Direct meshless local Petrov–Galerkin method for elastodynamic analysis

This article describes a new and fast meshfree method based on a generalized moving least squares (GMLS) approximation and the local weak forms for vibration analysis in solids. In contrast to the meshless local Petrov–Galerkin method, GMLS directly approximates the local weak forms from meshless nodal values, which shifts the local integrations over the low-degree polynomial basis functions rather than over the complicated MLS shape functions. Besides, if the method is set up properly, all local integrals have the same value if all local subdomains have the same shape. These features reduce the computational costs, remarkably. The new technique is called direct meshless local Petrov–Galerkin (DMLPG) method. In DMLPG, the stiff and mass matrices are constructed by integration against polynomials. This overcomes the main drawback of meshfree methods in comparison with the finite element methods (FEM). The Newmark scheme is adapted as a time integration method, and numerical results are presented for various dynamic problems. The results are compared with the exact solutions, if available, and the FEM solutions.     

用局部Petrov-Galerkin方法求解不可压超弹性材料问题

用一种修正的无网格局部Petrov-Galerkin方法求解了不可压超弹性材料平面应力问题。在建立求解方程过程中,采用径向基函数耦合多项式构造近似函数,并以Heaviside分段函数作为加权函数简化了刚度矩阵的域积分,引入平面应力假设避免了材料不可压引起的数值求解困难。数值算例表明:该文方法求解不可压超弹性材料平面应力问题具有稳定性好、精度高的特点。     

Comparisons of two meshfree local point interpolation methods for structural analyses

 As truly meshless methods, the local point interpolation method (LPIM) and the local radial point interpolation method (LR-PIM), are based on the point interpolations and local weak forms integrated in a local domain of very simple shape. LPIM and LR-PIM are examined and compared with each other. They are also compared with the established FEM and the meshless local Petrov-Galerkin (MLPG) method. The numerical implementations of these two methods are discussed in detail. Parameters that influence the performance of them are detailedly studied. The convergence and efficiency of them are thoroughly investigated. LPIM and LR-PIM formulations are developed for structural analyses of 2-D elasto-dynamic problems and 1-D Timoshenko beam problems in the first time. It is found that LPIM and LR-PIM are very easy to implement, and very efficient obtaining numerical solutions to problems of computational mechanics. Received 31 August 2001 / Accepted 04 March 2002     

SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations

We propose a new and simple technique called the Symmetric Smoothed Particle Hydrodynamics (SSPH) method to construct basis functions for meshless methods that use only locations of particles. These basis functions are found to be similar to those in the Finite Element Method (FEM) except that the basis for the derivatives of a function need not be obtained by differentiating those for the function. Of course, the basis for the derivatives of a function can be obtained by differentiating the basis for the function as in the FEM and meshless methods. These basis functions are used to numerically solve two plane stress/strain elasto-static problems by using either the collocation method or a weak formulation of the problem defined over a subregion of the region occupied by the body; the latter is usually called the Meshless Local Petrov–Galerkin (MLPG) method. For the two boundary-value problems studied, it is found that the weak formulation in which the basis for the first order derivatives of the trial solution are derived directly in the SSPH method (i.e., not obtained by differentiating the basis function for the trial solution) gives the least value of the L²-error norm in the displacements while the collocation method employing the strong formulation of the boundary-value problem has the largest value of the L²-error norm. The numerical solution using the weak formulation requires more CPU time than the solution with the strong formulation of the problem. We have also computed the L²-error norm of displacements by varying the number of particles, the number of Gauss points used to numerically evaluate domain integrals appearing in the weak formulation of the problem, the radius of the compact support of the kernel function used to generate the SSPH basis, the order of complete monomials employed for constructing the SSPH basis, and boundary conditions used at a point on a corner of the rectangular problem domain. It is recommended that for solving two-dimensional elasto-static problems, the MLPG formulation in which derivatives of the trial solution are found without differentiating the SSPH basis function be adopted.     

