Galerkin Boundary Node Method Research Articles

Improved complex variable moving least squares approximation for three-dimensional problems using boundary integral equations

The improved complex variable moving least squares approximation is an efficient method to generate meshless approximation functions. In the past, the approximation has been used only for 2D problems. In this paper, the approximation is developed to solve 3D problems. Theoretical error estimation of the approximation is given. Then, incorporating the approximation into boundary integral equations, a symmetric and boundary-only meshless method, the complex variable Galerkin boundary node method, is developed and analyzed theoretically for 3D potential, Helmholtz and Stokes problems. Numerical results demonstrate the accuracy and efficiency of the developed method.

A meshless complex variable Galerkin boundary node method for potential and Stokes problems

In this study, combining the boundary integral equations (BIEs) with the complex variable moving least squares (CVMLS) approximation, a symmetric and boundary-only meshless method, the complex variable Galerkin boundary node method (CVGBNM), is developed. Numerical applications and theoretical error estimates of the CVGBNM are derived for BIEs, potential problems and Stokes problems. Finally, numerical examples are given to demonstrate the efficacy of the method.

Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases

This paper first discusses the inherent instability of the interpolating moving least squares (IMLS) method. In the original IMLS method, non-scaled polynomial bases are used. Theoretical and numerical results indicate that the stability of the original IMLS method decreases as the separation distance decreases. Then, using shifted and scaled polynomial bases, a stabilized algorithm of the IMLS method is proposed and analyzed. As an application, the stabilized IMLS method is finally introduced into the meshless Galerkin boundary node method (GBNM) to produce a stabilized GBNM for potential problems and Stokes problems. Numerical examples are given to demonstrate the stability and convergence of the presented stabilized algorithms.

Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving least-square approximation

This paper combines variational formulations of boundary integral equations (BIEs) and the improved moving least-square (IMLS) approximation to develop a symmetric meshless method, the improved Galerkin boundary node method (IGBNM) for boundary-only analysis of boundary value problems in potential theory and viscous fluid flow. The IMLS approximation is used in the IGBNM to construct meshless shape functions. In the IMLS approximation, the system of algebra equations can be solved without the inverse matrix. Compared with other MLS-based meshless methods, the IGBNM is a direct numerical method in which the basic unknown quantities are the actual nodal values. Besides, boundary conditions in the IGBNM are implemented directly and easily, and the resulting ‘stiffness’ matrices conserve the symmetry and positive definiteness of the variational formulations. Thus, it gives a higher computational efficiency. Total details of numerical implementation and error analysis of the IGBNM are first given for general BIEs. Then, taking potential problems and viscous fluid flow problems as examples, we set up a framework for numerical implementation and asymptotic error estimates of the IGBNM. Finally, some numerical tests are presented to demonstrate the efficiency of the method.

A meshless interpolating Galerkin boundary node method for Stokes flows

Combining an improved interpolating moving least-square (IIMLS) scheme and a variational formulation of boundary integral equations, a symmetric and boundary-only meshless method, which is called the interpolating Galerkin boundary node method (IGBNM), is developed in this paper for 2D and 3D Stokes flow problems. The IIMLS is used to form shape functions with delta function property. So unlike the Galerkin boundary node method (GBNM), the IGBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables. Besides, to obtain uniqueness of unknown boundary functions and to retain symmetry of system matrices, a Lagrange multiplier is introduced and then a variational formulation with side conditions is gained. Consequently, in the IGBNM, boundary conditions can be applied directly and easily, and the resulting system matrices are symmetric. Thus, the IGBNM gives greater computational precision than the GBNM. The numerical formulae are valid for 2D and 3D Stokes flows and also valid for both interior and exterior problems simultaneously. The capability of the IGBNM is illustrated and assessed by some numerical examples.

