Numerical Solution Of Two-dimensional Problem Research Articles

Analytical and numerical solutions of two-dimensional problems of heat conduction and electronic optics

This work is devoted to a comparative analysis of the effectiveness of various methods for the numerical solution of a non-stationary two-dimensional problem of heat conduction in a cylindrical coordinate system in the absence of axial symmetry. The analysis is carried out on the basis of the obtained analytical solution of the two-dimensional problem of heat conduction in a steady state. The results of this work are of great applied importance in calculating the processes of gas transportation through offshore pipelines, as well as in calculating the electrostatic field of focusing systems during transportation of charged particle beams in problems of electronic optics.

MATHEMATICAL MODELING OF UNSTEADY FILTRATION OF ELASTIC LIQUID IN AN INHOMOGENEOUS RESERVOIR

The article considers the possibility of numerical solution of two-dimensional problem of unsteady filtration in an inhomogeneous elastic liquid reservoir. The problem of finding the distribution of the pressure p(x,y,t) in the process of exploitation of deposits is reduced to the solution of a differential equation of parabolic type with variable coefficients. The problem is solved approximately by using the method of finite differences.

A Review of Water Hammer Theory and Practice

Hydraulic transients in closed conduits have been a subject of both theoretical study and intense practical interest for more than one hundred years. While straightforward in terms of the one-dimensional nature of pipe networks, the full description of transient fluid flows pose interesting problems in fluid dynamics. For example, the response of the turbulence structure and strength to transient waves in pipes and the loss of flow axisymmetry in pipes due to hydrodynamic instabilities are currently not understood. Yet, such understanding is important for modeling energy dissipation and water quality in transient pipe flows. This paper presents an overview of both historic developments and present day research and practice in the field of hydraulic transients. In particular, the paper discusses mass and momentum equations for one-dimensional Flows, wavespeed, numerical solutions for one-dimensional problems, wall shear stress models; two-dimensional mass and momentum equations, turbulence models, numerical solutions for two-dimensional problems, boundary conditions, transient analysis software, and future practical and research needs in water hammer. The presentation emphasizes the assumptions and restrictions involved in various governing equations so as to illuminate the range of applicability as well as the limitations of these equations. Understanding the limitations of current models is essential for (i) interpreting their results, (ii) judging the reliability of the data obtained from them, (iii) minimizing misuse of water-hammer models in both research and practice, and (iv) delineating the contribution of physical processes from the contribution of numerical artifacts to the results of waterhammer models. There are 134 refrences cited in this review article.

Optimal control model for the critical state in superconductors

Grounded on a variational principle, we present a generalization of the standard critical-state approach in type-II superconductors. The free energy is minimized with the constraint $|\mathbf{J}(\mathbf{r})|<~{J}_{c}$ for the macroscopic current density, posing the problem in the framework of the optimal control theory. The application of this mathematical tool allows us to determine the critical state in which the system organizes itself. This permits to confirm the critical-state hypothesis for an idealized one-dimensional geometry and to deal with multicomponent field situations, for which additional constitutive laws are provided. A geometrical picture of the field penetration process has been developed and we obtain both analytical and numerical solutions for two-dimensional problems under an applied parallel field and superimposed transport current.

An integral equation formulation for boundary element analysis of acoustic radiation problems in viscous fluids

This paper presents an integral equation formulation for acoustic radiation in viscous fluids, and its implementation to the numerical solution of two-dimensional problems. The thin acoustic boundary layer approximation has been made in determining fluid vorticity in the vicinity of solid boundaries. Numerical examples have been given to show the effect of viscosity on sound radiation. The paper also includes a discussion of the effect of viscosity on the non-uniqueness problem.

Numerical solution of two-dimensional problem of gas flow from a chamber with flat walls

A study is made of the classical problem of steady flow of ideal gas from an infinite two-dimensional chamber with straight wall generator making angle θw with the symmetry plane of the flow. Specification of θw, the pressure ratio ɛa-Pa/Po (Po is the stagnation pressure of the gas in the chamber, Pα is the pressure in the ambient medium), and the specific-heat ratio χ completely determines the flow of gas from the chamber.

An approach to the numerical solution of two-dimensional problems of plate and shell theory with variable parameters

Here Zi = 2~ (a, ~) (a I ~ a ~ a2; ~, ~ ~ ~ ~_) are the desired resolving functions, r is a linear function in ~kzj/~flk; fi(~ ' fl) are free terms and ~, fl are orthogonal curvilinear coordinates. For open shells (plates) boundary conditions on the contour lines ~ = coast; fi = coast are appended. For shelis closed in one direction the boundary conditions'in this direction are replaced by periodicity conditions. Before going over to an exposition of the proposed approach, let us examine the fundamental aspects of the method of integral relations in application to the solution of the calls of problems under consideration (2, 3, 9). The solution of the boundary value problem consists of two stages. In the first stage the initial scheme of the partial differential equations is approximated by a system of ordinary differential equations satisfying the boundary conditions on a part of the shell contour.