A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods

The essential features of the Meshless Local Petrov-Galerkin (MLPG) method, and of the Local Boundary Integral Equation (LBIE) method, are critically examined from the points of view of a non-element interpolation of the field variables, and of the meshless numerical integration of the weak form to generate the stiffness matrix. As truly meshless methods, the MLPG and the LBIE methods hold a great promise in computational mechanics, because these methods do not require a mesh, either to construct the shape functions, or to integrate the Petrov-Galerkin weak form. The characteristics of various meshless interpolations, such as the moving least square, Shepard function, and partition of unity, as candidates for trial and test functions are investigated, and the advantages and disadvantages are pointed out. Emphasis is placed on the characteristics of the global forms of the nodal trial and test functions, which are non-zero only over local sub-domains Ω_tr ^J and Ω_te ^I , respectively. These nodal trial and test functions are centered at the nodes J and I (which are the centers of the domains Ω_tr ^J and Ω_te ^I ), respectively, and, in general, vanish at the boundaries ∂Ω_tr ^J and ∂Ω_te ^I of Ω_tr ^J and Ω_te ^I , respectively. The local domains Ω_tr ^J and Ω_te ^I can be of arbitrary shapes, such as spheres, rectangular parallelopipeds, and ellipsoids, in 3-Dimensional geometries. The sizes of Ω_tr ^J and Ω_te ^I can be arbitrary, different from each other, and different for each J, and I, in general. It is shown that the LBIE is but a special form of the MLPG, if the nodal test functions are specifically chosen so as to be the modified fundamental solutions to the differential equations in Ω_te ^I , and to vanish at the boundary ∂Ω_te ^I . The difficulty in the numerical integration of the weak form, to generate the stiffness matrix, is discussed, and a new integration method is proposed. In this new method, the Ith row in the stiffness matrix is generated by integrating over the fixed sub-domain Ω_te ^I (which is the support for the test function centered at node I); or, alternatively the entry K _{I J} in the global stiffness matrix is generated by integrating over the intersections of the sub-domain Ω_tr ^J (which is the sub-domain, with node J as its center, and over which the trial function is non-zero), with Ω_te ^I (which is the sub-domain centered at node I over which the test function is non-zero). The generality of the MLPG method is emphasized, and it is pointed that the MLPG can also be the basis of a Galerkin method that leads to a symmetric stiffness matrix. This paper also points out a new but elementary method, to satisfy the essential boundary conditions exactly, in the MLPG method, while using meshless interpolations of the MLS type. This paper presents a critical appraisal of the basic frameworks of the truly meshless MLPG/LBIE methods, and the numerical examples show that the MLPG approach gives good results. It now apears that the MLPG method may replace the well-known Galerkin finite element method (GFEM) as a general tool for numerical modeling, in the not too distant a future. Received 15 January 1999     

Modified meshless local Petrov–Galerkin formulations for elastodynamics

This paper is concerned with the extension to the elastodynamic problems of the idea of analytical integrations in meshless local weak formulations. In this context, Taylor series expansions of the incognita fields are considered, and the related integrals of the meshless formulations are solved analytically, rendering a so‐called modified methodology. The moving least squares approximation is employed for the spatial variation of the displacement fields, and two variants of the meshless local Petrov–Galerkin (MLPG) formulation are discussed here, which are based on the use of Heaviside or Gaussian weight test functions. Once the spatial discretization is considered, the semi‐discretized ordinary differential equations for nodal unknowns that arise are treated in the time‐domain by the Houbolt's method. Considering the modified meshless methodology, it was found that the standard derivatives of the shape functions must also be modified to achieve numerically stable procedures, taking into account dynamic analyses. Thus, an efficient and easy‐to‐implement technique is developed to properly compute the shape function derivatives. As described in the paper, the proposed modified MLPG formulations are more effective than standard MLPG formulations, especially taking into account well refined large scale problems, providing much better computational efficiency, good accuracy and more robust convergence. Copyright © 2012 John Wiley & Sons, Ltd.     

Meshless local Petrov–Galerkin (MLPG) method in combination with finite element and boundary element approaches

The meshless local Petrov–Galerkin (MLPG) method is an effective truly meshless method for solving partial differential equations using moving least squares (MLS) interpolants. It is, however, computationally expensive for some problems. A coupled MLPG/finite element (FE) method and a coupled MLPG/boundary element (BE) method are proposed in this paper to improve the solution efficiency. A procedure is developed for the coupled MLPG/FE method and the coupled MLPG/BE method so that the continuity and compatibility are preserved on the interface of the two domains where the MLPG and FE or BE methods are applied. The validity and efficiency of the MLPG/FE and MLPG/BE methods are demonstrated through a number of examples. Received 6 June 2000     

An improved meshless Shepard and least squares method possessing the delta property and requiring no singular weight function

The meshless Shepard and least squares (MSLS) method and the meshless Shepard method are partition of unity based meshless interpolations which eliminate the problems by other meshless methods such as the difficulty in direct imposition of the essential boundary conditions. However, singular weight functions have to be used in both methods to enforce the approximation interpolatory, which leads to the loss of smoothness in approximation and locally oscillatory results. In this paper, an improved MSLS interpolation is developed by using dually defined nodal supports such that no singular weight function is required. The proposed interpolation satisfies the delta property at boundary nodes and the compatibility condition throughout the domain, and is capable of exactly reproducing the basis function. The computational cost of the present interpolation is much lower than the moving least-squares approximation which is probably the most widely used meshless interpolation at present.     

The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics

 The meshless local Petrov-Galerkin (MLPG) approach is an effective method for solving boundary value problems, using a local symmetric weak form and shape functions from the moving least squares approximation. In the present paper, the MLPG method for solving problems in elasto-statics is developed and numerically implemented. The present method is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation are imposed by a penalty method, as the essential boundary conditions can not be enforced directly when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present MLPG method. The numerical examples show that the present MLPG approach does not exhibit any volumetric locking for nearly incompressible materials, and that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.     

Two-dimensional stress analysis of functionally graded solids using the MLPG method with radial basis functions

The meshless local Petrov–Galerkin (MLPG) method is used for analysing two-dimensional (2D) static and dynamic deformations of functionally graded materials (FGMs) with material response modelled as either linear elastic or as linear viscoelastic. The multiquadric radial basis function (RBF) is employed to approximate the trial solution. Results are computed with two different choices of test functions, namely a fourth-order spline weight function, and a Heaviside step function, each having a compact support. No background mesh is used to numerically evaluate integrals appearing in the weak formulation of the problem, thus the method is truly meshless. A benefit of using RBFs is that they possess the Kronecker delta property; thus it is easy to satisfy essential boundary conditions. For five problems, the computed results are found to match well with those either from their analytical solutions or numerical solutions of other researchers who employed different algorithms. For a dynamic problem, the Laplace-transform technique is utilised. The numerical examples illustrate that displacements and stress distributions in a structure made of an FGM differ considerably from those at the corresponding points in the same structure made of a homogeneous material. Thus, the inhomogeneity in material properties can be exploited to optimise stress distribution, minimise deflection and reduce the maximum stress.     

A modified meshless local Petrov–Galerkin method to elasticity problems in computer modeling and simulation

A modified meshless local Petrov–Galerkin (MLPG) method is presented for elasticity problems using the moving least squares (MLS) approximation. It is a truly meshless method because it does not need a mesh for the interpolation of the solution variables or for the integration of the energy. In this paper, a simple Heaviside test function is chosen to overcome the computationally expensive problems in the MLPG method. Essential boundary conditions are imposed by using a direct interpolation method based on the MLPG method establishes equations node by node. Numerical results in several examples show that the present method yielded very accurate solutions. And the sensitivity of the method to several parameters is also studied in this paper.     

Application of meshless local integral equations to two dimensional analysis of coupled non-Fick diffusion–elasticity

This work presents the application of meshless local Petrov–Galerkin (MLPG) method to two dimensional coupled non-Fick diffusion–elasticity analysis. A unit step function is used as the test functions in the local weak-form. It leads to local integral equations (LIEs). The analyzed domain is divided into small subdomains with a circular shape. The radial basis functions are used for approximation of the spatial variation of field variables. For treatment of time variations, the Laplace-transform technique is utilized. Several numerical examples are given to verify the accuracy and the efficiency of the proposed method. The molar concentration diffuses through 2D domain with a finite speed similar to elastic wave. The propagation of mass diffusion and elastic waves are obtained and discussed at various time instants. The MLPG method has a high capability to track the diffusion and elastic wave fronts at arbitrary time instants in 2D domain. The profiles of molar concentration and displacements in two orthogonal directions are illustrated at various time instants.     

Meshless local Petrov–Galerkin (MLPG) method for wave propagation in 3D poroelastic solids

A meshless local Petrov–Galerkin (MLPG) method is applied to solve wave propagation problems of three-dimensional poroelastic solids with Biot's theory. The Laplace transform is used to eliminate the time dependence of the field variables for the transient elastodynamic case. A weak formulation with a unit step function transforms the set of governing equations into local integral equations on local subdomains. The meshless approximation based on the radial basis function (RBF) is employed for the implementation. Unknown Laplace-transformed quantities, including displacements of solid frame and pressure in the fluid, are computed from the local boundary integral equations. The time-dependent values are obtained by Durbin's inversion technique. In addition, a one-dimensional poroelasticity analytical solution is derived in this paper and introduced for comparison. Several numerical examples demonstrate the efficiency and accuracy of the proposed method.     

Meshless solutions of 2D contact problems by subdomain variational inequality and MLPG method with radial basis functions

A subdomain variational inequality and its meshless linear complementary formulation are developed in the present paper for solving two-dimensional contact problems. The subdomain variational inequality will be defined in detail. The meshless method is based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain and radial basis functions as trial functions for interpolation. Three different radial basis functions (RBFs), i.e. Multiquadrics (MQ), Gaussian (EXP) and Thin Plate Splines (TPS) are examined and the selection of their shape parameters is studied based on 2D solid stress problems with closed-form solutions. The developed meshless/linear complementary method is applied to solve two frictionless contact problems. For the RBFs, it has been found that the TPS shape parameter is not sensitive to nodal distance and a value of 4 is found as a good choice for TPS from this research.     

