On kinematic constraint in microplane theory

In this paper it is shown that the microplane formulation based on the volumetric-deviatoric split (VD split) possess the property of losing macro information during the transition from macro to micro level, i.e. during the projection of macroscopic strain components on microplanes with various orientations. However, it is also shown that the kinematic constraint principle that includes the microplane lateral strains preserves all the information related to the macroscopic stain tensor

An iterative algorithm for initializing the flow in a pipe system with more reservoirs

For the analysis of a pressure pipe system with multiple reservoirs, a numerical algorithm for initializing the flow in the system is developed. This algorithm finds real hydraulic head values in junctions of a pressure pipe system by using an iterative method. Also, it determines the orientation and amount of the flow rate for an arbitrary given pressure pipe system and a number of reservoirs. By comparing the presented method with a classical three-reservoir problem, it might be said that the proposed method performs a hydraulic head correction in junctions in an inverse way. The method is implemented in the computer code developed for the analysis of an arbitrary pressure pipe system

Numerical model of compliant tower oscillations

U svrhu provedbe dinamičke analize rešetkastih offshore konstrukcija temeljenih na morskom dnu (eng. compliant tower), razvijen je numerički algoritam za proračun polja pomaka konstrukcije uslijed hidrodinamičkog djelovanja mora. Stohastička narav rasprostiranja valova u moru je uvedena u analizu putem funkcije gustoće snage spektra valova. Opterećenje konstrukcije morskim valovima se potom definira koristeći Morisonov model djelovanja sile. Kinematički parametri valova su definirani linearnom teorijom. Utjecaj vjetra na raspodjelu strujanja mora u vertikalnom smjeru je uzet u obzir kroz kontekst Ekmanove spirale. Rešetkasta konstrukcija je modelirana prostornim Euler-Bernoulijevim konačnim elementima. Inicijalizacija dinamičkog proračuna započinje utvrđivanjem polja pomaka nastalog uslijed statičkog djelovanja uzgona, korisnog i stalnog tereta. Integracija jednadžbi gibanja se provodi Newmarkovom β metodom. U svrhu izrade numeričkih primjera, prikazani algoritam je implementiran u programsko okruženje MathCAD 15. Numerički primjeri uključuju analizu slobodnih i prisilnih oscilacija konstrukcije te FFT analizu zapisa pomaka težišta operativne površine konstrukcije.For the purpose of conducting a dynamic analysis of compliant towers, a numerical algorithm was developed for calculating the structure of the movement field caused by hydrodynamic action of the sea. The stochastic nature of wave spreading in the sea is introduced into the analysis through the function of power spectral density of waves. The structure’s wave load is then defined by applying the Morison model of force action. Kinematic parameters of waves are defined by the linear theory. The wind impact on the distribution of marine circulation in the vertical direction is taken into account in the context of the Ekman spiral. The compliant tower is modelled with spatial Euler–Bernoulli final elements. The initiation of the dynamic calculation starts with the determining the movement field resulting from static activity of buoyant force, dead and permanent loads. The integration of the equations of motion is performed by applying the Newmark-β method. To develop numeric parameters, the presented algorithm is implemented in the MathCAD 15 programming environment. The numerical parameters include the analysis of free and forced oscillations of the structure and the FFT analysis of the records of the centre of gravity shift of the structure’s operating surface

