Meshless Methods

Linear systems of equations and their reliable solution is a key part of nearly all computational systems and in a solution of many engineering problems. Mostly, the estimation of the matrix conditionality is used for an assessment of the solvability of linear systems, which are important for interpolation, approximation , and solution of partial differential equations especially for large data sets with large range of values. In this contribution, a new approach to the matrix conditionality and the solvabil-ity of the linear systems of equations is presented. It is based on the application of the geometric algebra with the projective space representation using homogeneous coordinates representation. However, the physical floating-point representation , mostly the IEEE 754-219, has to be strictly respected as many algorithms fail due to this.

The intrinsic concept of meshless methods may be found in many approaches in interpolation and numerical methods for partial differential equations. Given this common concept, the aim of the Euro-Mediterranean workshop is to provide an opportunity for researchers and practitioners to discuss recent research results that may support a wide applicability in meshless related approaches. To build a foundation for these discussions, a number of experts has been invited to talk about their research. The covered topics will range from theoretical analysis of meshless based methods to applications and aspects of implementation. Anybody interested to participate is cordially invited to join the discussions and to give a presentation on his/her own research.

The Lane-Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. In this paper, a new numerical method is applied to investigate some well-known classes of Lane-Emden type equations which are non-linear ordinary differential equations on the semi-infinite domain ½0; 1Þ. We will apply a meshless method based on radial basis function differential quadrature (RBF-DQ) method. In RBFs-DQ, the derivative value of function with respect to a point is directly approximated by a linear combination of all functional values in the global domain. The main aim of this method is the determination of weight coefficients. Here, we concentrate on Gaussian (GS) as a radial function for approximating the solution of the mentioned equation. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.

In the present work we present a meshless natural neighbor Galerkin method for the bending and vibration analysis of plates and laminates. The method has distinct advantages of geometric flexibility of meshless method. The compact support and the connectivity between the nodes forming the compact support are performed dynamically at the run time using the natural neighbor concept. By this method the nodal connectivity is imposed through nodal sets with reduced size, reducing significantly the computational effort in construction of the shape functions. Smooth non-polynomial type interpolation functions are used for the approximation of inplane and out of plane primary variables. The use of nonpolynomial type interpolants has distinct advantage that the order of interpolation can be easily elevated through a degree elevation algorithm, thereby making them suitable also for higher order shear deformation theories. The evaluation of the integrals is made by use of Gaussian quadrature defined on background integration cells. The plate formulation is based on first order shear deformation plate theory. The application of natural neighbor Galerkin method formulation has been made for the bending and free vibration analysis of plates and laminates. Numerical examples are presented to demonstrate the efficacy of the present numerical method in calculating deflections, stresses and natural frequencies in comparison to the Finite element method, analytical methods and other meshless methods available in the literature.

A new enriched weight function for meshless methods is proposed for the numerical treatment of multiple arbitrary cracks in two dimensions. The main novelty consists in modifying the weight function with an intrinsic enrichment which is discontinuous over the finite length of the crack, represented by a segment, but continuous all around the crack tips. An analytical function is used to introduce discontinuities that are incorporated in the kernel in a simple, multiplicative manner.

Modelling of the expansion of 3-D single bubble using a multi-phase model has been developed for GIVE APPLICATION AREA with the potential of a meshless numerical simulation method, Smoothed Particle Hydrodynamics (SPH), and the consideration of the surface tension between phases and viscosity effect of the polymer melt surrounding the bubble. Mainly, bubble growth in the polymer material occurs because of the mass conversion (mass loss) from the polymer melt to gas due to heat such as fire. This mass conversion drives the expansion process of the gas bubble by increasing the pressure inside. To represent the mass transfer the from the polymer melt to the bubble, this paper proposes a novel algorithm to increase number of SPH gas particles inside the bubble during the simulation. The present paper aims to explain this new developed method including particle shifting scheme identifying the main challenges of dynamic and non-spherical bubble modelling which have a nonlinear multi-phase behaviour. In order to develop stable simulations for the multi-phase bubble growth in isothermal conditions in millimeter scale, surface tension effects have been scaled according to the Capillary number. The insertion of the new gas particles into the bubble centre has been performed at regular intervals to identify the influence of time period of particle insertion. The predicted results from the numerical study have been compared with the well-known analytical solution for single bubble growth for final bubble radius and bubble growth rate. Time step analysis has also been performed to show the numerical stability for this kind of bubble growth simulation. The importance of the particle shifting scheme has also been addressed for simulating bubble growth in this multi-phase problem.

