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.     

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

The 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.     

A new method for meshless integration in 2D and 3D Galerkin meshfree methods

A method for the evaluation of regular domain integrals without domain discretization is presented. In this method, a domain integral is transformed into a boundary integral and a 1D integral. The method is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method. The proposed technique results in truly meshless methods with better accuracy and efficiency in comparison with their original forms. Some examples, including linear and large-deformation problems, are also provided to demonstrate the usefulness of the proposed method.     

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 posteriori error estimator and an adaptive technique in meshless finite points method

In this work, an adaptive technique for application of meshless methods in one- and two-dimensional boundary value problems is described. The proposed method is based on the use of implicit functions for the geometry definition, fixed weighted least squares approximation and an error estimation by means of simple formulas and a robust strategy of refinement based on the own nature of the approximation sub-domains utilised. With all these aspects, the proposed method becomes an attractive alternative for the adaptive solutions to partial differential equations in all scopes of engineering. Numerical results obtained from the computational implementation show the efficiency of the present method.     

Development of a meshless hybrid boundary node method for Stokes flows

The meshless hybrid boundary node method (HBNM) is a promising method for solving boundary value problems, and is further developed and numerically implemented for incompressible 2D and 3D Stokes flows in this paper. In this approach, a new modified variational formulation using a hybrid functional is presented. The formulation is expressed in terms of domain and boundary variables. The moving least-squares (MLS) method is employed to approximate the boundary variables whereas the domain variables are interpolated by the fundamental solutions of Stokes equation, i.e. Stokeslets. The present method only requires scatter nodes on the surface, and is a truly boundary type meshless method as it does not require the ‘boundary element mesh’, either for the purpose of interpolation of the variables or the integration of ‘energy’. Moreover, since the primitive variables, i.e., velocity vector and pressure, are employed in this approach, the problem of finding the velocity is separated from that of finding pressure. Numerical examples are given to illustrate the implementation and performance of the present method. It is shown that the high convergence rates and accuracy can be achieved with a small number of nodes.     

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part I: Stress recovery and a posteriori error estimation

In this study, an adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D elastostatic problems is suggested. This adaptive refinement procedure is based on the Zienkiewicz and Zhu (ZZ) error estimator for the a posteriori error estimation and an adaptive finite point mesh generator for new point mesh generation. The presentation of the work is divided into two parts. In Part I, concentration will be paid on the stress recovery and the a posteriori error estimation processes for the RKPM. The proposed error estimator is different from most recovery type error estimators suggested previously in such a way that, rather than using the least-squares fitting approach, the recovery stress field is constructed by an extraction function approach. Numerical studies using 2D benchmark boundary value problems indicated that the recovered stress field obtained is more accurate and converges at a higher rate than the RKPM stress field. In Part II of the study, concentration will be shifted to the development of an adaptive refinement algorithm for the RKPM.     

A posteriori error estimates of mixed methods for miscible displacement problems

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is an elliptic equation of the pressure and the other is a parabolic equation of the concentration of one of the fluids. Since the pressure appears in the concentration only through its velocity field, we choose a mixed finite element method to approximate the pressure equation and for the concentration we use the standard Galerkin method. We shall obtain an explicit a posteriori error estimator in L²(L²) for the semi‐discrete scheme applied to the non‐linear coupled system. Copyright © 2007 John Wiley & Sons, Ltd.     

The hybrid boundary node method accelerated by fast multipole expansion technique for 3D potential problems

This paper presents a fast formulation of the hybrid boundary node method (Hybrid BNM) for solving problems governed by Laplace's equation in 3D. The preconditioned GMRES is employed for solving the resulting system of equations. At each iteration step of the GMRES, the matrix–vector multiplication is accelerated by the fast multipole method. Green's kernel function is expanded in terms of spherical harmonic series. An oct‐tree data structure is used to hierarchically subdivide the computational domain into well‐separated cells and to invoke the multipole expansion approximation. Formulations for the local and multipole expansions, and also conversion of multipole to local expansion are given. And a binary tree data structure is applied to accelerate the moving least square approximation on surfaces. All the formulations are implemented in a computer code written in C++. Numerical examples demonstrate the accuracy and efficiency of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method

In References 1 and 2 we showed that the error in the finite-element solution has two parts, the local error and the pollution error, and we studied the effect of the pollution error on the quality of the local error-indicators and the quality of the derivatives recovered by local post-processing. Here we show that it is possible to construct a posteriori estimates of the pollution error in any patch of elements by employing the local error-indicators over the mesh outside the patch. We also give an algorithm for the adaptive control of the pollution error in any patch of elements of interest.     

Virtual boundary meshless least square collocation method for calculation of 2D multi-domain elastic problems

A virtual boundary meshless least square collocation method is developed for calculation of two-dimensional multi-domain elastic problems in this paper. This method is different from the conventional virtual boundary element method (VBEM) since it incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Consequently, this method has the advantages of boundary-type meshless methods. In addition, it does not have to deal with singular integral and has the symmetric coefficient matrix, and the pre-processing operation is also very simple. This method can be used to analyze multi-domain composite structures with each subdomain having different materials or geometries. Since the configuration of virtual boundary has a certain preparability, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2D problems) to reduce the complicity and difficulty of calculus without loss of accuracy, while “Vertex Question” existing in BEM can be avoided. In the end, several numerical examples are analyzed using the proposed method and some other commonly used methods for verification and comparison purposes. The results show that this method leads to faster convergence and higher accuracy in comparison with the other methods considered in this study.     

