Vector Green's Functions Research Articles

Alternative flow models, vector Green functions and boundary integral flow representations in ship and offshore hydrodynamics

Wave diffraction-radiation by large bodies such as offshore structures and ships advancing in regular waves or in calm water is widely analyzed via panel methods based on a boundary integral flow representation and a Green function that satisfies the relevant free-surface boundary condition. The boundary integral flow representations solved in existing panel methods express the flow potential in terms of distributions of sources and dipoles, related to a Green function and its gradient, over the body surface. Other notable features of existing boundary integral flow representations are that they can be ill-posed for a set of irregular frequencies, and that – for a ship advancing in calm water or in regular waves – they involve a troublesome line integral around the ship waterline. Several alternative boundary integral flow representations that are well suited for wave diffraction-radiation by offshore structures or by ships advancing in regular waves or in calm water are given in this study. These representations, based on an alternative linear flow model (given previously) and an alternative vector Green function (not previously considered), are weakly singular and do not involve a waterline integral. The influence coefficients related to these new flow representations consist of Rankine components associated with complementary types of weakly singular Rankine singularities, and Fourier components defined by a Fourier superposition of elementary waves. Both the Rankine and Fourier components of the influence coefficients can be evaluated simply and efficiently. In particular, the Fourier component can be evaluated via the Fourier–Kochin (FK) method, in which the dominant computational task is independent of the number of panels that approximate the body surface. Thus, the new boundary integral flow representations given in this study and the FK method expounded previously provide a practical new basis for the evaluation of three main classes of wave diffraction-radiation by ships and offshore structures via panel methods.

Real-Time 3D Microwave Medical Imaging With Enhanced Variational Born Iterative Method.

In this paper, we present a new variational Born iterative method (VBIM) for real-time microwave imaging (MWI) applications. The S-parameter volume integral equation and waveport vector Green's function are implemented to utilize the measured signal of the MWI system. Meanwhile, the real and imaginary separation (RIS) approach is used at each iterative step to simultaneously reconstruct the dielectric permittivity and conductivity of unknown objects. Compared with the Born iterative method and distorted Born iterative method, VBIM requires less computational time to reach the convergence threshold. The graphics processing unit based acceleration technique is implemented for real-time imaging. To demonstrate the efficiency and accuracy of this VBIM-RIS method, synthetic analysis of a complex multi-layer spherical phantom is first conducted. Then, the algorithm is tested with measured data using our new MWI system prototype. Finally, a synthetic brain-tumor phantom model under a thermal therapy procedure is monitored to exemplify the real-time imaging with about 5 seconds per reconstruction frame.

Boundary-integral representation sans waterline integral for flows around ships steadily advancing in calm water

The boundary-integral flow-representation associated with the boundary-value problem, commonly called Neumann–Kelvin (NK) problem, that corresponds to linear potential flow around a ship steadily advancing in calm water involves an integral around the mean waterline of the ship. This ‘waterline integral’ is a notorious source of numerical difficulties and has been extensively studied. The waterline integral in the NK theory is largely – but not fully – eliminated in the modification, called Neumann–Michell (NM) theory, of the NK theory. Specifically, the NM theory includes a residual waterline-distribution of weak Rankine singularities, ignored in practical applications. A crucial element of the NM theory is a mathematical transformation that is based on a vector Green function, which is associated with the common scalar Green function used in the NK theory. This transformation is revisited in the present study. A rigorous analysis yields an answer to a fifty-year old puzzle: an exact boundary-integral flow representation that does not include a waterline integral. A remarkable feature of this new flow representation, which is a modification of the NM flow representation given previously, is that it explicitly determines the flow potential and the flow velocity at a ship hull surface in terms of the flow velocity at the hull surface, rather than in terms of the hull-surface potential as in usual boundary-integral flow representations obtained via Green’s classical identity.

Leveraging Prior Known Vector Green Functions in Solving Perturbed Dirac Equation in Clifford Algebra

Solving boundary value problems with boundary element methods requires specific Green functions suited to the boundary conditions of the problem. Using vector algebra, one often needs to use a Green function for the Helmholtz equation whereas it is a solution of the perturbed Dirac equation that is required for solving electromagnetic problems using Clifford algebra. A wealth of different Green functions of the Helmholtz equation are already documented in the literature. However, perturbed Dirac equation is only solved for the generic case and only its fundamental solution is reported. In this paper, we present a simple framework to use known Green functions of Helmholtz equation to construct the corresponding Green functions of perturbed Dirac equation which are essential in finding the appropriate kernels for integral equations of electromagnetic problems. The procedure is further demonstrated in a few examples.

