Approximate solutions to the scattering by nonlinear isotropic dielectric cylinders of circular cross sections under TM illumination

1995 ◽  
Vol 43 (11) ◽  
pp. 1262-1269 ◽  
Author(s):  
S. Caorsi ◽  
A. Massa ◽  
M. Pastorino
Keyword(s):  
Cross Sections ◽  
Approximate Solutions

An Asymptotic Analysis of Three-Dimensional Extrusion

1989 ◽  
Vol 56 (3) ◽  
pp. 519-526 ◽  
Author(s):  
N. Aravas ◽  
R. M. McMeeking
Keyword(s):  
Asymptotic Analysis ◽  
Cross Sections ◽  
Three Dimensional ◽  
Constitutive Models ◽  
Approximate Solutions ◽  
The Finite Element Method ◽  
Wide Range ◽  
Metal Extrusion ◽  
Asymptotic Perturbation ◽  
Generalized Plane Strain

A new method of analysis of three-dimensional metal extrusion using asymptotic perturbation methods is presented in this paper. The plasticity model used depends on the first and second invariants of the stress tensor and covers a wide range of constitutive models commonly used for the analysis of metal-forming operations. It is shown that the three-dimensional extrusion problem can be approximated, to leading order, by a problem of generalized plane-strain. The results of the asymptotic analysis together with the finite element method are used to obtain approximate solutions for extrusions of elliptic or square cross-sections from round billets.

The General theory of Cylindrical and Conical Tubes Under Torsion and Bending Loads

1949 ◽  
Vol 53 (461) ◽  
pp. 461-483 ◽  
Author(s):  
J. Hadji-Argyris ◽  
P. C. Dunne
Keyword(s):  
General Theory ◽  
Cross Sections ◽  
Approximate Solutions ◽  
Torsional Stiffness ◽  
Numerical Example ◽  
Bending Loads ◽  
Present Part

SummaryParts 1 to 5 of this paper (February, September and November 1947 issues of the JOURNAL) investigated the stresses and deformations of closed tubes in which the thicknesses were governed by the ts* and t* laws. In the present part, the analysis is extended to multi-cell tubes with openings, open tubes with or without St. Venant torsional stiffness, and to tubes formed by joining elements of different cross-sections. To illustrate the theory a numerical example of the stressing of a four-boom wing consisting of seven joined elements is fully worked out. Finally, an appendix gives practical methods of dealing with tubes which do not conform to the ts* and t* laws, and of finding approximate solutions for four-boom tubes with direct stress carrying covers.

scholarly journals Recovery of the Two-Dimensional Ion Distribution Function in a Magnetic Mirror from Measurements of Collective Thomson Scattering Spectra

2021 ◽  
Vol 47 (6) ◽  
pp. 503-517
Author(s):  
E. D. Gospodchikov ◽  
T. A. Khusainov ◽  
A. G. Shalashov
Keyword(s):  
Integral Equation ◽  
Distribution Function ◽  
Cross Sections ◽  
Measurement Data ◽  
Approximate Solutions ◽  
Thomson Scattering ◽  
Ion Distribution ◽  
Energetic Ions ◽  
Scattering Spectra ◽  
Collective Thomson Scattering

Abstract A method is proposed for tomography of the distribution function of energetic ions that are adiabatically trapped in an open magnetic trap, according to the diagnostic data by the method of collective Thomson scattering. This method is based on measurements of the scattering spectra from successive plasma cross sections corresponding to different values of the magnetic-field strength along a single line of force. It is shown that the problem of restoring the ion distribution function in the velocity space from the measurement data in this situation is reduced to an integral equation of the first kind that allows an analytical solution. Several ways to construct exact and approximate solutions of the resulting integral equation are considered.

