Null-field Boundary Integral Equation Research Articles

An analytical Green’s function for Laplace operator in an infinite plane with two circular holes using degenerate kernels

In this paper, Green’s function problem of an infinite plane containing two circular boundaries is analytically studied by using the boundary integral equation method (BIEM). The original problem is decomposed into two parts by introducing the superposition technique. The first part is a free field caused by the concentrated force. The second part is a Laplace problem subjected to the corresponding boundary conditions. The second part can be solved by using the null-field boundary integral equation in conjunction with the degenerate kernel. Since the geometry of problem under study is composed by two circular boundaries, the kernel function is naturally expanded to series form in terms of the bipolar coordinates. The Green’s function is analytically expressed in three regions instead of two regions for the eccentric domain. To the authors’ best knowledge, this is the first time that the Green’s function of the problem is presented in this form. To show the validity of the present method, the contour result of the present method is compared with those obtained by the image method, the null-field BIE in conjunction with the adaptive observer of polar degenerate kernel.

On the solution arising in two-cylinders electrostatics.

The electrostatics of two cylinders charged to the symmetrical or anti-symmetrical potential is investigated by using the null-field boundary integral equation (BIE) in conjunction with the degenerate kernel of the bipolar coordinates. The undetermined coefficient is obtained according to the Fredholm alternative theorem. The uniqueness of solution, infinite solution, and no solution are examined therein. A single cylinder (circle or ellipse) is also provided for comparison. The link to the general solution space is also done. The condition at infinity is also correspondingly examined. The flux equilibrium along circular boundaries and the infinite boundary is also checked as well as the contribution of the boundary integral (single and double layer potential) at infinity in the BIE is addressed. Ordinary and degenerate scales in the BIE are both discussed. Furthermore, the solution space represented by the BIE is explained after comparing it with the general solution. The present finding is compared to those of Darevski[2] and Lekner[4] for identity.

Analytical solutions for the Laplace problem of an eccentric annular domain

Boundary value problems including a source and no source for an eccentric annular domain are both revisited in this paper. Instead of using complex variables, the null-field boundary integral equation in conjunction with the degenerate kernel is employed to analytically solve the problem. Due to the geometry of the eccentric domain, the fundamental solution is expanded to the degenerate kernel under the bipolar coordinates. Once the degenerate kernel is found, the boundary integral equation is nothing more than a linear algebraic system. The solutions derived by using the present method have been compared with those done by using the complex variables. The potential field of the problems are plotted to show the validity of the present method.

Interaction between a screw dislocation and an elastic elliptical inhomogeneity by using the angular basis function

The behavior of displacement field due to a screw dislocation is similar to the angular basis function (ABF) Arg(z). It is different from the radial basis function (RBF) ln(r) that is used to describe the velocity potential of a sink or source. Nevertheless, the complex-valued fundamental solution ln(z) contains the two parts of RBF ln(r) and ABF Arg(z). In this paper, not only the RBF in the null-field boundary integral equation (BIE) but also the ABF for the screw dislocation are employed to study the interaction between a screw dislocation and an elastic elliptical inhomogeneity. This problem is decomposed into a free field with a screw dislocation and a boundary value problem containing an elliptical inhomogeneity. The boundary value problem is solved by using the RBF and the null-field BIE. Since the geometric shape is an ellipse, the degenerate kernel is expanded to a series form under the elliptical coordinates, while the unknown boundary densities are expanded to eigenfunctions. By combining the degenerate kernel and the null-field BIE, the boundary value problem can be easily solved. The inconsistency between Sendeckyj (In: Simmons JA, et al (eds) Fundamental aspects of dislocation theory. US National Bureau of Standards, Gaithersburg, pp 57–69, 1970) and Gong and Meguid (Int J Eng Sci 32(8):1221–1228, 1994) for the problem was also found by using the present approach. The error in Gong and Meguid (Int J Eng Sci 32(8):1221–1228, 1994) was also printed out. Finally, some examples are demonstrated to verify the validity of the present approach.

