Domain of convergence of an iterative procedure for an autonomous boundary-value problem

We find an estimate for the range of values of a small parameter for which the convergence of an iterative procedure for the construction of solutions of an autonomous weakly nonlinear Noether boundary-value problem for a system of ordinary differential equations in the critical case is preserved. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 416–432, July–September, 2006.     

Conditions for the existence of a solution of a Noether boundary-value problem for a second-order system

We consider a weakly nonlinear boundary-value problem for a system of second-order ordinary differential equations. We find a sufficient condition for the existence of at least one solution of this problem and propose a convergent iterative algorithm for the determination of its solution. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 368–375, July–September, 2006.     

The Domain-Boundary Element Method (DBEM) for hyperelastic and elastoplastic finite deformation: axisymmetric and 2D/3D problems

Summary  This paper presents the solution of geometrically nonlinear problems in solid mechanics by the Domain-Boundary Element Method. Because of the Total-Lagrange approach, the arising domain and boundary integrals are evaluated in the undeformed configuration. Therefore, the system matrices remain unchanged during the solution procedure, and their time-consuming computation needs to be performed only once. While the integral equations for axisymmetric finite deformation problems will be derived in detail, the basic ideas of the formulation in two and three dimensions can be found in [1]. The present formulation includes torsional problems with finite deformations, where additional terms arise due to the curvilinear coordinate system. A Newton–Raphson scheme is used to solve the nonlinear set of equations. This involves the solution of a large system of linear equations, which has been a very time-consuming task in former implementations, [1, 2]. In this work, an iterative solver, i.e. the generalized minimum residual method, is used within the Newton–Raphson algorithm, which leads to a significant reduction of the computation time. Finally, numerical examples will be given for axisymmetric and two/three-dimensional problems. Received 29 August 2000; accepted for publication 10 October 2000     

Formation of shock waves in flow of charge carriers in semiconductors

A hydrodynamic model of the physics of semiconductors is studied numerically. It is shown that the solution of the problem of an (n⁺-n-n⁺) ballistic diode has a shock wave. This problem is solved using an iterative method. An economical conservative semi-implicit difference scheme is developed for search of a numerical solution. Siberian State Geodetic Academy, Novosibirsk 630108. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 3–10, September–October, 1999.     

The inverse problem of aerodynamics with Cartesian coordinates as dependent variables

The gas-dynamic equations are written in Cartesian coordinates in the potential plane. This approach to the inverse problem makes it possible to seek a quasisolution within a set of closed contours. To solve this problem an iterative procedure is developed using successive approximations and taking into account the fact that aerodynamic airfoils are elongated in the streamwise direction. In the case where an arbitrary pressure distribution is prescribed, the procedure determines the airfoil providing the best approximation to the design data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 147–157, January–February, 1994.     

An inverse elastoplastic problem for plates

We study an inverse elastoplastic problem of determining the residual stresses, the plasticity zone, and the external loads for a plate for known residual deflections which occur after removal of these loads and elastic unloading. Assuming that the deformation theory of plasticity is valid at the active stage of deformation, we prove the theorem of unique solution. An iterative method of solution is proposed and a variational formulation of the problem is given. Some simple examples are considered. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 186–194, July–August, 1999.     

Large Eddy Simulations Using the Subgrid-Scale Estimation Model and Truncated Navier–Stokes Dynamics

We describe a procedure for large eddy simulations of turbulence which uses the subgrid-scale estimation model and truncated Navier–Stokes dynamics. In the procedure the large eddy simulation equations are advanced in time with the subgrid-scale stress tensor calculated from the parallel solution of the truncated Navier–Stokes equations on a mesh two times smaller in each Cartesian direction than the mesh employed for a discretization of the resolved quantities. The truncated Navier–Stokes equations are solved through a sequence of runs, each initialized using the subgrid-scale estimation model. The modeling procedure is evaluated by comparing results of large eddy simulations for isotropic turbulence and turbulent channel flow with the corresponding results of experiments, theory, direct numerical simulations, and other large eddy simulations. Subsequently, simplifications of the general procedure are discussed and evaluated. In particular, it is possible to formulate the procedure entirely in terms of the truncated Navier–Stokes equation and a periodic processing of the small-scale component of its solution. Received 27 April 2001 and accepted 16 December 2001     

A correlation for laminar mixed convection from vertical plates using transient experiments

