analyzing the 2d fracture problems via the enriched boundary element free method in this paper the enriched boundary element free method for two dimensional fracture problems is presented an improved moving least squares imls approximation in which the orthogonal function system with a weight function is used as the basis function is used to obtain the shape functions the imls approximation has greater computational efficiency and precision than the existing moving least squares mls approximation and does not lead to an ill conditioned system of equations combining the boundary integral equation bie method and the imls approximation a boundary element free method befm for two dimensional fracture problems is obtained for two dimensional fracture problems the enriched basis function is used at the tip of the crack and then the enriched befm is presented in comparison with other existing meshless boundary integral equation methods the 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 implemented easily which leads to a greater computational precision when the enriched befm is used the singularity of the stresses at the tip of the crack can be shown better than that in the befm for the purposes of demonstration some selected numerical examples are solved using the enriched befm

a point interpolation method for simulating dissipation process of consolidation abstract a biot s consolidation problem in foundation engineering is numerically investigated using improved point interpolation method pim a weak form of biot s theory is first developed to consider the unbalanced forces at previous time step and thus guarantees the global equilibrium at current step two independent variables in the weak form displacement and excess pore water pressure are approximated using the same shape functions through pim technique the pim technique constructs its interpolation functions through a cluster of scattered points in problem domain and its shape function is of delta properties thus implementation of essential boundary conditions is as easy as in conventional finite element method crank nicholson s integration scheme is used to discretize time domain finally examples are studied and compared with finite element methods to demonstrate its capability

error analysis in sobolev spaces for the improved moving least square approximation and the improved element free galerkin method the improved moving least square imls approximation is a method to form shape functions in meshless methods for the application of imls based meshless methods to the numerical solution of boundary value problems it is fundamental to analyze error of the imls approximation in sobolev spaces this paper begins by discussing properties of the imls shape function under appropriate assumption on weight functions error estimates for the imls approximation are then established in sobolev spaces in multiple dimensions the improved element free galerkin iefg method is a typical meshless galerkin method based on coupling the imls approximation and galerkin weak form error analysis of the iefg method is also provided numerical examples are finally presented to prove the theoretical error results

outer inverses and jacobi type identities abstract we consider matrices over a commutative ring and characterize the class of outer inverses for which jacobi type identities can be extended we obtain a necessary and sufficient condition for the existence of rao regular drazin inverse in terms of sum of principal minors of a k for some k also we obtain determinantal formula for the rao regular drazin inverse conjectures are formulated which give expressions for outer inverse and the conjectures are proved in some special cases

algebraic and proof theoretic characterizations of truth stressers for mtl and its extensions extensions of monoidal t norm logic mtl and related fuzzy logics with truth stresser modalities such as globalization and very true are presented here both algebraically in the framework of residuated lattices and proof theoretically as hypersequent calculi completeness with respect to standard algebras based on t norms embeddings between logics decidability and the finite embedding property are then investigated for these logics

boundary element free method befm for two dimensional elastodynamic analysis using laplace transform in this paper we present a direct meshless method of boundary integral equation bie known as the boundary element free method befm for two dimensional 2d elastodynamic problems that combines the bie method for 2d elastodynamics in the laplace transformed domain and the improved moving least squares imls approximation the formulae for the befm for 2d elastodynamic problems are given and the numerical procedures are also shown the befm is a direct numerical method in which the basic unknown quantities are the real solutions of the nodal variables and the boundary conditions can be implemented directly and easily that leads to a greater computational precision for the purpose of demonstration some selected numerical examples are solved using the befm copyright 2005 john wiley sons ltd

a hybrid boundary node method a new variational formulation for boundary node method bnm using a hybrid displacement functional is presented here the formulation is expressed in terms of domain and boundary variables and the domain variables are interpolated by classical fundamental solution while the boundary variables are interpolated by moving least squares mls the main idea is to retain the dimensionality advantages of the bnm and get a truly meshless method which does not require a boundary element mesh either for the purpose of interpolation of the solution variables or for the integration of the energy all integrals can be easily evaluated over regular shaped domains in general semi sphere in the 3 d problem and their boundaries numerical examples presented in this paper for the solution of laplace s equation in 2 d show that high rates of convergence with mesh refinement are achievable and the computational results for unknown variables are most accurate no further integrations are required to compute the unknown variables inside the domain as in the conventional bem and bnm copyright 2001 john wiley sons ltd

