Two-dimensional Elliptic Problems Research Articles

Toward a universal [formula omitted] adaptive finite element strategy part 3. design of [formula omitted] meshes

In this third paper in our series, the issue of designing h-p meshes which are optimal in some sense is addressed. Criteria for h-p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees-of-freedom. An optimization algorithm is developed based on these criteria and is applied to several model one-dimensional and two-dimensional elliptic boundary-value problems. Numerical results indicate that the approach can lead to exponential rates-of-convergence.

TWO-DIMENSIONAL PROBLEMS OF THE ANISOTROPIC ELASTIC SOLID WITH AN ELLIPTIC INCLUSION

The two-dimensional elliptic inclusion problems have been widely studied and many important results have been obtained. Solutions for which the inclusion as well as the matrix are of general anisotropic elastic materials are presented in this paper using the Stroh formalism. New identities which relate Stroh's complex eigenvalues and eigenvectors to real quantities obtainable directly from the elastic constants are derived, through which several solutions are expressed in real form. The special cases in which inclusion is rigid or a void are also studied

Discussion of a “New” Elliptic Linear Boundary Condition with Applications to Hydrodynamics

A new kind of linear boundary condition for two-dimensional elliptic problems is obtained. On part of the boundary, this condition is defined by two separate equations. For the second-order elliptic operator these two equations include: (1) the requirement of zero tangential derivative, (2) a given value of the integral of the normal derivative along the same part of the boundary. It is proven that this boundary condition leads to a self-adjoint operator possessing a unique bounded inverse operator. The application to channel flow is presented. The generalization for multidimensional cases or for higher-order differential equations is easily justified.

Spectral optimization of explicit iterative methods. I

Methods of constructing preconditioning of explicit iterative methods of solving systems of linear, algebraic equations with sparse matrices are considered in the work. The techniques considered can, first of all, be realized within the framework of the simplest data structures; secondly, the graph structures of the corresponding algorithms are well adapted to realization on parallel computers; thirdly, in conjunction with modifications of Chebyshev methods they make it possible to construct rather effective computational algorithms. Experimental data are presented which demonstrate the effect of the proposed techniques of preconditioning of the distribution of the eigenvalues of matrices of systems arising in discretization of two-dimensional elliptic boundary-value problems.

Finite element solution of flows through cascades of profiles in a layer of variable thickness

The paper is devoted to the numerical modelling of a subsonic irrotational nonviscous flow past a cascade of profiles in a variable thickness fluid layer. It leads to a nonlinear two-dimensional elliptic problem with nonstandard nonhomogeneous boundary conditions. The problem is discretized by the finite element method. Both theoretical and practical questions of the finite element implementation are studied; convergence of the method, numerical integration, iterative methods for the solution of the discrete problem and the algorithmization of the finite element solution. Some numerical results obtained by a multi-purpose program written by authors are presented.

Solving linear finite element systems by normalized approximate matrix factorization semi-direct methods

New normalized Extended to the Limit sparse factorization procedures in algorithmic form are derived yielding direct and iterative methods for the solution of finite element or finite difference systems of irregular structure. The proposed factorization procedures are chosen as the basis to yield normalized systems on which the Conjugate Gradient and Chebychev methods are implicitly applied. The application of the derived normalized implicit semi-direct methods on a two-dimensional elliptic boundary-value problem is discussed and numerical results are given.

Difference scheme for two-dimensional elliptic problems with an integral condition

Difference scheme for two-dimensional elliptic problems with an integral condition

A CALCULATION PROCEDURE FOR TWO-DIMENSIONAL ELLIPTIC SITUATIONS

A calculation method based on the control-volume approach has been developed for solving two-dimensional elliptic problems involving fluid flow and heat and mass transfer. The main features of the method include a power-law formulation for the combined convection-diffusion Influence, an equation-solving scheme that consists of a block-correction method coupled with a line-by-line procedure, and a new algorithm for handling the interlinkage between the momentum and continuity equations. Although the method is described for steady two-dimensional situations, its extension to unsteady flows and three-dimensional problems is very straightforward.

Jordan-Wachspress parameters in three dimensions

This paper describes a new technique which uses the Jordan-Wachspress parameters of the two-dimensional elliptic problem in the three-dimensional one. Thus A.D.I.-like schemes are obtained which, as is shown, converge faster than even the fastest ones known so far.

Singularities in the Finite Element Approximation of Two-Dimensional Diffusion Problems

The solution of a two-dimensional elliptic boundary value problem with piecewise smooth external boundaries, interfaces, and diffusion coefficients typical of nuclear reactor structures is known to contain a singular part. The presence of singular functions in the neighborhood of each angular point for a given geometric configuration has important consequences on the convergence orders for approximate solutions of the problem. These consequences are analyzed both theoretically and numerically, in the framework of the finite element method. Some means are described to overcome the damaging effects of the singular points. A thorough numerical study of various reactor configurations extending from liquid-metal fast breeder reactors to pressurized water reactors shows that in the latter case, the use of high-order polynomials is partially unjustified, given the severe limitations on the convergence orders.

The piece-by-piece solution of elliptic boundary value problems

A method of solution of a particular class of two-dimensional elliptic boundary value problems is developed. The method relies on splitting the region of the field into a number of sub-regions in each of which a field solution may be obtained in terms of well-known functions. Integral equations are formed for postulated source distributions on the interfaces between the sub-regions which are solved using Galerkin's method. The technique is particularly suitable for those cases where a functional of the field rather than the field itself is the required end product. An example is given of the calculation of the modification to the magnetic flux crossing an air gap due to the presence of slots.

Reduction of the two-dimensional elliptic restricted problem of three bodies to Hill's Equation .

view Abstract Citations (1) References (2) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Reduction of the two-dimensional elliptic restricted problem of three bodies to Hill's Equation . Erdi, Balint Abstract The differential equations of motion of the two-dimensional elliptic restricted problem are investigated. It is shown that a formal solution in the form of a power series of the eccentricity can be constructed by solving Hill's Equation for a given periodic solution of the circular problem. Publication: The Astronomical Journal Pub Date: May 1974 DOI: 10.1086/111592 Bibcode: 1974AJ.....79..653E full text sources ADS |

Periodic collision orbits in the elliptic restricted three-body problem

This article considers the two-dimensional elliptic restricted three-body problem, and in particular some of its aspects related to regularization and periodic collision orbits. The mechanism of regularization with Birkhoff coordinates and with theenergy differential equation is described. Then the initial conditions for collision orbits are established. The theory is illustrated with the description of a new family of symmetric periodic collision orbits. It is shown how this family is related to the work of Moulton, Darwin and Stromgren in the circular restricted problem, and also to the Earth-Moon mass ratio. In the high eccentricity ranges, some relations with the triple collision problem are pointed out. The differential corrections have been made with an automatic search technique which is described in the appendix.

Application of Point Matching Techniques to Two-Dimensional Solidification of Viscous Flow Over a Semi-Infinite Flat Plate

An analytical method is presented for the solution of two-dimensional solidification of fluid in motion over a semi-infinite plate. The method is applicable to other two-dimensional free boundary problems involving convection of the liquid phase. The solution is based upon an instantaneous source method which transforms the moving free boundary problem to a stationary domain. The transformed problem is solved by a Laplace transformation which results in a two-dimensional elliptic problem in an irregularly shaped region. An approximate point matching method is employed to solve the elliptic problem. Interface motion is obtained from the solution of a nonlinear integrodifferential equation of the Volterra type. Numerical solutions are presented to verify the validity of the model used in the analytical method and to examine the accuracy of the analytical solution.