A transverse aperture‐integral equation solution for edge diffraction by multiple layers of homogeneous material

The problems of SH-wave scattering, which is caused by half-space circular cavity near the bimaterial interface and crack of arbitrary length and arbitrary position, are studied in this paper based on the field of elastodynamics, and the methods of Green's Function, complex variables function, the method of crack-division and multi-polar coordinates are used here. Firstly, a suitable Green's function is constructed, which is the key technology of this problem. Two Green's functions are needed here: The first one is an essential solution of the displacement field for the elastic half-space containing a circular cavity and crack with arbitrary length and arbitrary position under the out-of-plane harmonic line-loads at an arbitrary point; The second Green' function is an essential solution of the displacement field for the elastic half space subjected to the out-of-plane harmonic line source loads at half space surface arbitrary point. The scattering of SH-wave by the crack at an arbitrary position and a circular cavity near the bimaterial interface is investigated, the problem can be regarded as harmony model: the bimaterial media is divided into two parts along the horizontal interface, one is an elastic half space with a circular cavity and a crack, and the other is a complete elastic half space. The horizontal surfaces of the two parts are loaded with unknown anti-plane forces in order to satisfy at the linking section. A series of Fredholm integral equations can be set up through continuity conditions. Then, the solution of the problem can be reduced to a series of algebraic equations and solved numerically by truncating the finite terms of the infinite integral equations, the analytical solution of displacement field and stress field of scattering of SH-wave by the crack and circular cavity near the bimaterial interface are given. Numerical examples are provided to show the influences of wave numbers, incident angle, the distance between the center of the circular cavity and horizontal surfaces, the distance between the center of the circular cavity and crack, the angle of crack, the length of crack, and parameter combinations of different media upon dynamic stress concentration factor (DSCF).

In the use of the time-domain integral equation (TDIE) method for the analysis of layered media, it is important to have the time-domain layered medium Green's function computed for many source-to-field distances /spl rho/ and time instants t. In this paper, a numerical method is used that computes the mixed potential Green's functions G/sub v/(/spl rho/,t) and G/sub A/(/spl rho/,t) for a multilayered medium for many /spl rho/'s and t's simultaneously. The method is applicable to multilayered media and for lossless or lossy dispersive media. Salient features of the method are: 1) the use of complex /spl omega/ so that the surface wave poles are lifted off the real k/sub /spl rho// axis such that pole extractions are not required; 2) the use of half-space extraction so that the integrand for the Sommerfeld integral decays exponentially along the k/sub /spl rho// axis to obtain fast convergence of the integral; and 3) the use of the fast Hankel transform so that the Green's function is calculated for many values of /spl rho/ simultaneously. For a four-layer medium, we illustrate the numerical results by a three-dimensional plot of /spl rho/G/sub v/(/spl rho/,t) versus /spl rho/ and t and demonstrate the space-time evolution of these Green's functions. For a maximum frequency range of 8 GHz, the method requires only a few CPU minutes to compute a table of 100 (points in /spl rho/) /spl times/ 168 (points in t) uniformly spaced values of G/sub v/(/spl rho/,t) on an 867-MHz Pentium PC.

Abstract: In this paper, a half space adaptive fast multipole boundary facemethod (FMBFM) is presented for solving the three-dimensional half space ex-terior acoustic problems. In the presented method, the Burton-Miller equationbased on the conventional boundary integral equation (CBIE) and its hyper-singularboundary integral equation (HBIE) is used to deal with the fictitious eigenfrequen-cies problem. The half space Green’s function is employed, thus the tree structurein the fast multipole method can be used only for the real domain. The higher orderelements and an adaptive tree structure are used to improve the efficiency of theFMBFM. This half space adaptive FMBFM for half space acoustic problems is anextension of the adaptive FMBFM for full space acoustic problems developed bythe authors. Numerical examples for half space acoustic problems in this paperdemonstrate the efficiency and validity of this method.Keywords: fast multipole boundary face method, Burton-Miller equation, acous-tic problems, half space, modified hyper-singular boundary integral equation1 IntroductionAcoustic problems in half space are a major subject in engineering. Many researchworks have been done on this aspect (Seybert et al., 1988; Secgin et al., 2010; Caoet al., 2010). The half space Green’s function (Seybert et al., 1988) can be appliedby adding the source solution at the image point to the original full-space Green’sfunction for dealing with half space problems. Sound source localization and iden-tification of a refrigerator were accomplished by using the half space boundaryelement method (BEM) (Secgin et al., 2010). Using the half space Green’s func-tion, the discretization of the rigid infinite plane is removed and only the boundaries