a point collocation method based on reproducing kernel approximations a reproducing kernel particle method with built in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations the design and implementation of a galerkin based reproducing kernel particle method however faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions in this paper we present a point collocation method based on reproducing kernel approximations we show that in a point collocation approach the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach the point collocation method based on reproducing kernel approximations however requires the calculation of higher order derivatives that would typically not be required in a galerkin method a correction function and reproducing conditions that enable consistency of the point collocation method are derived the point collocation method is shown to be accurate for several one and two dimensional problems and the convergence rate of the point collocation method is addressed copyright 2000 john wiley sons ltd

a stabilized mlpg method for steady state incompressible fluid flow simulation in this paper the meshless local petrov galerkin mlpg method is extended to solve the incompressible fluid flow problems the streamline upwind petrov galerkin supg method is applied to overcome oscillations in convection dominated problems and the pressure stabilizing petrov galerkin pspg method is applied to satisfy the so called babuska brezzi condition the same stabilization parameter t t s u p email protected p s p g is used in the present method the circle domain of support linear basis and fourth order spline weight function are applied to compute the shape function and bubnov galerkin method is applied to discretize the pdes the lid driven cavity flow backward facing step flow and natural convection in the square cavity are applied to validate the accuracy and feasibility of the present method the results show that the stability of the present method is very good and convergent solutions can be obtained at high reynolds number the results of the present method are in good agreement with the classical results it also seems that the present method which is a truly meshless is very promising in dealing with the convection dominated problems

the boundary node method for three dimensional linear elasticity the boundary node method bnm is developed in this paper for solving three dimensional problems in linear elasticity the bnm represents a coupling between boundary integral equations bie and moving least squares mls interpolants the main idea 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 problems in linear elasticity free rigid body modes in traction prescribed problems are typically eliminated by suitably restraining the body however an alternative approach developed recently for the boundary element method bem is extended in this work to the bnm this approach is based on ideas from linear algebra to complete the rank of the singular stiffness matrix also the bnm has been extended in the present work to solve problems with material discontinuities and a new procedure has been developed for obtaining displacements and stresses accurately at internal points close to the boundary of a body copyright 1999 john wiley sons ltd

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

dual reciprocity hybrid boundary node method for free vibration analysis abstract as a truly meshless method of boundary type the hybrid boundary node method hbnm has the advantages of both boundary element method bem and meshless method the main problem is that it is only suitable for the homogeneous problems now the dual reciprocity method drm is introduced into hbnm to deal with the integral for the inhomogeneous terms of the governing equations and the rigid body motion approach is employed to solve the hyper singular integrations a new meshless method named dual reciprocity hybrid boundary node method drhbnm is proposed and applied to solve free vibration problems in this method the solution composes into two parts i e the general solution and the particular solution the general solution is solved by hbnm and the particular one is obtained by drm drhbnm is a true boundary type meshless method it does not require the boundary element mesh either for the purpose of interpolation of the variables or for the integration of energy the points in the domain are only used to interpolate particular solution by the radial basis function finally the boundary variables are interpolated by the independent smooth boundary segments the q r algorithm and householder algorithm are applied to solve the eigenvalues and eigenvectors of the transformed matrix numerical examples for free vibration problems show that a good convergence with mesh refinement is achievable and the computational results for the natural circular frequencies and free vibration modes are very accurate furthermore the computation parameters have little influence on the results and can be chosen in a wide range it is shown that the present method is effective and can be widely applied in practical engineering