Self-force and the Huygens principle

We consider a relation between the Huygens Principle (HP) in gravity and the self-interaction force. We show that the self-force for an electric particle in the plane gravitational wave space-time has no tail term even the vector Green function does not obey the HP. The reason for this observation is that even vector potential does not obey the HP, the electromagnetic field does obey.

Statistical sources for broadband shock-associated noise using the Navier-Stokes equations.

A decomposition of the Navier-Stokes equations is used to identify the equivalent source term for broadband shock-associated noise (BBSAN). An analytical closed-form model to predict BBSAN is developed using an acoustic analogy based on the Navier-Stokes equations. The field-variables are decomposed into the base flow, aerodynamic fluctuations, and acoustic fluctuations. The spectral densities of fluctuating acoustic quantities are obtained by convolving the vector Green's function with the source terms involving the two-point cross-correlation of the aerodynamic quantities. The scaling of the source term with the off-design parameter β=(|Mj 2-Md 2|)1/2 is compared with experimental results. The base flow is obtained using a Reynolds-averaged Navier-Stokes solution, while the fluctuating statistical quantities are obtained using theoretical and experimental results. This paper identifies the equivalent source of BBSAN based on the scaling analysis and the physical mechanism of shock-associated noise. The identified source term resides within the Navier-Stokes equations without further rearrangement and correlates very highly with BBSAN. Predictions for BBSAN are made at multiple observer angles and nozzle pressure ratios using the identified source term, and these predictions compare favorably with the experimental results. Finally, identification of the source locations in the jet exhaust responsible for BBSAN at different Strouhal numbers is performed.

Modelling and prediction of the peak-radiated sound in subsonic axisymmetric air jets using acoustic analogy-based asymptotic analysis.

This paper uses asymptotic analysis within the generalized acoustic analogy formulation (Goldstein 2003 JFM 488, 315-333. (doi:10.1017/S0022112003004890)) to develop a noise prediction model for the peak sound of axisymmetric round jets at subsonic acoustic Mach numbers (Ma). The analogy shows that the exact formula for the acoustic pressure is given by a convolution product of a propagator tensor (determined by the vector Green's function of the adjoint linearized Euler equations for a given jet mean flow) and a generalized source term representing the jet turbulence field. Using a low-frequency/small spread rate asymptotic expansion of the propagator, mean flow non-parallelism enters the lowest order Green's function solution via the streamwise component of the mean flow advection vector in a hyperbolic partial differential equation. We then address the predictive capability of the solution to this partial differential equation when used in the analogy through first-of-its-kind numerical calculations when an experimentally verified model of the turbulence source structure is used together with Reynolds-averaged Navier-Stokes solutions for the jet mean flow. Our noise predictions show a reasonable level of accuracy in the peak noise direction at Ma = 0.9, for Strouhal numbers up to about 0.6, and at Ma = 0.5 using modified source coefficients. Possible reasons for this are discussed. Moreover, the prediction range can be extended beyond unity Strouhal number by using an approximate composite asymptotic formula for the vector Green's function that reduces to the locally parallel flow limit at high frequencies. This article is part of the theme issue 'Frontiers of aeroacoustics research: theory, computation and experiment'.

Green’s functions and Hadamard parametrices for vector and tensor fields in general linear covariant gauges

We determine the retarded and advanced Green's functions and Hadamard parametrices in curved spacetimes for linearized massive and massless gauge bosons and linearized Einstein gravity with a cosmological constant in general linear covariant gauges. These vector and tensor parametrices contain additional singular terms compared with their Feynman/de Donder-gauge counterpart. We also give explicit recursion relations for the Hadamard coefficients, and indicate their generalization to $n$ dimensions. Furthermore, we express the divergence and trace of the vector and tensor Green's functions in terms of derivatives of scalar and vector Green's functions, and show how these relations appear as Ward identities in the free quantum theory.

SOLVING SYSTEM OF HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS ON THE LEVEL OF OPERATORS

In this paper, we present a method for solving the system of higher-order linear differential equations (HLDEs) with inhomogeneous initial conditions on the level of operators. Using proposed method, we compute the matrix Green's operator as well as the vector Green's function of a fully-inhomogeneous initial value problems (IVPs). Examples are discussed to demonstrate the proposed method.

Qualitative Microwave Imaging With Scattering Parameters Measurements

Microwave imaging (MWI) systems extensively employ vector network analyzers for microwave measurements due to their high availability and accuracy. This is in contrast to theoretical models, which are naturally formulated in terms of scattered electric field vectors. Accordingly, experimental verification of MWI methods requires an intermediate step where measured scattering parameters are converted to scattered electric fields. In parallel to recent research, which formulates the Born iterative method in terms of scattering parameters, we develop formulations of two closely related qualitative inverse scattering methods-the linear sampling method and the factorization method-directly in terms of scattering parameters to avoid the intermediate conversion step. To this aim, we introduce vector S-parameters and we extend the vector Green's function for S-parameters to the dyadic case. There are certain advantages of these formulations over their electric field counterparts. First of all, the resulting formulations inherently incorporate the antenna radiation characteristics. Moreover, they reduce the measurement time since they do not require any pre- or post-measurement process. We experimentally verified the presented novel formulations against multi-frequency measurements performed inside an anechoic chamber. Obtained results show that the proposed methodologies can accurately reconstruct the shape of the targets by directly exploiting multifrequency measurements in the imaging process.

