016
Wydawca:
Instytut Naukowo-Wydawniczy TTS
Tematy:
continuity equation
Schwarz theorem
Cauchy method
Lagrangian method
równanie ciągłości
twierdzenie Schwarza
twierdzenie Lagrange'a
twierdzenie Cauchy'ego
Opis:
The authors analyze the method used by Cauchy and Lagrange to obtain the integral of continuity equation. The authors propose their own method of integration using Schwarz’ theorem. As a result, the authors obtain a greater number of possible solutions with a higher level of generality while also being able to identify the basic disadvantages of the Cauchy-Lagrangian method. Further, the authors conducted a detailed interpretation of the results of their solution.
Źródło:
TTS Technika Transportu Szynowego; 2016, 12; 77-81
1232-3829
2543-5728
Pojawia się w:
TTS Technika Transportu Szynowego
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena
Autorzy:
Besse, N.
Mauser, N. J.
Sonnendrücker, E.
Powiązania:
https://bibliotekanauki.pl/articles/929699.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Oficyna Wydawnicza
Tematy:
model Vlasova-Darwina
model Vlasova-Poiswell'a
metoda Lagrangiana
niskoczęstotliwościowe zjawisko elektromagnetyczne
Vlasov-Darwin model
Vlasov-Poisswell model
semi-Lagrangian method
low-frequency electromagnetic
Opis:
We present a new numerical method to solve the Vlasov-Darwin and Vlasov-Poisswell systems which are approximations of the Vlasov-Maxwell equation in the asymptotic limit of the infinite speed of light. These systems model low-frequency electromagnetic phenomena in plasmas, and thus “light waves” are somewhat supressed, which in turn allows the numerical discretization to dispense with the Courant-Friedrichs-Lewy condition on the time step. We construct a numerical scheme based on semi-Lagrangian methods and time splitting techniques. We develop a four-dimensional phase space algorithm for the distribution function while the electromagnetic field is solved on a two-dimensional Cartesian grid. Finally, we present two nontrivial test cases: (a) the wave Landau damping and (b) the electromagnetic beam-plasma instability. For these cases our numerical scheme works very well and is in agreement with analytic kinetic theory.
Źródło:
International Journal of Applied Mathematics and Computer Science; 2007, 17, 3; 361-374
1641-876X
2083-8492
Pojawia się w:
International Journal of Applied Mathematics and Computer Science
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A direct and accurate adaptive semi-Lagrangian scheme for the Vlasov-Poisson equation
Autorzy:
Campos Pinto, M.
Powiązania:
https://bibliotekanauki.pl/articles/929689.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Oficyna Wydawnicza
Tematy:
metoda Lagrangiana
oszacowanie błędu
szybkość zbieżności
fully adaptive scheme
semi-Lagrangian method
Vlasov-Poisson equation
error estimate
convergence rates
optimal transport of adaptive multiscale meshes
Opis:
This article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1 + 1)- dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be said optimal in the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter \epsilon which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge in L\infty towards the exact ones as \epsilon and \delta t tend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.
Źródło:
International Journal of Applied Mathematics and Computer Science; 2007, 17, 3; 351-359
1641-876X
2083-8492
Pojawia się w:
International Journal of Applied Mathematics and Computer Science
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Evaluation of Pressure Distribution in Buried Pipes with Liquid in the Explosion
Autorzy:
Parviz, M.
Aminnejad, B.
Fiouz, A. R.
Powiązania:
https://bibliotekanauki.pl/articles/230958.pdf
Data publikacji:
2019
Wydawca:
Polska Akademia Nauk. Czytelnia Czasopism PAN
Tematy:
rura zakopana
metoda Lagrange’a-Eulera
podmuch
ciśnienie
Ls-Dyna
buried pipe
Lagrangian-Eulerian method
blast
pressure
Opis:
In this Paper, a parametric study on pipes buried in soil was performed illustrating the results of blast loading. Effects of various parameters such as the physical properties of water, oil, gas, air, soil, pipes, and TNT have been investigated. The arbitrary Lagrangian-Eulerian (ALE) method was employed using LS-DYNA software. The maximum pressure in a buried pipe explosive was observed at an angle of about 0° to 45° and the minimum pressure occurred at an angle of about 45° to 90°. Therefore, all figures in this study illustrate that fluid pressure levels in buried pipes can help in their stabilization. In generally, by increasing the 1.23 times of liquid density under the explosion, the pressure levels in the soil decreased by 1.3 percent. The gas pressure has been increasing more than oil and water pipes 39.73 and 40.52 percent, respectively.