on the convergence analysis stability and implementation of meshless local radial point interpolation on a class of three dimensional wave equations summary in this article the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form utt ut u uxx uyy uzz f x y z t xt x y z rd d 2 3 t 0 subject to given initial and dirichlet boundary conditions the main difficulty of the great number of methods in full 3 d problems is the large computational costs in meshless local radial point interpolation method it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions the point interpolation method with the help of radial basis functions is proposed to construct shape functions that have kronecker delta function property a weak formulation with the heaviside step function converts the set of governing equations into local integral equations on local subdomains a two step time discretization method is employed to evaluate the time derivatives this suggests crank nicolson technique to be applied on the right hand side of the equation the convergence analysis and stability of the method are fully discussed three illustrative examples are presented and satisfactory agreements are achieved it is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones copyright 2015 john wiley sons ltd

boundary cloud method a combined scattered point boundary integral approach for boundary only analysis abstract a boundary cloud method bcm for boundary only analysis of partial differential equations is presented in this paper for solving potential equations in two dimensions 2 d the bcm combines a weighted least squares approach for construction of interpolation functions with a boundary integral formulation for the governing equations given a set of scattered points on the surface of an arbitrary object hermite type interpolation functions are constructed by developing a weighted least squares approach we also introduce truncated hermite type interpolation functions the boundary integrals are evaluated by using a cell structure we propose various schemes for evaluating the singular nearly singular and nonsingular integrals we also propose a true meshless approach for boundary only analysis of potential equations numerical results comparing the classical boundary element method and the bcm are presented for several 2 d potential problems

a fast boundary cloud method for exterior 2d electrostatic analysis an accelerated boundary cloud method bcm for boundary only analysis of exterior electrostatic problems is presented in this paper the bcm uses scattered points instead of the classical boundary elements to discretize the surface of the conductors the dense linear system of equations generated by the bcm are solved by a gmres iterative solver combined with a singular value decomposition based rapid matrix vector multiplication technique the accelerated technique takes advantage of the fact that the integral equation kernel 2d green s function in our case is locally smooth and therefore can be dramatically compressed by using a singular value decomposition technique the acceleration technique greatly speeds up the solution phase of the linear system by accelerating the computation of the dense matrix vector product and reducing the storage required by the bcm copyright 2002 john wiley sons ltd

evaluation of nearly singular integrals in boundary element contour and node methods for three dimensional linear elasticity this article addresses the issue of nearly singular integrals that arise in boundary element methods bems when one tries to evaluate displacements and stresses at points inside a body that lie close to its bounding surface an efficient new approach is proposed for the evaluation of these integrals the same issue is then discussed in the context of the boundary contour method bcm and the boundary node method bnm finally numerical results from both the bcm and the bnm are presented for an illustrative problem

element free galerkin methods for static and dynamic fracture abstract element free galerkin efg methods are presented and applied to static and dynamic fracture problems efg methods which are based on moving least square mls interpolants require only nodal data no element connectivity is needed the description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks this makes the method particularly attractive for growing crack problems since only minimal remeshing is needed to follow crack growth in moving least square interpolants the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method

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

complex variable moving least squares method a meshless approximation technique based on the moving least squares mls approximation we propose a new approximation method the complex variable moving least squares cvmls approximation with the cvmls approximation the trial function of a two dimensional problem is formed with a one dimensional basis function the number of unknown coefficients in the trial function of the cvmls approximation is less than in the trial function of the mls approximation and we can thus select fewer nodes in the meshless method that is formed from the cvmls approximation than are required in the meshless method of the mls approximation with no loss of precision the meshless method that is derived from the cvmls approximation also has a greater computational efficiency from the cvmls approximation we propose a new meshless method for two dimensional elasticity problems the complex variable meshless method cvmm and the formulae of the cvmm for two dimensional elasticity problems are obtained compared with the conventional meshless method the cvmm has a greater precision and computational efficiency for the purposes of demonstration some selected numerical examples are solved using the cvmm