The Scaling of Broadband Shock-Associated Noise with Increasing Temperature

A physical explanation for the saturation of broadband shock-associated noise (BBSAN) intensity with increasing jet stagnation temperature has eluded investigators. An explanation is proposed for this phenomenon with the use of an acoustic analogy. To isolate the relevant physics, the scaling of BBSAN peak intensity level at the sideline observer location is examined. The equivalent source within the framework of an acoustic analogy for BBSAN is based on local field quantities at shock wave – shear layer interactions. Propagation of sound through the jet shear layer is predicted with an adjoint vector Green's function solver of the linearized Euler equations. The combination of the equivalent source and adjoint vector Green's function allows for correct predictions of the saturation of BBSAN with increasing stagnation pressure and stagnation temperature. The sources and vector Green's function have arguments involving the steady Reynolds-Averaged Navier-Stokes solution of the jet. It is proposed that saturation of BBSAN with increasing jet temperature occurs due to a balance between the amplification of the sound propagation through the shear layer and the source term scaling.

The prediction of jet noise ground effects using an acoustic analogy and a tailored Green's function

Abstract An assessment of an acoustic analogy for the mixing noise component of jet noise in the presence of an infinite surface is presented. The reflection of jet noise by the ground changes the distribution of acoustic energy and is characterized by constructive and destructive interference patterns. The equivalent sources are modeled based on the two-point cross-correlation of the turbulent velocity fluctuations and a steady Reynolds-Averaged Navier–Stokes (RANS) solution. Propagation effects, due to reflection by the surface and refraction by the jet shear layer, are taken into account by calculating the vector Green's function of the linearized Euler equations (LEE). The vector Green's function of the LEE is written in relation to that of Lilley's equation; that is, it is approximated with matched asymptotic solutions and Green's function of the convective Helmholtz equation. The Green's function of the convective Helmholtz equation in the presence of an infinite flat plane with impedance is the Weyl–van der Pol equation. Predictions are compared with measurements from an unheated Mach 0.95 jet. Microphones are placed at various heights and distances from the nozzle exit in the peak jet noise direction above an acoustically hard and an asphalt surface. The predictions are shown to accurately capture jet noise ground effects that are characterized by constructive and destructive interference patterns in the mid- and far-field and capture overall trends in the near-field.

Reduction of Volume Integrals to Nonsingular Surface Integrals for Matrix Elements of Tensor and Vector Green Functions of Maxwell Equations

A procedure for the evaluation of matrix elements of tensor and vector Green functions with linear basis functions appearing in volumetric integral equations in electromagnetics is presented. The procedure reduces six-dimensional volumetric integrals with singular integrands to four-dimensional surface integrals with nonsingular integrands-hence, it offers simplicity and accuracy achieved at low computational cost.

Solution of the one-dimensional transport equation by the vector Green function method: Error bounds and simulation

In this work we solve the general anisotropic transport equation for an arbitrary source with semi-reflexive boundary conditions. First we present a complete existence theory for this problem in the space of continuous functions and in the space of α-Hölder continuous functions. As a result of our analysis we construct integral operators which we discretize in a finite dimensional functional space, yielding a new robust numerical method for the transport equation, which we call Green’s function decomposition method (GFD). As well, we demonstrate a convergence theorem providing error bounds for the reported method. Finally we provide numerical results and applications.

Space–time fractional Cauchy problem in spaces of generalized functions

We prove the theorem on existence and uniqueness of solution to the Cauchy problem $$ \begin{array}{*{20}{c}} {u_t^{{\left( \beta \right)}}+{a^2}{{{\left( {-\varDelta } \right)}}^{{\alpha /2}}}u=F\left( {x,t} \right),\quad \left( {x,t} \right)\in {{\mathbb{R}}^n}\times (0,T],\quad a=\mathrm{const}\hbox{,}} \\ {u\left( {x,0} \right)={u_0}(x),\quad x\in {R^n},} \\ \end{array} $$ where $ u_t^{{(\beta )}} $ is the Riemann–Liouville fractional derivative of order β ∈ (0, 1) and u 0 and F belong to spaces of generalized functions. A representation of this solution is obtained by using the vector Green function. We also establish the character of singularities of the solution for t = 0 depending on the order of singularity of a given generalized function in the initial condition and the character of power singularities of the function on the right-hand side of the equation. In this case, the fractional n-dimensional Laplace operator is defined by using the Fourier transformation $ \mathfrak{F}\left[ {{{{\left( {-\varDelta } \right)}}^{{{\alpha \left/ {2} \right.}}}}\psi (x)} \right]={{\left| \lambda \right|}^{\alpha }}\mathfrak{F}\left[ {\psi (x)} \right] $ .