A systematic approach for potentials on closely packed cells using the null-field boundary integral equation in conjunction with the degenerate kernel and eigenfunction expansion

A systematic approach based on the null-field integral formula is used to determine the electric potential of a tissue with many cells stimulated by remote electric fields. When the cells are very close to each other, the problem becomes nearly singular and the accuracy of the solution deteriorates. However, in the proposed approach, the highly accurate results are obtained because the separable kernel (degenerate kernel) and eigenfunction expansion are introduced to capture the geometry property in the integral formulation. Only boundary nodes are required instead of boundary elements to satisfy the boundary conditions or interface conditions. The proposed approach could be seen as one kind of meshless and semi-analytical methods. In addition, the error just stems from the number of truncation terms of the eigenfuntion expansion and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. For the problem of closely packed cells, the boundary density of sharp variation could be accurately simulated or captured by increasing the number of terms of eigenfunctions. Finally, the acceptable results are shown to see the efficiency and accuracy of the proposed approach by the given numerical examples including one, three and twenty cells.

Scattering problems of the SH Wave by Using the Null-Field Boundary Integral Equation Method

The scattering problem of seismic waves is an important issue for studying earthquake engineering. In this paper, the null-field boundary integral equation approach was used in conjunction with degenerate kernels and eigenfunction expansion to solve the SH-wave scattering problem of a circular or an elliptical-arc hill. The original problem is divided into subdomains by taking a free-body diagram. One region is an interior boundary value problem. The other is a canyon scattering problem. For the boundary value problem, not only a simply connected domain (elliptical-arc hill problem) but also a doubly connected domain (a circular-arc hill problem containing a circular tunnel or a circular inclusion) is considered. The canyon scattering problem may be addressed in an infinite domain with an artificial boundary of a full plane such that the degenerate kernel can be fully utilized. The null-field integral equation method is used to match boundary conditions. Numerical results are compared favorably with the available data.

Evaluation of the degenerate scale in Laplace equation by using Newton iteration method

This paper provides a numerical solution for the degenerate scale in antiplane elasticity or Laplace equation by using the null field boundary integral equation (BIE). When the used scale coincides with the degenerate scale, the relevant influence matrix is singular. The determinant of the influence matrix is expressed by a function of scale. Therefore, the problem for finding the degenerate scale can be reduced to a problem for finding the zeros of the function. Newton iteration method is used to find the zeros. Three numerical examples are provided in the paper.

Focusing phenomenon and near-trapped modes of SH waves

In this study, the null-field boundary integral equation method (BIEM) and the image method are used to solve the SH wave scattering problem containing semi-circular canyons and circular tunnels. To fully utilize the analytical property of circular geometry, the polar coordinates are used to expand the closed-form fundamental solution to the degenerate kernel, and the Fourier series is also introduced to represent the boundary density. By collocating boundary points to match boundary condition on the boundary, a linear algebraic system is constructed. The unknown coefficients in the algebraic system can be easily determined. In this way, a semi-analytical approach is developed. Following the experience of near-trapped modes in water wave problems of the full plane, the focusing phenomenon and near-trapped modes for the SH wave problem of the half-plane are solved, since the two problems obey the same mathematical model. In this study, it is found that the SH wave problem containing two semi-circular canyons and a circular tunnel has the near-trapped mode and the focusing phenomenon for a special incident angle and wavenumber. In this situation, the amplification factor for the amplitude of displacement is over 300.

Green’s Function Problem of Laplace Equation with Spherical and Prolate Spheroidal Boundaries by Using the Null-Field Boundary Integral Equation

A systematic approach of using the null-field integral equation in conjunction with the degenerate kernel and eigenfunction expansion is employed to solve three-dimensional (3D) Green’s functions of Laplace equation. The purpose of using degenerate kernels for interior and exterior expansions is to avoid calculating the principal values. The adaptive observer system is addressed to employ the property of degenerate kernels in the spherical coordinates and in the prolate spheroidal coordinates. After introducing the collocation points on each boundary and matching boundary conditions, a linear algebraic system is obtained without boundary discretization. Unknown coefficients can be easily determined. Finally, several examples are given to demonstrate the validity of the present approach.