dynamic fracture using element free galerkin methods the element free galerkin method for dynamic crack propagation is described and applied to several problems this method is a gridless method which facilitates the modelling of growing crack problems because it does not require remeshing the growth of the crack is modelled by extending its surfaces the essential feature of the method is the use of moving least squares interpolants for the trial and test functions in these interpolants the dependent variable is obtained at any point by minimizing a weighted quadratic form involving the nodal variables within a small domain surrounding the point the discrete equations are obtained by a galerkin method the procedures for modelling dynamic crack propagation based on dynamic stress intensity factors are also described

time dependent fractional advection diffusion equations by an implicit mls meshless method recently many new applications in engineering and science are governed by a series of fractional partial differential equations fpdes unlike the normal partial differential equations pdes the differential order in a fpde is with a fractional order which will lead to new challenges for numerical simulation because most existing numerical simulation techniques are developed for the pde with an integer differential order the current dominant numerical method for fpdes is finite difference method fdm which is usually difficult to handle a complex problem domain and also hard to use irregular nodal distribution this paper aims to develop an implicit meshless approach based on the moving least squares mls approximation for numerical simulation of fractional advection diffusion equations fade which is a typical fpde the discrete system of equations is obtained by using the mls meshless shape functions and the meshless strong forms the stability and convergence related to the time discretization of this approach are then discussed and theoretically proven several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation it is concluded that the present meshless formulation is very effective for the modeling and simulation of the fade

an adaptive orthogonal improved interpolating moving least square method and a new boundary element free method abstract the improved interpolating moving least square iimls method has been widely used in data fitting and meshfree methods and the obtained shape functions have the property of the delta function compared with those obtained by the moving least square mls method however the moment matrix in iimls may be singular or ill conditioned because of the ill quality of the point sets used in this paper the weighted orthogonal basis functions are applied in iimls to obtain a diagonal moment matrix which can overcome the difficulty caused by directly inversing singular or ill conditioned matrices however the weighted orthogonal basis functions cannot change the nature of the singular or ill conditioned moment matrix since the diagonal elements of the new moment matrix may be zero or close to zero thus an adaptive scheme is further employed to resolve this problem by ignoring the contribution from the zero or very small diagonal elements in the diagonal moment matrix combined with shifted and scaled polynomial basis functions a stabilized adaptive orthogonal iimls sao iimls approximation is obtained based on this approximation a new boundary element free method is proposed for solving elasticity problems numerical results for curve fitting surface fitting and the new boundary element free method have shown that the proposed sao iimls approximation is accurate stable and performs well for ill quality point sets

a semi analytical algorithm for the evaluation of the nearly singular integrals in three dimensional boundary element methods the nearly singular integrals occur in the boundary integral equations when the source point is close to an integration element as compared to its size but not on the element in this paper the concept of a relative distance from a source point to the boundary element is introduced to describe possible influence of the singularity of the integrals then a semi analytical algorithm is proposed for evaluating the nearly strongly singular and hypersingular integrals in the three dimensional bem by using integration by parts the nearly singular surface integrals on the elements are transformed to a series of line integrals along the contour of the element the singular behavior which appears as factor is separated from remaining regular integrals consequently standard numerical quadrature can provide very accurate evaluation of the resulting line integrals the semi analytical algorithm is applied to analyzing the three dimensional elasticity problems such as very thin walled structures meanwhile the displacements and stresses at the interior points very close to its bounding surface are also determined efficiently the results of the numerical investigation demonstrate the accuracy and effectiveness of the algorithm

application of radial basis functions to linear and nonlinear structural analysis problems the basic characteristic of the techniques generally known as meshless methods is the attempt to reduce or even to eliminate the need for a discretization at least not in the way normally associated with traditional finite element techniques in the context of numerical solutions for boundary and or initial value problems the interest in meshless methods is relatively new and this is why despite the existence of various applications of meshless techniques to several problems of mechanics as well as to other fields these techniques are still relatively unknown to engineers furthermore and compared to traditional finite elements it may be difficult to understand the physical meaning of the variables involved in the formulations as an attempt to clarify some aspects of the meshless techniques and simultaneously to highlight the ease of use and the ease of implementation of the algorithms applications are made in this work to structural analysis problems the technique used here consists of the definition of a global approximation for a given variable of interest in this case components of the displacement field by means of a superposition of a set of conveniently placed in the domain and on the boundary radial basis functions rbfs in this work various types of one dimensional problems are analyzed ranging from the static linear elastic case free vibration and linear stability analysis for a beam on elastic foundation to physically nonlinear damage models problems to further complement the range of problems analysed the static analysis of a plate on elastic foundation was also addressed several error measures are used to numerically establish the performance of both symmetric and nonsymmetric approaches for several global rbfs the results obtained show that rbf collocation leads to good approximations and very high convergence rates

