The Interaction of Oblique Flexural Gravity Waves With a Small Bottom Deformation on a Porous Ocean-Bed： Green＇s Function Approach

The interaction of oblique incident water waves with a small bottom deformation on a porous ocean-bed is examined analytically here within the framework of linear water wave theory. The upper surface of the ocean is assumed to be covered by an infinitely extended thin uniform elastic plate, while the lower surface is bounded by a porous bottom surface having a small deformation. By employing a simplified perturbation analysis, involving a small parameter c^（〈〈l ）, which measures the smallness of the deformation, the governing Boundary Value Problem （BVP） is reduced to a simpler BVP for the first-order correction of the potential function. This BVP is solved using a method based on Green＇s integral theorem with the introduction of suitable Green＇s function to obtain the first-order potential, and this potential function is then utilized to calculate the first-order reflection and transmission coefficients in terms of integrals involving the shape function c（x） representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number propagating just below the elastic plate and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the surface below the elastic plate. Again, for small angles of incidence, the reflected wave energy is more as compared to the other angles of incidence. It is also observed that the reflected wave energy is somewhat sensitive to the changes in the flexural rigidity of the elastic plate, the porosity of the bed and the ripple wave numbers. The main advantage of the present study is that the results for the values of reflection and transmission coefficients obtained are found to satisfy the energy-balance relation almost accurately.

Characteristics of unsteady flow in porous media while considering threshold pressure gradient with Green＇s function

The flow behavior in porous media with threshold pressure gradient(TPG) is more complex than Darcy flow and the equations of motion, and outer boundary and inner boundary with TPG are also different from Darcy flow for unsteady flow of a producing well in a reservoir. An analytic method to solve this kind of problem is in a need of reestablishment. The classical method of Green's function and Newman product principle in a new way are used to solve the unsteady state flow problems of various shapes of well and reservoir while considering the TPG. Four Green's functions of point, line, band and circle while considering the TPG are achieved. Then, two well models of vertical well and horizontal well are built and simultaneously the function to calculate the moving boundary of each well model is provided. The results show that when considering TPG the pressure field is much different, which has a sudden pressure change, with a moving boundary in it. And the moving boundary of each well model increases with time but slows down rapidly, especially when the TGP is large.

Study on the 3D Green＇s functions of the fluid and piezoelectric bimaterials

Because most piezoelectric devices have interfaces with fluid in engineering, it is valuable to study the coupled field between fluid and piezoelectric media. As the fundamental problem, the 3D Green＇s functions for point forces and point charge loaded in the fluid and piezoelectric bimaterials are studied in this paper. Based on the 3D general solutions expressed by harmonic functions, we constructed the suitable harmonic functions with undetermined constants at first. Then, the couple field in the fluid and piezoelectric bimaterials can be derived by substitution of harmonic functions into general solutions. These constants can be obtained by virtue of the compatibility, boundary, and equilibrium conditions. At last, the characteristics of the electromechanical coupled fields are shown by numerical results.

Location and moment tensor inversion of small earthquakes using 3D Green＇s functions in models with rugged topography： application to the Longmenshan fault zone

With dense seismic arrays and advanced imaging methods, regional three-dimensional （3D） Earth models have become more accurate. It is now increasingly feasible and advantageous to use a 3D Earth model to better locate earthquakes and invert their source mechanisms by fitting synthetics to observed waveforms. In this study, we develop an approach to determine both the earthquake location and source mechanism from waveform information. The observed waveforms are filtered in different frequency bands and separated into windows for the individual phases. Instead of picking the arrival times, the traveltime differences are measured by cross-correlation between synthetic waveforms based on the 3D Earth model and observed waveforms. The earthquake location is determined by minimizing the cross-correlation traveltime differences. We then fix the horizontal location of the earthquake and perform a grid search in depth to determine the source mechanism at each point by fitting the synthetic and observed waveforms. This new method is verified by a synthetic test with noise added to the synthetic waveforms and a realistic station distribution. We apply this method to a series of Mw3.4-5.6 earthquakes in the Longmenshan fault （LMSF） zone, a region with rugged topography between the eastern margin of the Tibetan plateau and the western part of the Sichuan basin. The results show that our solutions result in improved waveform fits compared to the source parameters from the catalogs we used and the location can be better constrained than the amplitude-only approach. Furthermore, the source solutions with realistic topography provide a better fit to the observed waveforms than those without the topography, indicating the need to take the topography into account in regions with rugged topography.

