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 biharmonic problems

A Galerkin boundary node method (GBNM) is developed in this paper for solving biharmonic problems. The GBNM combines an equivalent variational form of boundary integral formulations for governing equations with the moving least-squares approximations for construction of the trial and test functions. In this approach, only a nodal data structure on the boundary of a domain is required. In addition, boundary conditions can be implemented accurately and the system matrices are symmetric. The convergence of this method and numerical examples are given to show the efficiency.     

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 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.     

A hybrid radial boundary node method based on radial basis point interpolation

The hybrid boundary node method (HBNM) retains the meshless attribute of the moving least squares (MLS) approximation and the reduced dimensionality advantages of the boundary element method. However, the HBNM inherits the deficiency of the MLS approximation, in which shape functions lack the delta function property. Thus in the HBNM, boundary conditions are implemented after they are transformed into their approximations on the boundary nodes with the MLS scheme.This paper combines the hybrid displacement variational formulation and the radial basis point interpolation to develop a direct boundary-type meshless method, the hybrid radial boundary node method (HRBNM) for two-dimensional potential problems. The HRBNM is truly meshless, i.e. absolutely no elements are required either for interpolation or for integration. The radial basis point interpolation is used to construct shape functions with delta function property. So unlike the HBNM, the HRBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables, and boundary conditions can be applied directly and easily, which leads to greater computational precision. Some selected numerical tests illustrate the efficiency of the method proposed.     

Meshless local boundary integral equation method for 2D elastodynamic problems

A new meshless method for solving transient elastodynamic boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation (MLS), is proposed in this paper. The LBIE with the MLS is applied to both transient and steady‐state (Laplace transformed) elastodynamics. Applying the MLS approximation for spatially dependent terms in the first approach, the LBIEs are transformed into a system of ordinary differential equations for nodal unknowns. This system of ordinary differential equations is solved by the Houbolt finite difference scheme. In the second formulation, the time variable is eliminated by using the Laplace transformation. Unknown Laplace transforms of displacements and traction vectors are computed from the LBIEs with the MLS approximation. The time‐dependent values are obtained by the Durbin inversion technique. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary element‐free method (BEFM) and its application to two‐dimensional elasticity problems

In this study, we first discuss the moving least‐square approximation (MLS) method. In some cases, the MLS may form an ill‐conditioned system of equations so that the solution cannot be correctly obtained. Hence, in this paper, we propose an improved moving least‐square approximation (IMLS) method. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill‐conditioned system of equations. Combining the boundary integral equation (BIE) method and the IMLS approximation method, a direct meshless BIE method, the boundary element‐free method (BEFM), for two‐dimensional elasticity is presented. Compared to other meshless BIE methods, BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied easily; hence, it has higher computational precision. For demonstration purpose, selected numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd.     

Analyzing interaction between coplanar square cracks using an efficient boundary element‐free method

This paper examines the interaction between coplanar square cracks by combining the moving least‐squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element‐free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Implementation of meshless LBIE method to the 2D non‐linear SG problem

In this paper the meshless local boundary integral equation (LBIE) method for numerically solving the non‐linear two‐dimensional sine‐Gordon (SG) equation is developed. The method is based on the LBIE with moving least‐squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. The approximation functions are constructed entirely using a set of scattered nodes, and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes. A time‐stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non‐linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of method to deal with the unsteady non‐linear problems in large domains. Copyright © 2009 John Wiley & Sons, Ltd.     

A MLPG4 (LBIE) Formulation in Elastostatics

Very recently, Vavourakis, Sellountos and Polyzos (2006) ({CMES: Computer Modeling in Engineering {\&} Sciences, vol. 13, pp. 171--184}) presented a comparison study on the accuracy provided by five different elastostatic Meshless Local Petrov-Galerkin (MLPG) type formulations, which are based on Local Boundary Integral Equation (LBIE) considerations. One of the main conclusions addressed in this paper is that the use of derivatives of the Moving Least Squares (MLS) shape functions decreases the solution accuracy of any MLPG(LBIE) formulation. In the present work a new, free of MLS-derivatives and non-singular MLPG(LBIE) method for solving elastic problems is demonstrated. This is accomplished by treating displacements and stresses as independent variables through the corresponding local integral equations and considering nodal points located only internally and externally and not on the global boundary of the analyzed elastic structure. The MLS approximation scheme for the interpolation of both displacements and stresses is exploited. The essential displacement and traction boundary conditions are easily satisfied via the corresponding displacement and stress local integral equations. Representative numerical examples that demonstrate the achieved accuracy of the proposed MLPG(LBIE) method are provided.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Use of the Constrained Optimization Algorithms in Some Problems of Fracture Mechanics

In this paper we consider an algorithm of constrained optimization which arises from boundary variational principles of elastodynamics for bodies with cracks and unilateral constraints on the cracks edges. Variational formulation of unilateral contact problems with friction was considered, boundary variational functionals used with boundary integral equations were obtained and algorithm for solution of the unilateral contact problem with friction was developed. Some numerical results for 3-D elastodynamic unilateral contact problem for bodies with cracks are presented.     