Electromagnetic fields created in a homogeneous medium by a known distribution of electric current are determined by integral equations involving free space Green's functions. In actual situations, however, current distributions are not known beforehand and must be determined, in such a way that the total fields satisfy the boundary conditions on the conductors. The integral equation technique was extended to treat also heterogeneous structures, in order to analyze and design multilayer printed circuits and antennas, such as the broadband strip-slot-foam-inverted patch printed antenna. Sommerfeld integrals provide Green's functions for stratified media, which are evaluated by means of specifically developed, efficient computer algorithms. Structures including coaxial or line feeds and coupling through slots can now be easily analyzed with commercially available software packages.

A note on the elastodynamic load transfer problem

In this paper, we present the multilevel Green's function interpolation method (MLGFIM) for analyses of three-dimensional doubly periodic structures consisting of dielectric media and conducting objects. The volume integral equation (VIE) and surface integral equation (SIE) are adopted, respectively, for the inhomogeneous dielectric and conducting objects in a unit cell. Conformal basis functions defined on curvilinear hexahedron and quadrilateral elements are used to solve the volume/surface integral equation (VSIE). Periodic boundary conditions are introduced at the boundaries of the unit cell. Computation of the space-domain Green's function is accelerated by means of Ewald's transformation. A periodic octary-cube-tree scheme is developed to allow adaptation of the MLGFIM for analyses of doubly periodic structures. The proposed algorithm is first validated by comparison with published data in the open literature. More complex periodic structures, such as dielectric coated conducting shells, folded dielectric structures, photonic bandgap structures, and split ring resonators (SRRs), are then simulated to illustrate that the MLGFIM has a computational complexity of O(N) when applied to periodic structures.

The spatial frequency technique (SFT) was developed for solving the integral equation (IE) in electromagnetic scattering and radiation problems and shown to work faster than many existing solution methods. In this paper, the new formulation, which is based on the efficient FFT algorithm, is analyzed further for analytically solving the IE for certain types scattering and radiation problems. It is shown that the SFT description of the symmetric matrix which represents the Green's function may be decomposed into an orthogonal basis using Takagi factorization. Transforming the Green's function into this orthogonal basis has the potential to invert the Green's function analytically. The SFT is also compared with the Fast Multipole Method (FMM) and shown to have similar underlying properties which may also be utilized to analytically invert the Green's function using FMM.

A method for reconstructing dielectric objects buried in a half space is proposed. The approach is based on the inverse scattering formulation. The Green's function for the half space is used as the kernel of the integral equations relating the dielectric properties of the object to the samples of the scattered electric field. The inverse problem is recast as a global optimization problem by constructing a functional to be minimized. A differential evolution method is used to obtain a global minimum of this functional. The paper provides the mathematical formulation of the approach. Moreover, numerical results are reported.

The array Green's function (AGF) represents the basic constituent for the full wave description of electromagnetic radiation from rectangular periodic arrays and scattering from periodic surfaces. In modeling the performance of such structures: one of the main objectives is the reduction of the often prohibitive numerical effort that accompanies the full-wave analysis based on an integral equation, which is structured around the ordinary individual element Green's function. For a periodic array, this array Green's function is composed of the sum of the individual dipole radiations. Under certain assumptions the integral equation can be restructured around the active Green's function, which is the field collectively radiated by an array of elementary dipoles. A series of papers have investigated the description of the AGF in terms of propagating and evanescent Floquet waves (FWs) together with corresponding FW-modulated diffracted fields which arise from FW scattering at the array edges and vertices. The above mentioned formulation, being based on an asymptotic evaluation of the constituent radiation integrals, fails when observing too close to the edge and vertices. The present work is intended to illustrate a complementary strategy for treating the observation point also close to the array contour, as implicated by a full-wave scheme.

