Frequency-domain Green Function Research Articles

P- and S-wave simulation using a Cole-Cole model to incorporate thermoelastic attenuation and dispersion.

In thermoelastic wave attenuation, such as that caused by heterogeneities much smaller than the wavelength, e.g., Savage theory of spherical pores, the shape of the relaxation peak differs from that of the Zener (or standard linear solid) mechanical model. In these effective homogeneous media, the anelastic behavior is better represented by a stress-strain relation based on fractional derivatives; particularly, P- and S-wave dispersion and attenuation is well described by a Cole-Cole equation. We propose a time-domain algorithm for wave propagation based on the Grünwald-Letnikov numerical derivative and the Fourier pseudospectral method to compute the spatial derivatives. As an example, we consider Savage theory and verify the algorithm by comparison with the analytical solution in homogeneous media based on the frequency-domain Green function. Moreover, we illustrate the modeling performance with wave propagation in a two half-space medium where one section is lossless and the other is a Cole-Cole medium. This apparently simple example, which does not have an analytical solution, shows the complexity of the wavefield that characterizes a single flat interface.

Diffusive nonreciprocity and thermal diode

Wave propagation and diffusion in linear materials preserve local reciprocity in terms of a symmetric Green's function. For wave propagations, the relation between the fields entering and leaving a system is more relevant than the detailed information about the fields inside it. In such cases, the global reciprocity of the scattering off a system through several ports is more important, which is defined as the symmetric transmission between the scattering channels. When a two-port system supports nonreciprocal (electromagnetic, acoustic) wave propagation, it is a (optical, phonon) diode directly following the definition. However, to date no concrete definition or discussion has been made on the global reciprocity of diffusive processes through a multiple-port system. It thus remains unclear what are the differences and relations between the three concepts, namely, local nonreciprocity, global nonreciprocity, and diode effect in diffusion. Here, we provide theoretical analysis on the frequency-domain Green's function and define the global reciprocity of heat diffusion through a two-port system, which has a different setup from that of a thermal diode. We further prove the equivalence between a heat transfer system with broken steady-state global reciprocity and a thermal diode, assuming no temperature-dependent heat generation. The validities of some typical mechanisms in breaking the diffusive reciprocity and making a thermal diode have been discussed. Our results set a general background for future studies on symmetric and asymmetric diffusive processes.

Representation theorem for viscoelastic waves with a non-symmetric stiffness matrix

In an elastic medium, it was proved that the stiffness tensor is symmetric with respect to the exchange of the first pair of indices and the second pair of indices, but the proof does not apply to a viscoelastic medium. In this paper, we thus derive the representation theorem for viscoelastic waves in a medium with a non-symmetric stiffness matrix. The representation theorem expresses the wave field at a receiver, situated inside a subset of the definition volume of the viscoelastodynamic equation, in terms of the volume integral over the subset and the surface integral over the boundary of the subset. For the given medium, we define the complementary medium corresponding to the transposed stiffness matrix. We define the frequency-domain complementary Green function as the frequency-domain Green function in the complementary medium. We then derive the provisional representation theorem as the relation between the frequency-domain wave field in the given medium and the frequency-domain complementary Green function. This provisional representation theorem yields the reciprocity relation between the frequency-domain Green function and the frequency-domain complementary Green function. The final version of the representation theorem is then obtained by inserting the reciprocity relation into the provisional representation theorem.

Numerical study of trimaran motion and wave load prediction based on time-domain Rankine-Green matching method

The theoretical prediction of trimaran motion and load in complex environment has always been a hot issue in the field of ship and ocean engineering. Based on the calculation principle of Rankine source and free-surface Green function for solving ship-wave interaction, a new kind of nonlinear three-dimensional time-domain Rankine-Green matching method for simulating the trimaran motion and wave load in head seas is studied. In the numerical simulation, the influence of factors such as wave slamming, green water, panel space and density, are investigated systematically. And the elevation of surrounding wave surface caused by speed effect and side hull interference is also considered to modify the instantaneous impact load. In order to verify the reliability of Rankine-Green matching method, a kind of segmented trimaran model with water jet propulsion system was designed and taken in regular and irregular waves. Finally, the numerical forecasting values based on Rankine-Green matching method are compared with the results of model test and traditional frequency-domain Green function method to illustrate its advantages and application scope.

Green's function of the Lord–Shulman thermo-poroelasticity theory

SUMMARY The Lord–Shulman thermoelasticity theory combined with Biot equations of poroelasticity, describes wave dissipation due to fluid and heat flow. This theory avoids an unphysical behaviour of the thermoelastic waves present in the classical theory based on a parabolic heat equation, that is infinite velocity. A plane-wave analysis predicts four propagation modes: the classical P and S waves and two slow waves, namely, the Biot and thermal modes. We obtain the frequency-domain Green's function in homogeneous media as the displacements-temperature solution of the thermo-poroelasticity equations. The numerical examples validate the presence of the wave modes predicted by the plane-wave analysis. The S wave is not affected by heat diffusion, whereas the P wave shows an anelastic behaviour, and the slow modes present a diffusive behaviour depending on the viscosity, frequency and thermoelasticity properties. In heterogeneous media, the P wave undergoes mesoscopic attenuation through energy conversion to the slow modes. The Green's function is useful to study the physics in thermoelastic media and test numerical algorithms.

