Null-field Integral Equation Research Articles

Semi-analytical approach for torsion problems of a circular bar containing multiple holes and/or cracks

A semi-analytical approach of the null-field integral equation in conjunction containing the degenerate kernels is used to deal with the torsion problems of a circular bar with circular or elliptic holes and/or line cracks. In order to fully capture the elliptic geometry, the use of the addition theorem in terms of the elliptic coordinates plays an important role to expand the fundamental solution into the degenerate form. The boundary densities are expressed by using the eigenfunction expansion for the elliptic boundary. It is worthy of noting that the model of elliptic hole in companion with the limiting process of approaching the length of the semi-minor axis to zero is adopted to simulate the line crack. Besides, we also make the length of the semi-major axis close to the length of the semi-minor axis to approximate the circular boundary. By collocating the observation point exactly on the real boundary and matching the boundary conditions, a linear algebraic system is easily constructed to determine the unknown eigenfucntion coefficients. This approach can be seen as a semi-analytical manner since error purely attributes to the truncation of eigenfunction expansions and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. Finally, several numerical examples of a circular bar with circular or elliptic holes and/or line cracks are employed to show the validity of the proposed approach. Not only the torsional rigidity but also the stress intensity factors are calculated to compare with the available results in the literature.

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.

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.

Multipole expansion of Green’s function for guided waves in a transversely isotropic plate

The multipole expansion of Green’s function in a transversely isotropic plate is derived based on the eigenfunction expansion method. For the derivation, Green’s function is expressed in a bilinear form composed of the regular and singular Lamb-type (or shear-horizontal) wave eigenfunctions. The specific form of the derived Green’s function facilitates the handling of general scattering problems in an elastic plate when numerical methods such as the methods of the null-field integral equations are employed. In the derivation, the integral transform of an arbitrary guided wave field is first constructed by the Lamb-type and shear horizontal wave eigenfunctions that work as the kernel functions. After showing that the thickness-dependent parts of the eigenfunctions are orthogonal to each other in the transformed space, Green’s function is explicitly derived by using the orthogonality. As an application of the derived Green’s function, a scattering problem is solved by the transition matrix method.

Singular boundary method for modified Helmholtz equations

This study makes the first attempt to apply a recent strong-form boundary collocation method using the singular fundamental solutions, namely the singular boundary method (SBM), to 2D and 3D modified Helmholtz equations. By the desingularization of subtracting and adding-back technique, the corresponding nonsingular SBM formulations are derived based on null-field integral equations and an inverse interpolation technique. Numerical demonstrations show the feasibility and accuracy of the present SBM in some benchmark problems.

A new formulation of regularized meshless method applied to interior and exterior anisotropic potential problems

This paper proposes a new formulation of regularized meshless method (RMM), which differs from the traditional RMM in that the traditional formulation generates the diagonal elements of influence matrix via null-field integral equations, while our new one directly employs the boundary integral equations at the domain point to evaluate the diagonal elements. We test the present RMM formulation to two-dimensional anisotropic potential problems in finite and infinite domains in comparison with the traditional RMM. Numerical results show that the present RMM sharply outperforms the traditional RMM in the solution of interior problems, while the latter is clearly superior for exterior problems. A rigorous theoretical analysis of circular domain case also corroborates such numerical experiment observations and is provided in the appendix of this paper.

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.

Water wave interaction with surface-piercing porous cylinders using the null-field integral equations

Following the successful experiences of solving water wave scattering problems for multiple impermeable cylinders by the authors' group, we extend the null-field integral formulation in conjunction with the addition theorem and the Fourier series to deal with the problems of surface-piercing porous cylinders in this paper. In the implementation, the null-field point can be exactly located on the real boundary free of calculating the Cauchy and Hadamard principal values, thanks to the introduction of degenerate kernels (or separable kernels) for fundamental solutions. This method is a semi-analytical approach, since errors attribute from the truncation of the Fourier series. Not only a systematic approach is proposed but also the effect on the near-trapped modes due to porous cylinders and disorder of layout is examined. Several advantages such as mesh-free generation, well-posed model, principal value free, elimination of boundary-layer effect and exponential convergence, over the conventional boundary element method (BEM) are achieved. It is found that the disorder has more influence to suppress the occurrence of near-trapped modes than the porosity. The free-surface elevation is consistent with the results of William and Li and those using the conventional BEM. Besides, the numerical results of the force on the surface of cylinders agree well with those of William and Li. Besides, the present method is a semi-analytical approach for problems containing circular and elliptical shapes at the same time.

