Boundary Knot Research Articles

A boundary knot method for harmonic elastic and viscoelastic problems using single-domain approach

The boundary knot method is a promising meshfree, integration-free, boundary-type technique for the solution of partial differential equations. It looks for an approximation of the solution in the linear span of a set of specialized radial basis functions that satisfy the governing equation of the problem. The boundary conditions are taken into account through the collocation technique. The specialized radial basis function for harmonic elastic and viscoelastic problems is derived, and a boundary knot method for the solution of these problems is proposed. The completeness issue regarding the proposed set of radial basis functions is discussed, and a formal proof of incompleteness for the circular ring problem is presented. In order to address the numerical performance of the proposed method, some numerical examples considering simple and complex domains are solved.

Free Vibration Analysis of Arbitrary Shaped Plates by Boundary Knot Method

The boundary knot method (BKM) is a truly meshless boundary-type radial basis function (RBF) collocation scheme, where the general solution is employed instead of the fundamental solution to avoid the fictitious outside boundary of the physical domain of interest. In this study, the BKM is first used to calculate the free vibration of free and simply-supported thin plates. Compared with the analytical solution and ANSYS (a commercial FEM code) results, the present BKM is highly accurate and fast convergent.

Investigation of regularized techniques for boundary knot method

AbstractThis study investigates regularization techniques for the boundary knot method (BKM). We consider three regularization methods and two approaches for the determination of the regularization parameter. Our numerical experiments show that Tikhonov regularization in conjunction with generalized cross‐validation approach outperforms the other regularization techniques in the BKM solution of Helmholtz and modified Helmholtz problems. Copyright © 2009 John Wiley & Sons, Ltd.

Cobordism of disk knots

We study cobordisms and cobordisms rel boundary of PL locally-∞at disk knots D ni2 ,! D n . Any two disk knots are cobordant if the cobordisms are not required to flx the boundary sphere knots , and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of J. Levine on the cobordism of sphere knots, we deflne disk knot Seifert matrices and show that two higher-dimensional disk knots with isotopic boundaries are cobordant rel boundary if and only if their disk knot Seifert matrices are algebraically cobordant. We also ask which algebraic cobordism classes can be realized given a flxed boundary knot and provide a complete classiflcation when the boundary knot has no 2-torsion in its middle-dimensional Alexander module. In the course of this classiflcation, we establish a close connection between the Blanchfleld pairing of a disk knot and the Farber-Levine torsion pairing of its boundary knot (in fact, for disk knots satisfying certain connectivity assumptions, the disk knot Blanchfleld pairing will determine the boundary Farber-Levine pairing). In addition, we study the dependence of disk knot Seifert matrices on choices of Seifert surface, demonstrating that all such Seifert matrices are rationally S-equivalent, but not necessarily integrally S-equivalent. 2000 Mathematics Subject Classiflcation: Primary 57Q45; Secondary 57Q60, 11E39, 11E81

Boundary knot method based on geodesic distance for anisotropic problems

The radial basis function (RBF) collocation techniques for the numerical solution of partial differential equation problems are increasingly popular in recent years thanks to their striking merits being inherently meshless, integration-free, and highly accurate. However, the RBF-based methods have markedly been limited to handle isotropic problems due to the use of the isotropic Euclidean distance. This paper makes the first attempt to use the geodesic distance with the RBF-based boundary knot method (BKM) to solve 2D and 3D anisotropic Helmholtz-type and convection-diffusion problems. This approach is mathematically simple and easy to implement, and spectral convergence is numerically observed for problems under complex-shaped boundary. Numerical results show that the BKM based on the geodesic distance can produce highly accurate solutions of anisotropic problems with a relatively small number of knots. This study provides a promising strategy for the RBF-based methods to effectively solve anisotropic problems.

Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation

In this paper, the boundary knot method is extended to the solution of inhomogeneous equations, and it is applied to the Cauchy problem associated with the inhomogeneous Helmholtz equation. Here, we assume that the boundary condition is specified only on a part of the boundary, and the boundary conditions on the remaining part of the boundary are to be determined with the assistance of additional data. Since the resulting matrix equation is highly ill-conditioned, a regularized solution is obtained by employing the truncated singular value decomposition to solve the matrix equation arising from the boundary knot method, with the regularization parameter determined by the L-curve method. Numerical results are presented for several examples with smooth and piecewise smooth boundaries. The numerical verification shows that the proposed numerical scheme is accurate, stable with respect to data noise, and convergent with respect to decreasing the amount of noise in the data.

Boundary knot method for Poisson equations

The boundary knot method is a recent truly meshfree boundary-type radial basis function (RBF) collocation scheme, where the nonsingular general solution is used instead of the singular fundamental solution to evaluate the homogeneous solution, while the dual reciprocity method is employed to the approximation of particular solution. Despite the fact that there are not nonsingular RBF general solutions available for Laplace-type problems, this study shows that the method can successfully be applied to these problems.