Perturbations of an extremal Kerr spacetime: Analytic framework and late-time tails

We develop a complete and systematic analytical approach to field perturbations of extremal Kerr spacetime based on the formalism of Mano, Suzuki and Takasugi (MST) for the Teukolsky equation. Analytical expressions for the radial solutions and frequency-domain Green function in terms of infinite series of special functions are presented. As an application, we compute, for the first time, the leading late-time behavior due to the branch point at zero frequency of scalar, gravitational, and electromagnetic field perturbations on and off the event horizon. We also use the MST method to compute the leading behavior of the Green function modes near the branch point at the superradiant bound frequency and show that this behavior agrees with existing results in the literature using a different method.

A new ordinary differential equation for the evaluation of the frequency-domain Green function

Clément (2013) derived a second order ordinary differential equation (ODE) satisfied by the free-surface Green function in the frequency domain. Since then, similar ODEs for the gradient of the Green function have been developed. Unfortunately, all these ODEs degenerate at zero frequency. Therefore, it is not possible to initialize the numerical solution of these ODEs from this zero frequency. Alternative methods based on the shifting of the initial condition to frequencies strictly greater than zero have then been developed.The present paper describes an alternative approach to address this issue. It involves a new function which is the solution of a modified ODE which can be solved from the zero frequency.Finally, comparisons with evaluations of the Green function using the classical direct integration method are provided. They show that the new ODE can provide accurate estimates of the Green function.

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.

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.

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.

Modeling Extreme-Event Precursors with the Fractional Diffusion Equation

Abstract Extreme catastrophic events such as earthquakes, terrorism and economic collapses are difficult to predict. We propose a tentative mathematical model for the precursors of these events based on a memory formalism and apply it to earthquakes suggesting a physical interpretation. In this case, a precursor can be the anomalous increasing rate of events (aftershocks) following a moderate earthquake, contrary to Omori's law. This trend constitute foreshocks of the main event and can be modelled with fractional time derivatives. A fractional derivative of order 0 < v < 2 replaces the first-order time derivative in the classical diffusion equation. We obtain the frequency-domain Green's function and the corresponding time-domain solution by performing an inverse Fourier transform. Alternatively, we propose a numerical algorithm, where the time derivative is computed with the Grünwald-Letnikov expansion, which is a finitedifference generalization of the standard finite-difference operator to derivatives of fractional order. The results match the analytical solution obtained from the Green function. The calculation requires to store the whole field in the computer memory since anomalous diffusion “remembers the past”.

Motion Performance Analysis of Deep Water Pipe-Laying Vessel in Regular Wave

Based on 3-D potential flow theory, sources and sinks are brought in, distributed upon the wet surface of ship hull. Three-dimensional non-speed frequency domain Green function is also introduced to determine the flow field velocity potential. As a result, added mass, damping coefficient and ship motion response of this deep water pipe-laying vessel are calculated under the regular wave condition. After compared these calculation results to model tank test data, we can see that the theoretical calculation fit well with experimental data. These calculations could provide prediction and guidance for vessel design and marine engineering operation.

Theory and simulation of time-fractional fluid diffusion in porous media

We simulate a fluid flow in inhomogeneous anisotropic porous media using a time-fractional diffusion equation and the staggered Fourier pseudospectral method to compute the spatial derivatives. A fractional derivative of the order of 0 < ν < 2 replaces the first-order time derivative in the classical diffusion equation. It implies a time-dependent permeability tensor having a power-law time dependence, which describes memory effects and accounts for anomalous diffusion. We provide a complete analysis of the physics based on plane waves. The concepts of phase, group and energy velocities are analyzed to describe the location of the diffusion front, and the attenuation and quality factors are obtained to quantify the amplitude decay. We also obtain the frequency-domain Green function. The time derivative is computed with the Grünwald–Letnikov summation, which is a finite-difference generalization of the standard finite-difference operator to derivatives of fractional order. The results match the analytical solution obtained from the Green function. An example of the pressure field generated by a fluid injection in a heterogeneous sandstone illustrates the performance of the algorithm for different values of ν. The calculation requires storing the whole pressure field in the computer memory since anomalous diffusion ‘recalls the past’.

A 3D time-domain method for predicting the wave-induced forces and motions of a floating body

This study presents a 3D time-domain Green function method for predicting the wave-induced forces and motions of a floating body. The problem is considered in the cases of both infinite and finite water depths. Based on convolution theory, the convolution of the time-domain Green function and the velocity potential is replaced by the product of their Fourier transformations. A recursive formula for the convolution is proposed in terms of the frequency-domain Green function. This formula provides an efficient way to perform a long-time numerical simulation because the associated computational cost is constant over time. Furthermore, higher-order panels are used to give precise and continuous representations of the body geometry and the velocity potential over the body hull. Computed results for a hemisphere and a Wigley hull are compared with published results, and the comparisons indicate that the various results are in good agreement.

