On Green's functions for the reduced wave equation in a circular annular domain with dirichlet, Neumann and radiation type boundary conditions

Summary In this paper Green's functions for the reduced wave equation (Helmholtz equation) in a circular annular domain with the Dirichlet, the radiation, and Neumann boundary conditions are derived. The convergence of the series representing Green's functions is then established. Finally it is shown that these functions reduce to Green's function for the exterior of a circle as given by Franz and Etiènne when the outer radius is moved towards infinity.     

Nonlinear two- and three-dimensional free surface flows due to moving disturbances

Boundary integral equation methods for computing two- and three-dimensional nonlinear free surface flows are presented. In two dimensions, integral formulations can be derived by using complex variables or Green's functions. Both formulations are shown to yield the same level of accuracy. The formulation based on Green's functions is extended to three dimensions by following Forbes [J. Comput. Phys. 82 (1989) 330–347] and accurate numerical results are presented for moving distributions of pressure and moving submerged disturbances.     

Green''s functions for transversely isotropic piezoelectric solids

Closed-form expressions are obtained for the infinite-body Green's functions for a transversely isotropic piezoelectric medium. The four Green's functions represent the coupled elastic and electric response to an applied point force or point charge. The Green's functions are obtained using a formulation where the three displacements and the electric potential are derivable from two potential functions. When piezoelectric coupling is absent, the results reduce to those for uncoupled elasticity and electrostatics.     

A Green's function approach to the deflection of a FGM plate under transient thermal loading

Summary  Green's function approach is adopted for analyzing the deflection and the transient temperature distribution of a plate made of functionally graded materials (FGMs). The governing equations for the deflection and the transient temperature are formulated into eigenvalue problems by using the eigenfunction expansion theory. Green's functions for solving the deflection and the transient temperature are obtained by using the Galerkin method and the laminate theory, respectively. The eigenfunctions of Green's function for the deflection are approximated in terms of a series of admissible functions that satisfy the homogeneous boundary conditions of the plate. The eigenfunctions of Green's function for the temperature are determined from the continuity conditions of the temperature and the heat flux at interfaces. Received 9 October 2000; accepted for publication 3 April 2001     

Axisymmetric boundary-value problems in a piezoelectric medium

Based on the general solution of three-dimensional problems in piezoelectric medium, with the method of Green's functins^[2], axisymmetric boundary-value problems are discussed. The purpose of this research is for analyzing the effective on mechanics and electricity of the piezoelectric ceramics caused by voids and inclusions. The displacement, traction and electric Green's functions corresponding to circular ring loads acting in the interior of a piezoelectric ceramic are obtained. A cylindrical coordinate system is employed and Hankel transform are applied with respect to radial coordinates. Explicit solutions for Green's functions are presented in terms of infinite integrals of Lipshitz-Hankel type. By solving a traction boundary-value problem, the solution scheme is illustrated. Supported by the National Natural Science Foundation of China and the Foundation of the Open Laboratory of Solid Mechanics.     

A Radiation Condition for Layered Elastic Media

This study aims to establish a generalized radiation condition for time-harmonic elastodynamic states in a piecewise-homogeneous, semi-infinite solid wherein the “bottom” homogeneous half-space is overlain by an arbitrary number of bonded parallel layers. To consistently deal with both body and interfacial (e.g. Rayleigh, Love and Stoneley) waves comprising the far-field patterns, the radiation condition is formulated in terms of an integral over a sufficiently large hemisphere involving elastodynamic Green's functions for the featured layered medium. On explicitly proving the reciprocity identity for the latter set of point-load solutions, it is first shown that the layered Green's functions themselves satisfy the generalized radiation condition. By virtue of this result it is further demonstrated that the entire class of layered elastodynamic solutions, admitting a representation in terms of the single-layer, double-layer, and volume potentials (distributed over finite domains), satisfy the generalized radiation condition as well. For a rigorous treatment of the problem, fundamental results such as the uniqueness theorem for radiating elastodynamic states, Graffi's reciprocity theorem for piecewise-homogeneous domains, and the integral representation theorem for semi-infinite layered media are also established.     

Generalized Born scattering in anisotropic media

The Born scattering approximation has been widely used in seismology to study scattered waves, and to linearize the propagation problem for inversion. The standard Born theory requires the model be separated into a smooth, reference model and a perturbation. Scattering occurs from the pertubation. In the distorted Born approximation, when the reference model is inhomogeneous, the reference Green's functions are normally not known exactly, but the error in these Green's functions is rarely quantified. In this paper, we generalize Born scattering theory to include the errors in the Green's functions explicitly, and obtain scattering integrals from these errors. For forward modelling, there is no need to separate the model into a reference and perturbation part - approximate Green's functions in the true model can be used to calculate the scattered signals.