The Galerkin boundary node method for magneto-hydrodynamic (MHD) equation

In this research, a variational reformulation for magneto-hydrodynamic (MHD) problem is derived. Existence and uniqueness of the weak solution are discussed. The Galerkin boundary node method is a boundary meshless method which uses MLS basis functions to approximate solution of problems. This paper tries to apply this method for a variational form of the magneto-hydrodynamic (MHD) problem. Numerical experiments reveal that the method is effective and convenient for this problem.

A meshless Galerkin method with moving least square approximations for infinite elastic solids

Combining moving least square approximations and boundary integral equations, a meshless Galerkin method, which is the Galerkin boundary node method (GBNM), for two- and three-dimensional infinite elastic solid mechanics problems with traction boundary conditions is discussed. In this numerical method, the resulting formulation inherits the symmetry and positive definiteness of variational problems, and boundary conditions can be applied directly and easily. A rigorous error analysis and convergence study for both displacement and stress is presented in Sobolev spaces. The capability of this method is illustrated and assessed by some numerical examples.

THE MESHLESS GALERKIN BOUNDARY NODE METHOD FOR TWO-DIMENSIONAL SOLIDS

The Galerkin boundary node method (GBNM) is developed for two-dimensional solid mechanics problems. The GBNM is a boundary only meshless method that combines an equivalent variational form of boundary integral formulations for governing equations with the moving least-squares (MLS) approximations for construction of the trial and test functions. In this method, boundary conditions can be implemented directly and easily despite the MLS shape functions lack the delta function property, and the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The optimal asymptotic error estimates of this approach for displacements and stresses are derived in detail in Sobolev spaces. Numerical tests are also given to demonstrate the developed algorithms.

Application of the meshless Galerkin boundary node method to potential problems with mixed boundary conditions

Abstract In this paper, the meshless Galerkin boundary node method is developed for boundary-only analysis of two- and three-dimensional potential problems with mixed boundary conditions of Dirichlet and Neumann type. This meshless algorithm leads to a symmetric and positive definite system of linear equations. Additionally, boundary conditions can be implemented directly and easily despite the fact that the employed meshless shape functions lack the delta function property. Theoretical error analysis and numerical results indicate that it is an efficient and accurate numerical method.

A priori and a posteriori analysis of the meshless Galerkin boundary node method for three-dimensional elasticity

The meshless Galerkin boundary node method is presented in this paper for boundary-only analysis of three-dimensional elasticity problems. In this method, boundary conditions can be implemented directly and easily despite the employed moving least-squares shape functions lack the delta function property, and the resulting system matrices are symmetric and positive definite. A priori error estimates and the consequent rate of convergence are presented. A posteriori error estimates are also provided. Reliable and efficient error estimators and an efficient and convergent adaptive meshless algorithm are then derived. Numerical examples showing the efficiency of the method, confirming the theoretical properties of the error estimates, and illustrating the capability of the adaptive algorithm, are reported.

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L2-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.

Adaptive meshless Galerkin boundary node methods for hypersingular integral equations

Adaptive refinement techniques are developed in this paper for the meshless Galerkin boundary node method for hypersingular boundary integral equations. Two types of error estimators are derived. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two consecutive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the numerical solution itself and its projection. These error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization scheme is presented to accomodate the non-local property of hypersingular integral operators for the needed computable local error indicators. The convergence of the adaptive meshless techniques is verified theoretically. To confirm the theoretical results and to show the efficiency of the adaptive techniques, numerical examples in 2D and 3D with high singularities are provided.

On error estimator and adaptivity in the meshless Galerkin boundary node method

In this article, a simple a posteriori error estimator and an effective adaptive refinement process for the meshless Galerkin boundary node method (GBNM) are presented. The error estimator is formulated by the difference between the GBNM solution itself and its L 2-orthogonal projection. With the help of a localization technique, the error is estimated by easily computable local error indicators and hence an adaptive algorithm for h-adaptivity is formulated. The convergence of this adaptive algorithm is verified theoretically in Sobolev spaces. Numerical examples involving potential and elasticity problems are also provided to illustrate the performance and usefulness of this adaptive meshless method.