Electromagnetic fields in the presence of cylindrical objects of arbitrary physical properties and cross sections

1970 ◽  
Vol 48 (15) ◽  
pp. 1789-1798 ◽  
Author(s):  
L. Shafai
Keyword(s):  
Physical Properties ◽  
Electromagnetic Fields ◽  
Cross Sections ◽  
Approximate Solutions ◽  
Scattering Matrix ◽  
Two Dimensional ◽  
Matrix Elements ◽  
Approximate Evaluation ◽  
Permittivity And Permeability ◽  
The Matrix

Approximate solutions for two-dimensional problems of electromagnetic fields in the presence of cylindrical objects have been found by approximate evaluation of a scattering matrix. The equations are derived for cylindrical objects of arbitrary physical properties and cross sections and a procedure for evaluation of the matrix elements is discussed. The elements of permittivity and permeability tensors are assumed to be analytic, but otherwise arbitrary functions of the transverse coordinates.

Multiple scattering by a linear array of conducting spheres

1990 ◽  
Vol 68 (10) ◽  
pp. 1157-1165 ◽  
Author(s):  
A-K. Hamid ◽  
I. R. Ciric ◽  
M. Hamid
Keyword(s):  
Multiple Scattering ◽  
Cross Sections ◽  
Linear Array ◽  
Plane Electromagnetic Wave ◽  
Spherical Wave ◽  
Approximate Solutions ◽  
Addition Theorem ◽  
Wave Functions ◽  
Coordinate Systems ◽  
Conducting Spheres

The problem of multiple scattering of a plane electromagnetic wave incident on N closely spaced perfectly conducting spheres is solved analytically by expanding the incident and scattering fields in terms of an appropriate set of vector spherical wave functions. To impose the boundary conditions, the scattered field from one sphere is expressed in coordinate systems attached to the others by using the translation addition theorem. An approximate solution is obtained to solve for the scattering by N small spheres. Numerical results for the normalized backscattering and bistatic cross sections for systems of spheres show that the agreement between the analytic and approximate solutions is better for larger electrical distances between neighbouring spheres.

scholarly journals An approximate formula to calculate the critical depth in circular culvert

2020 ◽  
Vol 71 (7) ◽  
pp. 840-852
Author(s):  
Binh Hoang Nam
Keyword(s):  
Approximate Solution ◽  
Cross Sections ◽  
Approximate Solutions ◽  
Maximum Error ◽  
Regression Equations ◽  
Critical Depth ◽  
Design Engineers ◽  
Hydraulic Design ◽  
Long Time ◽  
True Values

Critical depth is a depth of flow where a specific energy section is at a minimum value with a flow rate. Critical depth is an essential parameter in computing gradually varied flow profiles in open channels and in designing culverts. If cross-sections are rectangular or triangular, the critical depth can be computed by the governing equation. However, for other geometries such as trapezoidal, circular, it is totally difficult to find a solution, because the governing equations are implicit. Therefore, the approximate solution could be determined by trial, numerical or graphical methods. These methods tend to take a long time to find an approximate solution, so a simple formula will be more convenient for consultant hydraulic design engineers. The existing formulas are simple, but the relative error between the approximate solutions and true values can reach 9% or greater. This article presents new explicit regression equations for the critical depth in a partially full circular culvert. The proposed formula is quite simple, and the relative maximum error is 3.03%. It would be very useful as a reference for design in conduit engineering

Calculated Pressure Distributions and Shock Shapes on Thick Conical Wings at High Supersonic Speeds

1967 ◽  
Vol 18 (2) ◽  
pp. 185-206 ◽  
Author(s):  
L. C. Squire
Keyword(s):  
Integral Equation ◽  
Cross Sections ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Poor Agreement ◽  
Delta Wings ◽  
Newtonian Theory ◽  
Pressure Distributions ◽  
First Order Correction ◽  
Good Agreement

SummaryIn recent papers Messiter and Hida have proposed a first-order correction to simple Newtonian theory for the pressure distributions on the lower surfaces of lifting conical bodies with detached shocks. The method involves the solution of an integral equation which Messiter solved numerically for thin delta wings, while Hida gave an approximate solution for thick wings with diamond and bi-convex cross-sections. It is shown in the present paper that Hida’s approximate solutions give poor agreement with experiment, and a series of more precise numerical solutions of the equation are given for wings with diamond cross-sections. The pressures, and shock shapes, obtained from these solutions are in very good agreement with experiment at Mach numbers as low as 4·0.The method has also been extended to Nonweiler wings at off-design when the shock wave is detached from the leading edges. Again the agreement with experiment is good provided the integral equation is solved numerically.

