A coupled far-field formulation for time-periodic numerical problems in fluid dynamics

Consider uniform flow past an oscillating body generating a time-periodic motion in an exterior domain, modelled by a numerical fluid dynamics solver in the near field around the body. A far-field formulation, based on the Oseen equations, is presented for coupling onto this domain thereby enabling the whole space to be modelled. In particular, examples for formulations by boundary elements and infinite elements are described.     

The use of splitting of the equations of motion in the numerical solution of problems of the dynamics of elastic systems

Linear elastic systems with a finite number of degrees of freedom, the initial equations of motion of which are constructed using the finite element method or other discretization methods, are considered. Since, in applied dynamics problems, the motions are usually investigated in a frequency range with an upper bound, the degrees of freedom of the initial system of equations are split into dynamic and quasi-dynamic degrees. Finally, the initial system of equations is split into a small number of differential equations for the dynamic degrees of freedom and into a system of algebraic equations for determining the quasi-static displacements, represented in the form of a matrix series. The number of terms of the series taken into account depends on the accuracy required.     

Domain decomposition method and numerical analysis of a fluid dynamics problem

A two-dimensional problem obtained by time discretization and linearization of a viscous flow governed by the incompressible Navier-Stokes equations is considered. The original domain is divided into subdomains such that their interface is a smooth (nonclosed, self-avoiding) curve with the ends belonging to the boundary of the domain. A nonconforming finite element method is constructed for the problem, and the convergence rate of the discrete solution of the problem to the exact one is estimated in the L ₂(Ω _h ) norm.     

A consistent numerical scheme for self‐gravitating fluid dynamics

In this article, we present a numerical scheme for the 3‐D system of self‐gravitating fluid dynamics in the collisional case as well as in the non‐collisional case. Consistency in the sense of distributions is proved in 1‐D and in absence of pressure. In the other cases consistency is proved under the numerical assumptions of boundedness of the velocity field in the CFL condition and of boundedness of the gradient of the gravitation potential. In 2‐D and 3‐D, concentrations of matter in strings and points can cause a theoretical difficulty in the pressureless case although one observes that the scheme still works. The initial data are L^∞ functions in velocity and L¹ functions in density. Applications are given to numerical simulations of the role of dark matter and gravitational collapse in cosmology as well as Jeans theory. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Experimental analysis of convergence of the numerical solution to a generalized solution in fluid dynamics

Convergence of the approximate solution of fluid dynamics problems obtained using Godunov’s scheme to the discontinuous solution is investigated.     

Dispersion equations for layered elastic and fluid systems

Dispersion equations are written by means of the matrix method for layered-inhomogeneous elastic, fluid, and elastic-fluid media. Using these equations, it is possible to investigate interference waves both in the low-frequency and high-frequency regions. A special study is made for the case of high-frequency Rayleigh waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 189–211, 1974.     

On grid generation for numerical models of geophysical fluid dynamics

A simple geometric condition that defines the class of classical (stereographic, conic and cylindrical) conformal mappings from a sphere onto a plane is derived. The problem of optimization of computational grid for spherical domains is solved in an entire class of conformal mappings on spherical (geodesic) disk. The characteristics of computational grids of classical mappings are compared for different spherical radii of geodesic disk. For a rectangular computational domain, the optimization problem is solved in the class of classical mappings and respective area of the spherical domain is evaluated.     

Variational approaches to solving initial-boundary-value problems in the dynamics of linear elastic systems

Problems of the controlled motion of an elastic body are considered in the linear theory. Using the method of integrodifferential relations, a family of quadratic functionals is introduced, which define the state of the elastic body, and variational formulations of the initial-boundary-value problem of dynamics are given. Euler's equations and boundary and terminal relations corresponding to them are obtained from the condition for the functionals to be stationary. It is shown that there is a relation between the proposed formulations and the Hamilton variational principle in the case of boundary-value and time-periodic problems of dynamics. A numerical algorithm is developed for finding the motions of an elastic body, based on piecewise-polynomial approximations and a criterion is proposed for estimating the quality of the approximate solutions. An example of the calculation and analysis of the forced transverse motions of a rectilinear beam with a square cross section is given for the three-dimensional model.     

A numerical method for identifying the location of a fluid leak in a pipeline

On the basis of mathematical models for studying gas or liquid flows in long-distance pipelines, a numerical method is developed that allows us to determine the coordinates of the fluid leak points by measuring data for the flow parameters (pressure and flow rate) both in stationary and nonstationary cases. The method is a modification of the Newton method applied to minimizing the functional of the mean-square deviation of the calculated parameters from the measured data.     

Free boundary fluid flow in a trough: A numerical approach

This paper examines the slow flow of a viscous liquid in an open rectangular container, one side (the base) of which moves steadily along its own plane, thereby providing the driving force the liquid needs. Unlike the two vertical sides that are rigid and stationary, the top side is left open so that the upper part of the liquid is in contact with air and is being controlled by surface tension and gravity. A numerical procedure for obtaining solutions for the cases when the capillary numbers are small is provided and the curves of the free boundaries obtained here are presented for some flow parameters. The deviation of the shape of the free boundary is observed to be strongly dependent on the aspect ratio of the boundary (i.e., the ratio of the vertical to horizontal spread of the liquid) with its curvature changing sign in the interval [1, 1.5].     