A Semi-Analytical Approach to Green＇s Functions for Problems in Multiply-Connected Regions on a Spherical Surface

This work aims at potential fields generated by point sources in conductive perforated fragments of spherical shells. Such fields are interpreted as profiles of Green＇s functions of relevant boundary-value problems stated in multiply-connected regions for Laplace equation written in geographical coordinates. Those are efficiently computed by a modification of the method of functional equations, with closed analytical forms preliminary obtained for Green＇s functions for the corresponding simply-connected regions.

Singularity-free Green's function for EM sources embedded in a stratified medium

We present a method to unify the calculation of Green＇s functions for an electromagnetic（EM） transmitting source embedded in a homogeneous stratified medium.A virtual interface parallel to layer interfaces is introduced through the source location.The potentials for Green＇s function are derived by decomposing the partial wave solutions to Helmholtz＇s equations into upward and downward within boundaries.The amplitudes of the potentials in each stratum are obtained recursively from the initial amplitudes at the source level.The initial amplitudes are derived by coupling with the transmitting sources and following the discontinuity of the tangential electric and magnetic fields at the source interface.Only the initial terms are related to the transmitting sources and thus need to be modified for different transmitters,whereas the kernel connected with the stratified media stays unchanged.Hence,the present method can be easily applied to EM transmitting sources with little modification.The application of the proposed method to the marine controlled-source electromagnetic method（MCSEM） demonstrates its simplicity and flexibility.

Nonequilibrium Green＇s function method for quantum thermal transport

This review deals with the nonequilibrium Green＇s function （NEGF） method applied to the problems of energy transport due to atomic vibrations （phonons）, primarily for small jtmction systems. We present a pedagogical introduction to the subject, deriving some of the well-known results such as the Laudauer-like formula for heat current in ballistic systerms. The main aim of the review is to build the machinery of the method so that it can be applied to other situations, which are not directly treated here. In addition to the above, we consider a nmnber of applications of NEGF, not in routine model system calculations, but in a few new aspects showing the power and usefulness of the formalism. In partkaflar, we discuss the problems of multiple leads, coupled left-right-lead system, and system without a center. We also apply the method to the problem of full counting statisties. In the case of nonlinear svstems, we make general comments on the thermal expansion effect. phonon relaxation timv. and a certain class of mean-field approximations. Lastly, we examine the relationship between NEGF. reduced density matrix, and master equation approaches to thermal transport,

Study on Characteristics of 3-D Translating-Pulsating Source Green Function of Deep-Water Havelock Form and Its Fast Integration Method

The singularities, oscillatory performances and the contributing factors to the 3-＇D translating-pulsating source Green function of deep-water Havelock form which consists of a local disturbance part and a far-field wave-like part, are analyzed systematically. Relative numerical integral methods about the two parts are presented in this paper. An improved method based on LOBATTO rule is used to eliminate singularities caused respectively by infinite discontinuity and jump discontinuous node from the local disturbance part function, which makes the improvement of calculation efficiency and accuracy possible. And variable substitution is applied to remove the singularity existing at the end of the integral interval of the far-field wave-like part function. Two auxiliary techniques such as valid interval calculation and local refinement of integral steps technique in narrow zones near false singularities are applied so as to avoid unnecessary integration of invalid interval and improve integral accordance. Numerical test results have proved the efficiency and accuracy in these integral methods that thus can be applied to calculate hydrodynamic performance of floating structures moving in waves.

Green's functions for uniformly distributed loads acting on an inclined line in a poroelastic layered site

Based on one type of practical Biot＇s equation and the dynamic-stiffness matrices ofa poroelastic soil layer and half-space, Green＇s functions were derived for unitformly distributed loads acting on an inclined line in a poroelastie layered site. This analysis overcomes significant problems in wave scattering due to local soil conditions and dynamic soil-structure interaction. The Green＇s functions can be reduced to the case of an elastic layered site developed by Wolf in 1985. Parametric studies are then carried out through two example problems.

Existence of Positive Solutions of Second-order Periodic Boundary Value Problems with Sign-Changing Green＇s Function

In this paper, we consider the existence of positive solutions of second-order periodic boundary value problem where 0 〈 ε 〈 1/2, g ： [0, 2π] →R is continuous, f ： [0, ∞） →R is continuous and λ 〉 0 is a parameter.