The Prediction of Broadband Shock-Associated Noise Including Propagation Effects

An acoustic analogy is developed based on the Euler equations for broadband shock-associated noise (BBSAN) that directly incorporates the vector Green's function of the linearized Euler equations and a steady Reynolds-Averaged Navier-Stokes solution (SRANS) to describe the mean flow. The vector Green's function allows the BBSAN propagation through the jet shear layer to be determined. The large-scale coherent turbulence is modeled by two-point second order velocity cross-correlations. Turbulent length and time scales are related to the turbulent kinetic energy and dissipation rate. An adjoint vector Green's function solver is implemented to determine the vector Green's function based on a locally parallel mean flow at different streamwise locations. The newly developed acoustic analogy can be simplified to one that uses the Green's function associated with the Helmholtz equation, which is consistent with a previous formulation by the authors. A large number of predictions are generated using three different nozzles over a wide range of fully-expanded jet Mach numbers and jet stagnation temperatures. These predictions are compared with experimental data from multiple jet noise experimental facilities. In addition, two models for the so-called ‘fine-scale’ mixing noise are included in the comparisons. Improved BBSAN predictions are obtained relative to other models that do not include propagation effects.

Vector Green's function for S-parameter measurements of the electromagnetic volume integral equation

We present a variation of the electric-field volume integral equation (VIE) for inhomogeneous media that is tailored for microwave S-parameter measurements. The kernel of the standard VIE is a dyadic Green's function that can relate fields in the object domain to observed scattered fields outside the object domain. Here, we derive a kernel we call a vector Green's function that gives, instead of observed fields, observed S-parameter measurements that would be obtained in a bistatic antenna setup. This formulation directly links the object properties to microwave measurements in a way that is useful to many microwave applications. This derivation uses an antenna model based on the source-scattering matrix formulation. We show how this vector Green's function can be obtained from simulation and validate this model with several experiments.

Vector Green's function algorithm for radiative transfer in plane-parallel atmosphere

Green's function is a widely used approach for boundary value problems. In problems related to radiative transfer, Green's function has been found to be useful in land, ocean and atmosphere remote sensing. It is also a key element in higher order perturbation theory. This paper presents an explicit expression of the Green's function, in terms of the source and radiation field variables, for a plane-parallel atmosphere with either vacuum boundaries or a reflecting (BRDF) surface. Full polarization state is considered but the algorithm has been developed in such way that it can be easily reduced to solve scalar radiative transfer problems, which makes it possible to implement a single set of code for computing both the scalar and the vector Green's function.

Classification of modes in suspended-membrane, 19-missing-hole photonic-crystal microcavities

Suspended-membrane 19-missing-hole microcavities in triangular lattice photonic crystals are numerically modeled by a three-dimensional finite-difference time-domain method. The resonance frequencies and the quality factors are calculated by interpolation of the discrete Fourier transformation series of the field with a Padé polynomial. The numerical results are compared with the photoluminescent spectra measured on the cavity of a nearly identical dimension. The symmetry properties of the defect modes are analyzed with the group theory, and resonance modes in the photonic-crystal cavities are identified as irreducible representations of the C_6v point group. The far-field radiations of the identified modes in the free space are also calculated by use of a vector Green's function. It is found that the numerical results agree very well with the experimental measurement in various aspects.

The role of instability waves in predicting jet noise

There is a continuing debate about the role of linear instability waves in the prediction of jet noise. Parallel mean flow models, such as the one proposed by Lilley, usually neglect these waves because they cause the solution to become infinite. The result is then non-causal and can, therefore, be quite different from the more realistic causal solution, especially for the chaotic flows being considered here. The present paper solves the relevant acoustic equations for a non-parallel mean flow by using a vector Green's function approach and requiring that the mean flow be weakly non-parallel, i.e. requiring that the spread rate be small. It demonstrates that linear instability waves must be accounted for in order to construct a proper causal solution to the jet noise problem. Recent experimental results (e.g. see Tam, Golebiowski & Seiner 1996) show that the small-angle spectra radiated by supersonic jets are significantly different from those radiated at larger angles (say, at 90°) and even exhibit dissimilar frequency scalings (i.e. they scale with Helmholtz number as opposed to Strouhal number). The present solution is (among other things) able to explain this rather puzzling experimental result.