Evaluation of the power-law impulse response for liver in the time- and frequency-domains

Ultrasound attenuation in biological media often follows a frequency-dependent power-law relationship. Power-law attenuation is accompanied by dispersion in which phase velocity also changes as a function of frequency. The power-law wave equation has exact time- and frequency-domain Green's function solutions that are used in numerical evaluations of the Rayleigh–Sommerfeld diffraction integral. These Green's functions contain stable probability density functions that are evaluated in the time-domain using the STABLE toolbox and in the frequency-domain by evaluating the characteristic function of a stable distribution. The impulse response for a circular piston is evaluated on-axis in the time- and frequency-domain using values for human liver. The results show the accuracy of the time-domain impulse response calculation is dependent upon adequate spatial sampling of the piston face. The frequency-domain impulse response exhibits aliasing caused by wrap around of the heavy tail of the stable distribution. Frequency-domain accuracy is dependent upon the spatial sampling of the piston face and the density of the frequency samples. Numerical evaluations show that the frequency- and time-domain calculations converge to the same result. [This work was supported in part by NIH Grant R01 EB012079.]

A free surface frequency domain green function with viscous dissipation and partial reflections from side walls

The free-surface Green function method is widely used in solving the radiation or diffraction problems caused by a ship or ocean structure oscillating on the waves. In the context of inviscid potential flow, hydrodynamic problems such as multi-body interaction and tank side wall effect cannot be properly dealt with based on the traditional free-surface frequency domain Green function method, in which the water viscosity is omitted and the energy dissipation effect is absent. In this paper, an open-sea Green function with viscous dissipation was presented within the theory of visco-potential flow. Then the tank Green function with a partial reflection from the side walls in wave tanks was formulated as a formal sum of open-sea Green functions representing the infinite images between two parallel side walls of the source in the tank. The new far-field characteristics of the tank Green function is vitally important for improving the validity of side-wall effects evaluation, which can be used in supervising the tank model tests.

Verification and validation of ShipMo3D ship motion predictions in the time and frequency domains

This paper compares frequency domain and time domain predictions from the ShipMo3D ship motion library with observed motions from model tests and sea trials. ShipMo3D evaluates hull radiation and diffraction forces using the frequency domain Green function for zero forward speed, which is a suitable approach for ships travelling at moderate speed (e.g., Froude numbers up to 0.4). Numerical predictions give generally good agreement with experiments. Frequency domain and linear time domain predictions are almost identical. Evaluation of nonlinear buoyancy and incident wave forces using the instantaneous wetted hull surface gives no improvement in numerical predictions. Consistent prediction of roll motions remains a challenge for seakeeping codes due to the associated viscous effects.

Verification and validation of ShipMo3D ship motion predictions in the time and frequency domains

This paper compares frequency domain and time domain predictions from the ShipMo3D ship motion library with observed motions from model tests and sea trials. ShipMo3D evaluates hull radiation and diffraction forces using the frequency domain Green function for zero forward speed, which is a suitable approach for ships travelling at moderate speed (e.g., Froude numbers up to 0.4). Numerical predictions give generally good agreement with experiments. Frequency domain and linear time domain predictions are almost identical. Evaluation of nonlinear buoyancy and incident wave forces using the instantaneous wetted hull surface gives no improvement in numerical predictions. Consistent prediction of roll motions remains a challenge for seakeeping codes due to the associated viscous effects.

The Highly Efficient Calculation Method of 3-D Frequency-Domain Green Function

The traditional calculation method of frequency-domain Green function mainly utilizes series or asymptotic expansion to carry out numerical approximation. This method requires very careful divisions of the regions the variables of the integrand. Thus the computing process can be time consuming, which greatly affects the calculation efficiency. Gaussian quadrature is applied to the numerical calculation of the frequency-domain Green function and its partial derivatives. The variation trend of Green function and its partial derivatives along the coordinate plane is analyzed. Comparison of the calculation result with that in existing references is made, which shows that the present method has greatly simplified the computing process, reduced the level of divisions of the regions of variables, improved the computing efficiency and provided higher accuracy.

三维频域格林函数的高效率计算方法研究

The traditional calculation method of frequency-domain Green function mainly utilizes series or asymptotic expansion to carry out numerical approximation, however, this method requires very careful zoning, thus the computing process is complex with many cycles, which has greatly affected the computing efficiency. To improve the computing efficiency, this paper introduces Gaussian integral to the numerical calculation of the frequency-domain Green function and its partial derivatives. It then compares the calculation result with that in existing references. The comparison results demonstrate that, on the basis of its sufficient accuracy, the method has greatly simplified the computing process, reduced the zoning and improved the computing efficiency.