An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials

A new meshless method for computing the dynamic stress intensity factors (SIFs) in continuously non-homogeneous solids under a transient dynamic load is presented. The method is based on the local boundary integral equation (LBIE) formulation and the moving least squares (MLS) approximation. The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by a circle centered at the collocation point. The boundary-domain integral formulation with elastostatic fundamental solutions for homogeneous solids in Laplace-transformed domain is used to obtain the weak solution for subdomains. On the boundary of the subdomains, both the displacement and the traction vectors are unknown generally. If modified elastostatic fundamental solutions vanishing on the boundary of the subdomain are employed, the traction vector is eliminated from the local boundary integral equations for all interior nodal points. The spatial variation of the displacements is approximated by the MLS scheme.     

Coupled boundary-element scheme for eddy-current computation

The mathematical foundation of a symmetric boundary-element method for the computation of eddy currents in a linear homogeneous conductor which is exposed to an alternating magnetic field is presented. Starting from the A-based variational formulation of the eddy-current equations and a related transmission problem, the problem inside and outside the conductors is reformulated in terms of integral equations on the boundary of the conductors. Surface currents occur as new unknowns of this direct formulation. The integral equations can be coupled in a symmetric fashion using the transmission conditions for the vector potential A and the magnetic field H. The resulting variational problem is elliptic in suitable trace spaces. A conforming Galerkin boundary-element discretization is employed, which relies on surface edge elements and provides quasi-optimal discrete approximations for the tangential traces of A and H. Surface stream functions supplemented with co-homology vector fields ensure the vital zero divergence of the discrete equivalent surface currents. Simple expressions allow the computation of approximate total Ohmic losses and surface forces from the discrete boundary data.     

New implementation of MLBIE method for heat conduction analysis in functionally graded materials

The meshless local boundary integral equation (MLBIE) method with an efficient technique to deal with the time variable are presented in this article to analyze the transient heat conduction in continuously nonhomogeneous functionally graded materials (FGMs). In space, the method is based on the local boundary integral equations and the moving least squares (MLS) approximation of the temperature and heat flux. In time, again the MLS approximates the equivalent Volterra integral equation derived from the heat conduction problem. It means that, the MLS is used for approximation in both time and space domains, and we avoid using the finite difference discretization or Laplace transform methods to overcome the time variable. Finally the method leads to a single generalized Sylvester equation rather than some (many) linear systems of equations. The method is computationally attractive, which is shown in couple of numerical examples for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Regular hybrid boundary node method for biharmonic problems

The regular hybrid boundary node method (RHBNM) is a new technique for the numerical solutions of the boundary value problems. By coupling the moving least squares (MLS) approximation with a modified functional, the RHBNM retains the meshless attribute and the reduced dimensionality advantage. Besides, since the source points of the fundamental solutions are located outside the domain, ‘boundary layer effect’ is also avoided. However, an initial restriction of the present method is that it is only suitable for the problems which the governing differential equation is in second order.Now, a new variational formulation for the RHBNM is presented further to solve the biharmonic problems, in which the governing differential equation is in fourth order. The modified variational functional is applied to form the discrete equations of the RHBNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by a linear combination of fundamental solutions of both the biharmonic equation and Laplace’s equation. Numerical examples for some biharmonic problems show that the high accuracy with a small node number is achievable. Furthermore, the computation parameters have been studied. They can be chosen in a wide range and have little influence on the results. It is shown that the present method is effective and can be widely applied in practical engineering.     

Convolution quadrature method‐based symmetric Galerkin boundary element method for 3‐d elastodynamics

The boundary integral equations in 3‐d elastodynamics contain convolution integrals with respect to the time. They can be performed analytically or with the convolution quadrature method. The latter time‐stepping procedure's benefit is the usage of the Laplace‐transformed fundamental solution. Therefore, it is possible to apply this method also to problems where analytical time‐dependent fundamental solutions might not be known. To obtain a symmetric formulation, the second boundary integral equation has to be used which, unfortunately, requires special care in the numerical implementation since it involves hypersingular kernel functions. Therefore, a regularization for closed surfaces of the Laplace‐transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one. Supplementarily, a weakly singular formulation of the Laplace‐transformed elastodynamic double layer potential is presented. This results in a time domain boundary element formulation involving at least only weakly singular integral kernels. Finally, numerical studies validate this approach with respect to different spatial and time discretizations. Further, a comparison with the wider used collocation method is presented. It is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Dual boundary‐element method: Simple error estimator and adaptivity

This paper is concerned with the effective numerical implementation of the adaptive dual boundary‐element method (DBEM), for two‐dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single‐region analysis. Taking advantage on the use of non‐conforming parametric boundary‐elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM. Copyright © 2011 John Wiley & Sons, Ltd.     

Fracture analysis of cracks in magneto-electro-elastic solids by the MLPG

A meshless method based on the local Petrov–Galerkin approach is proposed for crack analysis in two-dimensional (2-D) and three-dimensional (3-D) axisymmetric magneto-electric-elastic solids with continuously varying material properties. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem in axial cross section. Stationary and transient dynamic problems are considered in this paper. The local weak formulation is employed on circular subdomains where surrounding nodes randomly spread over the analyzed domain. The test functions are taken as unit step functions in derivation of the local integral equations (LIEs). The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs. The accuracy of the present method for computing the stress intensity factors (SIF), electrical displacement intensity factors (EDIF) and magnetic induction intensity factors (MIIF) are discussed by comparison with numerical solutions for homogeneous materials.