Shing-Kai Kao

Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs

Abstract It is well known that BEM/BIEM results in degenerate scale for a two-dimensional Laplace interior problem subjected to the Dirichlet boundary condition. In such a case, there is nontrivial boundary normal flux even if the trivial boundary potential is specified. It is proved that the unit logarithmic capacity in the Riemann conformal mapping with respect to the unit circle results in a null field for the interior domain. The logarithmic capacity is defined as the leading coefficient of the linear term in the Riemann conformal mapping. First, the real-variable BIE is transformed to the complex variable BIE. By considering the analytical field and taking care of the path of the branch cut, we can prove that unit logarithmic capacity in the Riemann conformal mapping results in a degenerate scale. When the logarithmic capacity is equal to one, a trivial interior field can be obtained but an exterior field is derived to be nonzero using the logarithmic function. Two mapping functions, the Riemann conformal mapping for the geometry and the logarithmic function for the physical field, are both utilized. This matches well with the BEM result that an interior trivial field yields nonzero boundary flux in case of degenerate scale. Regarding the ordinary scale, BIE results in a null field in the exterior domain owing to the Green’s third identity. It is interesting to find that ordinary and degenerate scales result in a null field in the exterior and interior domains, respectively. A parameter study for the scaling constant and the leading coefficient of the z term in the Riemann conformal mapping is also done. To demonstrate this finding, different shapes were demonstrated. Theoretical derivation using the Riemann conformal mapping with the unit logarithmic capacity and the degenerate scale in the BEM/BIEM both analytically and numerically indicate the null field in the interior domain.

Regularization methods for ill-conditioned system of the integral equation of the first kind with the logarithmic kernel

The occurring mechanism of the ill-conditioned system due to degenerate scale in the direct boundary element method (BEM) and the indirect BEM is analytically examined by using degenerate kernels. Five regularization techniques to ensure the unique solution, namely hypersingular formulation, method of adding a rigid body mode, rank promotion by adding the boundary flux equilibrium (direct BEM), CHEEF method and the Fichera’s method (indirect BEM), are analytically studied and numerically implemented. In this paper, we examine the sufficient and necessary condition of boundary integral formulation for the uniqueness solution of 2D Laplace problem subject to the Dirichlet boundary condition. Both analytical study and BEM implementation are addressed. For the analytical study, we employ the degenerate kernel in the polar and elliptic coordinates to derive the unique solution by using five regularization techniques for any size of circle and ellipse, respectively. Full rank of the influence matrix in the BEM using Fichera’s method for both ordinary scale and degenerate scale is also analytically proved. In numerical implementation, the BEM programme developed by NTOU/MSV group is employed to see the validity of the above formulation. Finally, the circular and elliptic cases are numerically demonstrated by using five regularization techniques. Besides, a general shape of a regular triangle is numerically implemented to check the uniqueness solution of BEM.

Revisit of degenerate scales in the BIEM/BEM for 2D elasticity problems

ABSTRACT The boundary integral equation method in conjunction with the degenerate kernel, the direct searching technique (singular value decomposition), and the only two-trials technique (2 × 2 matrix eigenvalue problem) are analytically and numerically used to find the degenerate scales, respectively. In the continuous system of boundary integral equation, the degenerate kernel for the 2D Kelvin solution in the polar coordinates is reviewed and the degenerate kernel in the elliptical coordinates is derived. Using the degenerate kernel, an analytical solution of the degenerate scales for the elasticity problem of circular and elliptical cases is obtained and compared with the numerical result. Further, the triangular case and square case were also numerically demonstrated.

