Null-field Integral Equation Research Articles

Surface motion of multiple alluvial valleys for incident plane SH-waves by using a semi-analytical approach

In this paper, the degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve the exterior Helmholtz problems with alluvial valleys. The main gain of using degenerate kernels in integral equations is free of calculating the principal values for singular integrals by locating the null-field point exactly on the real boundary. An adaptive observer system is addressed to fully employ the property of degenerate kernels for circular boundaries in the polar coordinate. Image concept and technique of decomposition are utilized for half-plane problems. After moving the null-field point to the real boundary and matching the boundary conditions, a linear algebraic system is obtained without boundary discretization. The unknown coefficients in the algebraic system can be easily determined. The present method is treated as a “semi-analytical” solution since error only attributes to the truncation of Fourier series. Earthquake analysis for the site response of alluvial valley or canyon subject to the incident SH-wave is the main concern. Numerical examples including single and successive alluvial valleys are given to test our program. Limiting cases of a single canyon and two successive canyons are also addressed. Amplification of soft basin is also observed in this study. The validity of the semi-analytical method is verified. Our advantages, well-posed model, principal value free, elimination of boundary-layer effect and exponential convergence and mesh-free, by using the present method are achieved.

A semi-analytical approach for radiation and scattering problems with circular boundaries

In this paper, the radiation and scattering problems with circular boundaries are studied by using the null-field integral equations in conjunction with degenerate kernels and Fourier series to avoid calculating the Cauchy and Hadamard principal values. In implementation, the null-field point can be located on the real boundary owing to the introduction of degenerate kernels for fundamental solution. An adaptive observer system of polar coordinate is considered to fully employ the property of degenerate kernels. For the hypersingular equation, vector decomposition for the radial and tangential gradient is carefully considered. This method can be seen as a semi-analytical approach since errors attribute from the truncation of Fourier series. Neither hypersingularity in Burton and Miller approach nor the CHIEF concepts were required to deal with the problem of irregular frequencies. Four gains, well-posed model, singularity free, boundary-layer effect free and exponential convergence are achieved by using the present approach. A fast convergence rate in exponential order than algebraic one in BEM stems from the series expansions. Three examples were demonstrated to see the validity of the present formulation and show the better accuracy than BEM.

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.

Degenerate scale for multiply connected Laplace problems

The degenerate scale in the boundary integral equation (BIE) or boundary element method (BEM) solution of multiply connected problem is studied in this paper. For the mathematical analysis, we use the null-field integral equation, degenerate kernels and Fourier series to examine the solvability of BIE for multiply connected problem in the discrete system. Two treatments, the method of adding a rigid body term and CHEEF concept (Combined Helmholtz Exterior integral Equation Formulation), are applied to remedy the non-unique solution due to the critical scale. The efficiency and accuracy of the two regularizations are also addressed. For simplicity without loss of generality, the eccentric case is considered to demonstrate the occurring mechanism of degenerate scale.

Null-Field Approach for the Multi-inclusion Problem Under Antiplane Shears

In this paper, we derive the null-field integral equation for an infinite medium containing circular holes and/or inclusions with arbitrary radii and positions under the remote antiplane shear. To fully capture the circular geometries, separable expressions of fundamental solutions in the polar coordinate for field and source points and Fourier series for boundary densities are adopted to ensure the exponential convergence. By moving the null-field point to the boundary, singular and hypersingular integrals are transformed to series sums after introducing the concept of degenerate kernels. Not only the singularity but also the sense of principle values are novelly avoided. For the calculation of boundary stress, the Hadamard principal value for hypersingularity is not required and can be easily calculated by using series sums. Besides, the boundary-layer effect is eliminated owing to the introduction of degenerate kernels. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. The method is basically a numerical method, and because of its semi-analytical nature, it possesses certain advantages over the conventional boundary element method. The exact solution for a single inclusion is derived using the present formulation and matches well with the Honein et al.’s solution by using the complex-variable formulation (Honein, E., Honein, T., and Hermann, G., 1992, Appl. Math., 50, pp. 479–499). Several problems of two holes, two inclusions, one cavity surrounded by two inclusions and three inclusions are revisited to demonstrate the validity of our method. The convergence test and boundary-layer effect are also addressed. The proposed formulation can be generalized to multiple circular inclusions and cavities in a straightforward way without any difficulty.

Alternative derivations for the Poisson integral formula

Poisson integral formula is revisited. The kernel in the Poisson integral formula can be derived in a series form through the direct BEM free of the concept of image point by using the null-field integral equation in conjunction with the degenerate kernels. The degenerate kernels for the closed-form Green's function and the series form of Poisson integral formula are also derived. Two and three-dimensional cases are considered. Also, interior and exterior problems are both solved. Even though the image concept is required, the location of image point can be determined straightforward through the degenerate kernels instead of the method of reciprocal radii.

