Null Field Approach Research Articles

Point force excitation of surface waves along the doubly corrugated traction-free boundary of an elastic half-space

The excitation of generalized Rayleigh waves due to the effect of a buried time-harmonic source located below the infinite, doubly periodically corrugated free boundary of a homogenous, isotropic, and linear elastic half-space is considered. The full elastodynamic equations are solved using the null-field approach (also called the T matrix method), and the contribution of the surface waves to the total displacement field in the vicinity of the boundary is obtained for the sinusoidally corrugated geometry with equal periods of corrugation. For different corrugation heights and frequencies and both a vertically and a horizontally directed point force excitation, numerical results are presented for the angular dependence of the surface field far away from the source where the generalized Rayleigh mode contribution dominates. When dealing with the horizontally directed point force the direction of the excitation in the horizontal plane is an important factor and it is explored numerically as well. In general, the far-field amplitude increases both with increasing frequency and corrugation height. It is smaller for a horizontal point force than a vertical one with the same location. As expected, the angular dependence of the amplitude is closely connected to the corresponding resonance and damped structure of the surface modes. Also, the appearance of stopbands or damped parts in the surface mode is followed by a disappearance of the far-field amplitude. The numerical computations are affected by some numerical instabilities that seem difficult to circumvent. Instabilities seem to arise when a residue of an inverse matrix is computed numerically and appear for all frequencies and corrugation heights for surface waves propagating at 0°, 45°, and 90° to the corrugations. Others appear only for certain frequencies and corrugation heights at other directions of surface wave propagation and seem to be a consequence of a cross-over resonance between the surface modes.

Point force excitation of a thick-walled elastic infinite pipe with an embedded inhomogeneity

The propagation of elastic waves in thick-walled pipe with an embedded inhomogeneity is considered. The pipe is excited by a point force applied on its surface and the time harmonic problem is solved using the null field approach, a method whose main characteristics are surface integral representations and expansions in spherical and cylindrical vector wave functions. Entering in the expression for the scattered field are the transition matrix for the cavity, the reflection matrices for the inner and outer surfaces of the pipe, the transformation functions between the spherical and cylindrical vector wave functions and also the translation for the cylindrical waves. Numerical examples, both in the frequency and time domain, are presented for a spherical cavity and an open circular crack.

TRANSMISSION AND REFLECTION OF ELASTIC WAVES BY A SPHERICAL OBSTACLE IN AN INFINITE CIRCULAR CYLINDRICAL ROD

The null-field approach is applied to the problem of scattering of elastic wave by a spherical obstacle in an infinite circular cylindrical rod. The axially symmetric longitudinal and vertically polarized, transverse waves are considered and the boundary conditions are those of vanishing surface traction. Thus the scattered field is obtained as a complicated mode sum containing the transitiion matrix of the sphere, the reflection matrix of the cylinder and the transformation function between teh spherical and cylindrical vector wave functions

Acoustic scattering from a rough sphere

The scattering of a plane acoustic wave from a random rough sphere is studied using the null field approach. The starting point is an integral representation derived from the Helmholtz equation. The incident field, the scattered field, and the free-space Green’s function are all expanded in terms of suitably chosen basis functions. In this way a relation between the incident and the scattered fields is obtained, which is expressed by the transition matrix (T matrix). The scattered field is expanded in a power series of a small parameter, namely the product of the wave number and the root-mean-square height of the irregularities. Analytical expressions for the leading terms of the series have been calculated. In particular, ensemble averages of the far-field amplitude and the scattering cross section have been determined. As the analytical results are somewhat complicated, some numerical results are presented. Numerical computations of higher-order terms indicate that the convergence of the series is satisfactory as long as the root-mean-square height is small compared to a correlation length describing the average distance between the peaks of the surface.

Point force excitation of an elastic infinite circular cylinder with an embedded spherical cavity

In this paper the scattering of elastic waves in an infinite circular cylinder is considered. The elastic medium of the cylinder is assumed to be homogeneous, isotropic, and linear except for a finite inhomogeneity which, for simplicity, is chosen as a spherical cavity. By applying the null field approach (or T matrix method), the scattered field is obtained as a complicated expression containing the transition matrix of the cavity, the reflection matrix of the cylinder, and the transformation function between the spherical and cylindrical vector wave functions. Numerical results, both in the frequency and time domain, are presented.