a meshless singular boundary method for three dimensional elasticity problems summary this study documents the first attempt to extend the singular boundary method sbm a novel meshless boundary collocation method for the solution of three dimensional 3d elasticity problems the sbm involves a coupling between the regularized boundary element method bem and the method of fundamental solutions mfs the main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration free attributes of the later this makes it particularly attractive for problems in complex geometries and three dimensions four benchmark 3d problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method the advantages disadvantages and potential applications of the proposed method as compared with the fem bem and mfs are also examined and discussed this article is protected by copyright all rights reserved

an element free galerkin method for simulation of stationary two dimensional shallow water flows in rivers abstract in this paper the meshless method is introduced to the hydraulics an element free galerkin efg method for simulation of two dimensional shallow water flows in rivers is presented and its implementation is described in this method only the nodal data which may be the same as those used in the finite element methods fems and a description of the domain boundary geometry are necessary no element or grid connectivity is needed this makes the method particularly attractive for modelling shallow water flows in rivers for which the mesh generation is usually very difficult because of very irregular topography and strongly varied roughness of the river bottoms in the efg method the moving least squares interpolation is used to construct the trial functions the modelled domain is represented through the nodal points a galerkin method is applied to discretise the governing differential equations resulting in a simultaneous equation system an underlying cell structure for calculation of the integrals in galerkin equations is used the key advantages of the efg method in comparison with the fem are that the method is meshless and the independent variables and their gradients are continuous in the entire domain the efg method is also advantageous by changing or refining the nodal distribution in the domain in addition the babuska brezzi condition is satisfied

explicit form and efficient computation of mls shape functions and their derivatives this work presents a general and efficient way of computing both diffuse and full derivatives of shape functions for meshless methods based on moving least squares approximation mls and interpolation it is an extension of the recently introduced consistency approach based on lagrange multipliers which provides a general framework for constrained mls along with robust algorithms for the computation of shape functions and their diffuse derivatives the particularity of the proposed algorithms is that they do not involve matrix inversion or linear system solving the previous approach is limited to diffuse derivatives of the shape functions and not their full derivatives which are usually much more expensive to obtain in the present paper we propose to efficiently compute the full derivatives by a new algorithm based on the formal differentiation of the previous one in this way we obtain a unified low cost consistent methodology for evaluating the shape functions and both their diffuse and full derivatives in the second part of the paper we introduce explicit forms of mls shape functions in 1d 2d and 3d for an arbitrary number of nodes these forms are especially useful for comparing finite element and mls approximations finally we present a general architecture of an mls program copyright 2000 john wiley sons ltd

complex variable boundary element free method for two dimensional elastodynamic problems abstract we proposed a new direct meshless boundary integral equation technique the complex variable boundary element free method cvbefm based on the complex variable moving least squares cvmls approximation and the boundary element free method befm to study the two dimensional elastodynamic problems with the cvmls approximation the trial function of a two dimensional problem is formed with a one dimensional basis function the number of unknown coefficients in the trial function of the cvmls approximation is less than that in the trial function of the moving least squares mls approximation therefore it requires fewer nodes in the meshless method which formed from the cvmls approximation than that formed from the mls approximation with no lose of precision the laplace transform is used to formulate the boundary integral equations of the two dimensional elastodynamics and then the formulae of the cvbefm for two dimensional elastodynamic problems are derived the cvbefm is a direct numerical method in which the basic unknown quantities are the real solutions of the nodal variables moreover in the cvbefm the boundary conditions can be applied directly and easily that leads to a greater computational precision in this paper we selected a few numerical examples to illustrate the applicability of the cvbefm

