Three-dimensional Green Function Research Articles

Unified wavefront singularity characterization of three-dimensional elastodynamic time-domain half-space Green's function under impulsive boundary and internal loads

Founded on a novel analytical formulation that led to a rigorous yet compact path-integral representation of the time-domain elastodynamic half-space Green's function, a unified analysis of the possible occurrence of different singular wavefront behaviour in the response under arbitrary impulsive internal or surface point loads at arbitrary source-receiver locations is presented. With the decomposition of the general solution into distinct initiating and reflected wave group integrals that share a common factored format and simple contour definitions, the mathematical framework is shown to allow a straightforward identification of the specific conditions and the particular wave groups that are responsible for the singular wavefront phenomena without resorting to advanced analytic function theories or asymptotic methods. Analytic characterizations of the nature, strength and direction of all intrinsic singular wavefront behaviours of the three-dimensional Green's function in three canonical cases of source-receiver configurations are given in a dual integral-closed form format to facilitate their theoretical understanding as well as computational applications. Graphical illustrations of their variation with the source-receiver configuration and the medium's Poisson's ratio together with relevant comparison and clarifications of some classical treatments are included.

Varvec{J}-, M- and varvec{L}-integrals of line charges and line forces

In this work, the varvec{J}-, M- and varvec{L}-integrals of two line charges and two line forces in generalized plane strain are derived in the framework of three-dimensional (3D) electrostatics and three-dimensional compatible linear elasticity, respectively, in order to study the interaction between them. The key point in this derivation is achieved by expressing the varvec{J}-, M- and varvec{L}-integrals of point charges and point forces in three dimensions in terms of the corresponding three-dimensional Green functions, that is, the three-dimensional Green function of the Laplace operator and the three-dimensional Green tensor of the Navier operator, respectively. The major mathematical tool used in deriving the varvec{J}-, M- and L_3-integrals of line charges and line forces from the corresponding varvec{J}-, M- and varvec{L}-integrals of point charges and point forces is the method of embedding or method of descent of Green functions to two dimensions (2D) from the corresponding Green functions in 3D. The analytical expressions of varvec{J}-, M- and L_3-integrals of line charges and line forces in antiplane and plane strain are derived and discussed. The varvec{J}-integral is the electrostatic part of the Lorentz force (electrostatic interaction force) between two line charges in electrostatics and the Cherepanov force between two line forces in elasticity. The M-integral of two line sources (charges or forces) equals half the electrostatic interaction energy in electrostatics and half the elastic interaction energy in elasticity of these two line sources, respectively. The L_3-integral of two line sources (charges or forces) is the z-component of the configurational vector moment or the rotational moment representing the total torque about the z-axis caused by the interaction of the two line sources. An important outcome is that the varvec{J}-integral is twice the negative gradient of the M-integral, leading to the result that the {varvec{J}}-integral for line charges and line forces is a conservative force (with the corresponding interaction energy playing the role of the potential energy) and consequently an irrotational vector field. Finally, the obtained varvec{J}-, M- and L_3-integrals being functions of the distance and of the angle, are able to describe the physical behaviour of the interaction of two line sources (charges and forces), since they represent the fundamental and necessary quantities, that is, the interaction force, interaction energy and total torque produced by the interaction of these two line sources.

Efficient evaluation of three-dimensional Helmholtz Green's functions tailored to arbitrary rigid geometries for flow noise simulations

The Lighthill's wave equation provides an accurate characterization of the hydrodynamic noise due to the interaction between a turbulent flow and an obstacle, that is needed to get in many industrial applications. In the present study, to solve the Lighthill's equation expressed as a boundary integral equation, we develop an efficient numerical method to determine the three-dimensional Green's function of the Helmholtz equation in presence of an obstacle of arbitrary shape, satisfying a Neumann boundary condition. This so-called tailored Green's function is useful to reduce the computational costs to solve the Lighthill's equation. The first step consists in deriving an integral equation to express the tailored Green's function thanks to the free space Green's function. Then a Boundary Element Method (BEM) is used to compute tailored Green's functions. Furthermore, an efficient method is performed to compute the second derivatives needed for accurate flow noise determinations. The proposed approach is first tested on simple geometries for which analytical solutions can be determined (sphere, cylinder, half plane). In order to consider realistic geometries in a reasonable amount of time, fast BEMs are used: fast multipole accelerated BEM and hierarchical matrix based BEM. A discussion on the numerical efficiency and accuracy of these approaches in an industrial context is finally proposed.

