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1
Content available remote Fourier Image Methods for Evolution Equations of Deep-Water Waves
EN
The velocity potential of the fluid satisfies the Laplace equation with nonlocal boundary conditions on a free surface. This differential problem is transformed to an evolution equation in Fourier variables. The Fourier transform images of boundary functions are approximated by Picard's iterations and the method of lines on meshes related to roots of Hermite polynomials. Due to convolutions of sine and cosine functions the integral terms of Picard's iterations reveal unexpected instabilities for wave numbers in a neighborhood of zero.
EN
The presented paper is focused on the comparison of the numerical solution of the Laplace equation in a two-dimensional space with the results obtained with the use of the analytical method. The results of the numerical model are computed on the base of the Finite Element Method. The analytical solution of the considered equation is obtained using the Fourier series. Finally the results of both methods are compared in order to verify the accuracy of numerical implementation.
PL
W pracy zaprezentowano procedurę obliczania współczynników pojemnościowych trójwymiarowych układów elektrostatycznych, opartą na idei Iteracyjnej Metody Rozwiązań Fundamentalnych. Procedura ta pozwala na bezpośrednie wyznaczanie ładunków zgromadzonych na powierzchniach ciał przewodzących bez konieczności obliczania rozkładu pola elektrycznego. Umożliwia ona także ustalenie żądanej dokładności rozwiązania. W przeprowadzonym teście numerycznym stwierdzono bardzo dobrą zbieżność i efektywność metody.
EN
A procedure for calculating the capacitive coefficients of three-dimensional electrostatic systems, based on the idea of the Iterative Method of Fundamental Solutions is presented in the paper. This procedure allows direct determination of charges on the surfaces of conductive bodies without the need to calculate the distribution of the electric field. It also allows you to determine the desired accuracy of the solution. The numerical test performed showed very good convergence and efficiency of the method.
PL
(MES) na podstawie analizy rezultatów testów numerycznych przeprowadzonych na modelowym zagadnieniu elektrostatyki. Stwierdzono, że za pomocą IMRF można uzyskać rozwiązanie o danej dokładności w czasie wielokrotnie krótszym (nawet kilkadziesiąt razy) niż za pomocą MES.
EN
The article presents efficiency comparison of the iterative method of fundamental solutions (IMFS) and finite element method (FEM), based on the analysis of numerical tests results obtained for the model of electrostatics problem. It has been found, that applying the IMFS a solution of assumed accuracy could be obtained numerous times shorter (even several dozen times) than using the FEM.
EN
The aim of this paper is to create an optimal shape of the 2D domain that is described by the Non-Uniform Rational B-Splines (NURBS) curves. This work presents a method based on the topological derivative for the Laplace equation that determines the sensitivity of a given cost function to the change of its topology. As a numerical approach, the boundary element method is considered. To check the effectiveness of the proposed approach, the example of computations was carried out.
6
Content available remote Wektor Poyntinga w analizie oscylacji mocy biernej w sieciach energetycznych
PL
Niniejsza publikacja przeznaczona jest dla elektryków zainteresowanych, dyskutowanym ostatnio, zagadnieniem oscylacji mocy biernej w sieciach energetycznych. Niektóre z prac sugerują konieczność użycia wektora Poyntinga do analizy zjawiska. W tym artykule przeprowadza się Czytelnika w sposób przystępny poprzez wybrane podstawy teorii pola elektromagnetycznego, aby pokazać, dlaczego do analizy mocy, przesyłanej na częstotliwości 50 Hz wzdłuż jednorodnego wieloprzewodowego toru transmisyjnego, stosowanie wektora Poyntinga jest zbędne, natomiast przydatne do określania przestrzennego rozkładu gęstości mocy przenoszonej pomiędzy przewodami. Artykuł nie analizuje wpływu źródeł i obciążeń. Ich wpływ można uwzględnić poprzez superpozycję rozwiązań tu uzyskanych [9].
EN
The paper is addressed to the electrical engineers who are interested in a non-active power analysis in power lines. The Poynting vector application, as a necessary tool to explain physical phenomenon, has been suggested in some papers. The reader is guided through the chosen problems of the electromagnetic theory in an accessible way, in order to explain to him why in the analysis of 50 Hz power, which is transmitted along uniform multi-wire line, the application of the Poynting vector is unnecessary. Nevertheless, the Poynting vector could be useful in the analysis of the 3D density distribution of the power transferred between wires of the line. The influence of the sources and loadings is not analysed in this paper. It could be taken into account by creating a superposition of solutions obtained in this paper, which is discussed in an accompanying paper [9].
EN
The paper addresses a two-dimensional boundary identification (reconstruction) problem in steady-state heat conduction. Having found the solution to the Laplace equation by superpositioning T-complete functions, the unknown boundary of a plane region is approximated by polynomials of an increasing degree. The provided examples indicate that sufficient accuracy can be obtained with a use of polynomials of a relatively low degree, which allows avoidance of large systems of nonlinear equations. Numerical simulations for assessing the performance of the proposed algorithm show better than 1% accuracy after a few iterations and very low sensitivity to small data errors.
EN
In order to achieve the desired topology we often have to remove material of the area considered. This work presents the author's algorithm which can be used in the reconstruction of the boundary of domain after elimination of a certain amount of material. The paper introduces some details about the procedure that allows one to achieve the expected shape of a domain. The topological-shape sensitivity method for the Laplace equation is used to obtain an optimal topology, whereas numerical methodology utilizes the boundary element method. In the conclusion of the paper the example of computation is shown.