A meshless method for generalized linear or nonlinear Poisson-type problems

This paper presents a meshless method, based on coupling virtual boundary collocation method (VBCM) with the radial basis functions (RBF) and the analog equation method (AEM), to analyze generalized linear or nonlinear Poisson-type problems. In this method, the AEM is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator, instead of other complicated ones which are needed in conventional BEM, can be employed. While global RBF is used to approximate fictitious body force which appears when the analog equation method is introduced, and VBCM are utilized to solve homogeneous solution based on the superposition principle. As a result, a new meshless method is developed for solving nonlinear Poisson-type problems. Finally, some numerical experiments are implemented to verify the efficiency of the proposed method and numerical results are in good agreement with the analytical ones. It appears that the proposed meshless method is very effective for nonlinear Poisson-type problems.     

A Meshless Local Petrov-Galerkin Method for the Analysis of Cracks in the Isotropic Functionally Graded Material

A meshless local Petrov-Galerkin method (MLPG) [[Atluri and Zhu (1998)] for the analysis of cracks in isotropic functionally graded materials is presented. The meshless method uses the moving least squares (MLS) to approximate the field unknowns. The shape function has not the Kronecker Delta properties for the trial-function-interpolation, and a direct interpolation method is adopted to impose essential boundary conditions. The MLPG method does not involve any domain and singular integrals to generate the global effective stiffness matrix if body force is ignored; it only involves a regular boundary integral. The material properties are smooth functions of spatial coordinates and two interaction integrals [Rao and Rahman (2003a,b)] are used for the fracture analysis. Two numerical examples including both mode-I and mixed-mode problems are presented to calculated the stress intensity factors (SIFs) by the proposed method. Example problems in functionally graded materials are presented and compared with available reference solutions. A good agreement obtained show that the proposed method possesses no numerical difficulties.     

Meshless local Petrov–Galerkin method-higher Reynolds numbers fluid flow applications

The meshless local Petrov–Galerkin (MLPG) primitive variable based method is extended to analyze the incompressible laminar fluid flow within or over some different two-dimensional geometries. Although still in laminar regions, the Reynolds numbers considered in this study are in the ranges for which, in the literature, the MLPG primitive variable based method has never produced stable solutions and comparable results with those of the conventional methods. The considered test problems include, a steady lid-driven cavity flow with Reynolds numbers up to and including 10,000, a flow over a backward-facing step at 800 Reynolds number, and a transient fluid flow past a circular cylinder with Reynolds numbers up to and including 200. The present method solves the incompressible Navier–Stokes (N–S) equations in terms of the primitive variables using the characteristic-based split (CBS) scheme for discretization. The weighting function in the weak formulation of the governing equations is taken as unity, and the field variables are approximated using the moving least square (MLS) interpolation. For validation purposes, the obtained results are compared with those of the conventional numerical methods. The agreements of the compared results reveal a step forward towards further applications of the MLPG primitive variable based approach.     

A meshless local Petrov–Galerkin method for large deformation contact analysis of elastomers

A meshless method based on the local Petrov–Galerkin formulation is applied to the large deformation contact analysis of elastomeric components. Trial functions are constructed using the radial-basis function (RBF) coupled with a polynomial-basis function. The plane stress hypothesis and a pressure projection method are employed to overcome the incompressibility or nearly incompressibility in the plane stress and plane strain problems, respectively. Two different sets of equations are used for the nodes on the contact surface and nodes not on the contact surface, respectively, which is based on the meshless local Petrov–Galerkin method (MLPG) establishing equations node by node. Numerical results for several examples show that the present method is effective in dealing with large deformation contact problems.     

The natural neighbour Petrov–Galerkin method for elasto‐statics

In this paper, an efficient and accurate meshless natural neighbour Petrov–Galerkin method (NNPG) is proposed to solve elasto‐static problems in two‐dimensional space. This method is derived from the generalized meshless local Petrov–Galerkin method (MLPG) as a special case. In the NNPG, the local supported trial functions are constructed based on the non‐Sibsonian interpolation and test functions are taken as the three‐node triangular FEM shape functions. The local weak forms of the equilibrium equation and the boundary conditions are satisfied in local polygonal sub‐domains. These sub‐domains are constructed with Delaunay tessellations and domain integrals are evaluated over included Delaunay triangles by using Gaussian quadrature scheme. As this method combines the advantages of natural neighbour interpolation with Petrov–Galerkin method together, no stiffness matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Several numerical examples are presented and the results show the presented method is easy to implement and very accurate for these problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A point interpolation meshless method based on radial basis functions

A point interpolation meshless method is proposed based on combining radial and polynomial basis functions. Involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis. This non‐singularity is useful in constructing well‐performed shape functions. Furthermore, the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property. This makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least‐squares approximation. In addition, the partial derivatives of shape functions are easily obtained, thus improving computational efficiency. Examples on curve/surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high. Copyright © 2002 John Wiley & Sons, Ltd.