Null Field Approach Research Articles

Electromagnetic scattering by a bounded obstacle in a parallel plate waveguide

This paper concerns scattering of an electromagnetic wave by a bounded object located inside a parallel plate waveguide. The exciting field in the waveguide is either an arbitrary source located at a finite distance from the obstacle or a plane wave generated in the far zone. In the latter case, the generating field corresponds to the lowest propagating mode (TEM) in the waveguide. The analytic treatment of the problem relies on an extension of the null field approach, or the T-matrix method, originally developed by Peter Waterman, and later generalized to deal with objects close to an interface. The present paper generalizes this approach further to deal with obstacles inside a parallel plate waveguide. This problem shows features that reflect both the two-dimensional geometry and the three-dimensional scattering characteristics. The analysis is illustrated by several numerical examples.

The polarizability and the capacitance change of a bounded object in a parallel plate capacitor

A method for solving the change in capacitance (or charge) if an object is introduced in a parallel plate capacitor is developed. The integral representation of the potential is exploited in a systematic way to solve the potential everywhere inside the capacitor. In particular, the change in capacitance is extracted. The method shows similarities with the null field approach for solving dynamic problems.

Null-field approach for the antiplane problem with elliptical holes and/or inclusions

In this paper, we extend the successful experience of solving an infinite medium containing circular holes and/or inclusions subject to remote shears to deal with the problem containing elliptical holes and/or inclusions. Arbitrary location, different orientation, various size and any number of elliptical holes and/or inclusions can be considered. By fully employing the elliptical geometry, fundamental solutions were expanded into the degenerate kernel by using an addition theorem in terms of the elliptic coordinates and boundary densities are described by using the eigenfunction expansion. The difference between the proposed method and the conventional boundary integral equation method is that the location point can be exactly distributed on the real boundary without facing the singular integral and calculating principal value. Besides, the boundary stress can be easily calculated free of the Hadamard principal values. It is worthy of noting that the Jacobian terms exist in the degenerate kernel, boundary density and contour integral; however, these Jacobian terms would cancel each other out and the orthogonal property is preserved in the process of contour integral. This method belongs to one kind of meshless methods since only collocation points on the real boundary are required. In addition, the solution is regarded as semi-analytical form because error purely attributes to the number of truncation term of eigenfunction. An exact solution for a single elliptical inclusion is also derived by using the proposed approach and the results agree well with Smith’s solutions by using the method of complex variables. Several examples are revisited to demonstrate the validity of our method.

Wave propagation in 2D elastic composites with partially debonded fibres by the null field approach

Time-harmonic plane wave propagation in a two-dimensional (2D) elastic matrix with partially debonded elastic fibres of nonclassical cross-section is investigated. The modified null field approach, taking into account the asymptotic behaviour of the solution at the interface crack-tips, is exploited to obtain the numerical results for a single scatterer. The effective medium approach based on Foldy's approximation is applied to estimate the average dynamic parameters of the composites containing randomly distributed partially debonded fibres of dilute concentration. Numerical results concern the longitudinal wave dispersion and attenuation owing to scattering by both randomly oriented and aligned fibres. The effects of the fibre shape, debonding (interface crack) size and direction of wave incidence on the effective P-wave velocity and attenuation coefficient are analysed.

Conformal mapping and bipolar coordinate for eccentric Laplace problems

AbstractBoundary value problems on the eccentric annulus are quite complex and cannot directly be solved analytically using cartesian or polar coordinates. Many mathematical techniques have been used to solve such a problem by using conformal mapping and bipolar coordinate. In the literature, Carrier and Pearson [Partial differential equation‐theory and technique. New York, Academic Press, 1976, pp 68–71], Muskhelishvili [Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen; 1953, pp 175–179], Ling [Torsion of an eccentric circular tube, Technical Report, No. 1, Chinese Bureau of Aeronautical Research, 1940], Timoshenko and Goordier [Theory of Elasticity. New York, McGraw‐Hill; 1972, pp 196–202], Shen [Null‐field approach for Laplace problems with circular boundaries using degenerate kernels, Master thesis, National Taiwan Ocean University, Keelung, Taiwan, 2005], Lebedev et al. [Worked Problems in Applied Mathematics. New York, Dover; 1965] have solved this kind of problems using similar techniques. By using transformation in a transformed plane in the complex variable theory, we can obtain the analytical solution easily. We focus on the connection between conformal mapping and curvilinear coordinates, and figure out the relation to take integration by way of mapping in the complex plane. All the transformations and curvilinear coordinates can be unified using the viewpoint of conformal mapping. Their relationship among available methods can be constructed by translation, stretching, rotation and inversion. Finally, an example of eccentric domain is solved by using various mappings and curvilinear coordinates and their relations are linked. Not only geometry transformation is addressed but also the solution of the Laplace equation is obtained. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 17: 314–322, 2009; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20208

