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1

Mathematical modelling of filtration processes in drainage systems with different depths of drainage

Havryliuk Volodymyr, Bomba Andrii, Pinchuk Oleg, Gerasimov Ievgenii, Klimov Serhii, Tkachuk Mykola, Turcheniuk Vasyl

Journal of Water and Land Development

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2021

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no. 50

74--78

EN

The article discusses the option for the application of the methodology for the solution of boundary value problems on the conformal mapping for the calculation of filtration process in the horizontal systematic drainage, provided that the drain is installed at a different depth. In particular, the case of methods combining fictitious areas and quasiconformal mappings for solving nonlinear boundary conditions problems for calculating filtration regimes in soils with free sections of boundaries (depression curves) and intervals of the “drainage” type. As an example, the authors designed a hydrodynamic flow grid, determined the values of the flows to the drain, established a section line and elicited other process characteristics. The algorithm for the numerical solution of model nonlinear boundary conditions problems of quasiconformal reflection in areas bounded by two equipotential lines and two flow lines, when for one of the sections, the boundary is an unknown (free) curve with fixed and free ends. The conducted numerical calculations prove that the problems and algorithms of their numerical solution, with a relatively small iterations number (k = 141) suggested in the paper, can be applied in the simulation of nonlinear filtration processes that arise in horizontal drainage systems. Total filtration flow obtained Q = 0.9 dm3∙s–1; flow for drains Q1 = 0.55 dm3∙s–1 and Q2 = 0.35 dm3∙s–1 are quite consistent with practically determined values.

2

Uniform stress field inside a non-parabolic open inhomogeneity interacting with a mode III crack

Wang X., Schiavone P.

Archives of Mechanics

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2021

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Vol. 73, nr 1

67--81

EN

Using conformal mapping techniques, analytic continuation and the theory of Cauchy singular integral equations, we prove that a non-parabolic open inhomogeneity embedded in an elastic matrix subjected to a uniform remote anti-plane stress nevertheless admits an internal uniform stress field despite the presence of a finite mode III crack in its vicinity. Our analysis indicates that: (i) the internal uniform stress field is independent of the specific shape of the inhomogeneity and the presence of the finite crack; (ii) the existence of the finite crack plays a key role in the non-parabolic open shape of the inhomogeneity and in the non-uniform stresses in the surrounding matrix; (iii) the two-term asymptotic expansion at infinity of the stress field in the matrix is independent of the presence of the finite crack. Detailed numerical results are presented to demonstrate the proposed theory.

3

Uniform field within a non-elliptical inhomogeneity in the vicinity of a nearby non-circular Eshelby inclusion

Wang X., Schiavone P.

Archives of Mechanics

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2021

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Vol. 73, nr 5-6

541--555

EN

We rigorously prove that a non-elliptical inhomogeneity continues to permit an internal uniform stress field despite the presence of a nearby non-circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. Here, we adopt a specific representation of the non-circular Eshelby inclusion as a Booth’s lemniscate inclusion. Our analysis indicates that the internal uniform stress field inside the non-elliptical inhomogeneity is independent of the existence of the Booth’s lemniscate inclusion whereas the non-elliptical shape of the inhomogeneity is attributed entirely to its presence. Representative numerical examples are presented to demonstrate the feasibility of the proposed method of general solution.

4

A screw dislocation located outside, inside or on the interface of a parabolic elastic inhomogeneity

Wang X., Schiavone P.

Archives of Mechanics

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2021

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Vol. 73, nr 3

219--235

EN

Using conformal mapping techniques, superposition and analytic continuation, we derive analytic solutions to the problem of a screw dislocation interacting with a parabolic elastic inhomogeneity. The screw dislocation can be located anywhere either in the surrounding matrix or in the parabolic inhomogeneity or simply on the parabolic interface itself. We obtain explicit expressions for the two analytic functions in the image plane characterizing the elastic fields describing displacement and stresses in the two-phase composite. Using the Peach-Koehler formula, we also obtain the image force acting on the screw dislocation. The analytic function defined in the parabolic inhomogeneity in the physical plane can be interpreted in terms of real and image screw dislocations for any location of the real screw dislocation.