A two-stage numerical procedure using Chebyshev polynomials and trigonometric functions is proposed to approximate the source term of a given partial differential equation. The purpose of such numerical schemes is crucial for the evaluation of particular solutions of a large class of partial differential equations. Our proposed scheme provides a highly efficient and accurate approximation of multivariate functions and particular solution of certain partial differential equations simultaneously. Numerical results on the approximation of eight two-dimensional test functions and their derivatives are given. To demonstrate that the scheme for the approximation of functions can be easily extended to evaluate the particular solution of certain partial differential equations, we solve a modified Helmholtz equation. Near machine precision can be achieved for all these test problems.

The application of a new Material Point Method (MPM) approach to model the proppant distribution in a reservoir where hydraulic fractures interact with natural fractures is presented and validated with an Eagle Ford well. The new MPM approach uses particles to represent the slurry and its effects on the hydraulic and natural fractures. The particles are injected in the hydraulic fractures and their action causes the hydraulic fractures to propagate and interact with the natural fractures thus providing new pathways for the proppant to move away from the wellbore when optimal natural fracture orientations are encountered. Elementary tests, conducted with the new MPM approach show that fractures oriented in certain directions close to the hydraulic fracture orientation facilitate the proppant placement while other fracture orientations that are perpendicular to the hydraulic fracture or close to perpendicular direction appear to cause screen out. When using the new MPM approach on multiple interacting natural fractures with different orientations the same conclusions observed with one single fractures still hold and only those close to the orientation of the hydraulic fractures promote the placement of the proppant. The application of the new technology to an Eagle Ford well shows that the simulated proppant travels the farthest in the frac stage that is known from other methods and measurements to be the one with the best half fracture length. Furthermore, the examination of the simulated proppant concentration curve versus time shows striking similarities with the actual slurry concentration measured at the same well. These encouraging results show that the macroscopic modeling of proppant distribution using the new MPM technology could help improve our understanding of the complex hydraulic fracturing process and the resulting proppant distribution.

A concise overview is given of various numerical methods that can be used to analyse localization and failure in engineering materials. The importance of the cohesive-zone approach is emphasized and various ways to incorporate the cohesive-zone methodology in discretization methods are discussed. Numerical representations of cohesive-zone models suffer from a certain mesh bias. For discrete representations this is caused by the initial mesh design, while for smeared representations it is rooted in the ill-posedness of the rate boundary value problem that arises upon the introduction of decohesion. A proper representation of the discrete character of cohesive-zone formulations which avoids any mesh bias can be obtained elegantly when exploiting the partition-of-unity property of finite element shape functions. The effectiveness of the approach is demonstrated for some examples at different scales. Moreover, examples are shown how this concept can be used to obtain a proper transition from a plastifying or damaging continuum to a shear band with gross sliding or to a fully open crack (true discontinuum). When adhering to a continuum description of failure, higher-order continuum models must be used. Meshless methods are ideally suited to assess the importance of the higher-order gradient terms, as will be shown. Finally, regularized strain-softening models are used in finite element reliability analyses to quantify the probability of the emergence of various possible failure modes.

The main aim of this paper is the development of a refinement procedure able to operate in the context of the constrained natural element method (C-NEM). The C-NEM was proposed by the authors in a former work [Yvonnet J, Ryckelynck D, Lorong P, Chinesta F. A new extension of the natural element method for non-convex and discontinuous domains: the constrained natural element method (C-NEM). Int J Numer Methods Eng 2004;60:1451-1474] and its main meshless features, that allow to describe large domain changes as well as to handle fixed or moving discontinuities, were analyzed in [Yvonnet J, Chinesta F, Lorong P, Rynckelynck D. The constrained natural element method (C-NEM) for treating thermal models involving moving interfaces. Int J Thermal Sci 2005;44:559-569; Yvonnet J, Lorong P, Ryckelynck D, Chinesta F. Simulating dynamic thermo-elastoplasticity in large transformations with adaptive refinement in the natural element method: application to shear banding. Int J Forming Process 2005;8:347-363]. Sometimes, in order to improve the interpolation accuracy for describing boundary layers or an anisotropic behavior, new nodes must be added, removed or repositioned. The interpolation in the vast majority of meshless techniques is free of mesh quality requirement. Thus, introduction, elimination or repositioning of nodes is a trivial task, because no geometrical restrictions exist. In this way, nodes can be added without geometrical checks in the regions where the solution must be improved (identified by using an appropriate error indicator). For this purpose, in this paper an a posteriori error indicator will be proposed and tested in some linear elastostatic problems benchmarks involving different levels of difficulty (stress concentration, solution singularities, . . .) all of them with a known exact solution. The computational implementation of this error indicator is very simple, and when it is used in tandem with an efficient refinement procedure, which makes use of the meshless features of the C-NEM, provides an accurate adaptation procedure, specially appropriate in the C-NEM framework.

