Three-dimensional Green Research Articles

Modeling and Simulation of Procoagulant Circulating Tumor Cells in Flow

We describe a mathematical/computational model for thrombin concentration gradients generated by procoagulant circulating tumor cells (CTCs) in flow. We examine how CTCs enhance blood coagulation as they diffuse tissue factor-dependent coagulation enzymes in a flow environment with vessel walls. Concentration fields of various enzymes, such as prothrombin and thrombin, diffuse, to, and from CTCs, respectively, as they propagate through the bloodstream. The diffusion-dependent generation of these enzymes sets up complex time-dependent concentration fields. The CTCs are modeled as diffusing point particles in an incompressible fluid, and we exploit exact analytical solutions based on three-dimensional Green’s functions for unbounded domains with one wall for high resolution numerical simulations. Time-dependent gradient trackers are used to highlight that concentration fields build-up (i) near boundaries (vessel walls), (ii) in regions surrounding the diffusing particles, and (iii) in complex time-dependent regions of the flow where fields associated with different particles overlap. Two flow conditions are modeled: no flow, and unidirectional constant flow. Our results indicate that the CTC-generated thrombin diffuses to and persists at the blood vessel wall, and that the spatial distribution of CTCs in flow determines local thrombin concentration. The magnitude of the diffusion gradient and local thrombin concentration is dependent upon bulk solution concentrations of coagulation factors within normal reported concentration ranges. Therefore, our model highlights the potential to determine patient-specific risks for CTC-induced hypercoagulability as a function of CTC number and individual patient concentration of coagulation factors.

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.

Determining partial differential cross sections for low-energy electron photodetachment involving conical intersections using the solution of a Lippmann-Schwinger equation constructed with standard electronic structure techniques

A method for obtaining partial differential cross sections for low energy electron photodetachment in which the electronic states of the residual molecule are strongly coupled by conical intersections is reported. The method is based on the iterative solution to a Lippmann-Schwinger equation, using a zeroth order Hamiltonian consisting of the bound nonadiabatically coupled residual molecule and a free electron. The solution to the Lippmann-Schwinger equation involves only standard electronic structure techniques and a standard three-dimensional free particle Green's function quadrature for which fast techniques exist. The transition dipole moment for electron photodetachment, is a sum of matrix elements each involving one nonorthogonal orbital obtained from the solution to the Lippmann-Schwinger equation. An expression for the electron photodetachment transition dipole matrix element in terms of Dyson orbitals, which does not make the usual orthogonality assumptions, is derived.

Three-dimensional Green's functions for two-dimensional quasi-crystal bimaterials

Owing to their specific structure, which can neither be classified as crystalline nor amorphous, quasi-crystals (QCs) exhibit properties that are interesting to both material science and mathematical physics or continuum mechanics. Within the framework of a mathematical theory of elasticity, one major focus is on features evolving from the coupling of phonon and phason fields, which is not observed in classical crystalline or amorphous materials. This paper deals with the problems of combinations of point phonon forces and point phason forces, which are applied to the interior of infinite solids and bimaterial solids of two-dimensional hexagonal QCs. By using the general solution of QCs, a series of displacement functions is adopted to obtain the analytical results when the two half-spaces are supposed to be ideally bonded or to be in smooth contact. In the final expressions, we provide three-dimensional Green’s functions for infinite bimaterial QC solids in the closed form, which are very convenient to be used in the study of dislocations, cracks and inhomogeneities of the new solid phase. Furthermore, the paper is concluded by a discussion of some special cases, in which Green’s functions for infinite transversely isotropic solids and Green’s functions for a half-space with free or fixed boundary are given.

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.

Paraxial Fields of a Wedge with Anisotropic Impedance and Perfect Electric Conductor Faces Excited by a Dipole

A three-dimensional dyadic Green's function is derived for electromagnetic scattering due to a point electric current source radiating in the vicinity of a wedge with anisotropic impedance and perfect electric conductor faces. The anisotropic impedance face is characterized by surface impedances along directions parallel and orthogonal to the edge axis (principal anisotropy axis). Arbitrary surface impedance is assumed in one direction only (either parallel or orthogonal to principal anisotropy axis), and vanishing impedance is assumed on the other. Assuming reactive surface impedance, the derived forms involve a summation over an angular wavenumber and a longitudinal spectral integral, which may be evaluated asymptotically to describe the fields in paraxial region, where the source and observation points are in close proximity to the apex but widely separated. The final asymptotic result reveals three different scattering mechanisms: edge-guided waves, surface waves, and guided waves in the classical sense. Numerical simulations are performed for capacitive and inductive surfaces using the asymptotic results. Simulation results are also compared to those of a commercially available electromagnetic field solver, which assumes uniform surface impedance.

An estimate for the three-dimensional discrete Green's function and applications

In this paper we first introduce definitions of the regularized Green's function and the discrete Green's function in three dimensions. With complicated arguments estimates for these two functions are then derived. Finally we give an application of the estimate for the discrete Green's function to the high accuracy analysis of the three-dimensional block finite element approximation.