5

On approximate conformal mapping of a disk and an annulus with radial and circular slits onto multiply connected domains

Ivanshin Pyotr N., Shirokova Elena A.

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 2

73--84

EN

The method of boundary curve reparametrization is generalized to the case of multiply connected domains. We construct the approximate analytical conformal mapping of the unit disk with m circular slits and n-m radial slits and an annulus with (m-1) circular slits and n-m radial slits onto an arbitrary given (n+1) multiply connected finite domain with a smooth boundary. The method is based on extension of the Lichtenstein-Gershgorin equation to a multiply connected domain. The proposed method is reduced to the solution of a linear system with unknown Fourier coefficients. The approximate mapping function has the form of a Cauchy integral. Numerical examples demonstrate that the proposed method is effective in computations.

6

Mathematical modelling of filtration processes in drainage systems using conformal mapping

Bomba A., Tkachuk M., Havryliuk V., Kyrysha R., Gerasimov I., Pinchuk O.

Journal of Water and Land Development

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2018

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no. 39

11--15

EN

The situation when groundwater considerably rises above the “normal” level, water intake, lowering of groundwater levels and other relevant practical tasks require the drainage facilities. The most effective techniques of numerical studies of the corresponding boundary problems at present time are methods of dealing with inverse boundary value problems (conformal and quasi-conformal mappings). As basis of this research we used the case of combining the fictitious domain methods with quasi-conformal mappings of the solution of nonlinear boundary value problems for the calculation of filtration regimes in environments with free boundary areas (depression curves) and zones of “mountainous” areas. This paper reviews the stationary issue of flat-vertical stationary non-pressure liquid filtration to horizontal symmetric drainage. In the paper a practical methodology for solving boundary value problems on conformal mappings is suggested for the calculation of the filtration process in the horizontal symmetrical drainage. The idea of block iterative methods was used during the creation of the corresponding algorithm which is based on the alternating “freeze” of the anticipated conformance parameter, the internal and boundary connections of the curvilinear area. The results of the conducted numerical calculations confirmed the effectiveness of the suggested problem formulations and algorithms of their numerical solution and the possibility of their use in the modelling of nonlinear filtration processes occurring in horizontal drainage systems, as well as in the design of drainage facilities and optimizing other hydrosystems. Therefore these results are of great importance.

PL

W przypadku podwyższenia poziomu wody gruntowej znacząco ponad „normalną” wysokość pobór wody, obniżanie jej zwierciadła i inne działania wymagają urządzeń drenarskich. Obecnie najbardziej efektywnymi technikami numerycznych badań odpowiednich problemów brzegowych są metody odwrotnej analizy wartości brzegowych (odwzorowania konforemne i quasi-konforemne). Jako podstawę prowadzonych badań przyjęto przykład połączenia metod fikcyjnej domeny z quasi-konforemnym odwzorowaniem rozwiązania problemu nieliniowych wartości brzegowych do obliczenia reżimu filtracji w środowisku o wolnej powierzchni brzegowej (krzywe depresji) i w strefach „górskich”. W pracy dokonano przeglądu rozwiązań pionowej stacjonarnej filtracji płynów do horyzontalnego symetrycznego drenażu. Zaproponowano praktyczną metodologię rozwiązywania problemów wartości brzegowych w przekształceniu konforemnym do obliczania procesu filtracji w horyzontalnym symetrycznym drenażu. Wykorzystano zasadę metod iteracyjnych podczas tworzenia odpowiednich algorytmów opartą na naprzemiennym „zamrażaniu” przewidywanego parametru odwzorowania z wewnętrznymi i brzegowymi powiązaniami krzywoliniowego obszaru. Wyniki przeprowadzonych numerycznych obliczeń potwierdzają efektywność formułowania sugerowanego problemu i algorytmów ich numerycznego rozwiązania oraz możliwości ich zastosowania w modelowaniu nieliniowych procesów filtracji występujących w horyzontalnym systemie odwadniania, w projektowaniu urządzeń drenarskich i w optymalizacji innych systemów wodnych. Z tego powodu uzyskane wyniki mają ogromne znaczenie.

