scholarly journals Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding

2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Panayiotis Vafeas
Keyword(s):  
Three Dimensional ◽  
Low Frequency ◽  
Analytical Form ◽  
Wave Number ◽  
Closed Form Solutions ◽  
Rayleigh Approximation ◽  
Time Harmonic ◽  
Ellipsoidal Harmonic ◽  
Type Boundary ◽  
Complex Valued

This contribution deals with the scattering by a metallic ellipsoidal target, embedded in a homogeneous conductive medium, which is stimulated when a 3D time-harmonic magnetic dipole is operating at the low-frequency realm. The incident, the scattered, and the total three-dimensional electromagnetic fields, which satisfy Maxwell’s equations, yield low-frequency expansions in terms of positive integral powers of the complex-valued wave number of the exterior medium. We preserve the static Rayleigh approximation and the first three dynamic terms, while the additional terms of minor contribution are neglected. The Maxwell-type problem is transformed into intertwined potential-type boundary value problems with impenetrable boundary conditions, whereas the environment of a genuine ellipsoidal coordinate system provides the necessary setting for tackling such problems in anisotropic space. The fields are represented via nonaxisymmetric infinite series expansions in terms of harmonic eigenfunctions, affiliated with the ellipsoidal system, obtaining analytical closed-form solutions in a compact fashion. Until nowadays, such problems were attacked by using the very few ellipsoidal harmonics exhibiting an analytical form. In the present article, we address this issue by incorporating the full series expansion of the potentials and utilizing the entire subspace of ellipsoidal harmonic eigenfunctions.

scholarly journals Electromagnetic Low-Frequency Dipolar Excitation of Two Metal Spheres in a Conductive Medium

2012 ◽  
Vol 2012 ◽  
pp. 1-37 ◽  
Author(s):  
Panayiotis Vafeas ◽  
Polycarpos K. Papadopoulos ◽  
Dominique Lesselier
Keyword(s):  
Magnetic Dipole ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Low Frequency ◽  
Wave Number ◽  
Conductive Medium ◽  
Time Harmonic ◽  
Frequency Interaction ◽  
Minor Significance ◽  
Infinite Linear Systems

This work concerns the low-frequency interaction of a time-harmonic magnetic dipole, arbitrarily orientated in the three-dimensional space, with two perfectly conducting spheres embedded within a homogeneous conductive medium. In such physical applications, where two bodies are placed near one another, the 3D bispherical geometry fits perfectly. Considering two solid impenetrable (metallic) obstacles, excited by a magnetic dipole, the scattering boundary value problem is attacked via rigorous low-frequency expansions in terms of integral powers(ik)n, wheren≥0,kbeing the complex wave number of the exterior medium, for the incident, scattered, and total non-axisymmetric electric and magnetic fields. We deal with the static (n=0) and the dynamic (n=1,2,3) terms of the fields, while forn≥4the contribution has minor significance. The calculation of the exact solutions, satisfying Laplace’s and Poisson’s differential equations, leads to infinite linear systems, solved approximately within any order of accuracy through a cut-off procedure and via numerical implementation. Thus, we obtain the electromagnetic fields in an analytically compact fashion as infinite series expansions of bispherical eigenfunctions. A simulation is developed in order to investigate the effect of the radii ratio, the relative position of the spheres, and the position of the dipole on the real and imaginary parts of the calculated scattered magnetic field.

scholarly journals Stationary Dynamic Stress Solutions for a Rectangular Load Applied within a 3D Viscoelastic Isotropic Full-Space

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
E. Romanini ◽  
J. Labaki ◽  
E. Mesquita ◽  
R. C. Silva
Keyword(s):  
Fourier Transforms ◽  
Damping Ratio ◽  
Three Dimensional ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Wave Number ◽  
Influence Functions ◽  
Time Harmonic ◽  
Final Stress ◽  
Stress Influence

This paper presents stress influence functions for uniformly distributed, time-harmonic rectangular loads within a three-dimensional, viscoelastic, isotropic full-space. The coupled differential equations relating displacements and stresses in the full-space are solved through double Fourier integral transforms in the wave number domain, in which they can be solved algebraically. The final stress fields are expressed in terms of double indefinite integrals arising from the Fourier transforms. The paper presents numerical schemes with which to integrate these functions accurately. The article presents numerical validation of the synthesized stress kernels and their behavior for high frequencies and large distances from the excitation source. The influence of damping ratio on the dynamic results is also investigated. This article is complementary to previous results of the authors in which the corresponding displacement solutions were derived. Stress influence functions, together with their displacement counterparts, are a fundamental part of many numerical methods of discretization such the boundary element method.