This paper deals with a numerical solution of an incompressible Navier-Stokes flow on non-uniform domains. The numerical solution procedure comprises the Meshless Local Strong Form Method for spatial discretization, explicit time stepping, local pressure-velocity coupling and an algorithm for positioning of computational nodes inspired by Smoothed Particles Hydrodynamics method. The presented numerical approach is demonstrated by solving a lid driven cavity flow and backward facing step problems, first on regular nodal distributions up to 315,844 (562 × 562) nodes and then on domain filled with randomly generated obstacles. It is demonstrated that the presented solution procedure is accurate, stable, conver-gent, and it can effectively solve the fluid flow problem on complex geometries. The results are presented in terms of velocity profiles, convergence plots, and stability analyses.

SUMMARY It is now commonly agreed that the global radial basis functions method is an attractive approach for approximating smooth functions. This superiority does not come free; one must find ways to circumvent the associated problem of ill-conditioning and the high computational cost for solving dense matrix systems. We previously proposed different variants of adaptive methods for selecting proper trial subspaces so that the instability caused by inappropriate shape parameters were minimized. In contrast, the compactly supported radial basis functions are more relaxing on the smoothness requirements. By settling with algebraic order of convergence only, compactly supported radial basis functions method, provided the support radius are properly chosen, can approximate functions with less smoothness. The reality is that end-users must know the functions to be approximated a priori in order to decide which method to be used; this is not practical if one is solving a time evolving partial differential equation. The solution could be smooth at the beginning but the formation of shocks may come later in time. In this paper, we propose a hybrid algorithm making use of both global and compactly supported radial basis functions with other developed techniques for meshfree approximation with minimal fine tuning. The first contribution here is an adaptive node refinement scheme. Secondly, we apply the global radial basis functions (with adaptive subspace selection) on the adaptively generated data sites and lastly, the compactly supported radial basis functions (with adaptive support selection) that can be used as a blackbox algorithm for robust approximation to a wider class of functions and for solving partial differential equations.

We develop a polygonal mesh simplification algorithm based on a novel analysis of the mesh geometry. Particularly, we propose first a characterization of vertices as hyperbolic or non-hyperbolic depending upon their discrete local geometry. Subsequently, the simplification process computes a volume cost for each non-hyperbolic vertex, in analogy with spherical volume, to capture the loss of fidelity if that vertex is decimated. Vertices of least volume cost are then successively deleted and the resulting holes re-triangulated using a method based on a novel heuristic. Preliminary experiments indicate a performance comparable to that of the best known mesh simplification algorithms.

We discuss the solution of cornea curvature using a meshless method based on radial basis functions (RBFs). A full two-dimensional nonlinear thin membrane partial differential equation (PDE) model is introduced and solved using the multiquadratic (MQ) and inverse multiquadratic (IMQ) RBFs. This new approach does not rely on radial symmetry or other simplifying assumptions in respect of the cornea shape. It also provides an alternative to corneal topography modeling methods requiring accurate material parameter values, such as Young's modulus and Poisson ratio, that may not be available. The results show good agreement with published corneal data and allow back calculations for estimating certain physical properties of the cornea, such as tension, and elasticity coefficient. All calculations and generation of graphics were performed using the R language programming environment [Rct-15], and RStudio, the integrated development environment (IDE) for R [Rst-15], both of which are open source and free to download.
Part II of this paper demonstrates how the method has been used to provide a very accurate fit to a corneal measured data set.

