Domain Green Function Research Articles

Vertical line time domain green function and its applications in numerical simulation of ship seakeeping performance

In present study, a vertical line source time domain Green function is proposed and the subsequent numerical evaluation methods are described in detail. For which, Taylor expansion and Chebyshev approximation algorithms are originally extended from point source to line source for small parameters, as well as asymptotic expansions are extended for large parameters. Meanwhile, polynomials are originally proposed to replace the asymptotic expansions to improve the computational efficiency. The accuracy and stability of the proposed method are validated through several cases and good agreement can be obtained.Furthermore, a hybrid Green function method based on the vertical line source Green function and Rankine source is developed for time domain simulation of seakeeping performances of ships. With respect to the hybrid method, the flow domain is divided into inner part and outer part by an artificial control surface. The first-order Taylor Expansion Boundary Element Method (TEBEM) based on the Rankine panel distribution is applied in the inner part, while three-dimensional panel method based on the vertical line source Green function distribution is adopted in the outer part. Under this circumstance, motions of the single-ship (Slender Wigley and Blunt Wigley) and two-ship (two side-by-side ships) in head waves are numerically investigated. The numerical results agree well with the experimental data, which proved the feasibility and the accuracy of present hybrid Green function method for motion prediction of both single-ship and two-ship.

Design Finite Array with Non-linear Element Spacing

The final matrix presented here consists of identical matrix elements, located at random. The fixed patch antenna serves as the base element for the grille. Arrays are created by randomly placing a base element. Arrays of this type are difficult to model and can be resource-intensive. It will show you how you can construct matrices using the available finite matrix tools. When the array is built in this way, DFM (Domain Green Function Method) acceleration can be applied to reduce the required resources. 20 mV, 20 dB general orientation, 0 dB axis ratio

Enhancing cross correlations of ocean ambient noise in the time domain based on random matrix theory.

Cross correlations of diffuse noise can be utilized to recover the time domain Green's function (TDGF) between two points. This principle allows for a variety of practical applications, such as seismic tomography, ocean thermometry, passive localization, etc. However, in the real ocean environment, loud interference noise sources usually bias the travel time estimates of the TDGF or result in poor recovery quality. To deal with this issue, a diffuse noise reconstruction approach is proposed to eliminate the influence of the strong interference noise by utilizing the time domain statistical property of ocean ambient noise recorded on single hydrophones with the help of random matrix theory. Simulation and experimental data analysis indicate that this algorithm can effectively extract the diffuse noise component from the ocean ambient noise field and retrieve the TDGF with a higher signal-to-noise ratio when coherent accumulation of cross correlations of the reconstructed diffuse noise is performed.

Integral Equation Formulation for Planar Plasmonic Structures With Finite Thickness in Layered Media

A detailed Volume Integral Equation (VIE) formulation for planar plasmonic nano structures with finite thickness in flat multi-layers medium is presented. The boundary condition along the localized metallic objects is expressed in terms of the unknown polarization current flowing through these objects in the form of an integral equation, which is solved using the Method of Moments (MoM). The Green's functions associated with a layered medium of practical importance are expressed in the spectral domain. The corresponding spatial domain Green's functions are obtained using the Discrete Complex Images Method (DCIM). Special treatment for the spectral function's asymptote at high spectral values is performed. The presented formulation is applied on different plasmonic structures immersed inside layered media. The structures include nano-rod and nano-patch excited by an incident plane wave. In addition, a simple band-stop filter based on quarter-wavelength stubs is considered. This filter is fed with a couple of plasmonic transmission lines. The obtained current distributions and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> -parameters are compared with those obtained using a commercial full-wave electromagnetic simulator, namely CST Microwave Studio. A very good agreement is observed. The proposed integral equation formulation enjoys high degree of stability, numerical efficiency, and accuracy.