Effective plane wave propagation through a medium with spheroidal inclusions surrounded by thin interface layers

Plane wave propagation of both longitudinal and shear waves through a medium containing a random distribution of randomly oriented spheroidal inclusions has been considered. In particular, the effect of thin interface layers between the inclusions and the matrix is investigated. The scattering problem for one inclusion with or without an interface layer is solved first by the null field approach and the effective phase velocity and attenuation are then obtained following the simple approach of Foldy. Numerical results are given for SiC inclusions of different aspect ratios in an Al matrix and the presence of the interface layers is seen to give rise to measurable effects.

Probe model implementation in the null field approach to crack scattering

A model of a compressional ultrasonic transducer is implemented into a T-matrix method-based solution to a crack scattering problem. The probe can act both as a receiver and as a transmitter, and it is modeled as an acoustic piston-like source. A previous solution by the null field approach is applied and is here used to model a crack that is partly closed due to an external background pressure. Numerical calculations of the signal response when the crack is penny-shaped are performed and compared with results from a program based on the Geometrical Theory of Diffraction, and the agreement is generally found to be very good.

The null-field approach to electromagnetic resonance of composite objects

In the present article, the null-field approach to electromagnetic resonance properties of three-dimensional composite objects is reviewed. In this approach the resonance problem is solved by searching zeroes of the determinant of the total Q-matrix for the composite object. The derivation of the Q-matrix for three main classes of composite objects is given. It is noted that for each class, several alternative null-field approaches are usually available. The numerical implementation of the Q-matrix for homogeneous and composite objects is discussed, with special emphasis on consistency checks for the results. It is found that numerical convergence is obtained in a frequency interval that often contains ten or more resonance modes. The resonance frequencies and quality factors for some axially symmetric composite objects are compared with published numerical and experimental data whenever possible. In general, good agreement is found in these comparisons.

Current distributions on a perfectly conducting elliptic disk (EM scattering)

Current distributions induced on a perfectly conducting elliptic disk are discussed within the framework of the null-field approach. The elliptic disk is treated as the zero thickness limit of a general ellipsoid. In this limit it is necessary to introduce modified expansion functions for the surface fields. It is shown that these expansion functions inherently satisfy the edge condition and no extra weight factors must be introduced. The first few terms in a low-frequency expansion of the surface currents induced by an incident plane wave are derived, and from this expansion the static limit is analyzed. The theoretical results are illustrated with numerical computations of the surface currents and charge distributions. >

THE NULL-FIELD APPROACH TO BOUNDARY-VALUE PROBLEMS FOR THE STOKES EQUATION

A number of boundary-value problems for the Stokes equation are considered. The problems are solved by means of a method employing integral representation theorems. The approach is an extension of the null-field approach to problems of creeping motion and rubber elasticity

Elastodynamic Scattering From Inclusions Surrounded by Thin Interface Layers

The scattering of elastic waves by elastic inclusions surrounded by interface layers is a problem of interest for nondestructive evaluation of interfaces in composites. In the present paper the scattering by a single elastic inclusion is studied. The scattering problem is solved by means of the null field approach and the properties of the interface layer enters through the boundary conditions on the inclusion. Various ways of doing this have been tried, from the simpler approach of just keeping the inertia of the layer, to using a membrane type of approximation or a more sophisticated method that includes all effects to first order in the layer thickness. The results obtained by using these different methods are compared numerically and with the exact solution for a layered sphere and with some recent results for a spheroid obtained using a hybrid finite element and wave function expansion technique. The numerical results show significant dependence on parameters containing the thickness and stiffness of the interface layer.

Elastodynamic Scattering From Penny-Shaped and Nonplanar Cracklike Flaws With Dissipation and Restoring Forces

Scattering from penny-shaped and nonplanar cracklike flaws, with a type of boundary condition which incorporates dissipation and restoring forces, is studied. Scattering cross-sections and energy absorption cross-sections are computed, and comparisons with open cracks are performed. A partially debonded inclusion, with the debonding in the form of a cracklike flaw, is also discussed. The method employed is a modified null field approach, incorporating some ideas previously used for treating scattering from nonplanar open or fluid-filled cracks. In the present context, the boundary conditions lead to great simplifications in the implementation of the method, in that part of the relevant matrices can be computed analytically.