The element-free Galerkin (EFG) method is probably the most widely used meshless method at present. In the EFG method, shape functions are derived from a moving least-squares approximation using a polynomial basis, a calculation involving the inversion of a small matrix. A new implementation of the EFG method was published soon after the original where an alternative approach using an orthogonal basis was proposed to avoid matrix inversion in the formulation of the shape functions. In this paper we revisit this topic and show that the difficulties associated with the use of a polynomial basis remain present in the orthogonal case. We also show that certain terms in the derivative expressions are omitted in the new implementation of the EFG, which can lead to errors. Finally, we propose a new approach that avoids inversion while maintaining accuracy. Copyright © 2009 John Wiley & Sons, Ltd.

The higher-order gradient plasticity theory is successful in explaining the size effects encountered at the micron and submicron length scale. Due to the incorporation of spatial gradients of one or more internal variables in these theories and the associated non-classical boundary conditions, special types of elements in the finite element method maybe necessary. This makes the numerical implementation of this higher-order theory not straightforward. In this paper, a robust but straightforward numerical implementation of higher-order gradient-dependent plasticity theories is presented. The novelty of this paper is in (1) the application of the meshless methods, particularly the moving weighted least square method, combined with the finite element method for the numerical computation of plastic strain gradients, and (2) the numerical implementation of different types of higher-order microscopic boundary conditions at internal/external surfaces, interfaces, and moving elastic-plastic boundaries. The proposed numerical implementation algorithms can be easily adapted in the implementation of any form of higher-order gradient-dependent constitutive models. Examples of applying the current numerical approach is demonstrated for capturing mesh-objective shear band formation and size effect and boundary layer formation in thin films on elastic substrates and metal matrix composites with embedded elastic inclusions through the consideration of stiff, intermediate, and soft interfaces.

A solid-shell MLPG approach for the numerical analysis of plates and shells is presented. A special attention is devoted to the transversal shear locking effect that appears in the structure thin limit. The theoretical origins of shear locking in the purely displacement-based approach are analyzed by means of the consistency paradigm. It is shown that the spurious constraints appear in the constrained strain field, which lead to the appearance of shear locking and sub-optimal convergence rates. The behaviour of the mixed MLPG approach in the thin limit is also considered. It is determined that in the mixed paradigm the Kirchhoff-Love conditions have to be satisfied only at the nodes to avoid the shear locking effects. The validity of the theoretical predictions is supported by the presented numerical examples, where good convergence and accuracy are obtained even if the low-order meshless approximations are used.

In this paper, a natural element method (NEM) is employed for the analysis of plates and laminates. The displacement field and strain field of plate are based on Reissner-Mindlin plate theory. Sibson interpolation [4] based on natural neighbor coordinates has been used for interpolation. The bending results of the isotropic and laminated composite plate based on natural neighbor method are validated with available finite element method as well analytical results taken from literature. Recovery based relative energy norm error estimator and an adaptive refinement strategy is proposed for the plates by NEM.

This paper presents an efficient meshless method in the formulation of the weak form of local Petrov-Galerkin method MLPG. The formulation is carried out by using an elliptic domain rather than conventional isotropic domain of influence. Therefore, the method involves an MLPG formulation in conjunction with an anisotropic weight function. In the elliptic weight function, each node has three characteristic indicated that were major radius, inner radius, and the direction of the local domain. Furthermore, the space that will be covered by the elliptical domain will be less than the area of the circle (isotropic) at the same main diameter. This means leaving many points of integration are not necessary. Therefore, the computational cost will be decreased. MLPG method with the elliptical domain is used in solving problems of linear elastic fracture mechanism LEFM. MATLAB and Fortran codes are used for obtaining the results of this research .The results were compared with those presented in the literature which shows a reduction in the computational cost up to 15%, and an error criteria enhancement up to 25%.

A meshless approach based on the reproducing kernel particle method is developed for the flexural, free vibration and buckling analysis of laminated composite plates. In this approach, the first-order shear deformation theory (FSDT) is employed and the displacement shape functions are constructed using the reproducing kernel approximation satisfying the consistency conditions. The essential boundary conditions are enforced by a singular kernel method. Numerical examples involving various boundary conditions are solved to demonstrate the validity of the proposed method. Comparison of results with the exact and other known solutions in the literature suggests that the meshless approach yields an effective solution method for laminated composite plates.