NONLINEAR EVOLUTION SYSTEMS AND GREEN’S FUNCTION

In this paper, we will introduce how to apply Green＇s function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen＇s principle. Then, for other nonlinear dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green＇s function of the linearized system and micro-local analysis, such as frequency decomposition and so on.

Green's function of multi-layered poroelastic half-space for models of ground vibration due to railway traffic

This study proposes a Green＇s function, an essential representation of water-saturated ground under moving excitation, to simulate ground borne vibration from trains. First, general solutions to the governing equations of poroelastic medium are derived by means of integral transform. Secondly, the transmission and reflection matrix approach is used to formulate the relationship between displacement and stress of the stratified ground, which results in the matrix of the Green＇s function. Then the Green＇s function is combined into a train-track-ground model, and is verified by typical examples and a field test. Additional simulations show that the computed ground vibration attenuates faster in the immediate vicinity of the track than in the surrounding area. The wavelength of wheel-rail unevenness has a notable effect on computed displacement and pore pressure. The variation of vibration intensity with the depth of ground is significantly influenced by the layering of the strata soil. When the train speed is equal to the velocity of the Rayleigh wave, the Mach cone appears in the simulated wave field. The proposed Green＇s function is an appropriate representation for a layered ground with shallow ground water table, and will be helpful to understand the dynamic responses of the ground to complicated moving excitation.

A fast integration method for translating-pulsating source Green＇s function in Bessho form

The singularities and oscillatory performance of translating-pulsating source Green＇s function in Bessho form were analyzed. Relative numerical integration methods such as Gaussian quadrature rule, variable substitution method （VSM）, and steepest descent integration method （SDIM） were used to evaluate this type of Green＇s function. For SDIM, the complex domain was restricted only on the 0-plane. Meanwhile, the integral along the real axis was computed by use of the VSM to avoid the complication of a numerical search of the steepest descent line. Furthermore, the steepest descent line was represented by the B-spline function. Based on this representation, a new self-compatible integration method corresponding to parametric t was established. The numerical method was validated through comparison with other existing results, and was shown to be efficient and reliable in the calculation of the velocity potentials for the 3D seakeeping and hydrodynamic performance of floating struc- tures moving in waves.

In-plane dynamic Green's functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic half-space

The dynamic stiffness method combined with the Fourier transform is utilized to derive the in-plane Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic（TI）half-space.The loaded layer is fixed to obtain solutions restricted in it and the corresponding reactions forces,which are then applied to the total system with the opposite sign.By adding solutions restricted in the loaded layer to solutions from the reaction forces,the global solutions in the wavenumber domain are obtained,and the dynamic Green’s functions in the space domain are recovered by the inverse Fourier transform.The presented formulations can be reduced to the isotropic case developed by Wolf（1985）,and are further verified by comparisons with existing solutions in a uniform isotropic as well as a layered TI halfspace subjected to horizontally distributed loads which are special cases of the more general problem addressed.The deduced Green’s functions,in conjunction with boundary element methods,will lead to significant advances in the investigation of a variety of wave scattering,wave radiation and soil-structure interaction problems in a layered TI site.Selected numerical results are given to investigate the influence of material anisotropy,frequency of excitation,inclination angle and layered on the responses of displacement and stress,and some conclusions are drawn.

Green＇s Relations on Semigroups of Transformations Preserving Two Equivalence Relations

Let Tx be the full transformation semigroup on a set X. For a non-trivial equivalence F on X, letTF（X） = {f ∈ Tx ： arbieary （x, y） ∈ F, （f（x）,f（y）） ∈ F}.Then TF（X） is a subsemigroup of Tx. Let E be another equivalence on X and TFE（X） = TF（X） ∩ TE（X）. In this paper, under the assumption that the two equivalences F and E are comparable and E lohtain in F, we describe the regular elements and characterize Green＇s relations for the semigroup TFE（X）.