scholarly journals Complex dispersion solutions for guided waves and properties of non-propagating waves in a piezoelectric spherical plate

2018 ◽  
Vol 10 (12) ◽  
pp. 168781401882069
Author(s):  
Xiaoming Zhang ◽  
Zhi Li ◽  
Jiangong Yu
Keyword(s):  
Guided Waves ◽  
Three Dimensional ◽  
Guided Wave ◽  
Wave Number ◽  
Polynomial Method ◽  
Propagating Waves ◽  
Complex Solutions ◽  
Spherical Plate ◽  
Complex Valued ◽  
Dispersion And Attenuation

The vibration modes of an elastic plate are usually divided into propagating and non-propagating (evanescent) kinds. Non-propagating wave modes are very important for guided wave inspection of defect size and shape. But it is difficult to obtain the complex solutions of the transcendental dispersion equation, corresponding to the non-propagating wave. In this article, we present an improved Legendre polynomial method to calculate the complex-valued dispersion and study properties of the non-propagating wave in a piezoelectric spherical plate. Comparisons with other related studies are conducted to validate the correctness of the presented method. The complete dispersion and attenuation curves are plotted in three-dimensional frequency-complex wave number space. The influences of material piezoelectricity and radius–thickness ratio on non-propagating waves in piezoelectric spherical plates are investigated. The amplitude distributions of the electric potential and displacement are also discussed in detail. All the results presented in this work can provide theoretical guidance for ultrasonic nondestructive evaluation and are promising to be applied to improve the resolution of piezoelectric transducers.

scholarly journals Experimental Study on Coherent Structures by Particles Suspended in Half-Zone Thermocapillary Liquid Bridges: Review

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 105
Author(s):  
Ichiro Ueno
Keyword(s):  
Coherent Structures ◽  
Three Dimensional ◽  
Wave Number ◽  
Flow Structures ◽  
Liquid Bridges ◽  
Thermocapillary Effect ◽  
Particle Accumulation ◽  
Azimuthal Wave Number ◽  
Experimental Approaches ◽  
Particle Behaviour

Coherent structures by the particles suspended in the half-zone thermocapillary liquid bridges via experimental approaches are introduced. General knowledge on the particle accumulation structures (PAS) is described, and then the spatial–temporal behaviours of the particles forming the PAS are illustrated with the results of the two- and three-dimensional particle tracking. Variations of the coherent structures as functions of the intensity of the thermocapillary effect and the particle size are introduced by focusing on the PAS of the azimuthal wave number m=3. Correlation between the particle behaviour and the ordered flow structures known as the Kolmogorov–Arnold—Moser tori is discussed. Recent works on the PAS of m=1 are briefly introduced.

scholarly journals The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
P. Kim ◽  
R. Jorge ◽  
W. Dorland
Keyword(s):  
Magnetic Field ◽  
Original Work ◽  
Three Dimensional ◽  
Radial Distance ◽  
Analytical Form ◽  
One Dimensional ◽  
Modern Notation ◽  
Magnetic Well ◽  
First Time ◽  
Dimensional Integral

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

A Three-Dimensional Finite Difference Solution for the Thermal Stresses in Railcar Wheels

1969 ◽  
Vol 91 (3) ◽  
pp. 891-896 ◽  
Author(s):  
G. E. Novak ◽  
B. J. Eck
Keyword(s):  
Computer Program ◽  
Thermal Stresses ◽  
Three Dimensional ◽  
Transient Temperature ◽  
Shear Stresses ◽  
Radial Temperature ◽  
Closed Form Solutions ◽  
Brake Application ◽  
History Of ◽  
Thin Disks

A numerical solution is presented for both the transient temperature and three-dimensional stress distribution in a railcar wheel resulting from a simulated emergency brake application. A computer program has been written for generating thermoelastic solutions applicable to wheels of arbitrary contour with temperature variations in both axial and radial directions. The results include the effect of shear stresses caused by the axial-radial temperature gradients and the high degree of boundary irregularity associated with this type of problem. The program has been validated by computing thermoelastic solutions for thin disks and long cylinders; the computed values being in good agreement with the closed form solutions. Currently, the computer program is being extended to general stress solutions corresponding to the transient temperature distributions obtained by simulated drag brake applications. When this work is completed, it will be possible to synthesize the thermal history of a railcar wheel and investigate the effects of wheel geometry in relation to thermal fatigue.