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.

Null-field Integral Equations for Stress Field around Circular Holes under Antiplane Shear

In this paper, we derive the null-field integral equation for a medium containing circular cavities with arbitrary radii and positions under uniformly remote shear. To fully capture the circular geometries, separate expressions of fundamental solutions in the polar coordinate and Fourier series for boundary densities are adopted. By moving the null-field point to the boundary, singular integrals are transformed to series sums after introducing the concept of degenerate kernels. The solution is formulated in a manner of a semi-analytical form since error purely attributes to the truncation of Fourier series. The two-hole problems are revisited to demonstrate the validity of our method. The bounded-domain approaches using either displacement or stress approaches are also employed. The proposed formulation has been generalized to multiple cavities in a straightforward way without any difficulty.

Null-Field Integral Equation Approach for Plate Problems With Circular Boundaries

Abstract In this paper, a semi-analytical approach for circular plate problems with multiple circular holes is presented. Null-field integral equation is employed to solve the plate problems while the kernel functions in the null-field integral equation are expanded to degenerate kernels based on the separation of field and source points in the fundamental solution. The unknown boundary densities of the circular plates are expressed in terms of Fourier series. It is noted that all the improper integrals are transformed to series sum and are easily calculated when the degenerate kernels and Fourier series are used. By matching the boundary conditions at the collocation points, a linear algebraic system is obtained. After determining the unknown Fourier coefficients, the displacement, slope, normal moment, and effective shear force of the plate can be obtained by using the boundary integral equations. Finally, two numerical examples are proposed to demonstrate the validity of the present method and the results are compared with the available exact solution, the finite element solution using ABAQUS software and the data of Bird and Steele.

Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods

In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches, continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace) and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations is also determined.

True and spurious eigensolutions for the problems with the mixed-type boundary conditions using BEMs

Boundary integral equations and boundary element methods were employed semi-analytically and numerically to study the occurrence of spurious eigenvalues for the interior Helmholtz equations with the mixed-type boundary conditions. The degenerate kernels of fundamental function and Fourier series were utilized in the null-field integral equation to derive the true and spurious eigenfunctions semi-analytically. The complex-valued boundary element method (BEM), the real-part BEM, the imaginary-part BEM and the multiple reciprocity method were utilized to solve the eigenproblem. The results are compared with those of FEM. A simple case of one-dimensional eigenproblem was also addressed. Moreover, the SVD updating techniques in conjunction with the Fredholm's alternative theorem were adopted to sort out the spurious eigenvalues and extract the true ones. It is emphasized that the occurrence of spurious eigenvalues depends on the adopted method (singular or hypersingular formulation) no matter what the given type of boundary conditions for the problem is. The illustrative examples were verified successfully and the numerical results matched well with those of FEM.

Fictitious frequency for the exterior Helmholtz equation subject to the mixed-type boundary condition using BEMs

Boundary integral equations and boundary element methods were employed analytically, semi-analytically and numerically to study the occurrence of fictitious frequency for the exterior Helmholtz equations subject to the mixed-type boundary conditions. A semi-infinite rod and a circular radiator of problems were addressed. Degenerate kernel of the fundamental solution and Fourier series for boundary density were utilized in the null-field integral equation to examine the occurrence of fictitious frequency semi-analytically. The BEM was utilized to solve the solution numerically. The CHIEF technique and Burton and Miller method were adopted to suppress the occurrence of the fictitious frequency. It is emphasized that the occurrence of fictitious frequency depend on the adopted method (singular or hypersingular formulation) no matter what the given type of boundary condition for the problem is. The illustrative examples were tested to verify this finding successfully.

Fictitious frequency for the exterior Helmholtz equation subject to the mixed-type boundary condition using BEMs

Boundary integral equations and boundary element methods were employed analytically, semi-analytically and numerically to study the occurrence of fictitious frequency for the exterior Helmholtz equations subject to the mixed-type boundary conditions. A semi-infinite rod and a circular radiator of problems were addressed. Degenerate kernel of the fundamental solution and Fourier series for boundary density were utilized in the null-field integral equation to examine the occurrence of fictitious frequency semi-analytically. The BEM was utilized to solve the solution numerically. The CHIEF technique and Burton and Miller method were adopted to suppress the occurrence of the fictitious frequency. It is emphasized that the occurrence of fictitious frequency depend on the adopted method (singular or hypersingular formulation) no matter what the given type of boundary condition for the problem is. The illustrative examples were tested to verify this finding successfully.