The applicability of the transient cooling experimental technique is studied for developing a correlation for Nusselt number with other pertinent parameters for laminar aiding mixed convection flows over an isothermal vertical plate. A heated aluminum plate, modeled as a lumped system, is allowed to cool in the mixed convection environment during which the transient response of the plate is recorded using a data acquisition system. The experimentally known transient response of the plate is then compared with the numerically computed transient response to estimate the square of the residual. The minimization of this residual using Levenberg–Marquardt iterative procedure retrieves the values of relevant parameters in the correlation, whose form is assumed a priori. The experiments were conducted for a temperature range of 410–305 K and a corresponding Richardson number range 20–0.04. The retrieved values of the parameters compare well with those available in literature.     

A Numerical Model of Transient Unsaturated Flow Under Centrifugation Based on Mass Balance

Numerical modeling of unsaturated flow in porous media under centrifugation is studied. A precise and numerically efficient approximation is presented for the mathematical model, based on Richards’ nonlinear and degenerate equation expressed in terms of effective saturation using the Van Genuchten–Mualem ansatz. The main difference with other methods is the utilization of a nonlocal condition based on mass balance. The method is suitable for determination of soil parameters, including the saturated conductivity, via the solution of an inverse problem in an iterative way. First, the fully saturated sample is centrifugated with a free outflow boundary during some time interval. Next, the output boundary is sealed and the sample is centrifugated for a prescribed time interval, or up to the creation of an equilibrium. Finally, the centrifugation is continued with a free outflow boundary. This procedure can be repeated to increase the information to drive the inverse problem. The application of the present method requires only non-intrusive, cheap measurements: rotational momentum and/or gravitational center of the sample, and optionally, the amount of expelled water.     

Convergent laminar fluid flow between two rotating disks

A stationary flow from the periphery to the center in a hollow between two coaxial, closely located rotating disks is studied by the iterative method of solving a system of equations of the dynamics of a viscous incompressible fluid. The existence and uniqueness of the approximate solution are shown. Moscow State University of the Food Industry, Moscow 125080. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 77–83, March–April, 2000.     

Investigation of even systems of nonlinear functional differential equations with restrictions

We establish consistency conditions for even systems of nonlinear functional differential equations with restrictions and substantiate the applicability of an iterative method to these problems. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 252–260, April–June, 2008.     

Negative Saturation Approach for Non-Isothermal Compositional Two-Phase Flow Simulations

This article deals with developing a solution approach, called the non-isothermal negative saturation (NegSat) solution approach. The NegSat solution approach solves efficiently any non-isothermal compositional flow problem that involves phase disappearance, phase appearance, and phase transition. The advantage of the solution approach is that it circumvents using different equations for single-phase and two-phase regions and the ensuing unstable procedure. This paper shows that the NegSat solution approach can also be used for non-isothermal systems. The NegSat solution approach can be implemented efficiently in numerical simulators to tackle modeling issues for mixed CO₂–water injection in geothermal reservoirs, thermal recovery processes, and for multicontact miscible and immiscible gas injection in oil reservoirs. We illustrate the approach by way of example to cold mixed CO₂–water injection in a 1D geothermal reservoir. The solution is compared with an analytical solution obtained with the wave-curve method (method of characteristics) and shows excellent agreement. A complete set of simulations is carried out, which identifies six bifurcations. The two main bifurcations are (1) when the most downstream compositional wave is replaced by a compositional shock and (2) when an extra Buckley–Leverett rarefaction appears. The plot of the useful energy (exergy) versus the CO₂ storage capacity shows a Z-shape. The top horizontal part represents a branch of high exergy recovery/relatively lower storage capacity, whereas the bottom part represents a branch of lower exergy recovery/higher storage capacity.     

Modeling of simultaneous heat and mass transfer processes in ammonia–water absorption systems from general correlations

This paper presents a general differential mathematical model to analyze the simultaneous heat and mass transfer processes that occur in different components of an ammonia–water absorption system: absorber, desorber, rectifier, distillation column, condenser and evaporator. Heat and mass transfer equations are considered, taking into account the heat and mass transfer resistances in the liquid and vapour phases. The model considers the different regions: vapour phase, liquid phase and an external heating or cooling medium. A finite difference numerical method has been considered to solve the resulting set of nonlinear differential equations and an iterative algorithm is proposed for its solution. A map of possible solutions of the mass transferred composition z is presented when varying the interface temperature, which enables to establish a robust implementation code. The analysis is focused on the processes presented in ammonia–water absorption systems. The model is applied to analyze the ammonia purification process in an adiabatic packed rectification column and the numerical results show good agreement with experimental data.     