A novel method for solving the displacement and stress fields of an infinite domain with circular holes and/or inclusions subject to a screw dislocation

In this paper, the degenerate kernel and superposition technique are employed to solve the screw dislocation problems with circular holes or inclusions. The problem is decomposed into the screw dislocation problem with several holes and the interior Laplace problems for several circular inclusions. Following the success of the null-field integral equation approach, the typical boundary value problems can be solved easily. The kernel functions and unknown boundary densities are expanded by using the degenerate kernel and Fourier series, respectively. To the authors’ best knowledge, the angle-type fundamental solution is first derived in terms of degenerate kernel in this paper. Finally, four examples are demonstrated to verify the validity of the present approach.

Analysis of mutiple-shepers radiation and scattering problems by using a null-field integral equation approach

A systematic approach using the null-field integral equation in conjunction with the degenerate kernel is employed to solve the multiple radiation and scattering problems. Our approach can avoid calculating the principal values of singular and hypersingular integrals. Although we use the idea of null-field integral equation, we can locate the point on the real boundary thanks to the degenerate kernel. The proposed approach is seen as one kind of semi-analytical methods, since the error is attributed from the truncation of spherical harmonics. Finally, the numerical examples including one and two spheres are given to verify the validity of proposed approach.

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

The subject of scattering flexural wave in a thin plate with multiple circular inclusions under the incident flexural wave is studied in this paper. A semi-analytical approach is proposed to solve this problem which can be decomposed into several interior circular inclusion problems and an exterior plate problem subject to the incident wave. The scattered field in the associated exterior problem is solved by using the null-field integral formulation in conjunction with degenerate kernels, tensor transformation and Fourier series. All dynamic kernels of plate in the direct formulation are expanded into degenerate forms to avoid the integral singularity and further the rotated degenerate kernels have been derived to consider the general case of multiple circular inclusions. The proposed results for an infinite plate with one circular inclusion are compared with the available analytical solutions to verify the validity of the proposed method. To demonstrate the generality of the proposed method, the cases of multiple inclusions are studied and their quasi-static results are verified by static data of FEM using ABAQUS. Numerical results indicate that the DMCF of two inclusions is apparently larger than that of one when two inclusions are close to each other. Fictitious frequency appearing in the exterior problem can be suppressed by using the more number of Fourier series terms. Numerical results show that the space between scatterers has the opposite effect on the near-field DMCF in comparison with the far-field scattering pattern.

On the spurious eigenvalues for a concentric sphere in BIEM

In this paper, the null-field integral equation method is employed to study the occurring mechanism of spurious eigenvalues for a concentric sphere. By expanding the fundamental solution into degenerate kernels and expressing the boundary density in terms of spherical harmonics, all boundary integrals can be analytically determined. It is noted that our null-field integral formulation can locate the collocation point on the real boundary thanks to the degenerate kernel. In addition, the spurious eigenvalues are parasitized in the formulations while true eigensolutions are dependent on the boundary condition such as the Dirichlet or Neumann problem. By using the updating term and updating document of singular value decomposition (SVD) technique, true and spurious eigenvalues can be extracted out, respectively. Besides, true and spurious boundary eigenvectors are obtained in the right and left unitary vectors in the SVD structure of the influence matrices. This finding agrees with that of 2D cases.

COMPUTER ASSISTED PROOF OF SPURIOUS EIGENSOLUTION FOR ANNULAR AND ECCENTRIC MEMBRANES

Spurious eigensolutions in the boundary integral equation (BIE) or boundary e1ement method (BEM) for doub1yconnected domain problems, the eccentric and annular membranes, are studied analytically and numerically in this paper. For the mathematica1 ana1ysis, we employ the null-field integral equation, the degenerate kernels and the Fourier series to prove the existence of spurious eigensolutions in the continuous system. Examples of eccentric case, annular membrane and general shape are solved by using the null-field equation approach and BEM, respectively. Based on the numerical experiment, computer-assisted proof for the existence of spurious eigenvalues in companion with the trivial outer boundary data is given. The spurious eigenvalue is found to be the true eigenvalue of a circular membrane with the inner radius. The SVD structure for the four influence matrices is examined Also, the trivial outer boundary densities are found in case of spurious eigenva1ues and are shown in the bar chart.

Interaction of water waves with vertical cylinders using null-field integral equations

The scattering of water waves by bottom-mounted vertical circular cylinders is solved 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 the implementation, the null-field point can be exactly located on the real boundary owing to the introduction of degenerate kernels for fundamental solutions. An adaptive observer system of polar coordinates is considered to fully employ the properties of degenerate kernels. For the hypersingular equation, vector decomposition for the radial and tangential gradients 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 the Burton and Miller approach nor the CHIEF concept was required to deal with the problem of irregular frequencies. Five advantages of free of calculating principal value, well-posed algebraic system, convergence rate of exponential order, meshfree and elimination of boundary-layer effect, are achieved by using the present approach. Numerical results are given for the forces and free-surface elevation around the circular boundaries. Also, the near-trapped behavior arisen from the physical resonance is detected. A general-purpose program for water wave impinging several circular cylinders with arbitrary number, radii, and positions was developed. Several examples of water wave structure interaction by vertical circular cylinders were demonstrated to see the validity of the present formulation.