Elastic wave scattering from debonded fibers in composite materials

A modified null-field approach has been applied to study the elastic wave scattering response of an inclusion with an interface crack; e.g., a rotationally debonded fiber in a fiber-reinforced composite. A fictitious mathematical surface to include the debonded region is introduced. Appropriate field expansions are introduced in all regions of interest, and used in integral representations. After using the orthogonality properties of the expansion functions, a T matrix is obtained to relate the scattered field to the incident field. The range of validity is checked for computational purposes. Scattering cross sections of infinitely long debonded fibers are computed as functions of frequency and the extent of debonding for P-, SV-, and SH-plane wave incidence conditions.

Null-field computations of radar cross sections of composite objects

The application of the null-field approach to radar cross section computations for composite objects is reviewed. The scattering problem is solved by means of a determination of the total transition matrix for the composite scatterer. It is noted that alternative approaches are usually available, which lead to different expressions for the transition matrix and which have different numerical characteristics. Results from the numerical implementation of these formulas are illustrated in a number of examples, and it is found that useful convergence can be achieved for a frequency interval that often extends into the resonance region. A number of ways of checking the quality of the results are also indicated. >

Propagating, damped, and leaky surface waves on the corrugated traction-free boundary of an elastic half-space

The dispersion relation for surface waves on the corrugated boundary (periodic in one direction and constant in the other) of an elastic half-space is derived using a modal approach, and the result is shown to be equivalent to that derived by the null-field approach. The dispersion relation is solved numerically for roots on all Riemann sheets for varying corrugation heights, frequencies, and angles of propagation. Many of the roots move on several sheets, and this leads to zero, one, or two roots residing on the physical sheet.

The null field approach to electromagnetic scattering from composite objects: the case of concavo-convex constituents

Time-harmonic electromagnetic scattering from a composite body consisting of a (dielectric or metallic) core plus one or several dielectric coatings was studied using the null-field approach. Previously developed null-field approaches to scattering from composite bodies do not apply when these coatings are of concavo-convex shapes. The authors examine this case and develop alternative null-field approaches to such geometries. While the scattering problem is usually solved by determination of the total transition matrix, referring to spherical waves, for the composite scatterer, the authors' approaches lead to different algebraic expressions for the transition matrix. Two main alternatives are studied. One of these makes use of Q-matrices for open surfaces while the other is based on a limit procedure applied to a previously developed formalism for layered scatterers. The numerical accuracy of the results is less than that obtained for homogeneous scatterers of similar exterior shape and electrical size. The convergence of the numerical implementation of the equations is studied in terms of several indicators such as dependence on the truncation order, the accuracy with which general constraints such as symmetry and unitarity are fulfilled, and the influence of different choices of expansion functions obtained from a moment-method solution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The null field approach to electromagnetic scattering from composite objects

An improved null-field approach to scattering from composite objects is reported. Several alternative expressions for the transition matrix of a composite object are derived that partly or completely avoid the geometrical constraints inherent in previous null-field results for a composite scatterer. Some of the new versions make use of Q-matrices for the open surfaces which are defined by the interfaces between the different parts of the composite scatterer. The numerical performance of the various alternatives is investigated for a number of test cases. Comparisons between alternative versions for the same scattering problem are made to provide a measure of the absolute accuracy of the computed null-field results. Whenever possible, the results are compared with other computed or measured results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The null field approach to electromagnetic scattering from composite objects: the case with three or more constituents

A null-field approach to scattering from composite objects due to S. Strom and W. Zhengs (ibid., vol.36, no.3, p.376-82, 1988) is extended to scatterers consisting of three of more homogeneous parts. The numerical performance of this extension has been tested for a number of different types of scatterers, and the numerical results have been compared with other computed and measured results. In particular, the application of the extended formulation to scattering from elongated objects is described and illustrated by numerical computations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Electromagnetic waves in a cylindrical waveguide with infinite or semi-infinite wall corrugations

The electromagnetic waves inside a circular waveguide having a periodically varying radius with corrugations of infinite or semi-infinite extent are discussed. The infinitely corrugated waveguide is investigated by use of the null-field approach, and some plots of the axial wavenumbers are presented. For a junction between a straight and a corrugated waveguide, the reflection and transmission coefficients are determined by mode matching; some computations of these reflection coefficients are also given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Basic features of the null field method for dielectric scatterers

The null field approach to electromagnetic scattering from a dielectric body is presented in its general form, i.e., for a general choice of surface field expansions. Vector spherical harmonics are used to construct tangential fields which are shown to be complete for star‐shaped surfaces. The numerical performance of these sets is compared to that obtained when expansions in terms of different kinds of spherical vector waves are used. A demonstration of the uniqueness of the solutions to the null field equations for a permeable body is included in an appendix.