Transient Thermal Analysis of a Plate Fin Heat Sink with Green＇s Function

The accurate analyses for a plate fin heat sink with the ability to control the temperature of the avionics devices within a pre-set controllable temperature range are required both in the process of circuit design and for the real-time temperature monitoring purposes. In order to provide an insight into the behavior of the temperature of a plate fin heat sink subjected non-uniform heat density on the surfaces, it is necessary to obtain accurate analytical solutions yielding explicit formulas relating the dissipated power to the temperature rise at any point of avionics devices. This paper presents a method for thermal simulation of a plate fin heat sink using an analytical solution of the three-dimensional heat equation resulting from an appropriate plate fin heat sink transient thermal model. The entire solution methodology is illustrated in detail on the particular examples of the plate fin heat sink subjected non-uniform heat density on the surfaces. The transient temperature profiles are obtained for different positions at the surface of the plate fin heat sink. The analytical results are compared with measurements made on the surface of the cold plate and it is found that they are in good agreement with an error of less than 3 K.

TIME-HARMONIC DYNAMIC GREEN'S FUNCTIONS FOR ONE-DIMENSIONAL HEXAGONAL QUASICRYSTALS

Quasicrystals have additional phason degrees of freedom not found in conventional crystals. In this paper, we present an exact solution for time-harmonic dynamic Green＇s function of one-dimensional hexagonal quasicrystals with the Laue classes 6/mh and 6/mhmm. Through the introduction of two new functions φ and ψ, the original problem is reduced to the determination of Green＇s functions for two independent Helmholtz equations. The explicit expressions of displacement and stress fields are presented and their asymptotic behaviors are discussed. The static Green＇s function can be obtained by letting the circular frequency approach zero.

Decoupling coefficients of dilatational wave for Biot's dynamic equation and its Green's functions in frequency domain

Green＇s functions for Blot＇s dynamic equation in the frequency domain can be a highly useful tool for the investigation of dynamic responses of a saturated porous medium. Its applications are found in soil dynamics, seismology, earthquake engineering, rock mechanics, geophysics, and acoustics. However, the mathematical work for deriving it can be daunting. Green＇s functions have been presented utilizing an analogy between the dynamic thermoelasticity and the dynamic poroelasticity in the frequency domain using the u-p formulation. In this work, a special term ＂decoupling coefficient＂ for the decomposition of the fast and slow dilatational waves is proposed and expressed to present a new methodology for deriving the poroelastodynamic Green＇s functions. The correct- ness of the solution is demonstrated by numerically comparing the current solution with Cheng＇s previous solution. The separation of the two waves in the present methodology allows the more accurate evaluation of Green＇s functions, particularly the solution of the slow dilatational wave. This can be advantageous for the numerical implementation of the boundary element method （BEM） and other applications.

Solution of pavement temperature field in“Environment-Surface”system through Green's function

In order to simplify the boundary conditions of pavement temperature field,the "Environment-Surface" system which considered the natural environment and pavement surface was established.Based on this system,the partial differential equations of the one-dimensional heat conduction in the pavement were established on the basis of the heat transfer theory.Furthermore,the function forms of the initial and boundary conditions of the equations were created through the field experiments.The general solution of the pavement one-dimensional heat conduction partial differential equations was acquired by using Green's function,and the explicit expression of pavement temperature field under specific constraint conditions was derived.For the purpose of analysis,the pavement temperatures in different seasons were calculated using the explicit expression of pavement temperature field,and the calculation accuracy was analyzed through the comparison between measured and calculated values.Then,the relationship between fitting accuracy and calculation accuracy of pavement temperatures was analyzed.The analysis results show that: the usage of "Environment-Surface" system simplifies the calculation of pavement temperature field; the relative error between calculated and measured values is generally less than 7% and is seldom influenced by seasons; there is a positive correlation between the calculation accuracy and the fitting accuracy of pavement surface temperature; high fitting accuracy would result in less error of pavement temperature prediction.

ON GREEN'S FUNCTION FOR HYPERBOLIC-PARABOLIC SYSTEMS

We study the Green＇s function for a general hyperbolic-parabolic system, including the Navier-Stokes equations for compressible fluids and the equations for magnetohydrodynamics. More generally, we consider general systems under the basic Kawashima- Shizuta type of conditions. The first result is to make precise the secondary waves with subscale structure, revealing the nature of coupling of waves pertaining to different characteristic families. The second result is on the continuous differentiability of the Green＇s function with respect to a small parameter when the coefficients of the system are smooth functions of that parameter. The results significantly improve previous results obtained by the authors.