scholarly journals Evolution of a narrow-band spectrum of random surface gravity waves

2003 ◽  
Vol 478 ◽  
pp. 1-10 ◽  
Author(s):  
KRISTIAN B. DYSTHE ◽  
KARSTEN TRULSEN ◽  
HARALD E. KROGSTAD ◽  
HERVÉ SOCQUET-JUGLARD
Keyword(s):  
Three Dimensional ◽  
Low Frequency ◽  
Three Dimensions ◽  
Band Spectrum ◽  
Nls Equation ◽  
Random Surface ◽  
Narrow Bandwidth ◽  
Quasi Stationary State ◽  
The Stability ◽  
Three Dimensional Simulations

Numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth have been performed both for two and three dimensions. Simulations using the nonlinear Schrödinger (NLS) equation approximately verify the stability criteria of Alber (1978) in the two-dimensional but not in the three-dimensional case. Using a modified NLS equation (Trulsen et al. 2000) the spectra ‘relax’ towards a quasi-stationary state on a timescale (ε2ω0)−1. In this state the low-frequency face is steepened and the spectral peak is downshifted. The three-dimensional simulations show a power-law behaviour ω−4 on the high-frequency side of the (angularly integrated) spectrum.

Solutions to the Vector Tetrahedron Equation

1965 ◽  
Vol 87 (2) ◽  
pp. 228-234 ◽  
Author(s):  
Milton A. Chace
Keyword(s):  
Closed Form ◽  
Digital Computer ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Spherical Coordinates ◽  
Tetrahedron Equation ◽  
Position Determination ◽  
Closed Form Solutions

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

THREE‐DIMENSIONAL FILTERING WITH AN ARRAY OF SEISMOMETERS

Geophysics ◽  
1964 ◽  
Vol 29 (5) ◽  
pp. 693-713 ◽  
Author(s):  
John P. Burg
Keyword(s):  
Array Processing ◽  
Three Dimensional ◽  
Processing System ◽  
Seismic Signal ◽  
P Wave ◽  
Wave Number ◽  
Theoretical Problem ◽  
Least Mean Square ◽  
Optimum Frequency ◽  
Filtering Theory

The development of the Wiener linear least‐mean‐square‐error processing theory for seismic signal enhancement through use of a two‐dimensional array of seismometers leads to the theory of three‐dimensional filtering. The array processing system for this theory consists of applying individual frequency filters to the outputs of the seismometers in the array before summation. The basic design equations for the optimum frequency filters are derived from the Wiener multichannel theory. However, the development of the three‐dimensional frequency and vector‐wave‐number‐filtering theory results in a physical understanding of generalized linear array processing. The three‐dimensional filtering theory is illuminated by a theoretical problem of P‐wave enhancement in the presence of ambient seismic noise. An analysis of the results shows why optimum three‐dimensional filtering gives greater signal‐to‐noise ratio improvements than achieved by conventional array processing techniques.

Three-dimensional time-harmonic elastodynamic Green ’ s functions for anisotropic solids

1995 ◽  
Vol 449 (1937) ◽  
pp. 441-458 ◽  
Keyword(s):  
Differential Equations ◽  
Radon Transform ◽  
Transverse Isotropy ◽  
Three Dimensional ◽  
Surface Integral ◽  
Line Integral ◽  
Surface Integration ◽  
Time Harmonic ◽  
Inverse Radon Transform ◽  
Eigenvectors And Eigenvalues

A method based on the Radon transform is presented to determine the displacement field in a general anisotropic solid due to the application of a time-harmonic point force. The Radon transform reduces the system of coupled partial differential equations for the displacement components to a system of coupled ordinary differential equations. This system is reduced to an uncoupled form by the use of properties of eigenvectors and eigenvalues. The resulting simplified system can be solved easily. A back transformation to the original coordinate system and a subsequent application of the inverse Radon transform yields the displacements as a summation of a regular elastodynamic term and a singular static term. Both terms are integrals over a unit sphere. For the regular dynamic term, the surface integration can be evaluated numerically without difficulty. For the singular static term, the surface integral has been reduced to a line integral over half a unit circle. Reductions to the cases of isotropy and transverse isotropy have been worked out in detail. Examples illustrate applications of the method.

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