7

Neutrality of coated holes in the presence of screw dislocation dipoles or circular thermal inclusions

Wang X., Schiavone P.

Archives of Mechanics

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2018

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Vol. 70, nr 5

405--427

EN

In this paper, we consider the design of neutral coated holes in two particular cases when the thick coating itself is altered by the presence of some form of material imperfection. In the first case we consider anti-plane deformations of a linearly elastic solid when the thick coating applied to the hole includes a screw dislocation dipole. In the second case, we investigate the design of neutral coated holes in plane elasticity when the thick coating contains a circular thermal inclusion and the surrounding linearly elastic solid is subjected to uniform remote hydrostatic stresses. The design is achieved by constructing particular forms of the conformal mapping function for the coating itself. Several examples are presented to demonstrate the resulting solutions. Our numerical results show that the existence of the screw dislocation dipole or the circular thermal inclusion in the coating exerts a significant influence on the shape of the neutral coated hole.

8

Estimate of geopressures using conformal mapping in eccentric wells

Silva U. F., Júnior M. I. de Q., Lemos S., Silva A. H. F., Furriel G. P., Segatti N., Calixto W. P.

Transactions on Environment and Electrical Engineering

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2017

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Vol. 2, No. 2

1--10

EN

The purpose of this work is to calculate Surge and Swab pressures in eccentric wells. Analysis of the phenomenon, in which fluid is confined between two eccentric cylinders, are made. Conformal mapping calculations is used to lead the original eccentric domain into equivalent concentric domain, since usual models only make calculation for concentric geometries. The results of this study, using the proposed methodology, are presented and discussed.

9

Using a conformal mapping method for calculation of a multipolar system magnetic field

Zagirnyak M. V.

Przegląd Elektrotechniczny

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2012

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R. 88, nr 7b

98-101

EN

A conformal mapping method was used to obtain formulas for calculation of magnetic field strength in the operating areas of multipolar cylindrical systems of magnetic separators. The offered expressions allow calculation of the field at any point of the initial area in an explicit form according to the coordinates of this point.

PL

Metoda odwzorowań konforemnych została zastosowana do uzyskania wzorów na natężenie pola magnetycznego w obszarach aktywnych wielobiegunowych systemów cylindrycznych w separatorach magnetycznych. Przedstawione wzory pozwalają obliczać pole w każdym punkcie obszaru wyjściowego.

10

A kinematic approach to the optimal shape synthesis of electric field

Costamagna E., Di Barba P., Savini A.

Przegląd Elektrotechniczny

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2012

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R. 88, nr 7b

90-93

EN

A parameter-free and global-oriented method of optimal field synthesis is proposed. The key idea is to consider the boundary of the field region to synthesize as a moving boundary, whose velocity leading to the optimal solution is unknown. The proposed method does not require to solve the transport equation and the adjoint variable problem. The design optimisation of the guard ring in a high-voltage transformer is considered as the case study. Finite-element method and conformal mapping method are used for field analysis.

PL

W artykule zaproponowano otymalną syntezę pola przy użyciu metody globalnie zorientowanej ze szswobodnymi parametrami. Kluczową ideą jest rozważenie brzegu obszaru polowego, który ma podlegać syntezie, jako brzegu poruszającego się w kierunku optymalnego rozwiązania z nieznaną prędkością z prędkością. Proponowana metoda nie wymaga rozwiązywania równania transport ani problem sprzężonego. Optymalny projekt pierścienia ochronnego w transformatorze wysokonapięciowym stanowi przykładowe studium przypadku. Do analizy pola wykorzystano metodę elementów skończonych i metodę odwzorowań konforemnych.

11

Conformal mapping approach for permanent magnet synchronous machines: on the modeling of saturation

Hafner M., Franck D., Hameyer K.