Numerical Investigation on Motion Responses of a Floating Hemisphere Over a Sloping Bottom

Abstract In the last several decades, some numerical approaches have been proposed to deal with three-dimensional wave-body interaction problems in sloping bottom environment. Most of them either adopt the finite depth Green function or add numerical damping terms into the free surface condition to treat far field radiation condition, which certainly give rise to numerical errors. The hybrid model (Belibassakis, “A Boundary Element Method for the Hydrodynamic Analysis of Floating Bodies in Variable Bathymetry Regions,” Eng. Anal. Boundary Elements 32, pp. 796–810) adopting the consistent coupled-mode system for incident wave propagation problem combining with the three-dimensional bottom-dependent Green function to treat the diffraction and radiation problem is a complete formulation, as the latter function appropriately characterize the far field radiation wave pattern over a smoothly sloping bottom. However, this model has not been validated after its publication. In this connection, comparisons with computational fluid dynamic (CFD) results are presented to verify its accuracy. Application of this hybrid model is also performed to investigate the effects on the floating hemisphere by the sloping bottom.

Quasistasic modeling of edge fields in planar resonators

This paper discusses quasistatic waveguide model for planar resonator placed in multilayer dielectic media, which do not use effective parameters such as effective width of resonator and effective dielectric constant. It allows to use waveguide model in new approach. Edge field is modeled by inclusion of LC-lumped element circuit where values of inductances and capacitances we determined from the solving of electrostatic Laplace equation by the method of moment using the three-dimensional Green function for multilayer dielectric media. A computational experiment showed that the proposed model is effective in modeling planar resonators of complex shape and requires low computational resources.

Three-dimensional analysis of bolometers responsivity using Green functions

In the present work, a bolometer with meander line pattern has been investigated using a three-dimensional Green function model. By calculating the appropriate Green function, the temperature variation for such system under harmonic radiation and a constant bias current have been obtained numerically. Furthermore, the voltage response of a such bolometer to the radiation is calculated with the help of Ohm's law using numerical simulation. The results show that the responsivity is increased by decreasing the chopping frequency and substrate thickness that is in agreement with other experimental and theoretical works. Furthermore, results reveal that by ignoring the Joule heating effect, increasing the track width will enhance the responsivity which is comparable to our previous experimental work.

An accurate and efficient method for the piezoelectric coated functional devices based on the three-dimensional Green's functions under a tangential point force

An accurate and efficient method for the electro-mechanical coupling fields of piezoelectric coated structure is presented in this paper. First, the three-dimensional Green's function for a tangential point force on the surface of coated structure combined with orthotropic piezoelectric coating and orthotropic elastic substrate is presented in the form of elementary functions based on the general solution method. Then, the corresponding electro-mechanical coupling fields of this coated structure under arbitrary mechanical loads can be obtained by superposition principle and Gauss integration. At last, numerical simulations are given to show the usage of this method in the analyses of interface effect, interface debonding and tensile failure. And this method may suggest possible mechanisms by which layered structures can be designed for engineering applications.