Numerical model of compliant tower oscillations

U svrhu provedbe dinamičke analize rešetkastih offshore konstrukcija temeljenih na morskom dnu (eng. compliant tower), razvijen je numerički algoritam za proračun polja pomaka konstrukcije uslijed hidrodinamičkog djelovanja mora. Stohastička narav rasprostiranja valova u moru je uvedena u analizu putem funkcije gustoće snage spektra valova. Opterećenje konstrukcije morskim valovima se potom definira koristeći Morisonov model djelovanja sile. Kinematički parametri valova su definirani linearnom teorijom. Utjecaj vjetra na raspodjelu strujanja mora u vertikalnom smjeru je uzet u obzir kroz kontekst Ekmanove spirale. Rešetkasta konstrukcija je modelirana prostornim Euler-Bernoulijevim konačnim elementima. Inicijalizacija dinamičkog proračuna započinje utvrđivanjem polja pomaka nastalog uslijed statičkog djelovanja uzgona, korisnog i stalnog tereta. Integracija jednadžbi gibanja se provodi Newmarkovom β metodom. U svrhu izrade numeričkih primjera, prikazani algoritam je implementiran u programsko okruženje MathCAD 15. Numerički primjeri uključuju analizu slobodnih i prisilnih oscilacija konstrukcije te FFT analizu zapisa pomaka težišta operativne površine konstrukcije.For the purpose of conducting a dynamic analysis of compliant towers, a numerical algorithm was developed for calculating the structure of the movement field caused by hydrodynamic action of the sea. The stochastic nature of wave spreading in the sea is introduced into the analysis through the function of power spectral density of waves. The structure’s wave load is then defined by applying the Morison model of force action. Kinematic parameters of waves are defined by the linear theory. The wind impact on the distribution of marine circulation in the vertical direction is taken into account in the context of the Ekman spiral. The compliant tower is modelled with spatial Euler–Bernoulli final elements. The initiation of the dynamic calculation starts with the determining the movement field resulting from static activity of buoyant force, dead and permanent loads. The integration of the equations of motion is performed by applying the Newmark-β method. To develop numeric parameters, the presented algorithm is implemented in the MathCAD 15 programming environment. The numerical parameters include the analysis of free and forced oscillations of the structure and the FFT analysis of the records of the centre of gravity shift of the structure’s operating surface

Water mass oscillations in a generic surge chamber

Numerički modeli zasad se najčešće koriste u svrhu prognoziranja oscilacija vodnih masa u sustavu akumulacija – dovodni tunel – vodna komora. Pritom, neovisno o usvojenoj metodi diskretizacije vladajućih jednadžbi, najčešće se susreće da su ti modeli razvijeni pod pretpostavkom da je vodna komora okarakterizirana kružnim i konstantnim poprečnim presjekom. Da bi se zaobišla ta ograničavajuća pretpostavka, predložen je numerički algoritam kojim se mogu analizirati oscilacije razine vode u vodnoj komori općenitog oblika.Numerical models are currently most often used to simulate water mass oscillations inside the system formed of the reservoir, pressure tunnel, and surge chamber. At the same time, regardless of the method used for discretisation of governing equations, the numerical models are most often developed under assumption that the surge chamber is characterized by the constant and circular cross section. To omit this restrictive assumption, a numerical algorithm is proposed to enable analysis of water level oscillations in a generic surge chamber

Numerical analysis of hysteresis in rating curves for open channel flow

For the purpose of studying the hysteresis in rating curves for unsteady flow regime in open channels, a numerical analysis of water wave propagation is performed by a numerical integration of one-dimensional (1D) Saint Venant differential equations. By applying two different boundary conditions, which are specified as the time change in water level h at the upstream boundary of the flow domain, it is shown that the downstream rating curves can obtain the same shapes in the Q-h plane. However, to emphasize the expected difference in rating curves, a third, time axis, is added. Accordingly, the rating curves obtain a spatial shape from which the dynamics of evolution in the relation between the stage h and related discharge Q can be evidenced. Apart from the description of the used numerical formulation (based on the method of characteristics), a numerical example is also presented in the support of the given statements

A physical model of convective-dispersive transport in an intergranular porous material

A physical model for tracer transport in an intergranular porous material is presented. Particularly, the measured temporal variations of a tracer mass concentration inside the physical model are compared to the one predicted for the same boundary and initial conditions by the 1D and 2D analytical solutions of the governing differential equation. A non-reactive tracer and steady flow conditions are considered. The Péclet number for the considered flow is such that the molecular diffusion can be neglected

Algorithm for Optimal Distribution of Diffuser Nozzle Areas of a Submarine Outfall