a meshless discrete galerkin mdg method for the numerical solution of integral equations with logarithmic kernels abstract this paper describes a computational method for solving fredholm integral equations of the second kind with logarithmic kernels the method is based on the discrete galerkin method with the shape functions of the moving least squares mls approximation constructed on scattered points as basis the mls methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least square polynomial fitting the numerical scheme developed in the current paper utilizes the non uniform gauss legendre quadrature rule for approximating logarithm like singular integrals and so reduces the solution of the logarithmic integral equation to the solution of a linear system of algebraic equations the proposed method is meshless since it does not require any background mesh or domain elements the error analysis of the method is provided the scheme is also applied to a boundary integral equation which is a reformulation of a boundary value problem of laplace s equation with linear robin boundary conditions finally numerical examples are included to show the validity and efficiency of the new technique

finite cloud method a true meshless technique based on a fixed reproducing kernel approximation we introduce fixed moving and multiple fixed kernel techniques for the construction of interpolation functions over a scattered set of points we show that for a particular choice of nodal volumes the fixed moving and multiple fixed kernel approaches are identical to the fixed moving and multiple fixed least squares approaches a finite cloud method which combines collocation with a fixed kernel technique for the construction of interpolation functions is presented as a true meshless technique for the numerical solution of partial differential equations numerical results are presented for several one and two dimensional problems including examples from elasticity heat conduction thermoelasticity stokes flow and piezoelectricity copyright 2001 john wiley sons ltd

compact rbf meshless methods for photonic crystal modelling meshless methods based on compact radial basis functions rbfs are proposed for modelling photonic crystals phcs when modelling two dimensional phcs two generalised eigenvalue problems are formed one for the transverse electric te mode and the other for the transverse magnetic tm mode conventionally the band diagrams for two dimensional phcs are calculated by either the plane wave expansion method pwem or the finite element method fem here the eigenvalue equations for the two dimensional phcs are solved using rbfs based meshless methods for the tm mode a meshless local strong form method rbf collocation is used while for the tricker te mode a meshless local weak form method rbf galerkin is used so that the discontinuity of the dielectric function e x can naturally be modelled the results obtained from the meshless methods are found to be in good agreement with the standard pwem thus the meshless methods are proved to be a promising scheme for predicting photonic band gaps

on the stability of the moving least squares approximation and the element free galerkin method in this paper the stability of the moving least squares mls approximation and a stabilized mls approximation is analyzed theoretically and verified numerically it is shown that the stability of the mls approximation deteriorates severely as the nodal spacing decreases while the stability of the stabilized mls approximation is not affected by the nodal spacing the stabilized mls approximation is introduced into the element free galerkin efg method to produce a stabilized efg method theoretical error analysis of the stabilized efg method is provided for boundary value problems with mixed boundary conditions of dirichlet and robin type numerical examples confirm the theoretical results and show that the stabilized efg method has higher computational precision and better stability than the original efg method

boundary knot method for 2d and 3d helmholtz and convection diffusion problems under complicated geometry the 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

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

two dimensional linear elasticity by the boundary node method abstract this paper presents a further development of the boundary node method bnm for 2 d linear elasticity in this work the boundary integral equations bie for linear elasticity have been coupled with moving least square mls interpolants this procedure exploits the mesh less attributes of the mls and the dimensionality advantages of the bie as a result the bnm requires only a nodal data structure on the bounding surface of a body a cell structure is employed only on the boundary in order to carry out numerical integration in addition the mls interpolants have been suitably truncated at corners in order to avoid some of the oscillations observed while solving potential problems by the bnm mukherjee and mukherjee 1997a numerical results presented in this paper including those for the solution of the lame and kirsch problems show good agreement with analytical solutions

