Modeling of the scattering of electromagnetic waves by a thin dielectric coating on a cylinder

Double-sided boundary conditions containing only tangential components of a diffracted field are used to model the interaction of electromagnetic waves with a cylinder of arbitrary cross section covered with a thin dielectric layer. The obtained boundary-value problem is reduced to a system of two singular integral equations of the second kind with kernels whose structure is similar to the kernels of integral equations of the first kind for a perfectly conducting scatterer. The numerical solution of the integral equations of the problem is obtained by the method of mechanical quadratures. The scattering properties of an elliptic cylinder with different dielectric coatings are studied in the superhigh-frequency band. It is shown that the coating strongly affects the diffraction properties of the cylinder. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 1, pp. 96–104, January–February, 2006.     

Multidomain analytical-numerical solution for a rotating magneticfield with a finite-length conducting cylinder

This paper treats the rotating magnetic field produced by a cylindrical sheet of temporally and azimuthally periodic axial electric current with a finite-length, electrically conducting cylinder surrounded by an electrical insulator inside the current sheet. As the frequency is increased, the magnetic field produced by the induced current in the cylinder cancels progressively more of the current-sheet magnetic field in the interior of the cylinder. This paper presents results for several different frequencies and for several different length-to-diameter ratios for the cylinder. It compares the results at the midplane of a long cylinder to the analytical solution for an infinitely long cylinder. It also compares results for a high frequency to the predictions from the first two terms in an asymptotic expansion for high frequency     

Current distribution in a parallel set of conducting strips

The current distribution in a parallel set of thin conducting sheets due to an external applied source is investigated. All sheets are placed in one plane. The source, and all excited fields, are time-harmonic. The frequency is low enough to allow for an electro quasi-static approximation (neglecting the displacement current). The conducting sheets are infinitely long and the current is uniform in the longitudinal direction of the sheets. The sheets have a thin rectangular cross-section, so thin that the current can be assumed uniform in the thickness-direction. Hence, the current distribution only depends on the transverse coordinate. Due to the mutual induction between the sheets, the current distribution over the width of the cross-section becomes non-uniform: it accumulates at the edges of the sheets. It is especially this so-called edge-effect, and its dependence on the applied frequency and the distances between the sheets, that is the aim of this investigation. From the Maxwell equations, a set of integral equations for the current distribution in the sheets is derived. These integral equations are solved, as far as possible by analytical means, by writing the current distribution in each sheet as a series of Legendre polynomials. The general method is worked out for N (N 1) sheets, but explicit results are presented for N=1 and 3. It turns out that the edge-effect becomes stronger for increasing frequencies. For this solution, only a very restricted number of Legendre polynomials are needed.     

A hypersingular integral equation for current density on the surface of a microstrip vibrator

The problem of finding the current-density distribution on the surface of a microstrip vibrator (MV) in the framework of a thin ideally conducting strip, deposited on a dielectric substrate with one-sided metallization, is reduced to solution of a hypersingular integral equation (HSIE). The dependences of the input impedance on the vibrator length and substrate thickness are presented.     

Stress intensity factor for a Griffith crack interacting with a coated inclusion

Stress investigation for the interaction problem between a coated circular inclusion and a near-by line crack has been carried out. The crack and the coated inclusion (a coated fiber) are embedded in an infinitely extended isotropic matrix, with the crack being along the radial direction of the inclusion. Two loading conditions, namely, the tensile and shear loading ones are considered. During the solution procedure, the crack is treated as a continuous distribution of edge dislocations. By using the solution of an edge dislocation near a coated fiber as the Green's function, the problem is formulated into a set of singular integral equations which are solved by Erdogan and Gupta (1972) method. The expressions for the stress intensity factors of the crack are then obtained in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Several numerical examples are given for various material and geometric parameters. The solutions obtained from the integral equations have been checked and confirmed by the finite element analysis results.     

Fast and accurate radar cross-section computation over a broad frequency band using the best uniform rational approximation

The method of moments (MOM) in conjugation with the best uniform rational approximation is applied to predict the broad-band radar cross-section of arbitrarily shaped three-dimensional (3D) bodies. The surface integral equations are solved using MOM to obtain the equivalent surface currents at the frequency points corresponding to the Chebyshev nodes, and then the surface current within the given frequency band is represented by the Chebyshev series. To improve the accuracy, the Chebyshev series is matched via the Maehly approximation to a rational function, which can be as good as the best uniform rational approximation. Using the rational function, the surface current is obtained at any frequency point within the frequency band. Numerical results for 3D arbitrarily shaped perfectly electric conducting objects and homogenous dielectric bodies are considered. Compared with the asymptotic waveform evaluation technique and the model-based parameter estimation, the proposed technique is accurate in much broader frequency band with lower memory required.     