Scattering of flexural wave in thin plate with multiple holes by using the null-field integral equation approach

In this paper, a semi-analytical approach is proposed to solve the scat- tering problem of flexural waves and to determine dynamic moment concentration factors (DMCFs) in an infinitethinplatewithmultiplecircularholes. Thenull-field integral formulation is employed in conjunction with degenerate kernels, tensor transformation and Fourier series. In the proposed direct formulation, all dynamic kernels of plate are expanded into degenerate forms and further the rotated degen- erate kernels have been derived for the general exterior problem. By uniformly collocating points on the real boundary, a linear algebraic system is constructed. The results of dynamic moment concentration factors for the plate with one hole are compared with the analytical solution to verify the validity of the proposed method. For the cases of small wave number, the quasi-staticresults of a plate with one or multiple circular holes are compared with the static data of finite element method (FEM) using ABAQUS. Numerical results indicate that the DMCF of two holes is apparently larger than that of one hole when two holes are close to each other. Fictitious frequency appeared in the external problem can be suppressed by using the more number of Fourier series terms. The effect of distance between the centers of holes on dynamic moment concentration factors is also investigated by using the proposed method. Keyword: scattering,flexuralwave, dynamicmoment concentration,biHelmholtz equation, null-field boundary integral equation, degenerate kernel, Fourier series

Wave Scattering from a Slightly Wavy Interface Between two Anisotropic Media

Wave scattering from a periodic interface separating two anisotropic layers in a thick elastic plate is studied in the two-dimensional case. The problem is solved by replacing the exact boundary conditions, i.e. continuous displacement and traction on the wavy interface, by approximate first order conditions on a flat reference surface. Numerical results are presented for a number of cases and compared to the exact solution obtained by the null field approach. The conclusion is that the approximate method gives reasonably accurate results as long as the slope of the surface is small and the amplitude of the wavy surface is not too large compared to the wavelength of the incident wave.

Null-field approach for piezoelectricity problems with arbitrary circular inclusions

In this paper, we derive the null-field integral equation for piezoelectricity problems with arbitrary piezoelectric circular inclusions under remote anti-plane shears and in-plane electric fields. Separable expressions of fundamental solutions and Fourier series for boundary densities are adopted to solve the piezoelectric problem with circular inclusions. Four gains are obtained: (1) well-posed model, (2) singularity free, (3) boundary-layer effect free and (4) exponential convergence. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. Two piezoelectric problems with two piezoelectric circular inclusions are revisited and compared with the Chao and Chang's solutions to demonstrate the validity of our method. The limiting case that the two inclusions separate far away leads to the Pak's solution of a single inclusion. Stress and electric field concentrations are calculated and are dependent on the distance between the two inclusions, the mismatch in the material constants and the magnitude of mechanical and electromechanical loadings. The results for the shear and electric loadings in two directions are also compared well with the Wang and Shen's results.

Analysis of Circular Torsion Bar with Circular Holes Using Null-field Approach

In this paper, we derive the null-field inte- gral equation for a circular bar weakened by circular cav- ities with arbitrary radii and positions under torque. To fully capture the circular geometries, separate forms of fundamental solution in the polar coordinate and Fourier series for boundary densities are adopted. The solution is formulated in a manner of a semi-analytical form since error purely attributes to the truncation of Fourier series. Torsion problems are revisited to demonstrate the validity of our method. Torsional rigidities for different number of holes are also discussed. keyword: Torsion, Null-field integral equation, Fourier series, Circular hole, Torsional rigidity.