A study on hydrodynamic analysis of ship with forward-speed based on a time domain boundary element method

The hydrodynamic analysis for ships is studied based on the three-dimensional (3D) time domain Green function (TDGF) method. The precise integration method (PIM) with variable dimension matrix is developed to evaluate the TDGF and its derivatives. The numerical convergence analysis for TDGF evaluation is carried out. Based on the TDGF method, the radiation problem and diffraction problem are solved by using impulsive response function. The hydrodynamic coefficients, wave exciting forces and motions for Wigley Ⅰ hull form ship are computed to verify the reliability and accuracy of the three-dimensional time domain program. The numerical results obtained by the proposed method show a good agreement with the previous experimental data and other published results.

Time-domain analysis of side-by-side floating bodies by the three-dimensional hybrid Green function method

ABSTRACT In the present research, the three-dimensional (3D) time-domain hybrid Green function method is developed for simulating the coupled motions of two floating bodies with the side-by-side arrangement in waves. The fluid domain is decomposed into an inner and an outer domain by an imaginary control surface. Then, the Rankine source is adopted in the inner domain, while the transient time domain Green function is adopted in the outer domain. The side-by-side problem is numerically solved by means of the continuity condition adopted on the control surface. A Wigley hull ship and a rectangular barge with the side-by-side arrangement in beam wave are investigated, while the hydrodynamic interactions between them are taken into consideration. Based on that, the coupled motions in waves between a supply ship and frigate both with large flare, as well as with same forward speed and side-by-side arrangement are also investigated. A good agreement is obtained between numerical and experimental results for all case studies, which indicates that the present method and developed time-domain numerical solver is accurate enough and available to handle the side-by-side problem in ship and ocean engineering.

Validation of a distributed simulation of ship replenishment at sea with model tests

A simulation of ship motions during replenishment at sea was validated using dedicated model test data. Motions of the supply ship and receiving ship are influenced by hydrodynamic forces and forces from replenishment gear used to transfer payloads. Simulated entities include rudders and propellers, with associated controllers. The composite simulation was implemented as a distributed simulation using the high-level architecture. An experimental model test program was developed for a tanker and destroyer conducting replenishment in head seas at a speed of 15 knots. The simulation gives generally good agreement with the model tests for heave, roll, and pitch motions, with some noted exceptions. For dynamic positioning of the destroyer relative to the tanker, there is very good agreement for mean values but underprediction of lower frequency sway and yaw motions. The variation of heave and roll motion accuracy is likely due to the hydrodynamic force prediction method using the frequency domain Green function for zero forward speed, thus not accurately modelling the propagation of radiated and diffracted waves between vessels. Prediction of viscous roll forces using semi-empirical methods likely contributed to differences between simulated and model test roll motions. The underprediction of lower frequency sway and yaw motions for the destroyer in waves could be caused by the ship motion predictions not including second-order wave drift forces.

Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs

In a previous work [Andrade \textit{et al.}, Phys. Rep. \textbf{647}, 1 (2016)], it was shown that the exact Green's function (GF) for an arbitrarily large (although finite) quantum graph is given as a sum over scattering paths, where local quantum effects are taken into account through the reflection and transmission scattering amplitudes. To deal with general graphs, two simplifying procedures were developed: regrouping of paths into families of paths and the separation of a large graph into subgraphs. However, for less symmetrical graphs with complicated topologies as, for instance, random graphs, it can become cumbersome to choose the subgraphs and the families of paths. In this work, an even more general procedure to construct the energy domain GF for a quantum graph based on its adjacency matrix is presented. This new construction allows us to obtain the secular determinant, unraveling a unitary equivalence between the scattering Schr\"odinger approach and the Green's function approach. It also enables us to write a trace formula based on the Green's function approach. The present construction has the advantage that it can be applied directly for any graph, going from regular to random topologies.