The theory is developed for inhomogeneous, anisotropic media. Asymptotic ray theory results are suitable approximate Green's functions for the generalized Born scattering theory. The error terms are simple, easily calculated and included in the scattering integrals. Various applications of generalized Born scattering theory have already appeared in the literature, e.g. quasi-shear ray coupling, and this paper is restricted to an improved and more complete theoretical development. Further applications will appear elsewhere.     

BOUNDARY VALUE PROBLEMS OF TWO-DIMENSIONAL ANISOTROPIC BODY WITH A PARABOLIC BOUNDARY

Based upon Stroh formalism we derive a novel and convenient scheme for determiningthe elastic fields of a two-dimensional anisotropic body with a parabolic boundary subject to two kindsof boundary conditions, which are free surface and rigid surface, respectively. The correspondingGreen's functions are found by using the conformal mapping method. When the parabolic curve de-generates into a half-infinite crack or rigid inclusion, the singular stress fields near the tip of defectsare obtained. In particular, those Green's functions for a concentrated moment M_0 applied at a pointon the parabolic curve are also studied. It is easily found that arbitrary parabolic boundary value prob-lems can be solved by using these Green's functions and associate integrals.     

On Green's function for the reduced wave equation in a spherical annular domain with Dirichlet's boundary conditions

Summary In this study Green's function for the reduced wave equation (Helmholtz equation) in a spherical annular domain with Dirichlet's boundary conditions is derived. The convergence of the series solution representing Green's function is then established. Finally it is shown that Green's function for the Dirichlet problem reduces to Green's function for the exterior of a sphere as given by Franz and Etiènne, when the outer radius is moved towards infinity, and when a special position of the coordinate system is chosen.     

Dyadic Green's functions for cylindrical waveguides with moving media

The dyadic Green's function for cylindrical waveguides of circular or rectangular cross section with a moving, isotropic, homogeneous medium is developed using the method of eigenfunction expansion. The orthogonality properties of the vector mode functions are discussed. In contrast to waveguides with a stationary medium, it is seen that the normalization factor in the case of the E mode introduces a pole in the integral representation for the Green's function which must be excluded from the integration contour.     

Two-dimensional Green's functions in anisotropic multiferroic bimaterials with a viscous interface

We derive, by virtue of the unified Stroh formalism, the extremely concise and elegant solutions for two-dimensional and (quasi-static) time-dependent Green's functions in anisotropic magnetoelectroelastic multiferroic bimaterials with a viscous interface subjected to an extended line force and an extended line dislocation located in the upper half-plane. It is found for the first time that, in the multiferroic bimaterial Green's functions, there are 25 static image singularities and 50 moving image singularities in the form of the extended line force and extended line dislocation in the upper or lower half-plane. It is further observed that, as time evolves, the moving image singularities, which originate from the locations of the static image singularities, will move further away from the viscous interface with explicit time-dependent locations. Moreover, explicit expression of the time-dependent image force on the extended line dislocation due to its interaction with the viscous interface is derived, which is also valid for mathematically degenerate materials. Several special cases are discussed in detail for the image force expression to illustrate the influence of the viscous interface on the mobility of the extended line dislocation, and various interesting features are observed. These Green's functions can not only be directly applied to the study of dislocation mobility in the novel multiferroic bimaterials, they can also be utilized as kernel functions in a boundary integral formulation to investigate more complicated boundary value problems where multiferroic materials/composites are involved.     

From imaging to material identification: A generalized concept of topological sensitivity

To establish a compact analytical framework for the preliminary stress-wave identification of material defects, the focus of this study is an extension of the concept of topological derivative, rooted in elastostatics and the idea of cavity nucleation, to 3D elastodynamics involving germination of solid obstacles. The main result of the proposed generalization is an expression for topological sensitivity, explicit in terms of the elastodynamic Green's function, obtained by an asymptotic expansion of a misfit-type cost functional with respect to the nucleation of a dissimilar elastic inclusion in a defect-free “reference” solid. The featured formula, consisting of an inertial-contrast monopole term and an elasticity-contrast dipole term, is shown to be applicable to a variety of reference solids (semi-infinite and infinite domains with constant or functionally graded elastic properties) for which the Green's functions are available. To deal with situations when the latter is not the case (e.g. finite reference bodies or those with pre-existing defects), an adjoint field approach is employed to derive an alternative expression for topological sensitivity that involves the contraction of two (numerically computed) elastodynamic states. A set of numerical results is included to demonstrate the potential of generalized topological derivative as an efficient tool for exposing not only the geometry, but also material characteristics of subsurface material defects through a local, point-wise identification of “optimal” inclusion properties that minimize the topological sensitivity at sampling location. Beyond the realm of non-invasive characterization of engineered materials, the proposed developments may be relevant to medical diagnosis and in particular to breast cancer detection where focused ultrasound waves show a promise of superseding manual palpation.     