Comparison of different spatial grids for numerical schemes of geophysical fluid dynamics

The generation of computational grids is an important component contributing to the efficiency of numerical schemes of atmosphere/ocean dynamics. In this study the problem of construction of the most uniform grids based on conformal mappings of spherical domains is considered. Stereographic, cylindrical and conic grids for computational rectangles are developed and their uniformity is compared. Numerical experiments with two schemes approximating shallow water equations are performed in order to assess the practical efficiency of the constructed grids and to compare the numerical results with analytical evaluations.     

A theorem on the stability of elastic systems

Summary Similar to the Ziegler's theorem [1] on the influence of nonworking constraints on the stability of elastic systems, a theorem on the influence of elastic supports (constraints), which do perform work during the motion of the system, is derived for conservative elastic systems. The theorem is applicable to both discrete and continuous systems.

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Zusammenfassung In Analogie zu einem Satz von Ziegler über den Einfluss arbeitsloser Bindungen auf die Stabilität elastischer Systeme wird ein Theorem über den Einfluss elastischer Einspannungen, die während der Bewegung Arbeit leisten, für konservative Systeme hergeleitet. Es ist sowohl für diskrete als auch für kontinuierliche Systeme anwendbar.

     

Direct numerical determination of the stress field in a thin elastic disk

Zusammenfassung Die gebräuchlichen Methoden der numerischen Spannungsbestimung in zwei- oder dreidimensionalen elastischen Kontinua haben indirekten und kinematischen Charakter. Das Verschiebungsfeld wird durch eine Linearkombination lokalisierter Verschiebungsfelder angenähert, deren Intensitäten nach dem Prinzip der minimalen potentiellen Energie bestimmt werden. Angenäherte Spannungen werden dann erhalten durch Anwendung des Hookeschen Gesetzes auf die Verformungen des angenäherten Verschiebungsfelds. Im Gegensatz zu diesem Vorgehen entwickelt die vorliegende Arbeit eine statische Methode zur unmittelbaren Bestimmung der Spannungen in einer dünnen elastischen Scheibe. Das Spannungsfeld wird durch eine Linearkombination von lokalisierten Feldern stückweise konstanter Spannung angenähert, in denen nur die zur Trennungslinie benachbarter Felder konstanter Spannung parallele Normalspannung unstetig ist. Die Intensitäten dieser lokalisierten Spannungsfelder werden nach dem Prinzip der minimalen Ergänzungsarbeit bestimmt. Die sich ergebenden linearen Gleichungen für diese Intensitäten werden besprochen und durch numerische Beispiele illustriert.

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This research was supported by the Advanced Research Projects Agency (Project DEFENDER) and was monitored by the U.S. Army Research Office-Durham under Contract DA-31-124-ARO-D-257     

Prediction-control problem for some systems of equations of fluid dynamics

By methods of the theory of degenerate semigroups of operators, we investigate inverse problems for some systems of equations of fluid dynamics containing the incompressibility equation ? · v = 0. Namely, we obtain well-posedness criteria for inverse problems for the system of Sobolev equations, which describes the dynamics of small internal motions of a stratified fluid in an equilibrium state under some additional assumptions, and for the system of Oskolkov equations, which, in the linear approximation, describes the dynamics of a viscoelastic incompressible Kelvin-Voigt fluid of order 1.     

A study on preconditioning iterative methods and relativity of pressure in the numerical calculation of fluid flow

This article presents the effect of preconditioning iterative methods on boundary conditions of the pressure‐correction in the numerical computation of fluid flow with known velocity components on all boundaries using the SIMPLE algorithm. In such computation, a set of solutions of the pressure‐correction is indefinite, because only the Neumann condition is imposed on all boundaries. However, solutions become unique if the value of pressure‐correction is fixed at least on one boundary point, and the Dirichlet condition is additionally imposed. Though both conditions must give exactly the same velocity and temperature fields, this problem arises from the relativity of the pressure. The mathematical illustration for this problem is provided using the numerical computation of the natural convection in an enclosure. It is concluded that the preconditioner adopted and the condition that only the Neumann condition on all boundaries is given are effective to reduce the number of iterations in solving the linear system of equations of the pressure‐correction at the computation of the natural convection. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

A two-dimensional model of the dynamics of sharp bending of a non-linearly elastic plate

To describe the dynamics of the bending of a thin non-linearly elastic plate, a version of perturbation theory is proposed which correctly takes into account the non-linearity of the medium, the non-uniformity of the deformations along the plate thickness and the boundary conditions on its surface. An effective (2 + 1)-dimensional model is constructed which generalizes the static non-linearly geometrical Föppl-Karman equations. Two-dimensional solitons of the longitudinal deformation are obtained. The conditions for their existence and stability are investigated.