Revisit of the Indirect Boundary Element Method: Necessary and Sufficient Formulation

Although the boundary element method (BEM) has been developed over forty years, the single-layer potential approach is incomplete for solving not only the interior 2D problem in case of a degenerate scale but also the exterior problem with bounded potential at infinity for any scale. The indirect boundary element method (IBEM) is revisited to examine the uniqueness of the solution by using the necessary and sufficient boundary integral equation (BIE). For the necessary and sufficient BIE, a free constant and an extra constraint are simultaneously introduced into the conventional IBEM. The reason why a free constant and an extra constraint are both required is clearly explained by using the degenerate kernel. In order to complete the range of the IBEM lacking a constant term in the case of a degenerate scale, we provide a complete base with a constant. On the other hand, the formulation of the IBEM does not contain a constant field in the degenerate kernel expansion for the exterior problem. To satisfy the bounded potential at infinity, the integration of boundary density is enforced to be zero. Besides, sources can be distributed on either the real boundary or the auxiliary (artificial) boundary in this IBEM. The enriched IBEM is not only free of the degenerate-scale problem for the interior problem but also satisfies the bounded potential at infinity for the exterior problem. Finally, three examples, a circular domain, an infinite domain with two circular holes and an eccentric annulus were demonstrated to illustrate the validity and the effectiveness of the necessary and sufficient BIE.

Revisit of a degenerate scale

Boundary element method (BEM) has been employed in engineering analysis since 1956, it has been widely applied in the engineering. However, the BEM/BIEM may result in an ill-conditioned system in some special situations, such as the degenerate scale. The degenerate scale also relates to the logarithmic capacity in the modern potential theory. In this paper, three indexes to detect the degenerate scale and five regularization techniques to circumvent the degenerate scale are reviewed and a new self-regularization technique by using the bordered matrix is proposed. Both the analytical study and the BEM implementation are addressed. For the analytical study, we employ the Riemann conformal mapping of complex variables to derive the unit logarithmic capacity. The degenerate scale can be analytically derived by using the conformal mapping as well as numerical detection by using the BEM. In the theoretical aspect, we prove that unit logarithmic capacity in the Riemann conformal mapping results in a degenerate scale. We revisit the Fredholm alternative theorem by using the singular value decomposition (SVD, the discrete system) and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. Besides, a zero index by using the free constant in Ficheras approach is also proposed to examine the degenerate scale. According to the relation between the SVD structure and Ficheras technique, we numerically provide a new self-regularization method in the matrix level. Finally, a semi-circular case and a special-shape case are designed to demonstrate the validity of six regularization techniques.

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.

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

Purpose The purpose of this paper is to detect the degenerate scale of a 2D bending plate analytically and numerically. Design/methodology/approach To avoid the time-consuming scheme, the influence matrix of the boundary element method (BEM) is reformulated to an eigenproblem of the 4 by 4 matrix by using the scaling transform instead of the direct-searching scheme to find degenerate scales. Analytical degenerate scales are derived from the boundary integral equation (BIE) by using the degenerate kernel only for the circular case. Numerical results of the direct-searching scheme and the eigen system for the arbitrary shape are also considered. Findings Results using three methods, namely, analytical derivation, the direct-searching scheme and the 4 by 4 eigen system, are also given for the circular case and arbitrary shapes. Finally, addition of a constant for the kernel function makes original eigenvalues (2 real roots and 2 complex roots) of the 4 by 4 matrix to be all real. This indicates that a degenerate scale depends on the kernel function. Originality/value The analytical derivation for the degenerate scale of a 2D bending plate in the BIE is first studied by using the degenerate kernel. Through the reformed eigenproblem of a 4 by 4 matrix, the numerical solution for the plate of an arbitrary shape can be used in the plate analysis using the BEM.

Three detecting indexes and five regularization techniques for degenerate scales in the BEM/BIEM

It is well known that BEM/BIEM results in degenerate scale for a twodimensional Laplace problem subjected to the Dirichlet boundary condition. In this paper, we reviewed three indexes for detecting the degenerate scale in BEM/BIEM and five regularization techniques to ensure the unique solution, the hypersingular formulation rank promotion by adding the boundary flux equilibrium, CHEEF method, (direct BEMs), Fichera’s method (indirect BEM) and method of adding a rigid body mode. In the numerical implementation, the BEM program developed by the NTOU/MSV group is employed to see the validity of the above formulation. Finally, a general shape of a regular triangle is numerically implemented to check the uniqueness solution of BEM.