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Elliptic coordinate system

About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.


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Journal ArticleDOI
TL;DR: In this paper, numerical methods are used to investigate the transient heat transfer from an elliptic cylinder to a steady stream of viscous, incompressible fluid, where the temperature of the cylinder is considered spatially uniform but not constant in time.

19 citations

Journal ArticleDOI
TL;DR: In this paper, the authors extended the successful experience of solving an infinite medium containing circular holes and/or inclusions subject to remote shears to deal with the problem containing elliptical holes and or inclusions.
Abstract: In this paper, we extend the successful experience of solving an infinite medium containing circular holes and/or inclusions subject to remote shears to deal with the problem containing elliptical holes and/or inclusions. Arbitrary location, different orientation, various size and any number of elliptical holes and/or inclusions can be considered. By fully employing the elliptical geometry, fundamental solutions were expanded into the degenerate kernel by using an addition theorem in terms of the elliptic coordinates and boundary densities are described by using the eigenfunction expansion. The difference between the proposed method and the conventional boundary integral equation method is that the location point can be exactly distributed on the real boundary without facing the singular integral and calculating principal value. Besides, the boundary stress can be easily calculated free of the Hadamard principal values. It is worthy of noting that the Jacobian terms exist in the degenerate kernel, boundary density and contour integral; however, these Jacobian terms would cancel each other out and the orthogonal property is preserved in the process of contour integral. This method belongs to one kind of meshless methods since only collocation points on the real boundary are required. In addition, the solution is regarded as semi-analytical form because error purely attributes to the number of truncation term of eigenfunction. An exact solution for a single elliptical inclusion is also derived by using the proposed approach and the results agree well with Smith’s solutions by using the method of complex variables. Several examples are revisited to demonstrate the validity of our method.

19 citations

Journal ArticleDOI
TL;DR: Reduced masses obtained by this program can be used as a decision tool for selecting the most appropriate internal coordinates for the considered vibrational problem and for the inclusion or omission of the kinetic coupling terms in the vibrational Hamiltonian.
Abstract: In this paper we present and analyze the most essential aspects of reduced masses along generalized internal coordinates. The definition of reduced masses in the internal coordinate formalism is established through the Wilson G-matrix concept and includes sophisticated relations between internal and Cartesian coordinates. Moreover, reduced masses in internal coordinates are, in general, no longer constant but coordinate-dependent. Based on the approach presented earlier [Stare, J.; Balint-Kurti, G. G. J. Phys. Chem. A 2003, 107, 7204-7214] and on our experience with reduced masses discussed in this paper, we have developed a robust program for the calculation of Wilson G-matrix elements and their functional coordinate dependence. The approach is based on the first principles and can be used in virtually any (internal) coordinate set. Since the program allows for projection of any kind of nuclear motion on the selected internal coordinates, the method is particularly suitable for ab initio or DFT potential energy functions calculated by partial geometry optimization. Moreover, reduced masses obtained by this program can be used as a decision tool for selecting the most appropriate internal coordinates for the considered vibrational problem and for the inclusion or omission of the kinetic coupling terms in the vibrational Hamiltonian.

19 citations

Journal ArticleDOI
TL;DR: In this article, a family of mass-scaled coordinate systems is presented for the triatom i75n which the kinetic energy operator for large-amplitude vibrational motion is diagonal (unlike that in current local mode theories).

19 citations

01 Aug 1976
TL;DR: In this paper, a theoretical analysis for the membrane and bending stresses around an arbitrarily oriented crack in a long, thin, circular cylindrical shell subjected to uniform axial tension, internal pressure and torsional loadings is presented.
Abstract: A theoretical analysis is presented for the membrane and bending stresses around an arbitrarily oriented crack in a long, thin, circular cylindrical shell subjected to uniform axial tension, internal pressure and torsional loadings. The method of analysis involves perturbation in a curvature parameter defining the crack size with respect to dimensions of the shell and the results are valid for cracks of small length. The problem is first solved completely in elliptic coordinates and the stress intensity factors are then determined by carrying out a transformation to polar coordinates with crack tip as the origin. Closed-form expressions are obtained for the membrane and bending components of the elastic stress intensity factors and numerical results are presented over the complete rage of the crack angle. For the two symmetric orientations, the solutions agree with known solutions.

19 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202211
202111
202010
201913
201810