Interaction of surface gravity wave motion with elastic bottom in three-dimensions

A general three-dimensional hydroelastic model is developed to study the effect of elastic bottom on surface gravity wave motion in three-dimensions under the action of uniform compressive force based on linearized theory of water wave in finite water depth. The elastic bottom bed is modeled as a thin plate theory. The progressive wave characteristics in different wave modes are analyzed in both the cases of deep and shallow water waves. Further, the linearized long wave equation under shallow water approximation in a direct manner is derived and compared the results obtained based on the small amplitude wave theory. Three-dimensional Green's function associated with surface gravity wave motion with elastic bed is derived using the fundamental source potentials. Fourier-type expansion formula and corresponding orthogonal mode-coupling relations are derived in finite depth and finite/semi-infinite width in three-dimensions. Utilizing the expansion formula, two classical problems (i) forced motion and (ii) wave reflection by a rigid wall in a channel of finite width and finite/semi-infinite length are illustrated which will play significant role in the analysis of wave-structure interaction problems arising in ocean engineering.The effects of elastic bottom on surface gravity waves are analyzed by presenting several numerical results on reflection wave amplitudes in different cases. Further, the behavior of free motions and oscillations in a basin of finite width and finite/semi-infinite length with elastic bed under the action of uniform compressive force are discussed.

The Green's function for the three-dimensional linear Boltzmann equation via Fourier transform

The linear Boltzmann equation with constant coefficients in the three-dimensional infinite space is revisited. It is known that the Green's function can be calculated via the Fourier transform in the case of isotropic scattering. In this paper, we show that the three-dimensional Green's function can be computed with the Fourier transform even in the case of arbitrary anisotropic scattering.

Cones of localized shear strain in incompressible elasticity with prestress: Green's function and integral representations.

The infinite-body three-dimensional Green's function set (for incremental displacement and mean stress) is derived for the incremental deformation of a uniformly strained incompressible, nonlinear elastic body. Particular cases of the developed formulation are the Mooney-Rivlin elasticity and the J2-deformation theory of plasticity. These Green's functions are used to develop a boundary integral equation framework, by introducing an ad hoc potential, which paves the way for a boundary element formulation of three-dimensional problems of incremental elasticity. Results are used to investigate the behaviour of a material deformed near the limit of ellipticity and to reveal patterns of shear failure. In fact, within the investigated three-dimensional framework, localized deformations emanating from a perturbation are shown to be organized in conical geometries rather than in planar bands, so that failure is predicted to develop through curved and thin surfaces of intense shearing, as can for instance be observed in the cup-cone rupture of ductile metal bars.

Green's Function for Transversely Isotropic Thermoelastic Diffusion Bimaterials

The aim of the present article is to study the Green's function in transversely isotropic thermoelastic diffusion bimaterial. With this objective, first the three-dimensional general solution in transversely isotropic thermoelastic diffusion bimaterial is derived. On the basis of general solution, Green's function, with a concentrated heat source in steady state, is completely solved using harmonic functions. The components of displacement, stress, temperature distribution, and mass concentration are expressed in terms of elementary functions. The resulting quantities are computed numerically and illustrated graphically. A particular case of three-dimensional Green function in transversely isotropic thermoelastic bimaterial has been deduced from the present investigation.

Nonlinear hull girder loads of a RoPax ship

Numerical and experimental studies of nonlinear wave loads are presented. A nonlinear time domain method has been developed and the theoretical background of the method are provided. The method is based on the source formulation expressed by means of the transient three-dimensional Green function. The time derivative of the velocity potential in Bernoulli's equation is solved with a similar source formulation to that of the perturbation velocity potential. The Wigley hull form is used to validate the calculation method in regular head waves. Model tests of a roll-on roll-off passenger ship with a flat bottom stern have been carried out. Model test results of ship motions, vertical shear forces and bending moments in regular and irregular head waves and calm water are given. The nonlinearities in ship motions and hull girder loads are investigated using the calculation method and the model test results. The nonlinearities in the hull girder loads have been found to be significant and the calculation method can predict the nonlinear loads for the model test ship.