Archives of Electrical Engineering

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2012

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Vol. 61, nr 2

211-220

EN

In the electromagnetic field simulation of modern servo drives, the computation of higher time and space harmonics is essential to predict torque pulsations, radial forces, ripple torques and cogging torque. Field computation by conformal map ping (CM) techniques is a time-effective method to compute the radial and tangential field components. In the standard CM approach, computational results of cogging torque simulations as well as overload operations observe deviations to nonlinear finite element (FE) simulations due to the neglection of slot leakage and saturation effects. This paper presents an extension of the classical CM. Additional CM parameters are computed from single finite element computations so as to consider both effects listed above in the model over a wide operation range of the electrical drive. The proposed approach is applied to a surface permanent magnet synchronous machine (SM-PMSM), and compared to numerical results obtained by finite element analysis (FEA). An accuracy similar to that of FE simulations is obtained with however the low computation time that is characteristic for analytical models.

12

O pewnym zastosowaniu odwzorowań konforemnych w lotnictwie

Bogowski F., Burniak C.

Zeszyty Naukowe / Wyższa Szkoła Oficerska Sił Powietrznych

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2011

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Nr 2

170--182

PL

W artykule autorzy omawiają podstawowe zasady odwzorowań konforemnych oraz przykład ich zastosowania w lotnictwie. Opisana jest metoda otrzymywania profilów skrzydeł samolotu.

EN

This articles discusses basic principles of conformal mapping and an example of their practical implementation. It describes the method of obtaining airplane wing profiles.

13

Resistance of Arbitrary Angle Corner

Cividjian G.A., Matagne E.

Przegląd Elektrotechniczny

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2010

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R. 86, nr 5

83-86

EN

The field modelling becomes simpler if strong field nonuniformities are solved theoretically. The early-obtained formulas for 2-D "corner" permeances, (capacitances or resistances) pertain to right angle configurations. In this paper, using a conformal mapping and analytical approximations, the field in the vicinity of arbitrary angle corner is studied and the corner resistance determined. Obtained results can be applied to a large range of electromagnetic and microelectronic devices. They agree with early-obtained values for right angle corner and with the FEM simulation.

PL

Modelowanie w przypadku silnie nierównomiernych pól jest prostsze przy zastosowaniu rozwiązań teoretycznych. Otrzymane wcześniej analityczne rozwiązanie dla naroża 2D (pojemność lub rezystancja) pozwala na konfiguracje naroża. Użyto mapowania i aproksymacji analitycznej. Obliczono pole, a następnie rezystancję naroża. Wyniki można zastosować do szeregu urządzeń elektromagnetycznych i mikroelektronicznych. Otrzymano zgodność z wynikami symulacji metodą elementów skończonych.

14

An approximate method for estimation of fluid motion in a lake containing islands

Klonowska-Prosnak M. E., Prosnak W. J., Cześnik P. P.

Archives of Hydro-Engineering and Environmental Mechanics

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2005

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Vol. 52, nr 4

335-349

EN

The physical phenomenon considered in the paper deals with flow in a lake, which contains one or several islands. The flow is generated by rivers, their inlets and outlets being distributed on the shoreline of the lake. The problem to be solved consists in determination of the velocity field - in the domain bounded by the shorelines of the lake and the islands. In fact, it is attempted to arrive solely at an estimation of the field. Consequently, a rather simple physical model of the phenomenon, as well as its simple mathematical description has been applied. In particular, plane, irrotational and steady flow of ideal liquid has been introduced, the inlets and outlets of the rivers being simulated by sources and sinks. Hence, the problem reduces to determination of a complex function, representing the complex velocity field, which satisfies the impermeability condition on all contours representing the shorelines, the field being generated by the singularities already mentioned. Unfortunately, the so formulated problem is "overconditioned" or "too stiff", what means that the impermeability condition on the outer contour cannot be satisfied. Nevertheless, we arrived at a simple method for circumventing this obstacle, the payoff consisting in some modification of this contour. We had this particular modification in mind, applying the word "approximate" in the title of the paper. The paper contains results - in the form of streamline patterns - for lakes containing from 1 to 3 islands. In the relevant figures the distances between the given and the modified exterior contours can be seen distinctly - allowing the reader to draw conclusions, whether the errors due to the modifications are admissible or not. Of course, it depends anyway on the point of view of the user of the results.