An Analytical Approach for the Green's Functions of Biharmonic Problems with Circular and Annular Domains

AbstractIn this paper, an analytical approach for deriving the Green's function of circular and annular plate was presented. Null-field integral equations were employed to solve the plate problems while kernel functions were expanded to degenerate kernels. The unknown boundary data of the displacement, slope, normal moment and effective shear force were expressed in terms of Fourier series. It was noticed that all the improper integrals were avoided when the degenerate kernels were used. After determining the unknown Fourier coefficients, the displacement, slope, normal moment and effective shear force of the plate could be obtained by using the boundary integral equations. The present approach was seen as an “analytical” approach for a series solution. Finally, several analytical solutions were obtained. To see the validity of the present method, FEM solutions using ABAQUS were compared well with our analytical solutions. The displacement, radial moment and shear variations of radial and angular positions were presented.

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

Derivation of Green’s function using addition theorem

Following the success of null-field integral equation to solve the BVP of the Laplace equation, this paper employs the addition theorem and superposition technique to revisit the Green’s function of Laplace problems with circular boundaries. The Green’s function is decomposed into two parts, one is the fundamental solution and the other is an infinite plane of circular boundaries subject to the specified boundary conditions derived from the addition theorem. After superimposing the two solutions, the governing equation and boundary condition are both satisfied. Some examples are demonstrated to see the validity of the present method.

Null-field integral equation approach for free vibration analysis of circular plates with multiple circular holes

In this paper, a semi-analytical approach for the eigenproblem of circular plates with multiple circular holes is presented. Natural frequencies and modes are determined by employing the null-field integral formulation in conjunction with degenerate kernels, tensor rotation and Fourier series. In the proposed approach, all kernel functions are expanded into degenerate (separable) forms and all boundary densities are represented by using Fourier series. By uniformly collocating points on the real boundary and taking finite terms of Fourier series, a linear algebraic system can be constructed. The direct searching approach is adopted to determine the natural frequency through the singular value decomposition (SVD). After determining the unknown Fourier coefficients, the cor- responding mode shape is obtained by using the boundary integral equations for domain points. The result of the annular plate, as a special case, is compared with the analytical solu- tion to verify the validity of the present method. For the cases of circular plates with an eccentric hole or multiple circu- lar holes, eigensolutions obtained by the present method are compared well with those of the existing approximate analyt- ical method or finite element method (ABAQUS). Besides, the effect of eccentricity of the hole on the natural frequency and mode is also considered. Moreover, the inherent prob- lem of spurious eigenvalue using the integral formulation is investigated and the SVD updating technique is adopted to suppress the occurrence of spurious eigenvalues. Excellent accuracy, fast rate of convergence and high computational

NULL-FIELD INTEGRAL EQUATION APPROACH FOR EIGENPROBLEMS WITH CIRCULAR BOUNDARIES

In this paper, the eigenproblems with circular boundaries of multiply-connected domain 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. 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 gradients is carefully considered in the polar coordinate system. Direct-searching scheme is employed to detect the eigenvalues by using the singular value decomposition (SVD) technique. Both the singular and hypersingular equations result in spurious eigenvalues which are the associated interior Dirichlet and Neumann problems of interior domain of inner circles, respectively. It is analytically verified that the spurious eigenvalue depends on the radius of any inner circle and numerical experiments support this point. Several examples are demonstrated to see the validity of the present formulation. More number of degrees of freedom of BEM is required to obtain the same accuracy of the present approach.

A Semi-Analytical Approach for Stress Concentration of Cantilever Beams with Holes Under Bending

AbstractIn the paper, the degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve bending problems of a circular beam with circular holes. The main gain of using degenerate kernels in integral equations is free of calculating the principal values for singular integrals. An adaptive observer system is addressed to fully employ the property of degenerate kernels for circular boundaries in the polar coordinate. After moving the null-field point to the 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. Stress concentration is also our concern. Finally, several examples, including two holes and four holes, are given to demonstrate the validity of the proposed method. The results are compared with those of Naghdi and Bird and Steele. Also, the position where the maximum concentration factor occurs is examined. The present formulation can be extended to handle beam problems with arbitrary number and various positions of circular holes by using the developed general-purpose program.