Plasmonic Shaping in Gold Nanoparticle Three-Dimensional Assemblies

When a large number of similar gold particles are organized into complex architectures, the dipolar plasmon spectrum of the individual plasmonic entities gives rise to a broader, red-shifted feature centered around 750 nm. In this work, we show that superstructures fabricated using the convective assisted capillary force assembly method (CA-CFA) and excited at that wavelength display a subwavelength patterning of their optical field intensity that results from the self-consistent coupling between the colloidal nanoparticles. First, we demonstrate the fabrication of shape-controlled three-dimensional assemblies of metallic nanocrystals using the CA-CFA method. In a second step, the absorption band resulting from the mutual coupling between the metallic building blocks is exploited to excite a coupled plasmon mode and map the two-photon luminescence (TPL) by scanning a tightly focused light beam. Highly resolved TPL images show that the morphology of the plasmonic particle assemblies has a strong impact on their optical response. A model based on a rigorous optical Gaussian beam implementation inside a generalized propagator derived from a three-dimensional Green dyadic function accurately reproduces the TPL maps revealing the influence of interparticle separation and thus coupling between the individual particles. Finally, we show that the spatial distribution of the electric field intensity can be controlled by tuning the linear polarization of the optical excitation.

Unique and Explicit Formulas for Green's Function in Three-Dimensional Anisotropic Linear Elasticity

Unique, explicit, and exact expressions for the first- and second-order derivatives of the three-dimensional Green's function for general anisotropic materials are presented in this paper. The derived expressions are based on a mixed complex-variable method and are obtained from the solution proposed by Ting and Lee (Ting and Lee, 1997,“The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids,” Q. J. Mech. Appl. Math. 50, pp. 407–426) which has three valuable features. First, it is explicit in terms of Stroh's eigenvalues pα (α=1,2,3) on the oblique plane with normal coincident with the position vector; second, it remains well-defined when some Stroh's eigenvalues are equal (mathematical degeneracy) or nearly equal (quasi-mathematical degeneracy); and third, they are exact. Therefore, both new proposed solutions inherit these appealing features, being explicit in terms of Stroh's eigenvalues, simpler, unique, exact and valid independently of the kind of degeneracy involved, as opposed to previous approaches. A study of all possible degenerate cases validate the proposed scheme.

Three-dimensional Green's functions for transversely isotropic thermoporoelastic bimaterials

Abstract Green's functions for transversely isotropic thermoporoelastic bimaterials are established in the paper. We first express the compact general solutions of transversely isotropic thermoporoelastic material in terms of harmonic functions and introduce eight new harmonic functions. The three-dimensional Green's function having a concentrated liquid source or a concentrated heat source in steady state is completely solved using these new harmonic functions. The analytical results show some new phenomena of the pore fluid pressure increment, thermal increment and stress distributions at the interface. In the two materials, the pore fluid pressure has the same distribution because of the common fluid permeability, but the situation is different for the thermal increment. Shear failure is most likely at the two sources due to the highly degenerated direction of shear stress contours.

Scalar self-energy for a charged particle in global monopole spacetime with a spherical boundary

We analyze combined effects of the geometry produced by a global monopole and a concentric spherical boundary on the self-energy of a point-like scalar charged test particle at rest. We assume that the boundary is outside the monopole’s core with a general spherically symmetric inner structure. An important quantity to this analysis is the three-dimensional Green function associated with this system. For both Dirichlet and Neumann boundary conditions obeyed by the scalar field on the sphere, the Green function presents a structure that contains contributions due to the background geometry of the spacetime and the boundary. Consequently, the corresponding induced scalar self-energy also presents a similar structure. For points near the sphere, the boundary-induced part dominates and the self-force is repulsive/attractive with respect to the boundary for Dirichlet/Neumann boundary condition. In the region outside the sphere at large distances from it, the boundary-free part in the self-energy dominates and the corresponding self-force can be either attractive or repulsive with dependence of the curvature coupling parameter for scalar field. In particular, for the minimal coupling we show the presence of a stable equilibrium point for the Dirichlet boundary condition. In the region inside the sphere, the nature of the self-force depends on the specific model for the monopole’s core. As illustrations of the general procedure adopted, we shall consider two distinct models, namely the flower-pot and the ballpoint-pen ones.