15

Closed-form 2D modeling of sub-100 nm MOSFETs in the subthreshold regime

Osthaug J., Fjeldly T.A., Iniguez B.

Journal of Telecommunications and Information Technology

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2004

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nr 1

70-79

EN

Closed-form 2D modeling of deep-submicron and sub-100 nm MOSFETs is explored using a conformal mapping technique where the 2D Poisson equation in the depletion regions is separated into a 1D long-channel case and a 2D Laplace equation. The 1D solution defines the boundary potential values for the Laplacian, which in turn provides a 2D correction of the channel potential. The model has been tested for classical MOSFETs with gate lengths in the range 200-250 nm, and for a super-steep retrograde MOSFET with a gate length of 70 nm. With a minimal parameter set, the present modeling reproduces both qualitatively and quantitatively the experimental data obtained for such devices.

16

Enhanced strength models for notched laminates with finite outer boundaries

Hufenbach W., Kroll L., Lepper M., Kostka P., Czulak M.

Kompozyty

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2003

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R. 3, nr 7

255-259

EN

For the practical design of notched fibre reinforced components, methods that consider finite anisotropic plates with a cutout are of special interest. The notch-induced stress concentrations lead to critical strains and frequently initiate catastrophic failure of the component. By selective fibre reinforcement of the matrix, a redistribution of the stress peaks relevant to failure can be achieved. In the course of this, the fibre orientation, besides the notch geometry, plays a decisive role. An anisotropic plate with finite dimensions and a hole in its center will be used here to analytically model stress concentrations. Unlike the infinite plate, this problem comprises a doubly connected outer area. A solution method has been developed for stress concentration problems of fibre-reinforced compounds based on the method of complex-valued stress functions combined with conformal mappings. Using the solution methods developed here, the whole calculation procedure was modified and extended in such a way, that even the influence of a finite outer boundary of the plate can be described. The consideration of the outer boundary is of high importance for practical problems, because then the essential influence of the notch size in dependence of the material, geometry and loading parameters can be also determined.

PL

Metody obliczeniowe, uwzględniające skończone wymiary płyt i obecność karbów mają szczególne znaczenie dla praktycznych zagadnień dotyczących konstruowania elementów z kompozytów włóknistych. Spiętrzenia naprężeń wywołane działaniem karbów prowadzą często do pojawienia się krytycznych odkształceń, inicjujących zniszczenie struktury nośnej. Poprzez celowe umocnienie osnowy włóknami możliwe jest osiągnięcie rozdziału lokalnie działających naprężeń i tym samym unikniecie uszkodzenia. Fakt ten świadczy o tym, że oprócz cech geometrycznych karbu również kierunek ułożenia włókien odgrywa istotną role. Przedstawiono analityczny model powstawania spiętrzenia naprężeń w anizotropowej płycie o skończonych wymiarach z centralnie usytuowanym otworem. W celu uwzględnienia skończonych wymiarów płyty obszar zewnętrzny traktuje się jako podwójnie połączony. Opisana metoda obliczania spiętrzenia naprężeń na skutek działania karbu łączy zastosowanie zespolonych funkcji opisujących naprężenia z odwzorowaniem konforemnym. Metoda ta umożliwia modyfikację i rozszerzenie procedury obliczeniowej w celu uwzględnienia wpływu skończonych wymiarów płyty. Możliwość uwzględnienia brzegu jest szczególnie ważna w zagadnieniach praktycznych, gdyż pozwala ona na ocenę działania krabu w zależności od materiału, cech geometrycznych oraz parametrów obciążenia.

17

On the wave propagation in generalised thermoelastic media

Sharma J. N., Sharma P. K., Gupta S. K.

International Journal of Applied Mechanics and Engineering

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2003

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Vol. 8, no 2

299-316

EN

In this paper, the analysis is based on the decoupled field equations of linear thermoelasticity without energy dissipation. These equations have been solved with the use of integral transforms. The dynamic behavior of an elastic half space due to (i) a line load and (ii) a mechanical load, acting on the boundary is also discussed. While the displacement in both of the cases is found to be continuous, the temperature, strain and stress are observed to be discontinuous at elastic and thermal wave fronts. The characteristic features of the underlying theory are analyzed in the light of these solutions and their counterparts in earlier works. The results obtained theoretically have been computed numerically and are represented graphically.