Źródło:
Archives of Civil Engineering; 2019, 65, 3; 3-14
1230-2945
Pojawia się w:
Archives of Civil Engineering
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
An analytical and numerical approach to a bilateral contact problem with nonmonotone friction
Autorzy:
Barboteu, M.
Bartosz, K.
Kalita, P.
Powiązania:
https://bibliotekanauki.pl/articles/330898.pdf
Data publikacji:
2013
Wydawca:
Uniwersytet Zielonogórski. Oficyna Wydawnicza
Tematy:
linearly elastic material
bilateral contact
nonmonotone friction law
hemivariational inequality
finite element method
error estimate
nonconvex proximal bundle method
quasi augmented Lagrangian method
Newton method
metoda elementów skończonych
szacowanie błędu
metoda Lagrangiana
metoda Newtona
Opis:
We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.
Źródło:
International Journal of Applied Mathematics and Computer Science; 2013, 23, 2; 263-276
1641-876X
2083-8492
Pojawia się w:
International Journal of Applied Mathematics and Computer Science
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The Coupled Eulerian-Lagrangian Analysis of the KOBO Extrusion Process
Autorzy:
Wójcik, Marta
Skrzat, Andrzej
Powiązania:
https://bibliotekanauki.pl/articles/2024076.pdf
Data publikacji:
2021
Wydawca:
Stowarzyszenie Inżynierów i Techników Mechaników Polskich
Tematy:
coupled Eulerian-Lagrangian method
KOBO extrusion
metal forming
finite element method
hardening parameters
cyclic loading
metoda sprzężona Eulera-Lagrangiana
wytłaczanie KOBO
obróbka plastyczna
metoda elementów skończonych
parametry hartowania
obciążenie cykliczne
Opis:
The KOBO extrusion is an unconventional elastic-plastic deformation process in which the phenomenon of chang- ing a path of plastic deformation due to die cyclic oscillations by a given angle and with a given frequency is applied. As the result of the application of the oscillating rotary motion of the die, the reduction of the extrusion force was obtained. The numerical study of the KOBO extrusion of metallic materials was presented in this paper. The three-dimensional coupled Eulerian-Lagrangian (CEL) analysis was applied. The relationship between the extrusion force and the punch displacement during the KOBO process was achieved. The effective plastic strain distribution in the butt was found. The results of the numerical computations were compared with the experimen- tal data. The influence of the material hardening parameters on the cyclic loading phenomena (ratcheting, mean stress relaxation) in terms of the course of the KOBO extrusion was also examined. The proper determination of the material hardening parameters can help to optimize the KOBO process in terms of the reduction the extrusion force and the choice of the amount of die oscillations.
Źródło:
Advances in Science and Technology. Research Journal; 2021, 15, 1; 197-208
2299-8624
Pojawia się w:
Advances in Science and Technology. Research Journal
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Goal-oriented mesh adaptivity for fluid-structure interaction with application to heart-valve settings
Metody adaptacji siatki w zagadnieniu oddziaływania płyn-struktura (FSI) w zastosowaniu do symulacji przepływu przez zastawkę serca
Autorzy:
Wick, T.
Powiązania:
https://bibliotekanauki.pl/articles/139712.pdf
Data publikacji:
2012
Wydawca:
Polska Akademia Nauk. Czytelnia Czasopism PAN
Tematy:
dynamika zastawki serca
fale elastyczne
metoda arbitrary Lagrangian-Eulerian
ALE
metoda elementów skończonych
oddziaływanie płyn-struktura
FSI
zorientowana na cel adaptacja siatki
heart-valve dynamics
elastic waves
arbitrary Lagrangian Eulerian method
finite element method
fluid-structure interaction
goal-oriented mesh adaption
Opis:
We apply a fluid-structure interaction method to simulate prototypical dynamics of the aortic heart-valve. Our method of choice is based on a monolithic coupling scheme for fluid-structure interactions in which the fluid equations are rewritten in the 'arbitrary Lagrangian Eulerian' (ALE) framework. To prevent the backflow of structure waves because of their hyperbolic nature, a damped structure equation is solved on an artificial layer that is used to prolongate the computational domain. The increased computational cost in the presence of the artificial layer is resolved by using local mesh adaption. In particular, heuristic mesh refinement techniques are compared to rigorous goal-oriented mesh adaption with the dual weighted residual (DWR) method. A version of this method is developed for stationary settings. For the nonstationary test cases the indicators are obtained by a heuristic error estimator, which has a good performance for the measurement of wall stresses. The results for prototypical problems demonstrate that heart-valve dynamics can be treated with our proposed concepts and that the DWR method performs best with respect to a certain target functional.
