Green's theorem and Green's functions for the steady-state cosmic-ray equation of transport

1977 ◽  
Vol 50 (1) ◽  
pp. 205-223 ◽  
Author(s):  
G. M. Webb ◽  
L. J. Gleeson
Keyword(s):  
Steady State ◽  
Green's Functions ◽  
Green’S Functions ◽  
Cosmic Ray ◽  
Green's Theorem ◽  
Green’S Theorem

scholarly journals Attenuation and amplification of the transient current in nanojunctions with time-varying gate potentials

2017 ◽  
Vol 31 (14) ◽  
pp. 1750105 ◽  
Author(s):  
Eduardo C. Cuansing
Keyword(s):  
Steady State ◽  
Green's Functions ◽  
Green’S Functions ◽  
Step Function ◽  
Time Varying ◽  
Nonequilibrium Green’S Functions ◽  
Bias Potential ◽  
Gate Potential ◽  
Drain Bias ◽  
Nonequilibrium Green's Functions

We study charge transport in a source-channel-drain system with a time-varying applied gate potential acting on the channel. We calculate both the current flowing from the source into channel and out of the channel into the drain. The current is expressed in terms of nonequilibrium Green’s functions. These nonequilibrium Green’s functions can be determined from the steady-state Green’s functions and the equilibrium Green’s functions of the free leads. We find that the application of the gate potential can induce current to flow even when there is no source-drain bias potential. However, the direction of the current from the source and the current to the drain are opposite, thereby resulting in no net current flowing within the channel. When a source-drain bias potential is present, the net current flowing to the source and drain can either be attenuated or amplified depending on the sign of the applied gate potential. We also find that the response of the system to a dynamically changing gate potential is not instantaneous, i.e., a relaxation time has to pass before the current settles into a steady value. In particular, when the gate potential is in the form of a step function, the current first overshoots to a maximum value, oscillates and then settles down to a steady-state value.

Boundary Element Methods for Steady-State Thermal-Mechanical Problems of Counterformal Contact

2004 ◽  
Vol 126 (3) ◽  
pp. 443-449
Author(s):  
Michael J. Rodgers ◽  
Shuangbiao Liu ◽  
Q. Jane Wang ◽  
Leon M. Keer
Keyword(s):  
Steady State ◽  
Boundary Element ◽  
Green's Functions ◽  
Green’S Functions ◽  
Boundary Element Methods ◽  
Triangular Element ◽  
Surface Load ◽  
Linear Element ◽  
Influence Coefficients ◽  
Constant Element

This paper presents a concise boundary integral equation framework for relating the thermal-mechanical surface load (the three traction components and the normal heat flux) to the thermal-mechanical response (the three quasi-static displacement components and the steady-state temperature). This uncoupled thermoelastic framework allows the simultaneous calculation of displacement and temperature—without subsurface discretization—because it is based on classical Green’s functions for displacement and for temperature and on newly derived Green’s functions for thermoelastic displacement. In general, the boundary element method (BEM) can be applied with this framework to finite geometry problems of steady-state thermal-mechanical contact. Here, example calculations are performed for counterformal contact problems, which can be modeled as contact on a halfspace. A linear element BEM is developed and compared with the constant element BEM for speed and accuracy. The linear element BEM uses newly derived influence coefficients for constant loads over an arbitrary triangular element, and these closed form expressions are used to improve the accuracy of the numerical algorithm. The constant element BEM uses the discrete convolution fast Fourier transform (DC-FFT) algorithm, which is based on influence coefficients for constant loads over rectangular elements. The quasi-static surface displacements and the steady-state surface temperature are calculated from an applied semi-ellipsoidal pressure with accompanying frictional heating effects. The surface thermal-mechanical behavior of the counterformal contact is shown in graphs vs. the radius, and the deviations from axisymmetry are highlighted.

Analysis of Spatial Steady-State Vibrations of a Layered Anisotropic Plate Using the Green’s Functions

2010 ◽  
Author(s):  
Alexander Karmazin ◽  
Evgenia Kirillova ◽  
Wolfgang Seemann ◽  
Pavel Syromyatnikov
Keyword(s):  
Fourier Transform ◽  
Steady State ◽  
Green's Functions ◽  
Anisotropic Plate ◽  
Fourier Transforms ◽  
Green’S Functions ◽  
Displacement Vector ◽  
Integral Representations ◽  
Surface Load ◽  
The Real

Spatial steady-state harmonic vibrations of a layered anisotropic plate excited by the distributed sources are considered. The work is based on the classical methods of the integral Fourier transforms and integral representations of the Green’s functions. In Fourier transform domain, the displacement vector is represented in terms of the Green’s matrix transform and the transform of the surface load vector. The two-dimensional inverse Fourier transform of the displacement vector is computed by reducing double integral to the iterated one with integrating along a contour, which deviates from the real axis while bypassing the real poles, and with subsequent integrating along the wave propagation angle. Three numerical algorithms of computing related iterated integrals are presented. The features of the application of these algorithms for the near- and far-field zones of the source are discussed. All of presented methods are compared for the numerical examples of vibrations on the surface of 24-layer symmetrical composite.

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