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1

A numerical method for viscous flow in a driven cavity with heat and concentration sources placed on its side wall

Ambethkar V., Basumatary L. R.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 3

17--30

EN

This paper proposes a method to numerically study viscous incompressible two-dimensional steady flow in a driven square cavity with heat and concentration sources placed on its side wall. The method proposed here is based on streamfunction-vorticity (Ψ-ξ) formulation. We have modified this formulation in such a way that it suits to solve the continuity, x and y-momentum, energy and mass transfer equations which are the governing equations of the problem under investigation in this study. No-slip and slip wall boundary conditions for velocity, temperature and concentration are defined on walls of a driven square cavity. In order to numerically compute the streamfunction Ψ, vorticityfunction ξ , temperature θ, concentration C and pressure P at different low, moderate and high Reynolds numbers, a general algorithm was proposed. The sequence of steps involved in this general algorithm are executed in a computer code, developed and run in a C compiler. We propose that, with the help of this code, one can easily compute the numerical solutions of the flow variables such as velocity, pressure, temperature, concentration, streamfunction, vorticityfunction and thereby depict and analyze streamlines, vortex lines, isotherms and isobars, in the driven square cavity for low, moderate and high Reynolds numbers. We have chosen suitable Prandtl and Schmidt numbers that enables us to define the average Nusselt and Sherwood numbers to study the heat ad mass transfer rates from the left wall of the cavity. The stability criterion of the numerical method used for solving the Poisson, vorticity transportation, energy and mass transfer has been given. Based on this criterion, we ought to choose appropriate time and space steps in numerical computations and thereby, we may obtain the desired accurate numerical solutions. The nature of the steady state solutions of the flow variables along the horizontal and vertical lines through the geometric center of the square cavity has been discussed and analyzed. To check the validity of the computer code used and corresponding numerical solutions of the flow variables obtained from this study, we have to compare these with established steady state solutions existing in the literature and they have to be found in good agreement.

2

On necessary and suﬃcient conditions for stability and quasistability in combinatorial multicriteria optimization

Kuzmin K. G., Nikulin Y. V., Makela M. M.

Control and Cybernetics

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2017

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Vol. 46, no. 4

361--382

EN

We consider a multiple objective combinatorial optimization problem with an arbitrary vector-criterion. The necessary and suﬃcient conditions for stability and quasistability are obtained for large classes of problems with partial criteria possessing certain properties of regularity.

3

Long term analysis of deformations in salt mines: Kłodawa Salt Mine case study, central Poland

Cała M., Tajduś A., Andrusikiewicz W., Kowalski M., Kolano M., Stopkowicz A., Cyran K., Jakóbczyk J.

Archives of Mining Sciences

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2017

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Vol. 62, no. 3

565--577

EN

Located in central Poland, the Kłodawa salt dome is 26 km long and about 2 km wide. Exploitation of the dome started in 1956, currently rock salt extraction is carried out in 7 mining fields and the 12 mining levels at the depth from 322 to 625 meters below sea level (m.b.s.l.). It is planned to maintain the mining activity till 2052 and extend rock salt extraction to deeper levels. The dome is characterised by complex geological structure resulted from halokinetic and tectonic processes. Projection of the 3D numerical analysis took into account the following factors: mine working distribution within the Kłodawa mine (about 1000 rooms, 350 km of galleries), complex geological structure of the salt dome, complicated structure and geometry of mine workings and distinction in rocks mechanical properties e.g. rock salt and anhydrite. Analysis of past mine workings deformation and prediction of future rock mass behaviour was divided into four stages: building of the 3D model (state of mine workings in year 2014), model extension of the future mine workings planned for extraction in years 2015-2052, the 3D model calibration and stability analysis of all mine workings. The 3D numerical model of Kłodawa salt mine included extracted and planned mine workings in 7 mining fields and 14 mining levels (about 2000 mine workings). The dimensions of the model were 4200 m × 4700 m × 1200 m what was simulated by 33 million elements. The 3D model was calibrated on the grounds of convergence measurements and laboratory tests. Stability assessment of mine workings was based on analysis of the strength/stress ratio and vertical stress. The strength/stress ratio analysis enabled to indicate endangered area in mine workings and can be defined as the factor of safety. Mine workings in state close to collapse are indicated by the strength/stress ratio equals 1. Analysis of the vertical stress in mine workings produced the estimation of current state of stress in comparison to initial (pre-mining) conditions. The long-term deformation analysis of the Kłodawa salt mine for year 2014 revealed that stability conditions were fulfilled. Local disturbances indicated in the numerical analysis were connected with high chambers included in the mining field no 1 and complex geological structure in the vicinity of mine workings located in the mining fields no 2 and 3. Moreover, numerical simulations that projected the future extraction progress (till year 2052) showed positive performance. Local weakness zones in the mining field no 7 are associated with occurrence of carnallite layers and intensive mining which are planned in the mining field no 6 at the end of rock salt extraction.