Oscillation of all solutions of iterative equations

The paper contains sufficient conditions for the oscillation of all solutions of linear functional iterative equations. Published in Neliniini Kolyvannya, Vol. 10, No. 3, pp. 348–364, July–September, 2007.     

In situ force-balance tensiometry

 Although a fundamental physical parameter, surface tension is difficult to measure. Common tensiometry inaccuracy comes from failure to control air–liquid–solid contact conditions, or account for liquid meniscus geometry and buoyancy corrections. This paper describes an in situ tensiometry technique, based on withdrawal of a thin-walled tube from the liquid interface, that enforces a known air–liquid–solid contact condition. This technique can be pursued at any level of experimental hygiene. Experimental results for filtered tap water, an alcohol–water solution, and a surfactant–water solution show that results repeatable to three significant digits are obtained with modest effort for a variety of geometrical parameters. Received: 7 October 1997/Accepted: 23 April 1998     

Identification of Material Damage Model Parameters: an Inverse Approach Using Digital Image Processing

A reliable prediction of ductile failure in metals is still a wide-open matter of research. Several models are available in the literature, ranging from empirical criteria, porosity-based models and continuum damage mechanics (CDM). One major issue is the accurate identification of parameters which describe material behavior. For some damage models, parameter identification is more or less straightforward, being possible to perform experiments for their evaluation. For the others, direct calibration from laboratory tests is not possible, so that the approach of inverse methods is required for a proper identification. In material model calibration, the inverse approach consists in a non-linear iterative fitting of a parameter-dependent load–displacement curve (coming from a FEM simulation) on the experimental specimen response. The test is usually a tensile test on a round-notched cylindrical bar. The present paper shows a novel inverse procedure aimed to estimate the material parameters of the Gurson–Tvergaard–Needleman (GTN) porosity-based plastic damage model by means of experimental data collected using image analysis. The use of digital image processing allows to substitute the load–displacement curve with other global quantities resulting from the measuring of specimen profile during loading. The advantage of this analysis is that more data are available for calibration thus allowing a greater level of confidence and accuracy in model parameter evaluation.     

A Generalized Piezoelectric Bernoulli–Navier Anisotropic Rod Model

We apply the asymptotic analysis procedure to the three-dimensional static equations of piezoelectricity, for a linear nonhomogeneous anisotropic thin rod. We prove the weak convergence of the rod mechanical displacement vectors and the rod electric potentials, when the diameter of the rod cross-section tends to zero. This weak limit is the solution of a new piezoelectric anisotropic nonhomogeneous rod model, which is a system of coupled equations, with generalized Bernoulli–Navier equilibrium equations and reduced Maxwell–Gauss equations.     

Electromagnetic and temperature fields in direct resistance heating of compound axisymmetric bodies

A nonstationary coupled problem of thermoelectrodynamics is formulated for resistance heating of dissimilar ferro- and paramagnetic bodies by an alternating current. An iterative algorithm for solving this problem by finite-difference methods is proposed. Temperature and electromagnetic-field distributions are obtained for the processes of direct resistance heating both in air and in the region of the molding tool. Perm' State Technical University, Perm' 614000. Translted from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 156–164, July–August, 2000.     

Thermal analysis of an infinite tube during quenching

 This paper deals with a numerical solution of the two-dimensional convection–diffusion equation in an infinite domain, arising out of quenching of an infinite tube. On the wetted side, upstream of the quench front, a constant heat transfer coefficient is assumed. The downstream of the quench front as well as the inside surface of the tube are assumed to be adiabatic. The solution gives the quench front temperature as a function of various model parameters such as Peclet number, Biot number and the radius ratio. The solution has been found to be in good agreement with the available analytical solutions and thus validates the numerical procedure suggested. Received on 10 July 2000     

On one iteration method for solving problems on the stress-strain state of thin shells

Decomposition methods are used in solving problems on the stress-starin state of thin shells. The problem is reduced to the iterative solution of a system of equations that have a simpler structure and permit constructing a computing parallel-integration algorithm. The stress-strain problem is solved by an example of a shallow spherical shell. The effect of the curvature of the shell on the number of iterations necessary for reaching the accuracy prescribed is studied. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 98–103, June, 2000.