a meshless integral method based on regularized boundary integral equation a meshless integral method based on the regularized boundary integral equation is developed and applied to two dimensional linear elasticity the governing integral equation is obtained from the weak form of elasticity over a local sub domain and the moving least squares approximation is used for meshless function approximation the method is built on the lbie method proposed by atluri and co workers and several key improvements are introduced in this work that significantly enhance the accuracy and robustness of the method the most critical improvement is the use of the subtraction technique to remove the strong singularity that results in a regularized governing integral equation the technique is straightforward and efficient and is much simpler and easier compared to other singularity removal techniques a special numerical integration is employed for the calculation of integrals with weak singularity which further improves accuracy the collocation method is employed to enforce the essential boundary conditions exactly which is simple and computationally efficient the natural boundary conditions are incorporated in the system governing equation and require no special handling numerical tests show that the meshless integral method is accurate and robust the effects of weight function support domain sub domain and monomial basis are investigated and discussed

elastoplastic meshless integral method abstract in this paper the meshless integral method based on regularized boundary integral equation a bodin j ma x j xin p krishnaswami a meshless integral method based on regularized boundary integral equation comput methods appl mech engrg 195 2006 6258 6286 has been extended to elastoplastic materials in the formulation the domain of interest is populated with a set of nodes using an automatic node generation algorithm the sub domain and support domain associated with each node are also generated automatically using algorithms that have been developed for this purpose the governing integral equation is obtained from the weak form of elastoplasticity over a local sub domain and the moving least squares approximation is used for meshless function approximation the constitutive law is the small deformation rate independent flow theory based on von mises yielding criterion with isotropic hardening the generalized collocation method is employed to enforce the essential boundary conditions exactly which is simple and computationally efficient natural boundary conditions are incorporated in the system governing equation and require no special handling the solution algorithm for elastoplastic analysis is discussed in detail the proposed method can handle any prescribed loading profile including unloading and reversed loading numerical examples show that the elastoplastic integral meshless method is accurate and robust

moving least square for systems of integral equations this paper aims at developing a meshless approximation based on the moving least square mls in addition to its application for solving a system of linear fredholm integral equations of the second kind for the mls nodal points are used to approximate the unknown functions these points can be selected as regular or random from the domain under study the method is a meshless one and since it uses a local shape function in the vicinity of each nodal point which is chosen from the support points it does not depend on the geometry of the domain in this method the unknown function is considered as a vector of functions of its kind an error analysis has also been provided for this new method a simple and efficient application of this method has also demonstrated through several numerical examples

a new implementation of the element free galerkin method abstract element free galerkin methods efg are methods for solving partial differential equations with moving least squares interpolants efg methods require only nodal data no element connectivity is needed in a previous implementation of the efg method lagrange multipliers were used to enforce the essential boundary condition however the use of lagrange multipliers increases the cost of solving the linear algebraic equations a new implementation is developed based on a modified variational principle in which the lagrange multipliers are replaced at the outset by their physical meaning so that the discrete equations are banded in addition weighted orthogonal basis functions are constructed so the need for solving equations at each quadrature point is eliminated numerical examples show that the present implementation effectively computes stress concentrations and stress intensity factors at cracks with very irregular arrangements of nodes the latter makes it very advantageous for modelling progressive cracking

the boundary node method for potential problems the element free galerkin efg method allows one to use a nodal data structure usually with an underlying cell structure within the domain of a body of arbitrary shape the usual efg combines moving least squares mls interpolants with a variational principle weak form and has been used to solve two dimensional 2 d boundary value problems in mechanics such as in potential theory elasticity and fracture this paper proposes a combination of mls interpolants with boundary integral equations bie in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter this new method called the boundary node method bnm only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself an underlying cell structure is again used for numerical integration in principle the bnm for 3 d problems should be extremely powerful since one would only need to put nodes points on the surface of a solid model for an object numerical results are presented in this paper for the solution of laplace s equation in 2 d dirichlet neumann and mixed problems have been solved some on bodies with piecewise straight and others with curved boundaries results from these numerical examples are extremely encouraging 1997 by john wiley sons ltd