PL

Wydobycie soli kamiennej w Kopalni Soli Kłodawa prowadzone jest obecnie w 7 polach eksploatacyjnych na 12 poziomach kopalniach (głębokość 322 - 625 m p.p.m.). Planowane przedłużenie eksploatacji do roku 2052, wiąże się z udostępnieniem zasobów na kolejnych (niższych) poziomach kopalnianych. Kopalnia Soli Kłodawa eksploatuje sól kamienną w wysadzie solnym, który charakteryzuje się skomplikowaną budową geologiczną. W modelu numerycznym Kopalni Soli Kłodawa odwzorowano przestrzenny układ wyrobisk kopalnianych (około 1000 komór i 350 km wyrobisk chodnikowych), złożoną geometrię wyrobisk, skomplikowaną budowę geologiczną wysadu oraz uwzględniono zróżnicowanie własności mechanicznych skał. Analizę numeryczną deformacji wyrobisk kopalnianych (dotychczas wybranych i planowanych do wybrania) podzielono na 4 etapy, które obejmowały: budowę modelu 3D kopalni (za bazowy przyjęto stan wyrobisk w roku 2014), rozszerzenie modelu o wyrobiska planowane do wykonania w latach 2015-2052, kalibrację modelu 3D, analizę stateczności. Model numeryczny 3D obejmował wycinek górotworu o wymiarach 4200 m × 4700 m × 1200 m. Model buduje około 33 miliony elementów. Kalibrację modelu wykonano na podstawie pomiarów konwergencji (prowadzonych w kopalni) oraz wyników testów laboratoryjnych skał budujących złoże. Ocenę stateczności wyrobisk kopalnianych przedstawiono na podstawie analizy stanu wytężenia oraz analizy stanu naprężenia w górotworze. Wyniki przeprowadzonej analizy numerycznej wykazały, że warunek stateczności wyrobisk Kopalni Soli Kłodawa jest spełniony dla stanu na rok 2014. Lokalnie występujące strefy wytężenia związane są z wysokimi komorami w polu eksploatacyjnym nr 1 oraz złożoną budową geologiczną w otoczeniu wyrobisk zlokalizowanych w polach eksploatacyjnych nr 2 i 3. Prognoza dla projektowanego stanu wyrobisk w roku 2052 (zakończenie eksploatacji) wykazała pozytywne rezultaty. Lokalne strefy osłabienia, które zidentyfikowano w polu eksploatacyjnym nr 7, związane są z występowaniem warstw karnalitu oraz intensywnymi pracami górniczymi, które w ostatniej fazie eksploatacji będą skoncentrowane w polu eksploatacyjnym nr 6.

4

Remarks on stability of magneto-elastic shocks

Domański W.

Biuletyn Wojskowej Akademii Technicznej

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2015

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Vol. 64, nr 4

241-256

EN

The problem of stability of plane shock waves for a model of perfect magneto-elasticity is investigated. Important mathematical properties, like loss of strict hyperbolicity and loss of genuine nonlinearity, and their consequences for the stability of magneto-elastic shocks are discussed. It is shown that some of these shocks do not satisfy classical Lax stability conditions. Both compressible and incompressible models of magneto-elasticity are discussed.

PL

Rozważono problem stabilności fal uderzeniowych dla modelu doskonałej magneto-sprężystości. Przedyskutowano ważne matematyczne własności takie jak utrata ścisłej hiperboliczności i istotnej nieliniowości oraz ich konsekwencje dla stabilności magneto-sprężystych fal uderzeniowych. Pokazano, że pewne z tych fal nie spełniają klasycznego warunku stabilności Laxa. Zanalizowano zarowno ściśliwe jak i nieściśliwe modele magneto-sprężystości.

5

Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for the asymptotic stability

Stanisławski R., Latawiec K. J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2013

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Vol. 61, nr 2

353--361

EN

This paper presents a series of new results on the asymptotic stability of discrete-time fractional difference (FD) state space systems and their finite-memory approximations called finite FD (FFD) and normalized FFD (NFFD) systems. In Part I, new, general, necessary and sufficient stability conditions are introduced in a unified form for FD/FFD/NFFD-based systems. In Part II, an original, simple, analytical stability criterion is offered for FD-based systems, and the result is used to develop simple, efficient, numerical procedures for testing the asymptotic stability for FFD-based and, in particular, NFFD-based systems. Consequently, the so-called f-poles and f-zeros are introduced for FD-based system and their closed-loop stability implications are discussed.

6

Wymiarowanie środników klasy 4. belek podsuwnicowych metodą naprężeń zredukowanych

Żmuda J., Skowrońska J.

Inżynieria i Budownictwo

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2012

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R. 68, nr 8

436-439

PL

Podano metodologię sprawdzania warunków stateczności środników klasy 4, nieużebrowanych i z żebrami podłużnymi, na podstawie postanowień zawartych w rozdziale 10 normy PN-EN 1993-1-5. W metodzie naprężeń zredukowanych nawiązuje się do wcześniejszych metod sprawdzania nośności belek podsuwnicowych zawartych w dawnej normie PN-B-03200:1976. Zamieszczono przykład liczbowy.