Ultrasonic wave propagation in an anisotropic cladding with a wavy interface

The propagation of ultrasonic waves in a thick plate with a cladding is investigated in the two-dimensional case. The surfaces of the plate are traction-free except where an ultrasonic probe is attached and emits waves into the plate. The plate is made of two different materials, the base material and the cladding, and these are both allowed to be anisotropic. The interface between the base material and the cladding is assumed to be wavy (sinusoidal) and this is common in practice for welded claddings. The null field approach is used to solve the wave propagation problem. Thus the starting point is the (surface) integral representations in the two materials. The Green’s tensors are chosen as the half space Green’s tensors as only the integrals along the interface then remain. The Green’s functions are expanded in Fourier transforms along the interface and the surface fields are likewise expanded. Applying the periodicity of the interface a discretized set of equations remains. For the sinusoidal interface all integrals can be computed analytically which leads to an efficient numerical scheme. Some numerical results show the influence of the anisotropy and the wavy interface.

Steady-state vibrations of a thin two-layer plate with delamination

We consider the problem of harmonic vibrations of a thin two-layer plate with horizontal crack. The problem is solved with the help of the null-field approach. The influence of the shape of the crack contour on the amplitude-frequency characteristics of plate vibrations is investigated.

Projection schemes in the null field method

Projection schemes for constructing an approximate solution to the exterior Maxwell boundary problem are presented. The methods are derived in the context of the null field approach by using the fundamental theorem of discrete approximation. We established the convergence and unique solvability of the linear system of equations for any system of functions which is complete on the particle surface. The convergence of the projection schemes is analyzed from a computational point of view.

Spherical wave blockage in reflector antennas

The effects of the spherical wave blockage in reflector antennas is investigated. This problem is likely to occur in axially symmetrical feed antennas of single- and dual-reflector type in both single- and dual-reflector configurations, owing to the presence of primary feeds and their supports including struts that are normally placed between the primary source and the main reflector. The main reflector blockage due to large obstacles is estimated by the well-known null-field technique that employs flat polygonal plate models of the masking structures to define the obscured area. Although this same approach may be used to predict the spherical wave blockage due to the struts, a more rigorous but yet efficient technique is also employed, which consists of superimposing to the primary field the high-frequency scattered field from the struts. This field is calculated by using scattering coefficients that are derived by locally approximating the actual structure, by an infinite circular cylinder. This latter formulation is compared with the null-field technique and validated by an experimental campaign. The measurement setup is particularly useful for isolating spherical wave-blockage effects. It consists of a single-reflector offset antenna where a single strut is mounted with its axis parallel to that of the focusing parabola, thus, practically enforcing the plane wave blockage to vanish. Comparisons with the measurements have shown that the null-field approach is adequate for predicting the secondary pattern for large polygonal obstacle, but it is unsatisfactory to treat the strut blockage. It is found that this latter can be successfully described with the more rigorous high-frequency approach.

Three-Dimensional Subterranean Target Identification by Use of Optimization Techniques

The identification of a subterranean metallic ore from scattering experiments, conducted on the surface of the ground or in a bore hole, is a classic geophysical problem. In general this problem is not well-posed. However, a priori information about the shape of the target provides enough regularization to make the problem numerically stable. The problem is solved by minimizing the mean-square error between an eleven parameter model, based on the null field approach, and the data. The optimization is done with a Newton technique in which a singular value decomposition of the model Jacobian is employed. The algorithm is very stable to noise and makes good reconstructions from feasible starting guesses, for realistically noise contaminated data. (Less)

Anti-plane elastic wave scattering from a rough volumetric crack

The scattering of elastic waves from cracks is a field of study which has a number of important applications in nondestructive testing (NDT) and characterization of materials. So far most theoretical studies have concerned smooth cracks. There is reason to believe that the influence of roughness will scatter the energy in a more diffuse way, thus complicating the interpretation of test results.The problem considered here is the scattering of anti-plane waves, i.e. “horizontally” polarized shear waves, from a crack in an otherwise homogeneous, isotropic, elastic solid. The geometry is taken to be two-dimensional, and the scatterer is a curved volumetric crack with a small random roughness, which is characterized by the RMS height and a correlation length which is related to the average distance between the peaks of the irregularities. The term “volumetric” refers to the fact that the crack surfaces are slightly separated. Effects due to overall curvature as well as effects due to surface roughness and separation of the crack surfaces are thus included in the analysis.The method of solution can be described as an extension of the null field approach where certain matrix elements are expanded in terms of a small parameter describing the deviation from the smooth, nonvolumetric crack. The ensemble averaged amplitude of the scattered field for an incident plane wave has been computed numerically. Some results are given for various values of the frequency, the RMS height, the correlation length and the maximum gap between the crack surfaces.