Numerical solution of two-dimensional problems of the pulse motion of floating bodies

An algorithm is given for the numerical solution by computer of plane problems on the pulse motion of an absolutely rigid body floating on the surface of an inviscid incompressible fluid. The solution is based on the method of discrete vortices, whose circulations are found from the boundary conditions for the potential or its normal derivative on the appropriate parts of the surface. Numerical results are quoted for problems on the impact of specific bodies either without breakaway of the fluid from the wetted surface or with formation of breakaway zones.

Numerical solution of the two-dimensional problem of shock wave propagation in outer space

A method for the numerical solution of the non-stationary two-dimensional problem of the propagation relative to a moving interplanetary medium of a perturbation caused by the emission of finite energy within the limits of a section of a cone, simulating the chromospheric region of the sun occupied by the flare is discussed. The effect of solar gravitation and of the radial component of the magnetic field strength on the motion of the gas is taken into account.

A numerical solution of two-dimensional problems involving heat and mass transfer

The finite element method is used for the solution of two-dimensional heat- and mass-transfer problems in porous media. The formulation is given in general terms and is not restricted to any particular type of element. It is demonstrated in the paper that the versatility of the technique results in a viable method of solution for new practical applications of the Luikov system of equations.

On the Accuracy of the Depletion Layer Approximation for Charge Coupled Devices

In this paper we examine the accuracy of a variant of the depletion layer approximation as used in the analysis of the electrostatic potential, in the absence of mobile charge, in various charge coupled devices. The approximation is a linearization of the nonlinear potential equations and is important in the numerical solution of two-dimensional problems. A one-dimensional model is solved both exactly and by means of the depletion layer approximation and the two results are compared. We conclude that this variant of the depletion layer approximation is excellent for buried channel devices, and adequate though not as good for surface devices. Some criteria are given which indicate the excellence of the approximation and can be used to estimate errors in two-dimensional calculations.

A Method of Solving Certain Plane Problems of Contained Elasto-Plasticity

Previous article Next article A Method of Solving Certain Plane Problems of Contained Elasto-PlasticityPhilip H. Francis and Kwan RimPhilip H. Francis and Kwan Rimhttps://doi.org/10.1137/0115073PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] L. A. Galin, A problem in plane elasto-plasticity: Plastic regions around circular apertures in plates and beams, Prikl. Mat. i Meh., 10 (1946), 367–386 Google Scholar[2] J. H. Michell, On the direct determination of stress in an elastic solid, with application to the theory of plates, Proc. London Math. Soc., 31 (1899), 100–124 CrossrefGoogle Scholar[3] P. H. Francis, Masters Thesis, Two-dimensional elastic and elasto-plastic analysis of infinite plates containing a hole, Doctoral dissertation, University of Iowa, Iowa City, 1965 Google Scholar[4] P. H. Francis and , K. Rim, A comparison of direct methods in plane elasticity, AIAA J., 5 (1967), 183–185 0149.22204 CrossrefGoogle Scholar[5] L. E. Hulbert, Masters Thesis, The numerical solution of two-dimensional problems of the theory of elasticity, Doctoral dissertation, Ohio State University, Columbus, 1963 Google Scholar[6] A. R. S. Ponter, On plastic torsion, Internat. J. Mech. Sci., 8 (1966), 227–235 10.1016/0020-7403(66)90038-5 0138.21403 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Amendment to the Galin Plane Elastoplastic SolutionJournal of Engineering Mechanics, Vol. 139, No. 11 | 1 Nov 2013 Cross Ref A variational method of solving an elastic-plastic problem for a body with a circular holeJournal of Applied Mathematics and Mechanics, Vol. 52, No. 1 | 1 Jan 1988 Cross Ref An analytical-experimental estimation of elastic linear zone for photoelastic mode I SIF extractionTheoretical and Applied Fracture Mechanics, Vol. 2, No. 2 | 1 Nov 1984 Cross Ref Topics in the Mathematical Theory of PlasticityMechanics of Solids | 1 Jan 1973 Cross Ref On the extent of plastic yielding surrounding a surface “inclusion” under remote tensionInternational Journal of Mechanical Sciences, Vol. 13, No. 5 | 1 May 1971 Cross Ref Volume 15, Issue 4| 1967SIAM Journal on Applied Mathematics767-1111 History Submitted:13 October 1966Published online:13 July 2006 InformationCopyright © 1967 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0115073Article page range:pp. 842-855ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics