Showing papers on "Meshfree methods published in 2011"

On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)

Abstract: By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25: 137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6): 859-877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non-polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). Copyright (C) 2011 John Wiley & Sons, Ltd.

Adaptive analysis using the node‐based smoothed finite element method (NS‐FEM)

Abstract: The paper presents an adaptive analysis within the framework of the node-based smoothed finite element method (NS-FEM) using triangular elements. An error indicator based on the recovery strain is used and shown to be asymptotically exact by an effectivity index and numerical results. A simple refinement strategy using the newest node bisection is briefly presented. The numerical results of some benchmark problems show that the present adaptive procedure can accurately catch the appearance of the steep gradient of stresses and the occurrence of refinement is concentrated properly. The energy error norms of adaptive models for both NS-FEM and FEM obtain higher convergence rate compared with the uniformly refined models, but the results of NS-FEM are better and achieve higher convergence rate than those of FEM. The effectivity index of NS-FEM is also closer and approaches to unity faster than that of FEM. The upper bound property in the strain energy of NS-FEM is always verified during the adaptive procedure. Copyright © 2009 John Wiley & Sons, Ltd.

Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling

Abstract: Fracture modelling using numerical methods is well-advanced in 2D using techniques such as the extended finite element method (XFEM). The use of meshless methods for these problems lags somewhat behind, but the potential benefits of no meshing (particularly in 3D) prompt continued research into their development. In methods where the crack face is not explicitly modelled (as the edge of an element for instance), two procedures are instead used to associate the displacement jump with the crack surface: the visibility criterion and the diffraction method. The visibility criterion is simple to implement and efficient to compute, especially with the help of level set coordinates. However, spurious discontinuities have been reported around crack tips using the visibility criterion, whereas implementing the diffraction method in 3D is much more complicated than the visibility criterion. In this paper, a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved. The formulation is based on the use of level set coordinates and the element-free Galerkin method, and is generally applicable for single or multiple crack problems in 2D or 3D. The paper explains the formulation and provides verification of the method against a number of 2D crack problems. Copyright © 2011 John Wiley & Sons, Ltd.

On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

Abstract: We investigate the influence of the shape parameter in the meshless Gaussian radial basis function finite difference (RBF-FD) method with irregular centres on the quality of the approximation of the Dirichlet problem for the Poisson equation with smooth solution. Numerical experiments show that the optimal shape parameter strongly depends on the problem, but insignificantly on the density of the centres. Therefore, we suggest a multilevel algorithm that effectively finds a near-optimal shape parameter, which helps to significantly reduce the error. Comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the Gaussian RBF is provided.

Thin shell analysis from scattered points with maximum-entropy approximants

Abstract: This is the accepted version of the following article: [Millan, D., Rosolen, A. and Arroyo, M. (2011), Thin shell analysis from scattered points with maximum-entropy approximants. Int. J. Numer. Meth. Engng., 85: 723–751. doi:10.1002/nme.2992], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.2992/abstract

A coupled ES-FEM/BEM method for fluid–structure interaction problems

Abstract: The edge-based smoothed finite element method (ES-FEM) developed recently shows some excellent features in solving solid mechanics problems using triangular mesh. In this paper, a coupled ES-FEM/BEM method is proposed to analyze acoustic fluid–structure interaction problems, where the ES-FEM is used to model the structure, while the acoustic fluid is represented by boundary element method (BEM). Three-node triangular elements are used to discretize the structural and acoustic fluid domains for the important adaptability to complicated geometries. The smoothed Galerkin weak form is adopted to formulate the discretized equations for the structure, and the gradient smoothing operation is applied over the edge-based smoothing domains. The global equations of acoustic fluid–structure interaction problems are then established by coupling the ES-FEM for the structure and the BEM for the fluid. The gradient smoothing technique applied in the structural domain can provide the important and right amount of softening effects to the “overly-stiff” FEM model and thus improve the accuracy of the solutions of coupled system. Numerical examples of acoustic fluid–structure interaction problems have been used to assess the present formulation, and the results show that the accuracy of present method is very good and even higher than those obtained using the coupled FEM/BEM with quadrilateral mesh.

A comparison of three explicit local meshless methods using radial basis functions

Abstract: In this paper, three kinds of explicit local meshless methods are compared: the local method of approximate particular solutions (LMAPS), the local direct radial basis function collocation method (LDRBFCM) which are both first presented in this paper, and the local indirect radial basis function collocation method (LIRBFCM). In all three methods, the time discretization is performed in explicit way, the multiquadric radial basis functions (RBFs) are used to interpolate either initial temperature field and its derivatives or the Laplacian of the initial temperature field. The five-noded sub-domains are used in localization. Numerical results of simple diffusion equation with Dirichlet jump boundary condition are compared on uniform and random node arrangement, the accuracy and stabilities of these three local meshless methods are asserted. One can observe that the improvement of the accuracy with denser nodes and with smaller time steps for all three methods. All methods provide a similar accuracy in uniform node arrangement case. For random node arrangement, the LMAPS and the LDRBFCM perform better than the LIDRBFCM.

A local radial basis functions—Finite differences technique for the analysis of composite plates

Abstract: Radial basis functions are a very accurate means of solving interpolation and partial differential equations problems. The global radial basis functions collocation technique produces ill-conditioning matrices when using multiquadrics, making the choice of the shape parameter a crucial issue. The use of local numerical schemes, such as finite differences produces much better conditioned matrices. However, finite difference schemes are limited to special grids. For scattered points, a combination of finite differences and radial basis functions would be a possible solution. In this paper, we use a higher-order shear deformation plate theory and a radial basis function—finite difference technique for predicting the static behavior of thin and thick composite plates. Through numerical experiments on square and L-shaped plates, the accuracy and efficiency of this collocation technique is demonstrated, and the numerical accuracy and convergence are thoughtfully examined. This technique shows great potential to solve large engineering problems without the issue of ill-conditioning.

Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures

Abstract: A dispersion analysis is carried out to study the dynamic behavior of the Hermite reproducing kernel (HRK) Galerkin meshfree formulation for thin beam and plate problems. The HRK approximation utilizes both the nodal deflectional and rotational variables to construct the meshfree approximation of the deflection field within the reproducing kernel framework. The discrete Galerkin formulation is fulfilled with the method of sub-domain stabilized conforming integration. In the dispersion analysis following the HRK Galerkin meshfree semi-discretization, both the deflectional and rotational nodal variables are expressed by harmonic functions and then substituted into the semi-discretized equation to yield the characteristic equation. Subsequently the numerical frequency and phase speed can be obtained. The transient analysis with full-discretization is performed by using the central difference time integration scheme. The results of dispersion analysis of thin beams and plates show that compared to the conventional Gauss integration-based meshfree formulation, the proposed method has more favorable dispersion performance. Thereafter the superior performance of the present method is also further demonstrated by several transient analysis examples.

A smoothed Hermite radial point interpolation method for thin plate analysis

Abstract: A smoothed Hermite radial point interpolation method using gradient smoothing operation is formulated for thin plate analysis. The radial basis functions augmented with polynomial basis are used to construct the shape functions that have the important Delta function property. The smoothed Galerkin weakform is adopted to discretize the governing partial differential equations, and a curvature smoothed operation is developed to relax the continuity requirement and achieve accurate bending solutions. The approximation based on both deflection and rotation variables make the proposed method very effective in enforcing the essential boundary conditions. The effects of different numbers of sub-smoothing-domains created based on the triangular background cell are investigated in detail. A number of numerical examples have been studied and the results show that the present method is very stable and accurate even for extremely irregular background cells.

3-D numerical analysis of orthogonal cutting process via mesh-free method

Abstract: In this paper, the applications of mesh-free SPH (Smoothed Particle Hydrodynamics) methodology to the simulation and analysis of 3-D hard machining process is presented. A Langrangian SPH based model is carried out using the Ls-Dyna software. Classical Lagrangian, Eulerian and ALE methods such as finite element methods (FEM) cannot resolve the large distortions very well. Conventional finite element analysis of metal cutting processes often breaks down due to severe mesh distortion. Recent developments in so called mesh-free or meshless methods provide alternates for traditional numerical methods in modeling the machining processes. SPH is a mesh-free approach, so large material deformations that happen in the analysis of cutting problem are easily managed and SPH contact control permits an “inherent” chip/workpiece separation. Because SPH combines the advantages of mesh-free, langrangian, particle methods, an alternative methodology, which appears to eliminate most of those difficulties is that of meshless methods. The orthogonal cutting process of AISI H13 steel material was modeled and analysed using SPH method. The developed SPH model gained its ability to correctly estimate the cutting forces, as illustrated in two orthogonal cutting situations. Cutting forces were compared for SPH, Langrangian explicit and experimental results. The predicted (SPH) cutting forces agree within 8.43% and 11.70% of the measured values for tangential and normal components respectively. A good agreement between predicted and experimental cutting forces was observed. Key words: Machining, cutting, mesh-free methods, meshless, SPH, Eulerian, ALE, Langrangian

A natural neighbour meshless method with a 3D shell-like approach in the dynamic analysis of thin 3D structures

Abstract: This work presents the dynamic analysis of three-dimensional plate and shell structures based on an improved meshless method, the Natural Neighbour Radial Point Interpolation Method (NNRPIM) using a shell-like formulation. In the NNRPIM, the nodal connectivity is imposed using the natural neighbours concept. An integration background mesh is constructed, totally node-dependent, and used in the numerical integration of the NNRPIM interpolation functions, which possess the delta Kronecker property. Several dynamic plate and shell problems are studied to demonstrate the effectiveness of the method.

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

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

A three-dimensional adaptive analysis using the meshfree node-based smoothed point interpolation method (NS-PIM)

Abstract: In this paper, a three-dimensional (3-D) adaptive analysis procedure is proposed using the meshfree node-based smoothed point interpolation method (NS-PIM) Previous study has shown that the NS-PIM works well with the simplest four-node tetrahedral mesh, which is easy to be implemented for complicated geometry In contrast to the displacement-based FEM providing lower bound solutions, the NS-PIM possesses the attractive property of providing upper bound solutions in strain energy norm In the present adaptive procedure, a novel error indicator is devised for NS-PIM settings, which evaluates the maximum difference of strain energy values among four nodes in each of the tetrahedral cells A simple h-type local refinement scheme is adopted and coupled with 3-D mesh automatic generator based on Delaunay technology Numerical results indicate that the adaptive refinement procedure can effectively capture the stress concentration and solution singularities, and carry out local refinement automatically The present adaptive procedure achieves much higher convergence in strain energy solution compared to the uniform refinement, and obtains upper bound solution in strain energy efficiently for force driven problems

An edge‐based smoothed finite element method (ES‐FEM) using 3‐node triangular elements for 3D non‐linear analysis of spatial membrane structures

Abstract: An edge-based smoothed finite element method using 3-node triangular membrane elements (ES-FEM-T3) is proposed to analyze three-dimensional (3D) spatial membrane structures under large deflection, rotation, and strain. In our 3D formulation, co-rotational local coordinate systems associated with the edge-based smoothing domains are constructed. Edge-based gradient smoothing for the spatial membrane structure is performed in global Cartesian coordinate system and transformed into the co-rotational local coordinate system. The smoothed strain and stress can then be properly evaluated in the co-rotational local coordinate system after the elimination of the rigid body motions simply by coordinates transformation. Explicit time integration scheme is used to compute the transient response of the 3D spatial membrane structure to time-domain excitations, and the dynamic relaxation method is employed to obtain steady-state solutions. The numerical results demonstrate that the proposed method produces much better solution accuracy and computational efficiency than the standard FEM using T3 membrane element. Copyright © 2010 John Wiley & Sons, Ltd.

MLPG Method for Transient Heat Conduction Problem with MLS as Trial Approximation in Both Time and Space Domains

Abstract: The meshless local Petrov-Galerkin (MLPG) method with an effi- cient technique to deal with the time variable are used to solve the heat conduction problem in this paper. The MLPG is a meshless method which is (mostly) based on the moving least squares (MLS) scheme to approximate the trial space. In this paper the MLS is used for approximation in both time and space domains, and we avoid using the time difference discretization or Laplace transform method to overcome the time variable. The technique is applied for continuously nonhomo- geneous functionally graded materials (FGM) in a finite strip and a hallow cylinder. This idea can be easily extended to all MLS based methods such as the element free Galerkin (EFG), the local boundary integral equation (LBIE) and etc.

Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions

Abstract: In this article a numerical solution of the time dependent, coupled system equations of magnetohydrodynamics (MHD) flow is obtained, using the strong-form local meshless point collocation (LMPC) method. The approximation of the field variables is obtained with the moving least squares (MLS) approximation. Regular and irregular nodal distributions are used. Thus, a numerical solver is developed for the unsteady coupled MHD problems, using the collocation formulation, for regular and irregular cross sections, as are the rectangular, triangular and circular. Arbitrary wall conductivity conditions are applied when a uniform magnetic field is imposed at characteristic directions relative to the flow one. Velocity and induced magnetic field across the section have been evaluated at various time intervals for several Hartmann numbers (up to 105) and wall conductivities. The numerical results of the strong-form MPC method are compared with those obtained using two weak-form meshless methods, that is, the local boundary integral equation (LBIE) meshless method and the meshless local Petrov---Galerkin (MLPG) method, and with the analytical solutions, where they are available. Furthermore, the accuracy of the method is assessed in terms of the error norms L 2 and L ?, the number of nodes in the domain of influence and the time step length depicting the convergence rate of the method. Run time results are also presented demonstrating the efficiency and the applicability of the method for real world problems.