Wave propagation according to higher order field equations I

Dyadic Green's Function Microstrip Circulator Theory for Inhomogeneous Ferrite with and without Penetrable Walls

We consider the method of evaluating matrix elements of electromagnetic volume integral equations with the help of suitably constructed Laplacian-type representations of singular kernels (Green functions) appearing in electromagnetic volume and surface integral equations. The method consists of representing the singular kernels (Green functions) in terms of a generalized Laplacian operator acting on some auxiliary functions. Such a representations allow us, by using Gauss divergence theorem, to convert volumetric and surface integrals representing matrix elements to integrals always involving only non-singular integrands. The task of of finding particular Laplacian representation of different kernels amounts to solving appropriate ordinary or partial inhomogeneous differential equations. We apply the Laplacian method to analytic evaluation of matrix elements of the volume integral equation by employing two different Laplacian representations of the integral equation kernels. The first representation allows us to reduce 6-dimensional volume integrals over tetrahedra to 2-dimensional surface integrals over tetrahedra faces. The second representation is subsequently used reducing surface integrals over tetrahedra faces to pairs of of 1-dimensional line integrals over tetrahedra edges. We also derive analytic expressions for the resulting line integrals given in terms of elementary functions.

We consider the motion of a thin elastic plate with non-uniform thickness. The plate is either submerged and has some inclination with the vertical or is floating on the upper surface of the water. Green's function arising from the fourth-order boundary condition for the non-uniform plate (which we refer to as plate Green's function) is determined using two different methods in terms of the vibrating modes of the plate. These, in turn, are derived from the modes of a plate with constant thickness. The problem is finally reduced to a boundary integral equation involving the plate Green's function and the fundamental Green's function. This equation is hypersingular in the case of a submerged plate. A numerical solution to the integral equation is used to find results for elastic plates with variable thicknesses. The results are validated by comparing them with those of an elastic plate with uniform thickness. We also present simulations of the time-domain motion when the plate–fluid system is subject to an incident wave pulse using Fourier transform.

We consider the motion of a thin elastic plate with non-uniform thickness. The plate is either submerged and has some inclination with the vertical or is floating on the upper surface of the water. Green's function arising from the fourth-order boundary condition for the non-uniform plate (which we refer to as plate Green's function) is determined using two different methods in terms of the vibrating modes of the plate. These, in turn, are derived from the modes of a plate with constant thickness. The problem is finally reduced to a boundary integral equation involving the plate Green's function and the fundamental Green's function. This equation is hypersingular in the case of a submerged plate. A numerical solution to the integral equation is used to find results for elastic plates with variable thicknesses. The results are validated by comparing them with those of an elastic plate with uniform thickness. We also present simulations of the time-domain motion when the plate–fluid system is subject to an incident wave pulse using Fourier transform.

The stochastic Green's matrix is calculated for a random rough surface with Gaussian statistics and a magnetic boundary condition. The techniques we use are similar to those developed for the scalar and elastic cases. The coupled surface integral equations which are derived are the Green's function version of the Franz formulas. These integral equations are represented in k-space in a certain gauge and a Feynman-diagram-like interpretation is given to each term in the equations. The diagram rules have many formal similarities with the scalar and elastic rules. By using partial summation techniques, the mean and second moment of this Green's function are shown to be solutions to Dyson and Bethe-Salpeter equations respectively. The Green's function is applied to a scattering problem. Some approximations and simple examples are presented. The lowest order approximations agree with the standard literature results. The main advantage of the diagram method, its systematic presentation of higher order approximations, is stressed.