Study on the 3D Green's functions of the fluid and piezoelectric bimaterials

Because most piezoelectric devices have interfaces with fluid in engineering,it is valuable to study the coupled field between fluid and piezoelectric media.As the fundamental problem,the 3 D Green's functions for point forces and point charge loaded in the fluid and piezoelectric bimaterials are studied in this paper.Based on the 3 D general solutions expressed by harmonic functions,we constructed the suitable harmonic functions with undetermined constants at first.Then,the couple field in the fluid and piezoelectric bimaterials can be derived by substitution of harmonic functions into general solutions.These constants can be obtained by virtue of the compatibility,boundary,and equilibrium conditions.At last,the characteristics of the electromechanical coupled fields are shown by numerical results.     

Sub-Additive and Asymptotically Sub-Additive Topological Pressure for \mathbb{Z }^d-Actions

We introduce the topological pressure for any sub-additive and asymptotically sub-additive potentials of $\mathbb{Z }^d$ -actions, and establish the variational principle for them.     

Determining transverse impact force on a composite laminate by signal deconvolution

Dynamic impact forces on a composite structure were recovered by using experimentally generated Green's functions and signal deconvolutions. The signal processing is straightforward. Extra windowing and filtering the recorded signals are unnecessary. The Green's functions account for boundary conditions, material properties and structure geometry. This approach can be applied to linearly elastic structures with different boundary conditions. It is realistic and convenient to use for the recovery of impact force on anisotropic or isotropic solid structures.     

Best constants in Sobolev inequalities for ultraspherical polynomials

A family of sharp Sobolev-type inequalities for functions on the classical measure spaces associated with the ultraspherical or Gegenbauer polynomials is obtained. These estimates generalize the Sobolev inequalities for the n-sphere S ⁿ given by Beckner, and are derived from a sharp Sobolev inequality for functions on the real line. Spectral considerations allow these estimates to be expressed as multiplier inequalities for functions which have expansions in terms of Gegenbauer polynomials.     

Thermoelectroelastic Green's function and its application for bimaterial of piezoelectric materials

Summary For a two-dimensional piezoelectric plate, the thermoelectroelastic Green's functions for bimaterials subjected to a temperature discontinuity are presented by way of Stroh formalism. The study shows that the thermoelectroelastic Green's functions for bimaterials are composed of a particular solution and a corrective solution. All the solutions have their singularities, located at the point applied by the dislocation, as well as some image singularities, located at both the lower and the upper half-plane. Using the proposed thermoelectroelastic Green's functions, the problem of a crack of arbitrary orientation near a bimaterial interface between dissimilar thermopiezoelectric material is analysed, and a system of singular integral equations for the unknown temperature discontinuity, defined on the crack faces, is obtained. The stress and electric displacement (SED) intensity factors and strain energy density factor can be, then, evaluated by a numerical solution at the singular integral equations. As a consequence, the direction of crack growth can be estimated by way of strain energy density theory. Numerical results for the fracture angle are obtained to illustrate the application of the proposed formulation. Received 10 November 1997; accepted for publication 3 February 1998     

SPLITTING OF THE SPECTRAL DOMAIN ELECTRICAL DYADIC GREEN’S FUNCTION IN CHIRAL MEDIA

IntroductionAchiralmediumisanewtypeofspecialmediummaterial.Broadapplicationprospectsforchiralmediainmicrowaves,millimeterwaves,electronicdevices,integratedoptics,andsoonfieldhaveattractedconsiderableattention .Theelectromagneticproblemwithchiralmediahasbeenahotresearchtopicoftheelectromagnetictheory .TheeigenfunctionexpansionproblemofthedyadicGreen’sfunctionfortheelectromagneticwavefieldinchiralmediahasbeendeeplyinvestigatedinRefs [1～ 6] .FromHelmholtztheorem ,anarbitraryvectorfieldfcouldb…     

波数-频率域内地基土表面位移Green函数的理论分析

建立了柱面坐标系下分层弹性半空间地基土模型。利用钟阳刚度矩阵法和Haskell-Thomson传递矩阵法推导出所有分层土体之间的振动传递关系;根据Helmholtz定理将土体的位移向量分解成势函数的形式,推导出弹性半空间表面应力与位移之间的关系;再将分层土体和半空间地基土通过位移与应力之间的关系进行耦合,得到分层弹性半空间地基土模型表面位移与应力之间的关系。结合单位脉冲荷载作用下地基土表面的边界条件,推导出波数-频率域内地基土表面位移Green函数的解析解,用Matlab程序语言对理论进行实现并通过算例对地基土表面位移Green函数的特征进行了分析和总结。     

Sharp Gagliardo–Nirenberg Inequalities and Mass Transport Theory

It is known that best constants and extremals of many geometric inequalities can be obtained via the Monge–Kantorovich theory of mass transport. But so far this approach has been successful for a special subclass of the Gagliardo–Nirenberg inequalities, namely, those for which the optimal functions involve only power laws. In this paper, we explore the link between Mass transport theory and all classes of the Gagliardo–Nirenberg inequalities. Sharp constants and optimal functions of all the Gagliardo–Nirenberg inequalities are obtained explicitly in dimension n = 1, and the link between these inequalities and Mass transport theory is discussed.