820 界面特性を考慮した3次元2相異方性弾性体のGreen関数(OS3-4 材料特性,異方性材料の力学的挙動,オーガナイズドセッション:3 表面・薄膜・接合部の力学特性と信頼性評価)

820 界面特性を考慮した3次元2相異方性弾性体のGreen関数(OS3-4 材料特性,異方性材料の力学的挙動,オーガナイズドセッション:3 表面・薄膜・接合部の力学特性と信頼性評価)

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.

Green's functions for semi-infinite transversely isotropic electro-magneto-thermo-elastic material

We use the compact harmonic general solutions of transversely isotropic electro-magneto-thermo-elastic material to construct the three-dimensional Green's function for a steady point heat source acted in the interior of a semi-infinite transversely isotropic electro-magneto-thermo-elastic material by five newly introduced harmonic functions. All components of coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results are given graphically by contours.

Three-dimensional Green's Functions for Two-phase Transversely Isotropic Piezothermoelastic Media

Green's functions play an important role in electroelastic analyses of piezoelectric media. However, most works available on this topic are on the case of identical temperature. Based on the compact general solution of transversely isotropic piezothermoelectric media, which is expressed in harmonic functions, and employing the trial-and-error method, the three-dimensional Green's function for a steady point heat source in a two-phase piezothermoelectric media is presented by eight newly induced harmonic functions. Numerical results are given graphically by contours.

A boundary element method for the hydrodynamic analysis of floating bodies in variable bathymetry regions

In this work, a hybrid technique is presented for the hydrodynamic analysis of floating bodies in variable bathymetry regions. Our method is based on the coupled-mode theory for the propagation of water waves in general bottom topography, developed by Athanassoulis and Belibassakis [Athanassoulis GA, Belibassakis KA. A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions. J Fluid Mech 1999;389:275–301.] and extended to 3D by Belibassakis et al. [Belibassakis KA, Athanassoulis GA, Gerostathis TP. A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry. Appl Ocean Res 2001;23:319–336.], which is free of any mild-slope assumption, in conjunction with a boundary integral representation of the near field in the vicinity of the floating body. Both 2D and 3D problems have been considered. In all cases the near field is represented by boundary integral representation involving simple (Rankine) sources. In the 2D case, the far-field is modelled by complete (normal-mode) series expansions derived by separation of variables in the constant-depth half-strips. In the 3D case, the far-field is modelled by an integral representation involving the appropriate Green's function for harmonic water waves over a bottom with different depths at infinity, developed by Belibassakis and Athanassoulis [Belibassakis KA, Athanassoulis GA. Three-dimensional Green's function for harmonic water waves over a bottom with different depths at infinity. J Fluid Mech 2004;510:267–302.]. The numerical solution is obtained by means of a low-order panel method materialising the hybrid technique. Numerical results are presented concerning floating bodies of simple geometry, lying over sloping and undulating seabeds. With the aid of systematic comparisons with benchmark solutions the convergence and accuracy of the present method in 3D has been studied, and the effects of bottom slope and curvature on the hydrodynamic characteristics (hydrodynamic coefficients and responses) of the floating bodies are illustrated and discussed.

Reply to "Comments on 'Comparative Analysis of Acceleration Techniques for 2-D and 3-D Green's Functions in Periodic Structures Along One and Two Directions'"

The authors wish to thank Dr. Skobelev for his interesting remark about the acceleration technique for the three-dimensional (3-D) Green's function of a two-dimensional (2-D) array first proposed by Ivanishin. It basically consists of a Kummer's decomposition of higher order with respect to those presented. The possibility of obtaining similar decompositions is mentioned, where details are shown only about the lowest-order extraction, and it has also been recalled in our paper, in connection with the first presented geometry (2-D Green's function of a one-dimensional array).

Some New Three-Dimensional Green's Functions in Antisotropic Piezoelectric Bimaterials

In this paper, we derive three-dimensional Green’s functions of point-force/pointcharge in anisotropic and piezoelectric bimaterials for six different interface models. Mechanically, the six interface models are either in perfect or smooth contact along the interface; electronically, they can be closed, open interface, or with continuous electrical potential and normal electrical displacement component along the interface. By introducing certain modified bimaterial Stroh matrices, along with the extended Stroh formalism and the Mindlin’s superposition method, the bimaterial Green’s functions for the six interface conditions are expressed in terms of a concise and mathematically similar uniform form. That is, the physical-domain bimaterial Green’s functions can all be expressed as a sum of a homogeneous full-space Green’s function in an explicit form and a complementary part in terms of simple line-integrals over [0, π] suitable for standard numerical integration. Furthermore, utilizing a direct connection between the 2D and 3D Stroh matrices observed in this paper, the corresponding 2D bimaterial Green’s functions are also derived, in exact-closed form, for the six interface conditions. Based on the bimaterial Green’s functions, the effects of different interface conditions on the mechanical and electrical fields are discussed. It is noted that only the complementary part of the solution contributes to the differences of the mechanical and electrical fields arising from different interface conditions. Also, numerical examples are presented for the Green’s functions in the bimaterials made of two half-spaces with two typical piezoelectric materials, quartz and ceramic. Certain new features are observed which could be of great interest to the design of piezoelectric composites and to the numerical modeling of strained quantum devices using the boundary element method.