An approximate formula to calculate the critical depth in circular culvert

2020 ◽  
Vol 71 (7) ◽  
pp. 840-852
Author(s):  
Binh Hoang Nam
Keyword(s):  
Approximate Solution ◽  
Cross Sections ◽  
Approximate Solutions ◽  
Maximum Error ◽  
Regression Equations ◽  
Critical Depth ◽  
Design Engineers ◽  
Hydraulic Design ◽  
Long Time ◽  
True Values

Critical depth is a depth of flow where a specific energy section is at a minimum value with a flow rate. Critical depth is an essential parameter in computing gradually varied flow profiles in open channels and in designing culverts. If cross-sections are rectangular or triangular, the critical depth can be computed by the governing equation. However, for other geometries such as trapezoidal, circular, it is totally difficult to find a solution, because the governing equations are implicit. Therefore, the approximate solution could be determined by trial, numerical or graphical methods. These methods tend to take a long time to find an approximate solution, so a simple formula will be more convenient for consultant hydraulic design engineers. The existing formulas are simple, but the relative error between the approximate solutions and true values can reach 9% or greater. This article presents new explicit regression equations for the critical depth in a partially full circular culvert. The proposed formula is quite simple, and the relative maximum error is 3.03%. It would be very useful as a reference for design in conduit engineering

Diffraction of a plane electromagnetic wave by a parallel plate grating with dielectric loading: the case of transverse magnetic incidence

1985 ◽  
Vol 63 (4) ◽  
pp. 453-465 ◽  
Author(s):  
Kazuya Kobayashi
Keyword(s):  
Cross Sections ◽  
Parallel Plate ◽  
Plane Electromagnetic Wave ◽  
Approximate Solutions ◽  
Transverse Magnetic ◽  
Transform Domain ◽  
Wave Scattering And Diffraction ◽  
Decomposition Procedure ◽  
Dielectric Loading ◽  
The Fourier Transform

Wave scattering and diffraction problems concerning objects with complex cross sections have been widely investigated so far with the advance of electronic computers. In this paper, a periodically placed parallel plate grating with dielectric loading is considered, and the problem of diffraction of a TM polarized plane wave is analyzed with the aid of the Wiener–Hopf technique. Introducing the Fourier transform pair for the unknown scattered field and applying boundary conditions in the transform domain, one can formulate this problem as the single Wiener–Hopf equation. This functional equation is then solved by a decomposition procedure and a rigorous solution is obtained. Furthermore, approximate solutions are derived by applying the modified residue calculus technique. Based on the above analysis, several numerical examples are given and the characteristics of this grating are discussed.

Pure Bending of an Incomplete Torus

1953 ◽  
Vol 20 (2) ◽  
pp. 215-226
Author(s):  
M. A. Sadowsky ◽  
E. Sternberg
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Stress Function ◽  
Circular Cross Section ◽  
Approximate Solutions ◽  
Function Approach ◽  
Initial Stresses ◽  
Pure Bending ◽  
Present Solution ◽  
In Series

Abstract An exact solution in series form is presented for the stresses and displacements in an incomplete torus of circular center line and circular cross section under conditions of pure bending. The solution is preceded by a formal treatment of the case in which the shape of the cross section is arbitrary. This more general problem is reduced to one of axisymmetry, which is in turn attacked by a modification of the stress-function approach of Timpe to rotationally symmetric problems in elasticity theory. The modified stress-function approach, which may be of interest beyond the present application, is referred to general orthogonal axisymmetric co-ordinates, and the solution of the specific problem considered here is based on the use of toroidal co-ordinates. The normal stresses acting on the circular cross sections of the torus are evaluated numerically in an illustrative example and are compared with the results of previous approximate solutions of the same problem. The adaptation of the present solution to the determination of the initial stresses in a complete torus from which a wedge-shaped portion has been removed is indicated.

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