Meshless Galerkin algorithms for boundary integral equations with moving least square approximations

In this paper, we first give error estimates for the moving least square (MLS) approximation in the H k norm in two dimensions when nodes and weight functions satisfy certain conditions. This two-dimensional error results can be applied to the surface of a three-dimensional domain. Then combining boundary integral equations (BIEs) and the MLS approximation, a meshless Galerkin algorithm, the Galerkin boundary node method (GBNM), is presented. The optimal asymptotic error estimates of the GBNM for three-dimensional BIEs are derived. Finally, taking the Dirichlet problem of Laplace equation as an example, we set up a framework for error estimates of the GBNM for boundary value problems in three dimensions.

Development of a meshless Galerkin boundary node method for viscous fluid flows

In this paper, a meshless Galerkin boundary node method is developed for boundary-only analysis of the interior and exterior incompressible viscous fluid flows, governed by the Stokes equations, in biharmonic stream function formulation. This method combines scattered points and boundary integral equations. Some of the novel features of this meshless scheme are boundary conditions can be enforced directly and easily despite the meshless shape functions lack the delta function property, and system matrices are symmetric and positive definite. The error analysis and convergence study of both velocity and pressure are presented in Sobolev spaces. The performance of this approach is illustrated and assessed through some numerical examples.

On a Galerkin boundary node method for potential problems

A Galerkin boundary node method (GBNM), for boundary only analysis of partial differential equations, is discussed in this paper. The GBNM combines an equivalent variational form of a boundary integral equation with the moving least-squares (MLS) approximations for generating the trial and test functions of the variational formulation. In this approach, only a nodal data structure on the boundary of a domain is required, and boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Formulations of the GBNM using boundary singular integral equations of the second kind for potential problems are developed. The theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method.

The meshless Galerkin boundary node method for Stokes problems in three dimensions

AbstractThe Galerkin boundary node method (GBNM) is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least‐squares (MLS) approximations for generating the trial and test functions. In this approach, boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Besides, the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The GBNM is developed in this paper for solving three‐dimensional stationary incompressible Stokes flows in primitive variables. The numerical scheme is based on variational formulations for the first‐kind integral equations, which are valid for both interior and exterior problems simultaneously. A rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in Sobolev spaces. The capability of the method is also illustrated and assessed through some selected numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.

Adaptive methodology for the meshless Galerkin boundary node method

The Galerkin boundary node method (GBNM) is a boundary-type meshless method that combines a variational form of boundary integral formulations for governing equations with the moving least-squares approximations for generation of the trial and test functions. In this paper, a posteriori error estimate and an effective adaptive h-refinement procedure are developed in conjunction with the GBNM. The error estimator is based on the difference between numerical solutions obtained using two successive nodal arrangements. The reliability and efficiency of this error estimator and the convergence of this adaptive meshless scheme are verified theoretically. Numerical examples are also given to show the efficiency of the adaptive methodology.

Meshless analysis of two-dimensional Stokes flows with the Galerkin boundary node method

In this paper, the Galerkin boundary node method (GBNM) is developed for the solution of stationary Stokes problems in two dimensions. The GBNM is a boundary only meshless method that combines a variational form of boundary integral formulations for governing equations with the moving least-squares (MLS) approximations for construction of the trial and test functions. Boundary conditions in this approach are included into the variational form, thus they can be applied directly and easily despite the MLS shape functions lack the property of a delta function. Besides, the GBNM keeps the symmetry and positive definiteness of the variational problems. Convergence analysis results of both the velocity and the pressure are given. Some selected numerical tests are also presented to demonstrate the efficiency of the method.

A Galerkin Boundary Node Method for Two-Dimensional Linear Elasticity

A Galerkin Boundary Node Method for Two-Dimensional Linear Elasticity