A three-dimensional sono-elastic method of ships in finite depth water with experimental validation

In order to conduct the comprehensive analysis of fluid-structure interactions, acoustic radiation and acoustic propagation, a three-dimensional sono-elastic method in the frequency domain is proposed in this paper. In this method, the three-dimensional hydroelasticity theory of floating structures is incorporated with the hydro-acoustic propagation theory by introducing frequency domain Green's functions appropriate to the ocean hydro-acoustic waveguide environment. The simple source boundary integral approach is adopted to obtain the sono-elastic response in the frequency domain. In order to deal with the irregular frequency problem in this approach, a Closed Virtual Impedance Surface (CVIS) method is put forward. At last, the sono-elastic method is validated by calculating the acoustic radiation of a floating elastic spherical shell and an experiment of the underwater acoustic radiation of two ring-stiffened cylindrical shell models.

A heterogeneous FMM for layered media Helmholtz equation I: Two layers in [formula omitted

In this paper, we introduce a new heterogeneous fast multipole method (H-FMM) for 2-D Helmholtz equation in layered media. To illustrate the main algorithm ideas, we focus on the case of two layers in this work. The key compression step in the H-FMM is based on a fact that the multipole expansion for the sources of the free-space Green's function can be used also to compress the far field of the sources of the layered-media or domain Green's function, and a similar result exists for the translation operators for the multipole and local expansions. The mathematical error analysis is shown rigorously by an image representation of the Sommerfeld spectral form of the domain Green's function. As a result, in the H-FMM algorithm, the multipole-to-multipole, multipole-to-local, and local-to-local translation operators are the same as those in the free-space case, allowing easy adaptation of existing free-space FMM. All the spatially variant information of the domain Green's function is collected into the multipole-to-local translations and therefore the FMM becomes heterogeneous. The compressed representation further reduces the cost of evaluating the domain Green's function when computing the local direct interactions. Preliminary numerical experiments are presented to demonstrate the efficiency and accuracy of the algorithm with much improved performance over some existing methods for inhomogeneous media.

BTZ Quasinormal Frequencies as Poles of Green’s Function

Based on the well known fact that the quasinormal frequencies are the poles of the frequency domain Green's function we describe a method that allows us to calculate exactly the quasinormal frequencies of the Klein-Gordon field moving in the three-dimensional rotating BTZ black hole. These quasinormal frequencies are already published and widely explored in several applications, but we use this example to expound the proposed method of computation. We think that the described procedure can be useful to calculate exactly the quasinormal frequencies of classical fields propagating in other backgrounds. Furthermore, we compare with previous results and discuss some related facts.

Extracting the time domain Green's function from ocean ambient noise using acoustic vector sensors.

A function that closely resembles the two-point time-domain Green's function (TDGF) representing the time delays associated with multipath between the two sensors can be recovered by correlating the noise field measured by two sensors. Here, a technique for extracting the TDGF from ambient ocean noise using acoustic vector sensors is presented. Experimental results suggest that the averaging time to extract TDGF is greatly reduced if sound pressure sensors (hydrophones) are replaced by acoustic vector sensors. The direct arrival and bottom bounce arrival were extracted successfully with only 1 min of vertical velocity data, while the bottom bounce arrival was not extracted with even 10 min of sound pressure data.

Mixed-Potential Integral Equation for Full-Wave Modeling of Grounding Systems Buried in a Lossy Multilayer Stratified Ground

The paper presents a comprehensive approach for full-wave modeling of grounding systems buried in a multilayer stratified ground. The dyadic spectral domain Green's function associated with the multilayer stratified ground is first derived. This Green's function is then evaluated by numerical solution of Sommerfeld's integrals along the deformed path in the complex plane. The proposed method is general as it can deal with arbitrarily shaped grounding electrodes buried in soils with any number of layers. The case of horizontal, vertical, and oblique electrodes buried in soils with different layers is analyzed. The accuracy of the proposed approach is first validated by comparing the results with those obtained by NEC-4 for a homogeneous soil, and with those obtained by finite difference time domain method for a multilayer soil.