W artykule przedstawiono analizę zagadnienia oddziaływania płyn-struktura (FSI) w komputerowej symulacji pracy zastawki serca. Przedstawiono monolityczne sformułowanie tego zagadnienia, w którym równania dla struktury i płynu rozwiązywane są w pełnym sprzężeniu, przy czym do opisu ruchu płynu stosowane jest podejście typu Arbitrary Lagrangian-Euelerian (ALE). Zaproponowano metodę eliminacji zjawiska niefizycznego odbicia fal odkształceń struktury, polegającą na wprowadzeniu sztucznej dyssypacji energii tych fal w części brzegu obszaru położonej za zastawkami. W celu zwiększenia efektywności obliczeniowej wprowadzono lokalną adaptację siatki. W szczególności, porównano heurystyczne techniki adaptacji siatki z techniką opartą na wykorzystaniu ważonego residuum sprzężonego (Dual Weighted Residual, DWR). Przedstawiono wyniki obliczeń testowych demonstrujące poprawność zaproponowanego podejścia oraz skuteczność metody adaptacyjnej DWR.
Źródło:
Archive of Mechanical Engineering; 2012, LIX, 1; 73-99
0004-0738
Pojawia się w:
Archive of Mechanical Engineering
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Derivative free optimal thrust allocation in ship dynamic positioning based on direct search algorithms
Autorzy:
Valčić, M.
Prpić-Oršić, J.
Powiązania:
https://bibliotekanauki.pl/articles/116991.pdf
Data publikacji:
2020
Wydawca:
Uniwersytet Morski w Gdyni. Wydział Nawigacyjny
Tematy:
algorithm
dynamic positioning
direct search algorithms
sequential quadratic programing (SQP)
Singular Values Decomposition (SVD)
Lagrangian Multiplier Method (LMM)
Direct Search (DS)
Mesh Adaptive Direct Search (MADS)
Opis:
In dynamic positioning systems, nonlinear cost functions, as well as nonlinear equality and inequality constraints within optimal thrust allocation procedures cannot be handled directly by means of the solvers like industry-standardized quadratic programing (QP), at least not without appropriate linearization technique applied, which can be computationally very expensive. Thus, if optimization requirements are strict, and problem should be solved for nonlinear objective function with nonlinear equality and inequality constraints, than one should use some appropriate nonlinear optimization technique. The current state-of-the-art in nonlinear optimization for gradient-based algorithms is surely the sequential quadratic programing (SQP), both for general applications and specific thrust allocation problems. On the other hand, in recent time, one can also notice the increased applications of gradient-free optimization methods in various engineering problems. In this context, the implementation of selected derivative free direct search algorithms in optimal thrust allocation is proposed and discussed in this paper, and avenues for future research are provided.
Źródło:
TransNav : International Journal on Marine Navigation and Safety of Sea Transportation; 2020, 14, 2; 309-314
2083-6473
2083-6481
Pojawia się w:
TransNav : International Journal on Marine Navigation and Safety of Sea Transportation
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Penalties, Lagrange multipliers and Nitsche mortaring
Autorzy:
Grossmann, Christian
Powiązania:
https://bibliotekanauki.pl/articles/729228.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
augmented Lagrangian
penalty method
domain decomposition
Nitsche mortaring
finite elements
Opis:
Penalty methods, augmented Lagrangian methods and Nitsche mortaring are well known numerical methods among the specialists in the related areas optimization and finite elements, respectively, but common aspects are rarely available. The aim of the present paper is to describe these methods from a unifying optimization perspective and to highlight some common features of them.
Źródło:
Discussiones Mathematicae, Differential Inclusions, Control and Optimization; 2010, 30, 2; 205-220
1509-9407
Pojawia się w:
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Pre-assembly for FEM 2D non-curvilinear quadrilateral Lagrangian elements
Autorzy:
Sowa, M.
Typańska, D.