Wave arrival singularities at cuspidal points in the acoustic wave surfaces of anisotropic solids and their unfolding under weak spatial dispersion

This paper is concerned with wave arrival singularities in the elastodynamic Green's functions of infinite anisotropic elastic solids, and their unfolding into smooth wave trains, known as quasi-arrivals, through spatial dispersion. The wave arrivals treated here are those occurring in (i) the displacement response to a suddenly applied point force or three-dimensional Green's function,, and (ii) the displacement response to an impulsive line force or two-dimensional Green's function,. These arrivals take on various analytical forms, including step function and logarithmic and power-law divergences. They travel outwards from the source at the group velocities in each direction, and their locus defines the three- and two-dimensional acoustic wave surfaces, respectively. The main focus of this paper is on the form of the wave arrivals in the neighbourhood of cuspidal points in the wave surfaces, and how these arrivals unfold into quasi-arrivals under the first onset of spatial dispersion. This regime of weak spatial dispersion, where the acoustic wavelength,λ, begins to approach the natural length scale,l, of the medium, is characterized by a correction to the phase velocity, which is quadratic in the wavevector,k, and the presence of fourth-order spatial derivatives of the displacement field in the wave equation. Integral expressions are established for the quasi-arrivals near to cuspidal points, involving the Airy function in the case ofand the Scorer function in the case of. Numerical results are presented, illustrating the oscillatory nature of the quasi-arrivals and the interference effects that occur near to cuspidal points in the wave surface.

Correlation Between Maximum Temperature Increase and Peak SAR with Different Average Schemes and Masses

This paper investigates the correlation between maximum temperature increases and peak spatial-average specific absorption rates (SARs), calculated by different average schemes and masses. For evaluating the effect of mass on the correlation properly, a three-dimensional Green's function is presented. From our computational investigation, no best average mass for peak spatial-average SAR exist from the aspect of the correlation with maximum temperature increase. This is attributed to the frequency dependent penetration depth of EM waves. Maximum temperature increase in the head including the pinna is reasonably correlated with peak spatial-average SARs for most average schemes and masses considered in this paper. Maximum temperature increase in the head only (excluding the pinna) is reasonably correlated with peak 10-g SARs for the average schemes considered in this paper. The rationale for this result is explained using the Green's function. The point to be stressed here is that the slope correlating them is largely dependent on the average scheme and mass. Additionally, good agreement is observed in the slopes obtained by using two head models, which have been developed at Osaka University and Nagoya Institute of Technology. However, weak correlation is observed for the brain, which is caused by the difference of the positions where peak SAR and maximum temperature increase appear. The 95th percentile values of the slope correlating maximum temperature increases in the head or brain and peak spatial-average SAR are quantified for different average schemes and masses

COMPUTATIONAL METHODS FOR GUIDED ULTRASONIC WAVES IN PLATES

Reducing the general problem of computing three-dimensional Green's function in a transversely isotropic plate to a finite summation of contributions from a series of planar problems can efficiently yield an accurate solution. Hence, solving the planar scattering problem, of the Pressure-Shear-Vertical (PSV) type or the Shear-Horizontal (SH) type, was performed by three different techniques: The boundary element method; the hybrid method; and the perfectly matched layer method. In the pursuit of these methods, the objective was to highlight their pros and cons in terms of accuracy and efficiency.

Green's function for the Helmholtz equation in a layered half-space§

In this article we construct the three-dimensional Green's function for Helmholtz's equation in a layered half-space.

Pulse Shape Analysis of a Segmented Planar Germanium Detector by Using the Green's Function Method and Rise-Time Correlations for Depth Information

We performed pulse shape analysis to extract depth information for an interaction point of γ rays in a 25-fold segmented germanium detector. With a 90° Compton spectrometer with collimators on both incidence and exit sides, we measured pulse shapes for the two interaction points, ( x , y , z ) = (2.5 cm, 2.5 cm, 0.5 cm) and (2.5 cm, 2.5 cm, 1.0 cm). Experimental pulse shapes were compared with calculated ones by the three-dimensional Green's function method. Analytical solutions for induced charges on each cathode and an anode were obtained for a hole and an electron as a function of time. It is suggested that a rise-time correlation between the anode and the central cathode signals, T 90 (anode) - T 50 (cathode), can be used as a useful means to determine the depth ( z ) of an interaction point given an in-plane position ( x , y ). It is shown that calculated rise-time correlations are in qualitative agreement with experimental ones obtained by uncollimated γ rays.

Position determination of interaction points inside a planar-type segmented germanium detector by pulse shape analysis using a three-dimensional Green's function

Pulse shape analysis was performed to determine three-dimensional positions of interaction points inside a planar-type segmented germanium gamma-ray detector. The crystal has surface dimensions of 5 cm × 5 cm with a thickness of 2 cm and is segmented into a total of 25 segments. Pulse shapes of a hole and an electron, respectively induced on each segment (cathode) and an anode, were analytically obtained in 1-ns intervals by using the three-dimensional Green's function method. For the depth position determination, we used the two-dimensional rise-time correlation between the anode and cathode signals.