Time domain Rankine-Green panel method for offshore structures

To solve the numerical divergence problem of the direct time domain Green function method for the motion simulation of floating bodies with large flare, a time domain hybrid Rankine-Green boundary element method is proposed. In this numerical method, the fluid domain is decomposed by an imaginary control surface, at which the continuous condition should be satisfied. Then the Rankine Green function is adopted in the inner domain. The transient free surface Green function is applied in the outer domain, which is used to find the relationship between the velocity potential and its normal derivative for the inner domain. Besides, the velocity potential at the mean free surface between body surface and control surface is directly solved by the integration scheme. The wave exciting force is computed through the convolution integration with wave elevation, by introducing the impulse response function. Additionally, the nonlinear Froude-Krylov force and hydrostatic force, which is computed under the instantaneous incident wave free surface, are taken into account by the direct pressure integration scheme. The corresponding numerical computer code is developed and first used to compute the hydrodynamic coefficients of the hemisphere, as well as the time history of a ship with large flare; good agreement is obtained with the analytical solutions as well as the available numerical results. Then the hydrodynamic properties of a FPSO are studied. The hydrodynamic coefficients agree well with the results computed by the frequency method; the influence of the time interval and the truncated time is investigated in detail.

Passive detection of scatterer using autocorrelation of surf noise

When a scatterer is located in a diffuse noise field, time domain Green's function between two different receivers can be extracted from cross-correlation of ambient noise which is recorded by the two receivers so that target detection can be implemented. However, the method based on cross-correlation strongly depends on timing synchronization of each receiver, otherwise there will be a time drift in the cross-correlation result, which can bring error in the positioning detection. Besides, two receivers that are far from each other must communicate with each other to implement cross-correlation in real-time data processing, but big data transmission is difficult in the ocean. Compared with cross-correlation, autocorrelation means that each receiver works independently and only the final autocorrelation result is to be transmitted. Actually, the scattered wave of target is always so weak that it is submerged in the autocorrelation result of the ambient noise. In this paper, we propose a method of processing the autocorrelation of the ambient noise. When the averaging noise autocorrelation of all receivers is subtracted from the autocorrelation result of the noise recorded by each receiver, the signalnoise ratio of the scattered wave will be significantly enhanced. With the help of Kirchhoff migration algorithm, detection of a scatterer can be implemented. We have conducted a scatterer passive detection experiment in Shilaoren beach, Qingdao, and accurately detected the position of a polyviny chloride pipe (about 8 m away from the nearest receiver) using only 12 min surf noise data. The experimental result shows that the processing of autocorrelation could replace cross-correlation in passive target detection when the ambient noise is time steady and the statistical characteristics of the background noise at different receivers are the same. Unlike Green's function extracted from cross-correlation of ambient noise, each receiver can work independently without considering the problems of massive data transmission and timing synchronization, which may be suitable for target detection using multi-receivers and mobile platform.

Exact Time Domain Solutions of 1-D Transient Dynamic Piezoelectric Problems with Nonlinear Damper Boundary Conditions

Novel exact solutions of one-dimensional transient dynamic piezoelectric problems for thickness polarized layers and disks, or length polarized rods, are obtained. The solutions are obtained using a time domain Green's function method that leads to an exact analytical recursive procedure which is applicable for a wide variety of boundary conditions including nonlinear cases. A nonlinear damper boundary condition is considered in more detail. The corresponding nonlinear relationship between stresses and velocities at a current time moment is used in the recursive procedure. In addition to the exact recursive procedure that is effective for calculations, some new practically important explicit exact solutions are presented. Several examples of the time behavior of the output electric potential difference are given to illustrate the effectiveness of the proposed exact approach.