Numerical solution for the degenerate scale in 2D Laplace equation for notch in half-plane using null field BIE

This paper provides a numerical solution for the degenerate scale in the antiplane elasticity with the traction free condition along the half-plane boundary. The image method is used in the formulation of the fundamental solution. After using the Betti׳s reciprocal theorem between the fundamental field and the physical field, the null field boundary integral equation (BIE) is obtained. After performing the coordinate transform, the BIE can be alternatively formulated in the normal scale, and a linear algebraic equation is obtained after discretization. The influence matrix in the normal scale is invertible. The degenerate scale can be evaluated by the invertible property of the influence matrix.

Evaluation of the degenerate scale in antiplane elasticity using null field BIE

This paper provides a numerical solution for the degenerate scale in antiplane elasticity using the null field boundary integral equation (BIE). With coordinate transformation, the BIE can be formulated in a normal scale, and a basic solution for the BIE in the normal scale is illustrated. The degenerate scale is easily obtained from the basic solution. Several numerical examples are given.

Eigenanalysis for a confocal prolate spheroidal resonator using the null-field BIEM in conjunction with degenerate kernels

In this paper, we employ the null-field boundary integral equation method (BIEM) in conjunction with degenerate kernels to solve eigenproblems of the prolate spheroidal resonators. To detect the spurious eigenvalues and the corresponding occurring mechanisms which are common issues while utilizing the boundary element method or the BIEM, we use angular prolate spheroidal wave functions and triangular functions to expand boundary densities. In this way, the boundary integral of a prolate spheroidal surface is exactly determined, and eigenequations are analytically derived. It is revealed that the spurious eigenvalues depend on the integral, representations and the shape of the inner boundary. Furthermore, it is interesting to find that some roots of the confocal prolate spheroidal resonator are double roots no matter that they are true or spurious eigenvalues. Illustrative examples include confocal prolate spheroidal resonators of various boundary conditions. To validate these findings and accuracy of the present approach, the commercial finite-element software ABAQUS is also applied to perform acoustic analyses. Good agreement is obtained between the acoustic results obtained by the null-field BIEM and those provided by the commercial finite-element software ABAQUS.

Null-field BIEM for solving a scattering problem from a point source to a two-layer prolate spheroid

In this paper, the acoustic scattering problem from a point source to a two-layer prolate spheroid is solved by using the null-field boundary integral equation method (BIEM) in conjunction with degenerate kernels. To fully utilize the spheroidal geometry, the fundamental solutions and the boundary densities are expanded by using the addition theorem and spheroidal harmonics in the prolate spheroidal coordinates, respectively. Based on this approach, the collocation point can be located on the real boundary, and all boundary integrals can be determined analytically. In real applications of a two-layer prolate spheroidal structure, it can be applied to simulate the kidney-stone biomechanical system. Here, we consider the confocal structure to simulate the kidney-stone system since its analytical solution can be analytically derived. The parameter study for providing some references in the clinical medical treatment is also considered. To check the validity of the null-field BIEM, a special case of the acoustic scattering problem of a point source by a rigid scatterer is also done by setting the density of the inner prolate spheroid to infinity. Results of the present method are compared with those obtained using the commercial finite element software ABAQUS.

A semi-analytical method for near-trapped mode and fictitious frequencies of multiple scattering by an array of elliptical cylinders in water waves

In this paper, we focus on the water wave scattering by an array of four elliptical cylinders. The null-field boundary integral equation method (BIEM) is used in conjunction with degenerate kernels and eigenfunctions expansion. The closed-form fundamental solution is expressed in terms of the degenerate kernel containing the Mathieu and the modified Mathieu functions in the elliptical coordinates. Boundary densities are represented by using the eigenfunction expansion. To avoid using the addition theorem to translate the Mathieu functions, the present approach can solve the water wave problem containing multiple elliptical cylinders in a semi-analytical manner by introducing the adaptive observer system. Regarding water wave problems, the phenomena of numerical instability of fictitious frequencies may appear when the BIEM/boundary element method (BEM) is used. Besides, the near-trapped mode for an array of four identical elliptical cylinders is observed in a special layout. Both physical (near-trapped mode) and mathematical (fictitious frequency) resonances simultaneously appear in the present paper for a water wave problem by an array of four identical elliptical cylinders. Two regularization techniques, the combined Helmholtz interior integral equation formulation (CHIEF) method and the Burton and Miller approach, are adopted to alleviate the numerical resonance due to fictitious frequency.