18

Argument Increment Stability Criterion for Linear Delta Models

Hofreiter M., Zitek P.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 4

485-491

EN

Currently used stability criteria for linear sampled-data systems refer to the standard linear difference equation form of the system model. This paper presents a stability criterion based on the argument increment rule modified for the delta operator form of the sampled-data model. For the asymptotic stability of this system form it is necessary and sufficient that the roots of the appropriate characteristic equation lie inside a circle in the left half of the complex plane, the radius of which is inversely proportional to the sampling period. Therefore the argument increment of the system characteristic polynomial of an asymptotically stable delta model has to increase by 2pi n if this circle has been run around in the counter-clockwise direction. The criterion developed based on this principle permits not only the proof of the system stability itself, but also the approximation of the dominant roots of its characteristic equation.

19

Conformal mapping of the Gulf of Gdańsk onto a canonical domain

Cześnik P. P., Prosnak W. J.

Oceanologia

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2002

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No. 44 (2)

179-207

EN

The paper deals with the conformal mapping of a plane, finite, simply connected domain, representing the southern segment of the Gulf of Gdansk, enclosed from the North by a parallel, tangential to Cape Rozewie. The segment contains the Hel Peninsula. The method of double decomposition, presented in Prosnak & Czesnik (2001), is applied to the transformation of such an original domain onto a canonical one, which consists of two separate unit discs. The first decomposition concerns the domain which is split into two adjacent subdomains by means of a segment of a straight line. The second decomposition involves two holomorphic functions, each one mapping a subdomain onto a separate disc. The decomposition consists in replacing the function by a sequence of simple ones, so that the mapping is performed step-wise. Each sequence starts with the Schwarz-Christoffel function, the last step consisting in an inversion that transforms an infinite circular domain onto a disc. The data for the problem is contained in the Annex, and is represented by two sets of rectangular coordinates defining consecutive discrete points of the contours bounding the subdomains. The solution to the problem consists of: - two sets of functions, consecutively transforming each of the subdomains; - the numerical values of the parameters of these functions; - a set of figures illustrating the consecutive transformations. The accuracy of the first, and of the penultimate transformation are given, because only in the case of these two functions do the unknown coefficients have to be determined by means of a suitable iterative process. The coefficients of all the remaining functions are evaluated from exact formulae. It should be recalled that the depth of the Gulf of Gdansk varies considerably - from a few to 110 metres - the gradients of the bottom being rather large. Therefore, the domain of the solution for any mathematical problem describing the hydrodynamical phenomena occurring in the Gulf is usually taken to be three-dimensional. Nevertheless, the paper deals with the transformation of the free surface of the Gulf, assumed as plane. It should be emphasized that this assumption does not mean that the whole domain of the solution has to be regarded as plane.

20

Tests on a new method for numerical solution to the Navier-Stokes equations. part II: plane unsteady flow around curvilinear contours

Klonowska-Prosnak M. E., Kołodziejczyk M.

International Journal of Applied Mechanics and Engineering

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2002

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Vol. 7, no 2

619-652

EN

Similarly as the preceding paper of the authors (Klonowska and Kołodziejczyk, 1999) also the present one deals with the new method (Prosnak and Kosma, 1991) for the determination of unsteady, plane flows of viscous incompressible fluids. The novelty of the method consists in elimination of pressure from the system of the Navier-Stokes equations by means of integration of the "total" differential of pressure. Consequently - the order of the system resulting from such an elimination is not increased in comparison with the original one, and no artificial, non-physical boundary conditions occur. The main difference between the present paper and the previous one consists in geometrical properties of the domain of solution. It is infinite, and bounded by a curvilinear contour. Three contours are considered in the paper: a flat plate, a circle, and the NACA 0012 airfoil. The results prove, that also in these cases the method works properly - at least for rather small values of the Reynolds number. Such a limitation is due to the efficiency of the computers being available at the time, when the calculations were made.