EN
In this work, the topological derivative for the Laplace equation is used to solve a design problem. This derivative describes the sensitivity of the problem when a small hole is formed at an arbitrary point of the domain. The goal of this work is to design topology of the domain when the Robin condition is imposed on the holes. Physically, the holes can be construed as cooling channels. For finding the solution of the governing equation the boundary element method is applied. The final part of the paper presents the design of the heat exchanger and results of computations.
EN
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 a given amount of material is removed. The objective of this work is to obtain an optimal topology of the domain considered. In the final part of the paper, the example of computations is shown.
EN
We present an accurate expression for the effective conductivity of a regular square-lattice arrangement of ideally conducting cylinders, valid for arbitrary concentrations. The formula smoothly interpolates between the two asymptotic expressions derived for low and high concentrations of the cylinders. Analogy with critical phenomena is suggested and taken to the extent of calculating the superconductivity critical exponent and the particle-phase threshold from the very long expansions in concentration. The obtained formula is valid for all concentrations including touching cylinders, hence it completely solves with high accuracy the problem of the effective conductivity for the square array.
EN
Comparison of several types of electrostatic micro-actuators is carried out, particularly with respect to their resultant force effects. The continuous mathematical model of such actuators is mostly described by the Laplace equation. In this paper, its numerical solution is performed by a fully adaptive higher-order finite element method, using a code developed by the authors. The methodology is illustrated by typical examples whose results are discussed.
EN
In this paper an application of the homotopy perturbation method for solving the steady state and unsteady state heat conduction problem is presented.
PL
W artykule przedstawiono zastosowanie homotopijnej metody perturbacyjnej do rozwiązania zagadnień ustalonego oraz nieustalonego przewodzenia ciepła.
14
Content available remote Shape sensitivity analysis : implicit approach using boundary element method
EN
The Laplace equation (2D problem) supplemented by boundary conditions is analyzed. To estimate the changes of temperature in the 2D domain due to the change of local geometry of the boundary, the implicit method of sensitivity analysis is used. In the final part of the paper, the example of numerical computations is shown.
15
Content available remote Identification of internal hole parameters on the basis of boundary temperature
EN
The Laplace equation describing temperature field in 2D domain with an internal hole of circle shape supplemented by adequate boundary conditions is considered. On the basis of known temperature at the fragment of boundary the position of circle center or its radius is identified. To solve the inverse problem discussed the least square criterion is formulated, and next the gradient method coupled with the boundary element method is applied. To determine sensitivity coefficients the shape sensitivity analysis is used. In the final part of the paper the examples of computations are shown.
16
Content available remote Level Set Method in Inverse Problem Solution
EN
The optimal shape design of the capacitor with Laplace equation of state and inverse problem solution in Electrical Impedance Tomography using level set method are presented in the paper. The inverse problem solution determines the positions of capacitor plates, which were optimized to achieve required potential distribution, while the inverse problem solution in EIT enables the identification of the size and the position of internal areas with different conductivity.
17
Content available remote Shape sensitivity analysis with respect to the parameters of internal hole
EN
The Laplace equation describing temperature field in 2D domain supplemented by adequate boundary conditions is considered. The aim of investigations is to estimate the changes of temperature due to changes of shape parameter (e.g. radius or position of internal hole). To solve the problem, the implicit differentiation method of shape sensitivity analysis coupled with the boundary element method is applied.
EN
In a large number of technological applications there are bodies submerged in various liquids. If they vibrate, their motion is significantly influenced by their interaction with the medium in the surrounding space. If amplitude of the vibration is small, the induced pressure field in the liquid can be described by a Laplace equation. Before performing its solution the governing equation is transformed into the curvilinear coordinates utilizing the Bézier surfaces and then a finite difference method is applied. As the hydraulic forces acting on the body are proportional to its acceleration, negatives of the coefficients of proportionality may be regarded as additional masses.
CS
V řadě průmyslových zařízení se nacházejí tělesa, která jsou ponořená v kapalinách. Jestliže kmitají, pak jejich pohyb je značně ovlivněn interakcí s okolním médiem. Jestliže amplituda kmitání je malá, pak tlakové pole vyvolané v kapalině lze popsat Laplaceovou rovnicí. Před provedením jejího řešení se transformuje do křivočarých souřadnic, k čemuž se využívají Bézierovy plochy, a pak se použije metoda konečných diferencí. Protože hydraulické síly působící na těleso jsou přímo úměrné jeho zrychlení, záporně vzaté koeficienty úměrnosti lze považovat za přídavné hmotnosti.
19
Content available remote Application of boundary element method to shape sensitivity analysis
EN
In the paper the boundary element method to shape sensitivity analysis is applied. The Laplace equation is analyzed and the aim of investigations is to estimate the changes of temperature in the 2D domain due to the change of local geometry of the boundary. Here the implicit differentiation method of shape sensitivity analysis is used. In the final part of the paper the example of numerical computations is shown.
20
Content available remote The Multipole method for the Laplace equation in domains with polyhedral corners
EN
A new analytic-numerical method has been developed for solving the Laplace equation in domains with cones of arbitrary base, in particular with polyhedral corners. The solution is represented as an expansion involving singular functions (the Multipoles), which play the role of basic functions. The method enables to find these functions explicitly and to compute efficiently their singularity exponents. The method possesses exponential rate of convergence and provides precise computation of the solution, its derivatives and intensity factors at the edges and at the corner point. In addition, an asymptotic expansion of the solution near the edges of polyhedral corner has been obtained.
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