Eddy-Current Nondestructive Inspection with Thin Spiral Coils: Long Cracks in Steel

The impedance of a cylindrical coil and a planar circular spiral coil carrying an alternating current above (i) a defect-free conducting magnetic half-space and (ii) a conducting magnetic half-space containing an infinitely long slot with uniform depth and width is examined in detail. Closed-form expressions for the coil impedance in these cases are presented, based on the theories of Dodd and Deeds and Harfield and Bowler. The validity of these expressions is tested by measurements using steel plates over the frequency range 100 Hz-10 MHz. The theoretical predictions are in good agreement with experiment, with the best agreement for the smallest slot width. The results confirm that thin, flexible spiral coils offer some attractive features for eddy-current detection of cracks in metals, particularly in terms of sensitivity and potential for unobtrusive permanent attachment to the material being inspected. Approximate expressions for a spiral coil above a defect-free magnetic half-space are also given to allow easy calculation in limiting cases.     

Griffith crack opened by a thin symmetric wedge in an elastic strip

We consider the problem of determining the distribution of stress in an infinitely long elastic strip containing a Griffith crack which is opened by a thin symmetric wedge. We assume that the strip is bonded to semi-infinite elastic planes on either side and that there is an internal pressure on the part of the crack not in touch with the wedge. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with cosine kernel and a weight function. These equations are solved using finite Hilbert transform techniques. The particular case of a rectangular wedge is considered in detail.     

Hybrid finite-element-boundary integral algorithm to solve the problem of scattering from a finite and infinite array of cavities with stratified dielectric coating

This work presents a hybrid finite-element-boundary integral algorithm to solve the problem of scattering from a finite and infinite array of two-dimensional cavities engraved in a perfectly electric conducting screen covered with a stratified dielectric layer. The solution region is divided into interior regions containing the cavities and the region exterior to the cavities. The finite-element formulation is applied only inside the interior regions to derive a linear system of equations associated with unknown field values. Using a two-boundary formulation, the surface integral equation employing the grounded dielectric slab Green's function in the spatial domain is applied at the opening of the cavities as a boundary constraint to truncate the solution region. Placing the truncation boundary at the opening of the cavities and inside the dielectric layer results in a highly efficient solution in terms of computational resources, which makes the algorithm well suited for the optimization problems involving scattering from grating surfaces. The near fields are generated for an array of cavities with different dimensions and inhomogeneous fillings covered with dielectric layers.     

Two coplanar griffith cracks under shear loading in an infinitely long elastic layer

We consider the problem of determining the stress distribution in an infinitely long isotropic homogeneous elastic layer containing two coplanar Griffith cracks which are opened by internal shear stress acting along the lengths of the cracks. The faces of the layer are rigidly fixed. The cracks are located in the middle plane of the layer parallel to its faces. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with a cosine kernel and a weight function. The triple integral equations are solved exactly. Closed form analytical expressions are derived for the stress intensity factors, shape of the deformed crack, and the crack energy. Solutions to some particular problems are derived as limiting cases. Numerical results are presented in the form of graphs.     

Analytical solution for functionally graded magneto-electro-elastic plane beams

This paper investigates the plane stress problem of generally anisotropic magneto-electro-elastic beams with the coefficients of elastic compliance, piezoelectricity, dielectric impermeability, piezomagnetism, magnetoelectricity, and magnetic permeability being arbitrary functions of the thickness coordinate. Firstly, partial differential equations governing stress function, electric displacement function and magnetic induction function are derived for plane problems of anisotropic functionally graded magneto-electro-elastic materials. Secondly, these functions are assumed in forms of polynomials in the longitudinal coordinate and can be acquired through a successive integral approach. The analytical expressions of axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced. Thirdly, problems of functionally graded magneto-electro-elastic plane beams are considered, with integral constants being completely determinable from boundary conditions. A series of analytical solutions are thus obtained, including the solutions for beams under tension and pure bending, for cantilever beams subjected to shear force applied at the free end, and for cantilever beams subjected to uniform load. These solutions can be easily degenerated into the solutions for homogenous anisotropic magneto-electro-elastic beams. Finally, a numerical example is presented to show the application of the proposed method.     

Electroelastic analysis for a Griffith crack interacting with a coated inclusion in piezoelectric solid

A Mode III Griffith crack interacting with a coated inclusion in piezoelectric media is investigated. The crack, the coated inclusion are embedded in an infinitely extended piezoelectric matrix media, with the crack being along the radial direction of the inclusion. In the study, three different piezoelectric material phases are involved: the inclusion, the coating layer, and the matrix. A far-field loading condition is considered. During the solution procedure, the crack is simulated as a continuous distribution of screw dislocations. By using the solution of a screw dislocation near a coated inclusion in piezoelectric media as the Green function, the problem is formulated into a set of singular integral equations, which are solved by numerical method. The stress and electric displacement intensity factors are derived in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Numerical examples are given for various material constants combinations and geometric parameters.     