Powiązania:
https://bibliotekanauki.pl/articles/97541.pdf
Data publikacji:
2014
Wydawca:
Politechnika Poznańska. Wydawnictwo Politechniki Poznańskiej
Tematy:
pre-assembly
finite element method
higher order elements
quadrilateral Lagrangian elements
Opis:
The paper concerns an idea of preparing auxiliary equations and data for the finite element method assembly. This is made for a specific element type and element order and is referred to as the pre-assembly. Most importantly the prepared auxiliary equations require that only the element’s placement (coordinates) needs to be known, while the remaining coefficients that are required are stored in data sheets. The pre-assembly allows programmers who implement the FEM to significantly reduce the effort put into the assembly algorithm through its replacement by a few simple equations and the application of the prepared data. These data sheets can be prepared and can be utilized by FEM programmers. The construction of these data sheets for non-curvilinear quadrilateral Lagrangian elements (of any selected order) is explained in this paper.
Źródło:
Computer Applications in Electrical Engineering; 2014, 12; 106-119
1508-4248
Pojawia się w:
Computer Applications in Electrical Engineering
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Coupled Thermomechanical Eulerian-Lagrangian Analysis of the KOBO Extrusion Process
Autorzy:
Wójcik, Marta
Skrzat, Andrzej
Powiązania:
https://bibliotekanauki.pl/articles/2134115.pdf
Data publikacji:
2022
Wydawca:
Polska Akademia Nauk. Czytelnia Czasopism PAN
Tematy:
KOBO extrusion
Chaboche model
coupled Eulerian-Lagrangian analysis
SPD method
plasticity
Opis:
Numerical simulations of the KOBO extrusion process are presented in this paper. The coupled thermomechanical Eulerian-Lagrangian approach was applied for the three-dimensional finite element model. The dynamic explicit Euler forward method was used in numerical calculations. The elastic-plastic Chaboche model assuming isotropic and kinematic hardening under variable temperature conditions was applied to describe the behaviour of the material under cyclic loading. In numerical computations Chaboche material model implemented in commercial software, as well as the proprietary one written as FORTRAN procedure were tested. The numerical results present the stress and strain distributions in the extruded material, as well as an increase of temperature due to the plastic work and friction. The shape of plastic strain zones was verified experimentally. The approach presented in the paper is a promising numerical tool to simulate the KOBO process.
Źródło:
Archives of Metallurgy and Materials; 2022, 67, 3; 1185--1193
1733-3490
Pojawia się w:
Archives of Metallurgy and Materials
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Lagrangian Finite Element Treatment of Transient Gravitational Waves in Compressible Viscous Fluids
Autorzy:
Staroszczyk, R.
Powiązania:
https://bibliotekanauki.pl/articles/241224.pdf
Data publikacji:
2007
Wydawca:
Polska Akademia Nauk. Instytut Budownictwa Wodnego PAN
Tematy:
Newtonian viscous fluid
gravitational wave
transient problem
Lagrangian formulation
finite element method
Opis:
In this paper the problem of transient gravitational wave propagation in a viscous compressible fluid is investigated. The problem is formulated in the Lagrangian description and is solved numerically by a finite element method. In computations either fixed in space or moving meshes that follow the material fluid particles are used with the purpose to compare their numerical performance. As illustrations, results of numerical simulations carried out for plane flows in a domain of simple geometry are presented. First, the finite element results are compared with available experimental data for the case of small-amplitude waves in order to validate the numerical model. Then, the problem of large-amplitude transient water wave propagation over a horizontal bottom, involving the wave reflection at a rigid wall, is considered. For the flow parameters typical of a laboratory flume, the evolution of the free-surface elevation and the time variations of the surface displacements at chosen locations are shown for a range of different moving wall amplitudes and excitation times.
Źródło:
Archives of Hydro-Engineering and Environmental Mechanics; 2007, 54, 4; 261-284
1231-3726
Pojawia się w:
Archives of Hydro-Engineering and Environmental Mechanics
Dostawca treści:
Biblioteka Nauki
Artykuł
Wyświetlanie 1-15 z 18

Ta witryna wykorzystuje pliki cookies do przechowywania informacji na Twoim komputerze. Pliki cookies stosujemy w celu świadczenia usług na najwyższym poziomie, w tym w sposób dostosowany do indywidualnych potrzeb. Korzystanie z witryny bez zmiany ustawień dotyczących cookies oznacza, że będą one zamieszczane w Twoim komputerze. W każdym momencie możesz dokonać zmiany ustawień dotyczących cookies