A posteriori error estimates and an adaptive scheme of least‐squares meshfree method

A posteriori error estimates and an adaptive refinement scheme of first‐order least‐squares meshfree method (LSMFM) are presented. The error indicators are readily computed from the residual. For an elliptic problem, the error indicators are further improved by applying the Aubin–Nitsche method. It is demonstrated, through numerical examples, that the error indicators coherently reflect the actual error. In the proposed refinement scheme, Voronoi cells are used for inserting new nodes at appropriate positions. Numerical examples show that the adaptive first‐order LSMFM, which combines the proposed error indicators and nodal refinement scheme, is effectively applied to the localized problems such as the shock formation in fluid dynamics. Copyright © 2003 John Wiley & Sons, Ltd.     

A posteriori error estimates for numerical solutions to inverse problems of elastography

The article deals with one of inverse problems of elastography: knowing displacement of compressed tissue finds the distribution of Young’s modulus in the investigated specimen. The direct problem is approximated and solved by the finite element method. The inverse problem can be stated in different ways depending on whether the solution to be found is smooth or discontinuous. Tikhonov regularization with appropriate regularizing functionals is applied to solve these problems. In particular, discontinuous Young’s modulus distribution can be found on the class of 2D functions with bounded variation of Hardy–Krause type. It is shown in the paper that a variant of Tikhonov regularization provides for such discontinuous distributions the so-called piecewise uniform convergence of approximate solutions as the error levels of the data vanish. The problem of practical a posteriori estimation of the accuracy for obtained approximate solutions is under consideration as well. A method of such estimation is presented. As illustrations, model inverse problems with smooth and discontinuous solutions are solved along with a posteriori estimations of the accuracy.     

The boundary node method for three‐dimensional problems in potential theory

The boundary node method (BNM) is developed in this paper for solving potential problems in three dimensions. The BNM represents a coupling between boundary integral equations (BIE) and moving least‐squares (MLS) interpolants. The main idea here is to retain the dimensionality advantage of the former and the meshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure for the field variables. A general BNM computer code for 3‐D potential problems has been developed. Several parameters involved in the BNM need to be chosen carefully for a successful implementation of the method. An in‐depth and systematic study has been carried out in this paper in order to better understand the effects of various parameters on the performance of the method. Numerical results for spheres and cubes, subjected to different types of boundary conditions, are extremely encouraging. Copyright © 2000 John Wiley & Sons, Ltd.     

A node enrichment adaptive refinement in Discrete Least Squares Meshless method for solution of elasticity problems

In this paper, an adaptive refinement procedure is proposed to be used with Discrete Least Squares Meshless (DLSM) method for accurate solution of planar elasticity problems. DLSM method is a newly introduced meshless method based on the least squares concept. The method is based on the minimization of a least squares functional defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries. A Moving Least Square (MLS) method is used to construct the shape function making the approach a fully least squares based approach. An error estimate and adaptive refinement strategy is proposed in this paper to increase the efficiency of the DLSM method. For this, a residual based error estimator is introduced and used to discover the region of higher errors. The proposed error estimator has the advantages of being available at the end of each analysis contributing to the efficiency of the proposed method. An enrichment method is then used by adding more nodes to the area of higher errors as indicated by the error estimator. A Voronoi diagram is used to locate the position of the nodes to be added to the current nodal configuration. Efficiency and effectiveness of the proposed procedure is examined by adaptively solving two benchmark problems. The results show the ability of the proposed strategy for accurate simulation of elasticity problems.     

A meshless local boundary integral equation method for solving transient elastodynamic problems

A new local boundary integral equation (LBIE) method for solving two dimensional transient elastodynamic problems is proposed. The method utilizes, for its meshless implementation, nodal points spread over the analyzed domain and employs the moving least squares (MLS) approximation for the interpolation of the interior and boundary variables. On the global boundary, displacements and tractions are treated as independent variables. The local integral representation of displacements at each nodal point contains both surface and volume integrals, since it employs the simple elastostatic fundamental solution and considers the acceleration term as a body force. On the local boundaries, tractions are avoided with the aid of the elastostatic companion solution. The collocation of the local boundary/volume integral equations at all the interior and boundary nodes leads to a final system of ordinary differential equations, which is solved stepwise by the -Wilson finite difference scheme. Direct numerical techniques for the accurate evaluation of both surface and volume integrals are employed and presented in detail. All the strongly singular integrals are computed directly through highly accurate integration techniques. Three representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.     

A posteriori error estimates and adaptivity for finite element solutions in finite elasticity

Methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems. In contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context. In the paper a brief outline of continuum theory of finite elasticity is first given. Using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally. The proposed error estimator is physically interpreted in the energy sense. We then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences.     

A boundary face method for potential problems in three dimensions

This work presents a new implementation of the boundary node method (BNM) for numerical solution of Laplace's equation. By coupling the boundary integral equations and the moving least‐squares (MLS) approximation, the BNM is a boundary‐type meshless method. However, it still uses the standard elements for boundary integration and approximation of the geometry, thus loses the advantages of the meshless methods. In our implementation, here called the boundary face method, the boundary integration is performed on boundary faces, which are represented in parametric form exactly as the boundary representation data structure in solid modeling. The integrand quantities, such as the coordinates of Gauss integration points, Jacobian and out normal are calculated directly from the faces rather than from elements. In order to deal with thin structures, a mixed variable interpolation scheme of 1‐D MLS and Lagrange Polynomial for long and narrow faces. An adaptive integration scheme for nearly singular integrals has been developed. Numerical examples show that our implementation can provide much more accurate results than the BNM, and keep reasonable accuracy in some extreme cases, such as very irregular distribution of nodes and thin shells. Copyright © 2009 John Wiley & Sons, Ltd.