In this paper, an adaptive refinement procedure is proposed to be used with Discrete Least Squares Meshless (DLSM) method to obtain accurate solution of planar elasticity problems. DLSM method is a newly introduced meshless method based on the least squares concept. The method leads the solution to a given problem that minimizes a least squares functional defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions. A moving least square is also 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 further increase the efficiency of 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 adaptive configuration contributing to the efficiency of the proposed process. An enrichment method is then used by adding more nodes in the vicinity of nodes with higher errors. 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 new mixed meshless formulation based on the interpolation of both strains and displacements has been proposed for the analysis of plate deformation responses. Kinematics of a three dimensional solid is adopted and discretization is performed by the nodes located on the upper and lower plate surfaces. The governing equations are derived by employing the local Petrov-Galerkin approach. The approximation of all unknown field variables is carried out by using the same Moving Least Squares functions in the in-plane directions, while linear polynomials are applied in the transversal direction. The shear locking effect is efficiently minimized by interpolating the strain field independently from the displacements. The Poisson's thickness locking phenomenon is eliminated by introducing a new procedure based on the modification of the nodal values for the normal transversal strain component. The numerical efficiency of the derived algorithm is demonstrated by the numerical examples.

A meshless computational method based on the local Petrov-Galerkin approach for the analysis of shell structures is presented. A concept of a three dimensional solid, allowing the use of completely 3-D constitutive models, is applied. Discretization is carried out by using both a moving least square approximation and polynomial functions. The exact shell geometry can be described. Thickness locking is eliminated by using a hierarchical quadratic approximation over the thickness. The shear locking phenomena in case of thin structures and the sensitivity to rigid body motions are minimized by applying interpolation functions of sufficiently high order. The numerical efficiency of the derived concept is demonstrated by numerical examples.

In this contribution, a mixed meshless formulation, based on the Local Petrov-Galerkin approach proposed in [1], is developed for analysis of plate and shell structures. In contrast to the available displacement based meshless methods for analysis of shell-like structures [2,3,4], both the strains and displacements are approximated independently. The concept of a three dimensional (3-D) solid is used, allowing the application of complete 3-D constitutive models. Shell geometry is described exactly by employing a parametric mapping technique. Discretization of the shell continuum is carried out by couples of nodes located on the upper and lower surfaces. In order to obtain the governing equations, Petrov-Galerkin principle is applied over the local subdomain I

This paper examines the numerical solution of the transient nonlinear coupled Burgers' equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large values of Reynolds number (Re) up to 10 3. The LRBFCM belongs to a class of truly meshless methods which do not need any underlying mesh but works on a set of uniform or random nodes without any a priori node to node connectivity. The time discretization is performed in an explicit way and collocation with the multiquadric radial basis functions (RBFs) are used to interpolate diffusion-convection variable and its spatial derivatives on decomposed domains. Five nodded domains of influence are used in the local support. Adaptive upwind technique [1,54] is used for stabilization of the method for large Re in the case of mixed boundary conditions. Accuracy of the method is assessed as a function of time and space discretizations. The method is tested on two benchmark problems having Dirichlet and mixed boundary conditions. The numerical solution obtained from the LRBFCM for different value of Re is compared with analytical solution as well as other numerical methods [15,47,49]. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Reynolds numbers.

This contribution focuses on the simulation of two-dimensional elastic
wave propagation in functionally graded solids and structures. Gradient volume
fractions of the constituent materials are assumed to obey the power law function
of position in only one direction and the effective mechanical properties of the
material are determined by the Mori–Tanaka scheme. The investigations are
carried out by extending a meshless method known as the Meshless Local
Petrov-Galerkin (MLPG) method which is a truly meshless approach to thermoelastic
wave propagation. Simulations are carried out for rectangular domains
under transient thermal loading. To investigate the effect of material composition
on the dynamic response of functionally graded materials, a metal/ceramic
(Aluminum (Al) and Alumina ( Al2O3 ) are considered as ceramic and metal
constituents) composite is considered for which the transient thermal field,
dynamic displacement and stress fields are reported for different material
distributions.

A meshless collocation method is developed for the static analysis of plane problems of functionally graded (FG) elastic beams and plates under transverse mechanical loads using the differential reproducing kernel (DRK) interpolation, in which the DRK interpolant is constructed by the randomly distributed nodes. A point collocation method based on this DRK interpolation is developed for the plane stress and strain problems of homogeneous and FG elastic beams and plates. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach with excellent accuracy and has a fast convergence rate.

