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 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 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.

The mellin integral transform in fractional calculus

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Yuri Luchko ◽  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Fourier Transforms ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Monotone Functions ◽  
Completely Monotone ◽  
Mellin Convolution ◽  
New Applications

AbstractIn Fractional Calculus (FC), the Laplace and the Fourier integral transforms are traditionally employed for solving different problems. In this paper, we demonstrate the role of the Mellin integral transform in FC. We note that the Laplace integral transform, the sin- and cos-Fourier transforms, and the FC operators can all be represented as Mellin convolution type integral transforms. Moreover, the special functions of FC are all particular cases of the Fox H-function that is defined as an inverse Mellin transform of a quotient of some products of the Gamma functions.In this paper, several known and some new applications of the Mellin integral transform to different problems in FC are exemplarily presented. The Mellin integral transform is employed to derive the inversion formulas for the FC operators and to evaluate some FC related integrals and in particular, the Laplace transforms and the sin- and cos-Fourier transforms of some special functions of FC. We show how to use the Mellin integral transform to prove the Post-Widder formula (and to obtain its new modi-fication), to derive some new Leibniz type rules for the FC operators, and to get new completely monotone functions from the known ones.

scholarly journals Frictional Contact Problem of One Dimensional Hexagonal Piezoelectric Quasicrystals Layer

2021 ◽  
Author(s):  
RuKai Huang ◽  
Sheng hu Ding ◽  
Xin Zhang ◽  
Xing Li
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Three Dimensional ◽  
Integral Transforms ◽  
Fourier Integral ◽  
General Solutions ◽  
One Dimensional ◽  
Frictional Contact Problem ◽  
Material Development ◽  
Influence Coefficients

Abstract Based on three-dimensional (3D) general solutions for one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs), this paper studied the frictional contact problem of 1D-hexagonal PEQCs layer. The frequency response functions (FRFs) for 1D-hexagonal PEQCs layer are analytically derived by applying double Fourier integral transforms to the general solutions and boundary conditions, which are consequently converted to the corresponding influence coefficients (ICs). The conjugate gradient method (CGM) is used to obtain the unknown pressure distribution, while the discrete convolution-fast Fourier transform technique (DC-FFT) is applied to calculate the displacements and stresses of phonon and phason, electric potentials and electric displacements. Numerical results are given to reveal the influences of material parameters and loading conditions on the contact behavior. The obtained 3D contact solutions are not only helpful further analysis and understanding of the coupling characteristics of phonon, phason and electric field, but also provide a reference basis for experimental analysis and material development.

On the Zap Integral Operators over Fourier Transforms

2021 ◽  
Author(s):  
Juan Manuel Velazquez Arcos ◽  
Ricardo Teodoro Paez Hernandez ◽  
Alejandro Perez Ricardez ◽  
Jaime Granados Samaniego ◽  
Alicia Cid Reborido
Keyword(s):  
Integral Equation ◽  
Quantum Mechanics ◽  
Fourier Transforms ◽  
Integral Operators ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Fredholm Equation ◽  
Projection Operators ◽  
Inhomogeneous Term ◽  
Generalized Projection Operators

We devote the current chapter to describe a class of integral operators with properties equivalent to a killer operator of the quantum mechanics theory acting over a determined state, literally killing the state but now operating over some kind of Fourier integral transforms that satisfies a certain Fredholm integral equation, we call this operators Zap Integral Operators (ZIO). The result of this action is to eliminate the inhomogeneous term and recover a homogeneous integral equation. We show that thanks to this class of operators we can explain the presence of two extremely different solutions of the same Generalized Inhomogeneous Fredholm equation. So we can regard the Generalized Inhomogeneous Fredholm Equation as a Super-Equation with two kinds of solutions, the resonant and the conventional but coexisting simultaneously. Also, we remember the generalized projection operators and we show they are the precursors of the ZIO. We present simultaneous academic examples for both kinds of solutions.

scholarly journals Stationary Dynamic Displacement Solutions for a Rectangular Load Applied within a 3D Viscoelastic Isotropic Full Space—Part I: Formulation

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Euclides Mesquita ◽  
Edivaldo Romanini ◽  
Josue Labaki
Keyword(s):  
Closed Form ◽  
Integral Transform ◽  
Physical Space ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Constant Load ◽  
Wave Number ◽  
Dynamic Displacement ◽  
Closed Form Solutions ◽  
Fourier Integral Transform

A dynamic stationary semianalytical solution for a spatially constant load applied over a rectangular surface within a viscoelastic isotropic full space is presented. The solution is obtained within the frame of a double Fourier integral transform. Closed-form solutions for general loadings within the full space are furnished in the transformed wave number domain. Expressions for three boundary value problems, associated to a normal and two tangential rectangular loadings in the original physical space, are given in terms of a double inverse Fourier integral. These inverse integral transforms must be evaluated numerically. In the second part of the present paper a strategy to evaluate these integrals is described, the procedure validated and a number of original results are reported.

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.

Extending the unified transform: curvilinear polygons and variable coefficient PDEs

2018 ◽  
Vol 40 (2) ◽  
pp. 976-1004 ◽  
Author(s):  
Matthew J Colbrook
Keyword(s):  
Fourier Transforms ◽  
Elliptic Boundary ◽  
Neumann Boundary ◽  
Integral Transforms ◽  
Variable Coefficients ◽  
Central Component ◽  
Unknown Boundary ◽  
Boundary Values ◽  
Global Relation ◽  
Chebyshev Interpolation

Abstract We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

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.

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