EN

The article presents the methodology for checking the stability conditions of class 4 unribbed webs and longitudinal ribs in accordance with the method of reduced stresses based on the provisions of Chapter 10 of the standard EN 1993-1-5. This method of reduced stress reference is made to previous method of testing capacity crane beams which no longer contained in the standard such as PN-B-03200:1976. Numerical example was prepared to determine the stability conditions of monosymmetrical beam class 4.

7

The Analysis of Stability of Weak Solutions of Circular Plate

Jama D., Czech D., Szopa M.

Machine Dynamics Problems

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2009

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Vol. 33, No. 3

25-41

EN

Stability of parametric vibrations of a circular plate has been discussed for equations which describe the dynamics of a circular plate in weak formulation. A weak form of dynamic equations of mechanical structures has been derived applying the calculus of variation. Nearly definite asymptotic balance stability of a circular plate, without prior digitizing, has been analyzed in accordance with direct Lyapunov's method. The method of stability analysis has been worked out for the dynamic problems of a circular plate taking into account boundary conditions corresponding to the simple supported, the clamped and free edges. Sufficient stability conditions have been determined without involving the third and the fourth derivatives of solutions.

8

Stability Analysis of Continuous Systems with Imperfect Boundary Conditions in a Weak Formulation

Tylikowski A.

Machine Dynamics Problems

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2008

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Vol. 32, No. 3

107-116

EN

Dynamics of continuous systems have been considered in a weak (variational) form. Dynamics equation of beam subjected to the axial stochastic force in the weak formulation has been derived. The weak form of dynamics equations of linear mechanical structures is obtained using variational calculus. The almost sure stochastic stability of beam equilibrium, without the previous discretization, has been analysed by means of direct Lyapunov method. The stability analysis method is developed for distributed dynamic problems with relaxed assumptions imposed on solutions. Sufficient stability conditions have been established for the imperfect boundary conditions and two limit cases: the simply supported beam and the clamped beam are obtained.

9

Teoria wielomodowej pracy lasera z ośrodkiem aktywnym w postaci kryształu fotonicznego o sieci kwadratowej

Koba M., Szczepański P.

Elektronika : konstrukcje, technologie, zastosowania

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2008

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Vol. 49, nr 9

69-78

PL

W artykule przedstawiono połklasyczny opis generacji promieniowania w laserze posiadającym ośrodek aktywny w postaci kryształu fotonicznego. Model uwzględnia pracę zarówno jednomodową jak i wielomodową. Przedstawiono warunki stabilności generacji dla struktury laserowej posiadającej aktywny kryształu fotonicznego o sieci kwadratowej.

EN

This article describes generation of radiation in photonic crystal laser with square lattice in semiclassical manner. Presented model takes into account singlemode as well as multimode processes. The stability conditions for both single and multimode operations are expressed for active photonic crystal with square lattice.

10

Charakterystyka stanów równowagi atmosfery w okresie IV-X 2002 roku dla Ursynowa-SGGW

Rozbicka K.

Przegląd Naukowy Inżynieria i Kształtowanie Środowiska

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2003

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Vol. 12, No. 1

105--113

EN

The paper presents atmospheric stability description by the use Richardson number (Ri). The calculation was done for two types o f weather: sunny days (24h) and cloudy days (24h). In accordance with Oke (1987) stability static was divided as follows: unstable stability a-fully forced convection, b-mixed convection, c-free convection; d-neutral stability; stable stability e-fully forced convection, f-damped forced convection, g-no convection. Richardson number accomplish negative value, which stands unstable static conditions (a, b) for more cases of day-time period both sunny and cloudy days. Stable static conditions, particularly f characterized by positive value of Ri appear for cloudless nights most frequently. A large diversity of static conditions was observed for night-time period with cloudy weather. The most frequent static conditions in considered period is stable (maximum in October), then unstable (maximum in summertime months).

11

Design and Stability of Fuzzy Logic Multi-Regional Output Controllers

Domański P., Brdyś M. A., Tatjewski P.

International Journal of Applied Mathematics and Computer Science

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1999

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Vol. 9, no 4

883-897

EN

Design and stability analysis of fuzzy multi-regional digital controllers is considered in the paper. The controllers are based on a notion of NARMAX systems, very similar to the Takagi-Sugeno fuzzy model. The nonlinear system is approximated by a number of linear subsystems. Linear controllers are designed for all subsystems. It can be made in a classical way due to the subsystems linearity. The controllers are blended into one controller by employing fuzzy logic, the result being the fuzzy multi-regional controller (FuMR). The stability analysis of nonlinear systems with FuMR controllers composed of dynamic output feedback local linear controllers is provided. Examples illustrate the design procedure and the meaning of the stability criterion.