Implementation of the Integrated Green’s Function Method for 3D Poisson’s Equation in a Large Aspect Ratio Computational Domain

The solution of Poisson’s Equation plays an important role in many areas, including modeling high-intensity and high-brightness beams in particle accelerators. For the computational domain with a large aspect ratio, the integrated Green’s function method has been adopted to solve the 3D Poisson equation subject to open boundary conditions. In this paper, we report on the efficient implementation of this method, which can save more than a factor of 50 computing time compared with the direct brute force implementation and its improvement under certain extreme conditions.

Existence of Monotone Positive Solution for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique.

Continuum Skyrme Hartree-Fock-Bogoliubov theory with Green’s function method for neutron-rich Ca,Ni,Zr,and Sn isotopes

The possible exotic nuclear properties in the neutron-rich Ca,Ni,Zr,and Sn isotopes are examined with the continuum Skyrme Hartree-Fock-Bogoliubov theory in the framework of the Green’s function method.The pairing correlation,the couplings with the continuum,and the blocking effects for the unpaired nucleon in odd-A nuclei are properly treated.The Skyrme interaction SLy4 is adopted for the ph channel and the density-dependentinteraction is adopted for the pp chan-nel,which well reproduce the experimental two-neutron separation energies S_(2n)and one-neutron separation energies Sn.It is found that the criterion S_(n)>0 predicts a neutron drip line with neutron numbers much smaller than those for S_(2n)>0.Owing to the unpaired odd neutron,the neutron pairing energies−E_(pair)in odd-A nuclei are much lower than those in the neighbor-ing even-even nuclei.By investigating the single-particle structures,the possible halo structures in the neutron-rich Ca,Ni,and Sn isotopes are predicted,where sharp increases in the root-mean-square(rms)radii with significant deviations from the traditional rA^(1∕3)rule and diffuse spatial density distributions are observed.Analyzing the contributions of various partial waves to the total neutron densityρlj(r)∕ρ(r)reveals that the orbitals located around the Fermi surface-particularly those with small angular momenta-significantly affect the extended nuclear density and large rms radii.The number of neutrons Nλ(N_(0))occupying above the Fermi surfacen(continuum threshold)is discussed,whose evolution as a function of the mass number A in each isotope is consistent with that of the pairing energy,supporting the key role of the pairing correlation in halo phenomena.

Searching for single-particle resonances with the Green’s function method

Single-particle resonances in the continuum are crucial for studies of exotic nuclei.In this study,the Green’s function approach is employed to search for single-particle resonances based on the relativistic-mean-field model.Taking^(120)Sn as an example,we identify singleparticle resonances and determine the energies and widths directly by probing the extrema of the Green’s functions.In contrast to the results found by exploring for the extremum of the density of states proposed in our recent study[Chin.Phys.C,44:084105(2020)],which has proven to be very successful,the same resonances as well as very close energies and widths are obtained.By comparing the Green’s functions plotted in different coordinate space sizes,we also found that the results very slightly depend on the space size.These findings demonstrate that the approach by exploring for the extremum of the Green’s function is also very reliable and effective for identifying resonant states,regardless of whether they are wide or narrow.

Practical Green’s function for the thermal stress field induced by a heat source in plane thermoelasticity

The classical Green’s functions used in the literature for a heat source in a homogeneous elastic medium cannot lead to ?nite remote thermal stresses in the medium,so that they may not work well in practical thermal stress analyses. In this paper, we develop a practical Green’s function for a heat source disposed eccentrically into an elastic disk/cylinder subject to plane deformation. The edge of the disk/cylinder is assumed to be thermally permeable and traction-free. The full thermal stress ?eld induced by the heat source in the disk/cylinder is determined exactly and explicitly via the Cauchy integral techniques. In particular, a very simple formula is obtained to describe the hoop thermal stress on the edge of the disk/cylinder, which may be conveniently useful for analyzing the thermal stresses in microelectronic components.

GREEN’S FUNCTION AND EFFECTIVE ELASTIC STIFFNESS TENSOR FOR ARBITRARY AGGREGATES OF CUBIC CRYSTALS

A closed but approximate formula of Green’s function for an arbitrary aggregate of cubic crystallites is given to derive the e?ective elastic sti?ness tensor of the polycrystal. This formula, which includes three elastic constants of single cubic crystal and ?ve texture coe?cients, accounts for the e?ects of the orientation distribution function (ODF) up to terms linear in the tex- ture coe?cients. Thus it is expected that our formula would be applicable to arbitrary aggregates with weak texture or to materials such as aluminum whose single crystal has weak anisotropy. Three examples are presented to compare predictions from our formula with those from Nishioka and Lothe’s formula and Synge’s contour integral through numerical integration. As an applica- tion of Green’s function, we brie?y describe the procedure of deriving the e?ective elastic sti?ness tensor for an orthorhombic aggregate of cubic crystallites. The comparison of the computational results given by the ?nite element method and our e?ective elastic sti?ness tensor is made by an example.