A Counterexample to a Conjecture by Błocki–Zwonek

ABSTRACTFor a bounded pseudoconvex domain and pluricomplex Green function gΩ(z, a) with pole at a ∈ Ω, it was conjectured by Błocki and Zwonek that β(t) = log λn({z ∈ Ω: gΩ(z, a) < t}) is a convex function on ( − ∞, 0). With Ω the annulus the Green function gΩ(z, a) with pole at a = 1 + 0i can be explicitly given in terms of Jacobi theta functions. We show numerically that in this case β is not convex.

Study on Havelock form translating–pulsating source Green's function distributing on horizontal line segments and its applications

Calculation of free surface Green's function is the fundamental element in numerically solving and analyzing hydrodynamic performance of ships or floating structures. Three dimensional translating and pulsating (3DTP) frequency domain Green's function in Havelock form is of explicit physical meaning, while it is difficult to be evaluated for its integrand has infinity multipliers k and high-frequency oscillatory behavior. To extend it to ship hydrodynamic problems, integration of 3DTP along line segment is outlined and analytically performed in advance. A θ-type integral of Havelock form Green's function distributing on a horizontal line segment is obtained. The integrals of both 3DTP source Green's function distributing on a horizontal line segment and partial derivatives are derived, as a result, both the order of infinity multiplier and the amplitude of oscillation function decrease, and the term with infinity number vanishes as well. It avoids the difficulties and also provides a new robust approach to obtain integral of Green's function over a panel accurately and stably.To calculate the 3DTP Green's function distributing on horizontal line segment and its partial derivatives. Fraction polynomial is used to evaluate complex exponential integral. Lobatto rule is applied to eliminate the effect of singularity at end points. The horizontal line segment is divided by critical points to make sure the original integrand is continuous on each line subsection. For the integration of 3DTP Green's function over a panel, the profiles along vertical direction can be obtained by accumulating a series of 3DTP source distributing on various horizontal line segments. The corresponding computing codes are developed and validated. 3DTP Green's function distributing on a horizontal line segments and its integral over a panel are performed by present method and Gaussian integral method. Numerical results show that present method is more efficient and accurate than normal Gaussian integral with point sources. By use of it, hydrodynamic of Wigley III and S175 container ship advancing in waves are calculated and analyzed. The calculated motions of ship are acquired and agree well with experimental data.

The time domain moving Green function of a railway track and its application to wheel–rail interactions

When dealing with wheel-rail interactions for a high-speed train using the time domain Green function of a railway track, it would be more reasonable to use the moving Green function associated with a reference frame moving with the train, since observed from this frame wheel/rail forces are stationary. In this paper, the time domain moving Green function of a railway track as an infinitely long periodic structure is defined, derived, discussed and applied. The moving Green function is defined as the Fourier transform, from the load frequency domain to the time domain, of the response of the rail due to a moving harmonic load. The response of the rail due to a moving harmonic load is calculated using the Fourier transform-based method. A relationship is established between the moving Green function and the conventional impulse response function of the track. Properties of the moving Green function are then explored which can largely simplify the calculation of the Green function. And finally, the moving Green function is applied to deal with interactions between wheels and a track with or without rail dampers, allowing non-linearity in wheel-rail contact and demonstrating the effect of the rail dampers.

Black hole binary inspiral: Analysis of the plunge

Binary black hole coalescence has its peak of gravitational wave generation during the "plunge," the transition from quasicircular early motion to late quasinormal ringing. Although advances in numerical relativity have provided plunge waveforms, there is still no intuitive or phenomenological understanding of plungecomparable to that of the early and late stages. Here we make progress in developing such understanding by focusing on the excitation of quasinormal ringing (QNR) during the plunge. We rely on insights of the linear mathematics of the particle perturbation model for the extreme mass limit. Our analysis, based on the Fourier domain Green function, and a simple initial model, point to the crucial role played by the kinematics near the "light ring" (the circular photon orbit) in determining the excitation of QNR. That insight is then shown to successfully explain Schwarzschild QNR found with evolution codes. Lastly, a phenomenological explanation is given for the underlying importance of the light ring.