True and spurious eigensolutions for the problems with the mixed-type boundary conditions using BEMs

Boundary integral equations and boundary element methods were employed semi-analytically and numerically to study the occurrence of spurious eigenvalues for the interior Helmholtz equations with the mixed-type boundary conditions. The degenerate kernels of fundamental function and Fourier series were utilized in the null-field integral equation to derive the true and spurious eigenfunctions semi-analytically. The complex-valued boundary element method (BEM), the real-part BEM, the imaginary-part BEM and the multiple reciprocity method were utilized to solve the eigenproblem. The results are compared with those of FEM. A simple case of one-dimensional eigenproblem was also addressed. Moreover, the SVD updating techniques in conjunction with the Fredholm's alternative theorem were adopted to sort out the spurious eigenvalues and extract the true ones. It is emphasized that the occurrence of spurious eigenvalues depends on the adopted method (singular or hypersingular formulation) no matter what the given type of boundary conditions for the problem is. The illustrative examples were verified successfully and the numerical results matched well with those of FEM.

3D transient eddy current fields using the u-v integral-eigenvalue formulation

The three-dimensional eddy current transient field problem is formulated using the u-v method. This method breaks the vector Helmholtz equation into two scalar Helmholtz equations. Null field integral equations and the appropriate boundary conditions are used to set up an identification matrix which is independent of null field point locations. Embedded in the identification matrix are the unknown eigenvalues of the problem representing its impulse response in time. These eigenvalues are found by equating the determinant of the identification matrix to zero. When the initial transient forcing function is Fourier decomposed into its spatial harmonics, each Fourier component can be associated with a unique eigenvalue by this technique. The true transient solution comes through a convolution of the impulse response, so obtained with the particular external field decay governing the problem at hand. The technique is applied to the FELIX (fusion electromagnetic induction experiments) medium cylinder experiment; computed results are compared with data. A pseudoanalytic confirmation of the eigenvalues so obtained is formulated to validate the procedure.

An integral eigenvalue approach to three-dimensional field problems--A low-frequency variation of SEM

The three-dimensional (3-D) eddy-current transient field problem is formulated first using the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u-v</tex> method. This method breaks the vector Helmholtz equation into two scalar Helmholtz equations. Null-field integral equations and the appropriate boundary conditions germane to the problem are used to set up an identification matrix which is independent of null-field point locations. Embedded in the identification matrix are the unknown eigenvalues of the problem representing its impulse response in time. These eigenvalues are found by equating the determinant of the identification matrix to zero. The eigenvalues, which can be equated with temporal response, are found to be intimately linked to the initial forcing function which triggers the transient in question. When this initial forcing function is Fourier decomposed into its respective spatial harmonics, it is possible to associate with each Fourier component a unique eigenvalue by this technique. The true transient solution comes through a convolution of the impulse response so obtained with the particular imposed external field governing the problem at hand. The technique is applied to the FELIX medium cylinder (a conducting cylinder placed in a collapsing external field) and compared to data. A pseudoanalytic confirmation of the eigenvalues so obtained is formulated to validate the procedure. The technique proposed is applied in the low-frequency regime where the near-field effects must be considered. Application of the technique to a high frequency follows directly if the Coulomb gauge is adopted to represent the vector potential.

An Integral Approach to Electromagnetic Scattering off a Dielectric Wedge

ABSTRACT The scattering of electromagnetic waves off a dielectric wedge is approached using an integral equation formulation. This formulation seeks to express the interfacial field as a combination of the geometric optics field and a diffraction field made up of a number of Bessel functions. Both null field and interfacial integral equations are used to overspecify the problem and thus, seek a least squares approximation to the interfacial field on the wedge interface. The optimum number of Bessel functions used to specify the diffraction field is sought through a minimization of additional interfacial fields of equations. The unboundedness of the wedge interface problem makes it particularly difficult; the use of global basis functions embodying limiting forms of quasi-analytic formulations appears to be an efficacious approach to computing the scattered field. The methodology is quite general, lending itself to the analysis of a wedge with any dielectric constant, having any arbitrary angle.

The use of null field integral equations in magnetic-field problems

The objective of this paper is to discuss the use of the null field integral equations in the solution of eddy current magnetic field problems. Their usage is approached through a development of the classical boundary integral equation technique as applied to irregularly shaped two-dimensional objects. Two immediate advantages are discussed, the first being the circumvention of singularities in the integrands. The second is the ability to improve the conditioning of the determination matrix by appropriate choice of null field points. The technique is tested on an irregularly shaped ellipsoid and results compared to those obtained from perturbation theory.

An orthonormalization method for solving the time harmonic acoustic scattering from finite, smooth, solid, elastic bodies

The null field integral equations [P. C. Waterman, J. Acoust. Soc. Am. 45, 1417–1429 (1969); A. Bostrom, J. Acoust. Soc. Am. 67, 390–398 (1980)] are solved using orthonormalization in the appropriate Hilbert Space. An interior limiting amplitude principle is developed to identify the unique interior elastic field in the long time limit. The result is an algorithm for computing the scattered acoustic and interior elastic field which requires no matrix inversions.