General solutions and fundamental solutions of varied orders to the vibrational thin, the Berger, and the Winkler plates

In this note, we derive the general and the fundamental solutions of varied orders of vibrational thin plate, Berger plate, and Winkler plate. These solutions are of important use in the multiple reciprocity BEM, dual reciprocity BEM, boundary particle method, boundary knot method, and a variety of radial basis function techniques.

Solving partial differential equations by BKM combined with DDM

The boundary knot method (BKM) has recently been developed as an inherently meshless, integration-free, boundary-type collocation technique for the numerical discretization of general partial differential equation systems. Unlike the method of fundamental solutions, the use of non-singular radial basis functions in the BKM avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain. But the condition number of the coefficient matrix of the linear algebraic equations which is derived from the homogeneous boundary value problem becomes very high as the total number of boundary knots increase, which in turn affects the accuracy of the computation result. Combined with domain decomposition method, we can avoid this problem easily. In this paper we will solve homogeneous and non-homogeneous partial differential equations using BKM combined with overlapped DDM. From the numerical experiments, we can find that it can solve PDEs accurately.

GROUPS OF LOCALLY-FLAT DISK KNOTS AND NON-LOCALLY-FLAT SPHERE KNOTS

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the "boundary knot" that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classification of all such group pairs in dimension ≥6. For more general knots, we also obtain complete classifications of these group pairs under certain restrictions on the singularities. Finally, we use spinning constructions to realize further examples of boundary knot groups.

Symmetric Boundary Knot Method for Membrane Vibrations under Mixed-type Boundary Conditions

Article Symmetric Boundary Knot Method for Membrane Vibrations under Mixed-type Boundary Conditions was published on December 1, 2005 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 6, issue 4).

Boundary knot method for some inverse problems associated with the Helmholtz equation

The boundary knot method is an inherently meshless, integration-free, boundary-type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill-posed Cauchy problem. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L-curve method. Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems. Copyright © 2005 John Wiley & Sons, Ltd.

Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection–diffusion problems

This paper concerns a numerical study of convergence properties of the boundary knot method (BKM) applied to the solution of 2D and 3D homogeneous Helmholtz, modified Helmholtz, and convection–diffusion problems. The BKM is a new boundary-type, meshfree radial function basis collocation technique. The method differentiates from the method of fundamental solutions (MFS) in that it does not need the controversial artificial boundary outside physical domain due to the use of non-singular general solutions instead of the singular fundamental solutions. The BKM is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method (DRM). Therefore, when applied to inhomogeneous problems, the error of the DRM confounds the BKM accuracy in approximation of homogeneous solution, while the latter essentially distinguishes the BKM, MFS, and boundary element method. In order to avoid the interference of the DRM, this study focuses on the investigation of the convergence property of the BKM for homogeneous problems. The given numerical experiments reveal rapid convergence, high accuracy and efficiency, mathematical simplicity of the BKM.

Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry

AbstractThe boundary knot method (BKM) of very recent origin is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the numerical discretization of general partial differential equation systems. Unlike the method of fundamental solutions, the use of non‐singular general solution in the BKM avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain. The purpose of this paper is to extend the BKM to solve 2D Helmholtz and convection–diffusion problems under rather complicated irregular geometry. The method is also first applied to 3D problems. Numerical experiments validate that the BKM can produce highly accurate solutions using a relatively small number of knots. For inhomogeneous cases, some inner knots are found necessary to guarantee accuracy and stability. The stability and convergence of the BKM are numerically illustrated and the completeness issue is also discussed. Copyright © 2003 John Wiley & Sons, Ltd.

Symmetric boundary knot method

The boundary knot method (BKM) is a recent boundary-type radial basis function (RBF) collocation scheme for general partial differential equations. Like the method of fundamental solution (MFS), the RBF is employed to approximate the inhomogeneous terms via the dual reciprocity principle. Unlike the MFS, the method uses a non-singular general solution instead of a singular fundamental solution to evaluate the homogeneous solution so as to circumvent the controversial artificial boundary outside the physical domain. The BKM is meshfree, super-convergent, integration-free, very easy to learn and program. The original BKM, however, loses symmetricity in the presence of mixed boundary. In this study, by analogy with Fasshauer's Hermite RBF interpolation, we developed a symmetric BKM scheme. The accuracy and efficiency of the symmetric BKM are also numerically validated in some 2D and 3D Helmholtz and diffusion-reaction problems under complicated geometries.

A meshless, integration-free, and boundary-only RBF technique

Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation technique for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while nonsingular general solution leads to a boundary-only RBF formulation for homogenous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions. The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the sole use of the nonsingular part of complete fundamental solution is also discussed.

Discrete and non‐discrete mixed methods for plate bending analysis

AbstractFor plate bending analysis, both discrete and non‐discrete mixed methods are presented by using spline functions in the Hellinger‐Reissner variational principle. Piecewise spline interpolation with local supports and overall spline interpolation with boundary knots are applied to each of two mixed functionals. The prescribed boundary conditions can be considered by multiple boundary knots. The convergence behaviour of the mixed methods presented is studied. Numerical examples show that a high degree of accuracy can be obtained due to continuous properties in the high derivatives of the employed shape functions. The proposed mixed methods are found to be very competitive with other numerical methods for plate bending problems.