A Hermite differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving fourth-order differential equations where the field variable and its first-order derivatives are regarded as the primary variables. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the Hermite DRK interpolation, without directly differentiating

The aim of this manuscript is to give a practical overview of meshless methods (for solid mechanics) based on global weak forms through a simple and well-structured MATLAB code, to illustrate our discourse. The source code is available for download on our website and should help students and researchers get started with some of the basic meshless methods; it includes intrinsic and extrinsic enrichment, point collocation methods, several boundary condition enforcement schemes and corresponding test cases. Several one and two-dimensional examples in elastostatics are given including weak and strong discontinuities and testing different ways of enforcing essential boundary conditions.

An efficient meshless formulation based on the Local Petrov-Galerkin approach for the analysis of shear deformable thick plates is presented. Using the kinematics of a three-dimensional continuum, the local symmetric weak form of the equilibrium equations over the cylindrical shaped local sub-domain is derived. The linear test function in the plate thickness direction is assumed. Discretization in the in-plane directions is performed by means of the moving least squares approximation. The linear interpolation over the thickness is used for the in-plane displacements, while the hierarchical quadratic interpolation is adopted for the transversal displacement in order to avoid the thickness locking effect. The numerical efficiency of the proposed meshless formulation is illustrated by the numerical examples. keyword: meshless formulation, thick plates, threedimensional solid concept, moving least squares approximation.

A comparison between weak form meshless local Petrov-Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.

We use a Meshless local Petrov-Galerkin method (MLPG) to analyse an elastostatic problem deformation of a homogeneous rectangular plate of two –dimensional. First the formulations of method local Petrov-Galerkin (MLPG) and Local radial point interpolation method (LRPIM) are obtained. The aim of this article is to study the convergence and accuracy of these methods: MLPG and LRPIM. Finally a comparative study of numerical results obtained is made.

Meshless and mesh-based methods are among the tools frequently applied in the numerical treatment of partial differential equations (PDEs). This paper presents a coupling of the meshless finite cloud method (FCM) and the standard (mesh-based) boundary element method (BEM), which is motivated by the complementary properties of both methods. While the BEM is appropriate for solving linear PDEs with constant coefficients in bounded or unbounded domains, the FCM is appropriate for either homogeneous, inhomogeneous or even nonlinear problems in bounded domains. Both techniques (FCM and BEM) use a collocation procedure in the numerical approximation. No mesh is required in the FCM subdomain and its nodal point distribution is completely independent of the BEM subdomain. The coupling approach is demonstrated for linear elasticity problems. Because both FCM and BEM employ tractiondisplacement relations, no transformations between forces and tractions (typical of BEM and finite element coupling) are needed. Several numerical examples are given to demonstrate the proposed approach.

This paper examines the numerical solution of the transient nonlinear coupled Burgers' equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large values of Reynolds number (Re) up to 10 3 . The LRBFCM belongs to a class of truly meshless methods which do not need any underlying mesh but works on a set of uniform or random nodes without any a priori node to node connectivity. The time discretization is performed in an explicit way and collocation with the multiquadric radial basis functions (RBFs) are used to interpolate diffusion-convection variable and its spatial derivatives on decomposed domains. Five nodded domains of influence are used in the local support. Adaptive upwind technique [1,54] is used for stabilization of the method for large Re in the case of mixed boundary conditions. Accuracy of the method is assessed as a function of time and space discretizations. The method is tested on two benchmark problems having Dirichlet and mixed boundary conditions. The numerical solution obtained from the LRBFCM for different value of Re is compared with analytical solution as well as other numerical methods . It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Reynolds numbers.

Purpose -The purpose of this paper is to explore the application of the mesh-free local radial basis function collocation method (RBFCM) in solution of coupled heat transfer and fluid-flow problems. Design/methodology/approach -The involved temperature, velocity and pressure fields are represented on overlapping five nodded sub-domains through collocation by using multiquadrics radial basis functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time stepping. Findings -The performance of the method is assessed on the classical two dimensional de Vahl Davis steady natural convection benchmark for Rayleigh numbers from 10 3 to 10 8 and Prandtl number 0.71. The results show good agreement with other methods at a given range. Originality/value -The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local mass continuity equation violation. This formulation does not require solution of pressure Poisson or pressure correction Poisson equations and thus much simplifies the previous attempts in the field.