U radu se predlaže iterativni numerički algoritam razvijen u svrhu određivanja optimalne raspodjele površina otvora sapnica duž difuzora podmorskog ispusta. Za zadanu geometriju toka, predloženi numerički postupak dovodi do raspodjele površina otvora sapnica potrebnih za osiguravanje jednolikog izlaznog protoka i nužnog početnog razrjeđenja u bliskom polju. Postupak omogućuje i identifikaciju površina otvora sapnica potrebnih za osiguravanjem nejednolikog rasporeda protoka koji je prethodno bio zadan. Nepoznate vrijednosti površina otvora i izlaznih brzina dobivaju se istodobno pomoću dvije međudjelujuće iterativne petlje. Za dane vrijednosti izlaznih brzina, u vanjskoj iterativnoj petlji, proračunavaju se površine sapnica i konstitutivne jednadžbe za gubitak tlaka. Dobivene vrijednosti se zatim koriste u unutarnjoj iterativnoj petlji, u kojoj se, pomoću Newton-Raphsonove metode, definira sljedeća aproksimativna vrijednost izlaznih brzina. Postupak se ponavlja sve dok oba konvergencijska kriterija nisu ispunjena. Osim teorijskog pregleda i pripadajućeg numeričkog algoritma, predstavljeno je i obrađeno nekoliko numeričkih primjera.The paper proposes an iterative numerical algorithm developed for determining optimal distribution of nozzle opening areas along the diffuser of a submarine outfall. For the given flow geometry, the proposed nu¬merical procedure results in a distribution of nozzle opening areas needed for ensuring a uniform outlet flow and the required initial dilution in the near field. The procedure also facilitates the determination of nozzle opening areas necessary for ensuring uneven distribution of the previously given flow. The unknown values of the nozzle opening areas and outlet velocities are obtained simultaneously by means of two interactive iterative loops. For the given values of the outlet velocities, nozzle areas and constitutive equations for pressure loss are calculated in the outer iterative loop. The obtained values are then used in the inner iterative loop where the next approximate value of the outlet velocities is determined by the Newton-Raphson method. The procedure is repeated until both convergence criteria are met. In addition to a theoretical overview and the associated numerical algorithm, several numerical examples are also presented

An Iterative Algorithm for Optimizing Pipe Diameter in Pressurized System

In order to achieve optimal flow velocity in stationary flow condition in pressurized pipe systems, it is necessary to determine an optimal value of pipe diameter. It should be noted that even for a simple pipe network with pipes connected in only few numbers of loops, the determination of the optimal pipe diameter is an unintuitive problem and from the mathematical point of view a nonlinear problem that requires some iterative process with a relatively complicated update procedure between iterations. Usually, the optimization algorithms are based on stochastic methods. A deterministic approach is proposed that satisfies the prescribed constrains in terms of the specified flow velocity in the pressurized system of pipes. The presented iterative algorithm for optimizing the pipe diameter is implemented in computer code PIPENET3D (written in FORTRAN90) and used for hydraulic analysis of steady flow in an arbitrary pressured pipe system

A mixed MOC/FDM numerical formulation for hydraulic transients

U radu je predložena miješana numerička formulacija, bazirana na metodi karakteristika (MK) i metodi konačnih razlika (MKR) te je namijenjena za provedbu računalnih simulacija hidrauličkih tranzijenata. U svrhu uključivanja utjecaja nestacionarnog trenja tj. doprinosa posmičnih naprezanja na stijenkama cijevi karakterističnih za nestacionarni tok, prikazan numerički algoritam uključuje Brunoneov modela trenja. Osnovne diferencijalne jednadžbe hidrauličkog udara su diskretizirane MK te je lokalna i konvektivna akceleracija u nestacionarnom modelu trenja diskretizirana MKR. Navedeno dovodi do eksplicitno-implicitne numeričke sheme za predikciju nepoznatih veličina piezometarskog potencijala i protoka iz poznatih početnih uvjeta. Dobiveni sustav jednadžbi je riješen iterativnim postupkom. Obzirom na relativno malu kompleksnost, numerički algoritam je atraktivan za računalnu implementaciju. Na kraju rada su prikazani ilustrativni numerički primjeri koji su uspoređeni sa eksperimentalno dobivenim podacima te je priložena diskusija.A mixed numerical formulation, based on the method of characteristics (MOC) and finite difference method (FDM), is proposed for the computational simulations of hydraulic transients. To include the effects of unsteady friction, i.e. the contribution of unsteady behavior of wall shear stress, the numerical algorithm includes the Brunone friction model. The governing water hammer equations are discretized by MOC and the discretization of the local and convective acceleration term in the unsteady friction model is conducted by FDM. The procedure results in an explicit-implicit time marching scheme used for predicting the unknown values of piezometric head and discharge from the known initial conditions. The resulting system of equations is resolved in an iterative fashion. Due to its relatively low complexity, the obtained numerical algorithm is attractive for computational implementation. At the end of the paper a few illustrative numerical examples are presented, compared with available experimental data, and discussed