Aeroacoustic Integrals Accelerated by Fast Multipole Method

The calculation of acoustic field solutions due to aeroacoustic sources is performed for a large number of observer locations. Sound is predicted by a hybrid method that uses direct calculation for near-field source computations and the Ffowcs Williams―Hawkings equation as the acoustic analogy formulation. The integrations of surface dipole and volume quadrupole source terms appearing in the Ffowcs Williams―Hawkings formulation are accelerated by a three-dimensional wideband multilevel adaptive fast multipole method. The three-dimensional wideband fast multipole method presented here applies a plane-wave expansion formulation with fast spherical filtering and interpolation and diagonal translation for calculations in the high-frequency regime and a partial-wave expansion formulation with rotational―coaxial translation in the low-frequency regime. The method is described for the solution of a three-dimensional Green's function that incorporates convective effects. The sound generated by the unsteady flow past a cylinder in the proximity of a NACA0012 airfoil is studied and the interaction of both geometries is analyzed for different cylinder positions. The developed numerical capability allows the analysis of each noise source individually. Therefore, the effects of dipoles and quadrupoles for each configuration are investigated. Results for acoustic field solutions obtained by the accelerated Ffowcs Williams―Hawkings formulation are up to 2 orders of magnitude faster than with the conventional computation of Ffowcs Williams―Hawkings equation.

A lifting line theory for a three-dimensional hydrofoil

Prandtl’s lifting line theory was generalized to the lifting problem of a three-dimensional hydrofoil in the presence of a free surface. Similar to the classical lifting theory, the singularity distribution method was utilized to solve two-dimensional lifting problems for the hydrofoil beneath the free surface at the air-water interface, and a lifting line theory was developed to correct three-dimensional effects of the hydrofoil with a large aspect ratio. Differing from the classical lifting theory, the main focus was on finding the three-dimensional Green function of the free surface induced by the steady motion of a system of horseshoe vortices under the free surface. Finally, numerical examples were given to show the relationship between the lift coefficient and submergence Froude numbers for 2-D and 3-D hydrofoils. If the submergence Froude number is small free surface effect will be significant registered as the increase of lift coefficient. The validity of these approaches was examined in comparison with the results calculated by other methods.

Induced self-energy on a static scalar charged particle in the spacetime of a global monopole with finite core

We analyze the induced self-energy and self-force on a scalar point-like charged test particle placed at rest in the spacetime of a global monopole admitting a general spherically symmetric inner structure to it. In order to develop this analysis we calculate the three-dimensional Green's function associated with this physical system. We explicitly show that for points outside the monopole's core the scalar self-energy presents two distinct contributions. The first one is induced by the non-trivial topology of the global monopole considered as a point-like defect and the second is a correction induced by the non-vanishing inner structure attributed to it. For points inside the monopole, the self-energy also present a similar structure, where now the first contribution depends on the geometry of the spacetime inside. As illustrations of the general procedure adopted, two specific models, namely flower-pot and the ballpoint-pen, are considered for the region inside. For these two different situations, we were able to obtain exact expressions for the self-energies and self-forces in the regions outside and inside the global monopole.

Three-dimensional Green's function and its derivative for materials with general anisotropic magneto-electro-elastic coupling

Explicit expressions of Green’s function and its derivative for three-dimensional infinite solids are presented in this paper. The medium is allowed to exhibit a fully magneto-electro-elastic (MEE) coupling and general anisotropic behaviour. In particular, new explicit expressions for the first-order derivative of Green’s function are proposed. The derivation combines extended Stroh formalism, Radon transform and Cauchy’s residue theory. In order to cover mathematical degenerate and non-degenerate materials in the Stroh formalism context, a multiple residue scheme is performed. Expressions are explicit in terms of Stroh’s eigenvalues, this being a feature of special interest in numerical applications such as boundary element methods. As a particular case, simplifications for MEE materials with transversely isotropic symmetry are derived. Details on the implementation and numerical stability of the proposed solutions for degenerate cases are studied.