Ambient Noise Tomography, Green’s Function and Earthquakes

Green’s function is well-known, among others, in the application of ambient noise tomography methodologies that may demonstrate the potential of hydrocarbon entrapment in the study area. Here it is also shown to be of key importance in identifying the fractal dimension in the unified scaling law for earthquakes as well as in studying an explicit relationship of a future strong earthquake epicenter to the average earthquake potential score. Such studies are now in progress.

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.

Lamb waves topological imaging combining with Green’s function retrieval theory to detect near filed defects in isotropic plates

A method of combining Green’s function retrieval theory and ultrasonic array imaging using Lamb waves is presented to solve near filed defects in thin aluminum plates.The defects are close to the ultrasonic phased array and satisfy the near field calculation formula.Near field acoustic information of defects is obscured by the nonlinear effects of initial wave signal in a directly acquired response using the full matrix capture mode.A reconstructed full matrix of inter-element responses is produced from cross-correlation of directly received ultrasonic signals between sensor pairs.This new matrix eliminates the nonlinear interference and restores the near-field defect information.The topological imaging method that was developed in recent ultrasonic inspection is used for displaying the scatterers.The experiments are conducted on both thin aluminum plates containing two and four defects, respectively.The results show that these defects are clearly identified when using a reconstructed full matrix.The spatial resolution is equal to about one wavelength of the selectively excited mode and the identifiable defect is about one fifth of the wavelength.However, in a conventional directly captured image,the images of defects overlap together and cannot be distinguished.The proposed method reduces the background noise and allows for effective topological imaging of near field defects.

METHOD OF GREEN’S FUNCTION OF CORRUGATED SHELLS

By using the fundamental equations of axisymmetric shallow shells of revolution, the nonlinear bending of a shallow corrugated shell with taper under arbitrary load has been investigated. The nonlinear boundary value problem of the corrugated shell was reduced to the nonlinear integral equations by using the method of Green＇s function. To solve the integral equations, expansion method was used to obtain Green＇s function. Then the integral equations were reduced to the form with degenerate core by expanding Green＇s function as series of characteristic function. Therefore, the integral equations become nonlinear algebraic equations. Newton＇ s iterative method was utilized to solve the nonlinear algebraic equations. To guarantee the convergence of the iterative method, deflection at center was taken as control parameter. Corresponding loads were obtained by increasing deflection one by one. As a numerical example,elastic characteristic of shallow corrugated shells with spherical taper was studied.Calculation results show that characteristic of corrugated shells changes remarkably. The snapping instability which is analogous to shallow spherical shells occurs with increasing load if the taper is relatively large. The solution is close to the experimental results.

A Green’s function model for ferromagnetism and spin excitations of (Ga, Mn)As diluted magnetic semiconductors

We study （Ga, Mn）As diluted magnetic semiconductors in terms of the Ruderman-Kittel-Kasuya-Yosida quantum spin model in Green＇s function approach. Random distributions of the magnetic atoms are treated by using an analytical average of magnetic configurations. Average magnetic moments and spin excitation spectra as functions of temperature can be obtained by solving self-consistent equations, and the Curie temperature TC is given explicitly. Tc is proportional to magnetic atomic concentration, and there exists a maximum for Tc as a function of carrier concentration. Applied to （Ga, Mn）As, the theoretical results are consistent with experiment and the experimental TC can be obtained with reasonable parameters. This modelling can also be applied to other diluted magnetic semiconductors.

Quasi-Green’s function method for free vibration of clamped thin plates on Winkler foundation

The quasi-Green＇s function method is used to solve the free vibration problem of clamped thin plates on the Winkler foundation. Quasi-Green＇s function is established by the fundamental solution and the boundary equation of the problem. The function satisfies the homogeneous boundary condition of tile problem. The mode-shape differential equation of the free vibration problem of clamped thin plates on the Winkler foundation is reduced to the Fredholm integral equation of the second kind by Green＇s formula. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The numerical results show the high accuracy of the proposed method.

Numerical Green’s function method:Application to quantifying ground motion variations of M7 earthquakes

The concept of ＂numerical Green’s functions＂ （NGF or Green’s function database） is developed. The basic idea is： a large seismic fault is divided into subfaults of appropriate size, for which synthetic Green’s functions at the surface （NGF） are calculated and stored. Consequently, ground motions from arbitrary kinematic sources can be simulated, rapidly, for the whole fault or parts of it by superposition. The target fault is a simplified, vertical model of the Newport-Inglewood fault in the Los Angeles basin. This approach and its functionality are illustrated by investigating the variations of ground motions （e.g. peak ground velocity and synthetic seismograms） due to the source complexity. The source complexities are considered with two respects： hypocenter location and slip history. The results show a complex behavior, with dependence of absolute peak ground velocity and their variation on source process directionality, hypocenter location, local structure, and static slip asperity location. We concluded that combining effect due to 3-D structure and finite-source is necessary to quan- tify ground motion characteristics and their variations. Our results will facilitate the earthquake hazard assessment projects.

Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green＇s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi- Green＇s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob- lem. The mode shape differential equations of the free vibration problem of a simply- supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa- tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green＇s function method.

Green’s Function Solution for the Dual-Phase-Lag Heat Equation

The present work is devoted to define a generalized Green’s function solution for the dual-phase-lag model in homogeneous materials in a unified manner .The high-order mixed derivative with respect to space and time which reflect the lagging behavior is treated in special manner in the dual-phase-lag heat equation in order to construct a general solution applicable to wide range of dual-phase-lag heat transfer problems of general initial-boundary conditions using Green’s function solution method. Also, the Green’s function for a finite medium subjected to arbitrary heat source and arbitrary initial and boundary conditions is constructed. Finally, four examples of different physical situations are analyzed in order to illustrate the accuracy and potentialities of the proposed unified method. The obtained results show good agreement with works of [1-4].

Adomian Decomposition Method with Green’s Function for Solving Tenth-Order Boundary Value Problems

In this paper, the Adomian decomposition method with Green’s function (Standard Adomian and Modified Technique) is applied to solve linear and nonlinear tenth-order boundary value problems with boundary conditions defined at any order derivatives. The numerical results obtained with a small amount of computation are compared with the exact solutions to show the efficiency of the method. The results show that the decomposition method is of high accuracy, more convenient and efficient for solving high-order boundary value problems.

Numerical Solution of Green’s Function for Solving Inhomogeneous Boundary Value Problems with Trigonometric Functions by New Technique

A numerical technique is presented for solving integration operator of Green’s function. The approach is based on Hermite trigonometric scaling function on [0,2π], which is constructed for Hermite interpolation. The operational matrices of derivative for trigonometric scaling function are presented and utilized to reduce the solution of the problem. One test problem is presented and errors plots show the efficiency of the proposed technique for the studied problem.

Some Universal Properties of the Green’s Functions Associated with the Wave Equation in Bounded Partially-Homogeneous Domains and Their Use in Acoustic Tomography

Direct and inverse scattering problems connected with the wave equation in non-homogeneous bounded domains constitute challenging actual subjects for both mathematicians and engineers. Among them one can mention, for example, inverse source problems in seismology, nondestructive archeological probing, mine prospecting, inverse initial-value problems in acoustic tomography, etc. In spite of its crucial importance, almost all of the available rigorous investigations concern the case of unbounded simple domains such as layered planar or cylindrical or spherical structures. The main reason for the lack of the works related to non-homogeneous bounded structures is the extreme complexity of the explicit expressions of the Green’s functions. The aim of the present work consists in discovering some universal properties of the Green’s functions in question, which reduce enormously the difficulties arising in various applications. The universality mentioned here means that the properties are not depend on the geometrical and physical properties of the configuration. To this end one considers first the case when the domain is partially-homogeneous. Then the results are generalized to the most general case. To show the importance of the universal properties in question, they are applied to an inverse initial-value problem connected with photo-acoustic tomography.

Spectral-domain dyadic Green’s functions in chiral media

A novel method of deriving the electromagnetic dyadic Green＇s functions in an unbounded, lossless, reciprocal and homogeneous chiral media described by the constitutive relations D = εE ＋ jγB and H = jγE ＋ μ^-1B - （ωε）^-1γJ is given. The divergenceless and irrotational splitting of dyadic Dirac 8 function and Fourier transformation are used to directly obtain the divergenceless and irrotational component of spectral-domain dyadic Green＇s functions in chiral media. This method avoids using the dyadic Green＇s function eigenfunction expansion technique. The method given here can be generalized to a source-free region and an achiral case.

Quasi-Dynamic Green’s Functions for Efficient Full-Wave Integral Formulations for Microstrip Interconnects

Integral formulations are widely used for full-wave analysis of microstrip interconnects. A weak point of these formulations is the inclusion of the proper planar-layered Green’s Functions (GFs), because of their computational cost. To overcome this problem, usually the GFs are decomposed into a quasi-dynamic term and a dynamic one. Under suitable approximations, the ?rst may be given in closed form, whereas the second is approximated. Starting from a general criterion for this decomposition, in this paper we derive some simple criteria for using the closed-form quasi-dynamic GFs instead of the complete GFs, with reference to the problem of evaluating the full-wave current distribution along microstrips. These criteria are based on simple relations between frequency, line length, dielectric thickness and permittivity. The layered GFs have been embedded into a full-wave transmission line model and the results are ?rst benchmarked with respect to a full-wave numerical 3D tool, then used to assess the proposed criteria.