Two coplanar cracks in an infinitely long elastic strip bonded to semi-infinite elastic planes

We consider the problem of determining the stress intensity factors and the crack energy in an infinitely long elastic strip containing two coplanar Griffith cracks. We assume that the strip is bonded to semi-infinite elastic planes on either side and that the cracks are opened by constant internal pressure. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations withcosine kernel and a weight function. These equations are solved using Finite Hilbert transform techniques. Analytical expressions upto the order off δ^?10 where 2δ denotes the thickness of the strip and δ is much greater than 1 are derived for the stress intensity factors and the crack energy.     

On the spectrum of the electric field integral equation and the convergence of the moment method

Existing convergence estimates for numerical scattering methods based on boundary integral equations are asymptotic in the limit of vanishing discretization length, and break down as the electrical size of the problem grows. In order to analyse the efficiency and accuracy of numerical methods for the large scattering problems of interest in computational electromagnetics, we study the spectrum of the electric field integral equation (EFIE) for an infinite, conducting strip for both the TM (weakly singular kernel) and TE polarizations (hypersingular kernel). Due to the self‐coupling of surface wave modes, the condition number of the discretized integral equation increases as the square root of the electrical size of the strip for both polarizations. From the spectrum of the EFIE, the solution error introduced by discretization of the integral equation can also be estimated. Away from the edge singularities of the solution, the error is second order in the discretization length for low‐order bases with exact integration of matrix elements, and is first order if an approximate quadrature rule is employed. Comparison with numerical results demonstrates the validity of these condition number and solution error estimates. The spectral theory offers insights into the behaviour of numerical methods commonly observed in computational electromagnetics. Copyright © 2001 John Wiley & Sons, Ltd.     

Transient magnetic and electric fields above a conducting ground plane of arbitrary thickness

The time-dependent Maxwell equations without displacement current terms are solved above, within, and below a doubly infinite slab of finite conductivity and arbitrary thicknessDwith a prescribed current in an infinitely long wire above and parallel to the slab. Closed analytic expressions for the magnetic and electric field above the ground plane are obtained by transform methods in terms of a Laplace transform variable representing time. For finiteDthe results in actual time are computed by numerical inversion of the Laplace transform; forD = infinthey are given analytically to within numerical quadratures. Asymptotic expressions valid for large time and/or large lateral distance from the wire are obtained both forD < infinandD = infin. A summary of numerical results is given.     

Applications and further developments of the eddy current program EDDYNET

EDDYNET is a computer code for solving eddy current problems using an integral equation method and a network (wire mesh) approach. The code can be applied to infinitely long prisms, thin plates, and thin shell. Preliminary results with a three-dimensional version are described. Application to a tokamak limiter experiencing a plasma disruption is also described.     

An interface integral equation method applied to a crack impinging upon a bimaterial,frictional interface

A crack impinging upon a frictional, bimaterial interface is studied theoretically. Specifically we consider the problem of an infinitely long, cracked, two-dimensional fiber, which is embedded in an infinite plane with distinct elastic properties. The composite is subjected to tensile loading parallel to the fiber. An interface integral equation method is developed to solve this problem. This method, involving to-be-determined distributions of line forces, reduces the specific problem considered here to four coupled integral equations which are solved numerically. The bimaterial effect appears to be significant with respect to the length of the slip zone along the interface and the interfacial shear stress. However, the blunting of the crack by the frictional interface is virtually independent of the bimaterial effect.     

Dynamic stress intensity factor of a cracked dielectric medium in a uniform electric field

Summary We consider the scattering of normally incident longitudinal waves by a finite crack in an infinite isotropic dielectric body under a uniform electric field. By the use of Fourier transforms, we reduce the problem to that of solving two simultaneous dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind having the kernel that is a finite integral. The dynamic stress intensity factor versus frequency is computed, and the influence of the electric field on the normalized values is displayed graphically.     

Boundary integral equations analysis of induction devices with rotational symmetry

The application of boundary integral equations (BIE's) for the analysis of linear induction devices with rotational symmetry is considered. One-dimensional Fredholm integral equations are derived for the tangential field components at the boundary of a conducting medium with constant scalar conductivity and permeability excited by a time-harmonic azimuthal current source. The important special case of a short right circular conducting cylinder (magnetic or nonmagnetic) coaxial with one or more short coils is treated in detail. The explicit form of the kernels and the numerical solution technique are presented. Numerical results are presented for typical induction heating applications where the load length as well as the coil length are finite. Results are also presented for the magnetostatic problem of finding the demagnetization factors for short magnetic rods. In each case the results are compared with published results and the accuracy and efficiency of the proposed approach are demonstrated.     

Inductance calculation of twisted conductors by the broken line approximation

The mutual inductance between two skew straight thin conductors is obtained as a function of two vectors corresponding to two current carrying line segments. Based on the obtained analytical expressions for the mutual inductance, the versatile calculation method for the self- and mutual inductances of various twisted conductors is studied by means of the broken line or polygonal curve approximation. In particular, it is confirmed that this numerical calculation is consistent with the analytical calculation of the self- and mutual inductances for coaxial helical conductors for the asymptotic form of the long axial length. Furthermore, for the inductances of general twisted conductors, the similar asymptotic forms of the length dependence are obtained.