Study on harbor resonance and focusing by using the null-field BIEM

Abstract In this paper, the resonance of a circular harbor is studied by using the semi-analytical approach. The method is based on the null-field boundary integral equation method in conjunction with degenerate kernels and the Fourier series. The problem is decomposed into two regions by employing the concept of taking free body. One is a circular harbor, and the other is a problem of half-open sea with a coastline subject to the impermeable (Neumann) boundary condition. It is interesting to find that the SH wave impinging on the hill can be formulated by the same mathematical model. After finding the analogy between the harbor resonance and hill scattering, focusing of the water wave inside the harbor as well as focusing in the hill scattering are also examined. Finally, two numerical examples, circular harbor problems of 60° and 180° opening entrance, are both used to verify the validity of the present formulation.

A Semianalytical Approach for a Nonconfocal Suspended Strip in an Elliptical Waveguide

A problem of a nonconfocal suspended strip in an elliptical waveguide is analyzed by using a semianalytical approach, which is the so-called null-field boundary integral-equation method (BIEM). The null-field BIEM is proposed by introducing the idea of null field, degenerate kernels, and eigenfunction expansion to improve the conventional dual boundary-element method (BEM). A closed-form fundamental solution can be expressed in terms of the degenerate kernel containing the Mathieu and modified Mathieu functions in the elliptical coordinates. Boundary densities are represented by using the eigenfunction expansion. By this way, the efficiency is promoted in three aspects: analytical boundary integral without numerical error, natural bases for boundary densities, and exact description of boundary geometry. Due to the semianalytical formulation, the null-field BIEM can fully capture the property of geometry and the error only occurs from the truncation of the number of the eigenfunction expansion terms in the real implementation. The present method is also a kind of meshless method since only boundary nodes are needed to construct influence matrices instead of using boundary elements. Both TE and TM cases are considered in this paper. To verify the validity of the present method, the dual BEM and finite-element method are also utilized to provide cutoff wavenumbers. Besides, the analytical solution of a confocal elliptical waveguide can be derived by using the present method. After comparing with published data, good agreement is made.

On the null and nonzero fields for true and spurious eigenvalues of annular and confocal elliptical membranes

Abstract In this paper, the dual boundary element method (BEM) and the null-field boundary integral equation method (BIEM) are both employed to solve two-dimensional eigenproblems. The positions of true and spurious eigenvalues for circular, elliptical, annular and confocal elliptical membranes are analytically examined in the continuous system and numerically studied in the discrete system. To analytically study eigenproblems, the polar and elliptical coordinates in conjunction with the Bessel functions, the Mathieu functions, the Fourier series and eigenfunction expansions are adopted. The fundamental solution is expanded into the degenerate kernel while the boundary densities of circular and elliptical boundaries are expanded by using the Fourier series and eigenfunction expansion, respectively. Dirichlet and Neumann eigenproblems are both considered as well as simply and doubly-connected domains are both addressed. By employing the singular value decomposition (SVD) technique in the discrete system, the common right unitary vectors corresponding to the true eigenvalues for the singular and hypersingular formulations are found while the common left unitary vectors corresponding to the spurious eigenvalues are obtained for the singular formulation or hypersingular formulation. True eigenvalues depend on the boundary condition while spurious eigenvalues depend on the approach, the singular formulation or hypersingular formulation of BEM/BIEM. Nonzero field in the domain are analytically derived and are numerically verified in case of the true eigenvalue while the interior null field and nonzero field for the complementary domain are obtained in case of the spurious eigenvalue. Four examples, circular, elliptical, annular and confocal elliptical membranes, are considered to demonstrate the finding of the present paper. After comparing with the analytical and numerical results, good agreements are made. The dual BEM displays the dual structure in the unitary vector and the null field.