THE NULL-FIELD APPROACH TO DISSIPATIVE SILENCERS OF ARBITRARY CROSS-SECTION

The wavenumbers and eigenmodes of an infinite cylindrical dissipative silencer of arbitrary cross-section are determined by using the null-field approach. This analytical technique provides an alternative to the numerical variational formulations used in the literature so far. The rate of convergence of the null-field approach is indeed very good for those cross-sections encountered in most practical applications, for instance the flat-oval cross-section of the typical dissipative automotive exhaust silencer. For more complicated geometries, complete determination of the eigenmodes may not be possible even though the wavenumbers can always be obtained. To show the applicability of the technique presented some numerical examples are given.

Modified null field method for P- and SV-wave scattering from a partially debonded fiber

The modified null field approach to elastic P- and SV-wave scattering (in plane strain) from a partially debonded fiber has been developed. The debonded region on the fiber surface is subjected to traction free boundary conditions, whereas the fiber is assumed to be in welded contact with the host medium elsewhere. Additional null field equations for an elliptical extension of the actual surface of the scatterer are introduced which require additional expansions for the field in the region between the extended mathematical boundary and the actual boundary of the scatterer. The numerical accuracy of the modified null field method has been tested for the case of a perfectly bonded fiber for P- and SV-wave incidence. Scattering cross-section plots are presented for different degrees of debonding.

THE POINT-MATCHING METHOD ON DISSIPATIVE SILENCERS OF ARBITRARY CROSS-SECTION

A most straightforward and simple numerical technique known as point-matching, or collocation, is used to obtain the wavenumbers and eigenmodes of an infinite cylindrical dissipative silencer of arbitrary cross-section in the case of negligible mean flow. This technique provides an alternative both to the different numerical variational formulations used in the literature so far, as well as to possible analytical techniques such as for instance the null-field approach. In principle the point-matching method does not suffer from any severe restrictions upon the cross-sectional shape of the silencer even though the rate of convergence is strongly related to the geometry. The grid used in the collocation is also found to be critical for the rate of convergence. Especially in the case of the points in the collocation separated by equal polar angles, the results are poor. To handle this complication a simple technique for choosing the grid is proposed. To show the applicability of the point-matching method, a commercial dissipative silencer is analyzed and comparisons with measurements are made.

Antiplane Elastic Wave Scattering From a Curved Randomly Rough Crack

The problem of scattering of antiplane elastic waves from a rough crack is solved by means of an extension of the null field approach. The extension consists of the expansion of certain matrix elements in terms of a small parameter followed by an analytical ensemble averaging. The geometry is taken to be two-dimensional, and the scatterer is a curved crack with a small random roughness. Both effects due to overall curvature and effects due to surface roughness are thus included in the analysis. It is assumed that the roughness is one where the height has less effect on the scattering than does the gradient (this is satisfied whenever the correlation length of the surface roughness is much smaller than the curvature of the crack), and some numerical results are given for this case. However, it is a straightforward extension of the analysis to consider cases where the height effects are included.

Guided modes in a layered elastic half-space with doubly corrugated surfaces

Time-harmonic Rayleigh, Love, and Rayleigh–Lamb-like waves in a layer on a half-space with both the free surface and the interface doubly periodically corrugated are considered. The dispersion relation for the full problem is derived using the null-field approach. The stopband and passband structure of the physical modes are investigated numerically for doubly sinusoidally corrugated geometries with equal periods and equal corrugation heights. First the Rayleigh-like surface waves that exist when there is no layer are investigated. The passbands, stopbands, and damped parts are presented as a function of frequency and angle of propagation for two different corrugations heights. Some dispersion curves of wave number versus frequency are also given. With the layer present similar dispersion curves are given for some of the lowest Love and Rayleigh–Lamb-like waves. In general, the modes have a considerable angular dependence. Their resonance structure with respect to the position as well as the number of resonances and the stopband widths are affected. The passband, stopband, and the damped part for a mode as well as the possibility of an appearance of new physical solutions are also dependent on the corrugation height. More surprising is that some inhomogeneous modes become involved in the interaction pattern for the layered half-space, thus changing the characteristic behavior of the homogeneous modes and generating new passbands, stopbands, and damped parts.