In part I we discussed the solution of corneal curvature using a 2D meshless method based on radial basis functions (RBFs). In Part II we use these methods to fit a full nonlinear thin membrane model to a measured data-set in order to generate a topological mathematical description of the cornea. In addition, we show how these results can lead to estimations for corneal radius of curvature and certain physical properties of the cornea; namely, tension and elasticity coefficient. Again all calculations and graphics generation were performed using the R language programming environment.

The model describes corneal topology extremely well, and the estimated properties fall well within the expected range of values. The method is straight forward to implement and offers scope for further analysis using more detailed 3D models that include corneal thickness.

Radial basis functions (RBF) are widely used in many areas especially for interpolation and approximation of scattered data, solution of ordinary and partial differential equations, etc. The RBF methods belong to meshless methods, which do not require tessellation of the data domain, i.e. using Delaunay triangu-lation, in general. The RBF meshless methods are independent of a dimensional-ity of the problem solved and they mostly lead to a solution of a linear system of equations. Generally, the approximation is formed using the principle of unity as a sum of weighed RBFs. These two classes of RBFs: global and local, mostly having a shape parameter determining the RBF behavior. In this contribution, we present preliminary results of the estimation of a vector of "optimal" shape parameters , which are different for each RBF used in the final formula for RBF approximation. The preliminary experimental results proved, that there are many local optima and if an iteration process is to be used, no guaranteed global optima are obtained. Therefore, an iterative process, e.g. used in partial differential equation solutions, might find a local optimum, which can be far from the global op-tima.

A particular meshless method, named meshless local Petrov -Galerkin is investigated. To treat the essential boundary condition problem, an alternative approach is proposed. The basic idea is to merge the best features of two different methods of shape function generation: the moving least squares (MLS) and the radial basis functions with polynomial terms (RBFp). Whereas the MLS has lower computational cost, the RBFp imposes in a direct manner the essential boundary conditions. Thus, dividing the domain into different regions a hybrid method has been developed. Results show that it leads to a good trade-off between computational time and precision.

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least-squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind-biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h-adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution-based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. resources and ever-challenging demands for practical and theoretical applications. Nowadays, there are two main types of numerical techniques for solving PDEs. On the one hand, there exist meshbased or conventional discretization methods; among them the classical finite differences (FD), finite volume (FV) and finite element (FE) methods are of singular interest. These techniques are mostly employed in practice due to their robustness, efficiency and high confidence gained through years of continuous use and enhancement. On the other hand, there exist meshless methods. Having their pros and cons, meshless methods offer an alternative to mesh-based techniques. Meshless methods are conceptually attractive; however, their practical implementations have not succeeded so far to prove their efficiency and this is a fact which can explain the comparatively little attention that has been devoted to these techniques. In spite of this, over the last 10 years, some difficulties that arose in conventional mesh-based methods when performing particular applications have brought meshless methods into the focus of attention.

In the proposed nearest-nodes finite element method (NN-FEM), finite elements are used only for numerical integration; while shape functions are constructed in a similar way as in meshless methods, i.e. by using a set of nodes that are the nearest to a concerned quadrature point. Some of the nodes may be from adjacent elements. A recently developed local multivariate Lagrange interpolation method is used to construct shape functions. In the above way, the proposed NN-FEM inherits most of the merits from both the finite element method and meshless methods. The major attractive features of NN-FEM include: analysis results are insensitive to element distortion; a crack can propagate along an arbitrary path; with element adjacency information, time spent on searching nearest nodes is much shorter than that used by a meshless method for identifying nodes covered under an influence domain.

The three-dimensional exposure method for the detection of the boundary of a set of overlapping spheres is presented. Like the two-dimensional version described in a previous paper, the three-dimensional algorithm precisely detects void opening or closure, and is optimally suited to the kernel-mediated interactions of smoothed-particle hydrodynamics, although it may be used in any application involving sets of overlapping spheres. The principle idea is to apply the two-dimensional method, on the surface of each candidate boundary sphere, to the circles of intersection with neighboring spheres. As the algorithm finds the exact solution, the quality of detection is independent of particle configuration, in contrast to gradient-based techniques. The observed CPU execution times scale as O(MN ), where M is the number of particles, N is the average number of neighbors of a particle, and is a problem-dependent constant between 1.6 and 1.7. The time required per particle is comparable to the amount of time required to evaluate a three-dimensional linear moving-least-squares interpolant at a single point.