Water wave problems using null-field boundary integral equations: ill-posedness and remedies

In this article, we focus on the hydrodynamic scattering of water wave problems containing circular and/or elliptical cylinders. Regarding water wave problems, the phenomena of numerical instability due to fictitious frequencies may appear when the boundary element method (BEM) is used. We examine the occurring mechanism of fictitious frequency in the BEM through a water wave problem containing an elliptical cylinder. In order to study the fictitious frequency analytically, the null-field boundary integral equation method in conjunction with degenerate kernels is employed to derive the analytical solution. The modal participation factor for the numerical instability of zero divided by zero can be exactly determined in a continuous system even though the circulant matrix cannot be obtained in a discrete system for the elliptical case. It is interesting to find that irregular values depend on the geometry of boundaries as well as integral representations and happen to be zeros of the mth-order (even or odd) modified Mathieu functions of the first kind or their derivatives. To avoid using the addition theorem to translate the Bessel functions to the Mathieu functions, the present approach can solve for the water wave problem containing circular and/or elliptical cylinders at the same time in a semi-analytical manner by using the adaptive observer system. The closed-form fundamental solution is expressed in terms of the degenerate kernel in the polar and elliptic coordinates for circular and elliptical cylinders, respectively. Three examples are considered to demonstrate the validity of the present approach, including an elliptical cylinder, two parallel identical elliptical cylinders and one circular cylinder and one elliptical cylinder. Finally, two regularization techniques, the combined Helmholtz interior integral equation formulation method and the Burton and Miller approach, are adopted to alleviate the numerical resonance due to fictitious frequency.

SH-wave scattering by a semi-elliptical hill using a null-field boundary integral equation method and a hybrid method

Following the success of seismic analysis of a semi-circular hill, the problem of SH-wave scattering by a semi-elliptical hill is revisited by using the null-field boundary integral equation method (BIEM). To fully use the analytical property in the null-field boundary integral equation approach in conjunction with degenerate kernels for solving the semi-elliptical hill scattering problem, the problem is decomposed to two regions to produce elliptical boundaries by using the technique of taking free body. One is the half-plane problem containing a semi-elliptical boundary. This semi-infinite problem is imbedded in an infinite plane with an artificial elliptical boundary such that degenerate kernel can be fully applied. The other is an interior problem bounded by an elliptical boundary. The degenerate kernel in the elliptic coordinates for two subdomains is used to expand the closed-form fundamental solution. The semi-analytical formulation in companion with matching boundary conditions yields six constraint equations. Instead of finding admissible wave-expansion bases, our null-field BIEM in conjunction with degenerate kernels has the five features over the conventional BIEM/BEM, (1) free of calculating principal values, (2) exponential convergence, (3) elimination of boundary-layer effect, (4) meshless and (5) well-posed system. All numerical results are compared well with those of using the hybrid method which is also described in this paper. It is interesting to find that a focusing phenomenon is also observed in this study.

On near-trapped modes and fictitious frequencies for water wave problems containing an array of circular cylinders using a null-field boundary integral equation

To avoid using the addition theorem to translate the Bessel function, scattering of water waves by an array of circular cylinders is solved by using the null-field boundary integral equations, in conjunction with the adaptive observer system. Both the near-trapped modes (physics) and fictitious frequencies (mathematics) are observed. To alleviate the resonance problem of fictitious frequency for multiple cylinders, Combined Helmholtz Interior integral Equation Formulation (CHIEF) approach and Burton and Miller formulation are both considered. Regarding the Burton and Miller approach, hypersingular integrals can be easily calculated by using series summability without any difficulty owing to the introduction of degenerate kernels. The highly rank-deficient matrices for equal radius of cylinders are numerically examined and the rank is improved by adding valid CHIEF constraints. Besides, the selection of location and number for CHIEF points is addressed instead of trial and error. Parameter study of the incident angle on the resultant force is investigated. The effect of spacing and radius of cylinders on the near-trapped mode and fictitious frequency is also discussed. Several examples of water wave interaction by circular cylinders were demonstrated to see the validity of the present formulation.