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

regularized moving least square method and regularized improved interpolating moving least square method with nonsingular moment matrices abstract the moving least square mls method has been popular applied in surface construction and meshless methods however the moment matrix in mls method may be singular for ill quality point sets and the computation of the inverse of the singular moment matrix is difficult to overcome this problem a regularized moving least square method with nonsingular moment matrix is proposed the shape functions obtained from the regularized mls method still do not have the delta function property and may result in difficulty for imposing boundary conditions in regularized mls based meshless method to overcome this problem a regularized improved interpolating moving least square iimls method based on the iimls method is also proposed compared with the regularized mls method the regularized iimls not only has nonsingular moment matrices but also obtains shape functions with delta function property shape functions of the proposed methods are compared in 1d and 2d cases and the methods have been applied in curve fitting surface fitting and meshless method in numerical examples

the meshless standard and hypersingular boundary node methods applications to error estimation and adaptivity in three dimensional problems the standard singular boundary node method bnm and the novel hypersingular boundary node method hbnm are employed for the usual and adaptive solutions of three dimensional potential and elasticity problems these methods couple boundary integral equations with moving least squares interpolants while retaining the dimensionality advantage of the former and the meshless attribute of the latter the hypersingular residuals developed for error estimation in the mesh based collocation boundary element method bem and symmetric galerkin bem by paulino et al are extended to the meshless bnm setting a simple a posteriori error estimation and an effective adaptive refinement procedure are presented the implementation of all the techniques involved in this work are discussed which includes aspects regarding parallel implementation of the bnm and hbnm codes several numerical examples are given and discussed in detail conclusions are inferred and relevant extensions of the methodology introduced in this work are provided copyright 2001 john wiley sons ltd

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

novel meshless method for solving the potential problems with arbitrary domain in this article a non singular and boundary type meshless method in two dimensions is developed to solve the potential problems the solution is represented by a distribution of the kernel functions of double layer potentials by using the desingularization technique to regularize the singularity and hypersingularity of the kernel functions the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined the main difficulty of the coincidence of the source and collocation points then disappears by employing the two point function the off diagonal coefficients of influence matrices are easily obtained the numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the results of exact solution conventional mfs and bem for the dirichlet neumann and mix type boundary conditions bcs of interior and exterior problems with simple and complicated boundaries good agreements with exact solutions are observed

a galerkin boundary node method and its convergence analysis the boundary node method bnm exploits the dimensionality of the boundary integral equation bie and the meshless attribute of the moving least square mls approximations however since mls shape functions lack the property of a delta function it is difficult to exactly satisfy boundary conditions in bnm besides the system matrices of bnm are non symmetric a galerkin boundary node method gbnm is proposed in this paper for solving boundary value problems in this approach an equivalent variational form of a bie is used for representing the governing equation and the trial and test functions of the variational formulation are generated by the mls approximation as a result boundary conditions can be implemented accurately and the system matrices are symmetric total details of numerical implementation and error analysis are given for a general bie taking the dirichlet problem of laplace equation as an example we set up a framework for error estimates of gbnm some numerical examples are also given to demonstrate the efficacity of the method

a meshless local petrov galerkin method for the time dependent maxwell equations abstract in this paper the meshless local petrov galerkin mlpg method is employed to solve the 2 d time dependent maxwell equations the mlpg method is a truly meshless method in which the trial and test functions are chosen from totally different functional spaces in the current work the moving least square reproducing kernel mlsrk scheme is chosen to be the trial function the method is applied for the unsteady maxwell equations in different media in the local weak form by employing the difference operator for evolution in time and simultaneously in time and space the semi discrete and fully discrete schemes are obtained respectively the error estimation is discussed for both the semi discrete and fully discrete numerical schemes for modelling the time dependent maxwell equations we show that provided that the time step size is sufficiently small the proposed scheme yields an error of o 2 m 1 2 in the l 2 norm for the square of error the new scheme is implemented and the numerical results are provided to justify our theoretical analysis

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