Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation

In this paper a numerical approach based on the truly meshless methods is proposed to deal with the second-order two-space-dimensional telegraph equation. In the meshless local weak–strong (MLWS) method, our aim is to remove the background quadrature domains for integration as much as possible, and yet to obtain stable and accurate solution. The MLWS method is designed to combine the advantage of local weak and strong forms to avoid their shortcomings. In this method, the local Petrov–Galerkin weak form is applied only to the nodes on the Neumann boundary of the domain of the problem. The meshless collocation method, based on the strong form equation is applied to the interior nodes and the nodes on the Dirichlet boundary. To solve the telegraph equation using the MLWS method, the conventional moving least squares (MLS) approximation is exploited in order to interpolate the solution of the equation. A time stepping scheme is employed to approximate the time derivative. Another solution is also given by the meshless local Petrov-Galerkin (MLPG) method. The validity and efficiency of the two proposed methods are investigated and verified through several examples.     

Finite volume meshless local Petrov–Galerkin method in elastodynamic problems

A finite volume meshless local Petrov–Galerkin (FVMLPG) method is presented for elastodynamic problems. It is derived from the local weak form of the equilibrium equations by using the finite volume (FV) and the meshless local Petrov–Galerkin (MLPG) concepts. By incorporating the moving least squares (MLS) approximations for trial functions, the local weak form is discretized, and is integrated over the local subdomain for the transient structural analysis. The present numerical technique imposes a correction to the accelerations, to enforce the kinematic boundary conditions in the MLS approximation, while using an explicit time-integration algorithm. Numerical examples for solving the transient response of the elastic structures are included. The results demonstrate the efficiency and accuracy of the present method for solving the elastodynamic problems.     

A novel meshless approach – Local Kriging (LoKriging) method with two-dimensional structural analysis

This paper develops a novel meshless approach, called Local Kriging (LoKriging) method, which is based on the local weak form of the partial differential governing equations and employs the Kriging interpolation to construct the meshless shape functions. Since the shape functions constructed by this interpolation have the delta function property based on the randomly distributed points, the essential boundary conditions can be implemented easily. The local weak form of the partial differential governing equations is obtained by the weighted residual method within the simple local quadrature domain. The spline function with high continuity is used as the weight function. The presently developed LoKriging method is a truly meshless method, as it does not require the mesh, either for the construction of the shape functions, or for the integration of the local weak form. Several numerical examples of two-dimensional static structural analysis are presented to illustrate the performance of the present LoKriging method. They show that the LoKriging method is highly efficient for the implementation and highly accurate for the computation.     

Efficient cubature formulae for MLPG and related methods

The paper introduces four kinds of compact, simple to implement Gaussian cubature formulae for approximating the domain integrals arising in the discrete local weak form (DLWF) of a governing partial differential equation solved by means of the meshless local Petrov–Galerkin method of type MLPG1. The integral weight functions are fixed to be the quartic‐spline weight function of the moving least squares (MLS) method and the function's gradient. The integration domain is a circle in 2D or a sphere in 3D. The fact that the DLWF test functions are directly incorporated into the formulae increases both their exactness degree and their computational efficiency. A number of numerical tests are carried out in order to asses the accuracy of the cubature formulae. For integrands involving MLS shape functions, the main factor controlling the integration accuracy is found to be the accuracy of the MLS‐approximation. Only a small number of cubature points is thus required to match that accuracy without a need for domain partitioning. The recommended approach for increasing the overall accuracy is by adding more MLS nodes and taking advantage of the computationally inexpensive cubature formulae. Copyright © 2005 John Wiley & Sons, Ltd.     

Meshless local Petrov-Galerkin method for continuously nonhomogeneous linear viscoelastic solids

A meshless method based on the local Petrov-Galerkin approach is proposed for the solution of quasi-static and transient dynamic problems in two-dimensional (2-D) nonhomogeneous linear viscoelastic media. A unit step function is used as the test functions in the local weak form. It is leading to local boundary integral equations (LBIEs) involving only a domain-integral in the case of transient dynamic problems. The correspondence principle is applied to such nonhomogeneous linear viscoelastic solids where relaxation moduli are separable in space and time variables. Then, the LBIEs are formulated for the Laplace-transformed viscoelastic problem. The analyzed domain is covered by small subdomains with a simple geometry such as circles in 2-D problems. The moving least squares (MLS) method is used for approximation of physical quantities in LBIEs.     

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.     

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.     

Isoparametric finite point method in computational mechanics

In this paper, a new meshless method, the isoparametric finite point method (IFPM) in computational mechanics is presented. The present IFPM is a truly meshless method and developed based on the concepts of meshless discretization and local isoparametric interpolation. In IFPM, the unknown functions, their derivatives, and the sub-domain and its boundaries of an arbitrary point are described by the same shape functions. Two kinds of shape functions that satisfy the Kronecker-Delta property are developed for the scattered points in the domain and on the boundaries, respectively. Conventional point collocation method is employed for the discretization of the governing equation and the boundary conditions. The essential (Dirichlet) and natural (Neumann) boundary conditions can be directly enforced at the boundary points. Several numerical examples are presented together with the results obtained by the exact solution and the finite element method. The numerical results show that the present IFPM is a simple and efficient method in computational mechanics.     

Three-Dimensional Meshfree Analysis of Interlocking Concrete Blocks for Step Seawall Structure

This study adapts the flexible characteristic of meshfree method in analyzing three-dimensional (3D) complex geometry structures, which are the interlocking concrete blocks of step seawall. The elastostatic behavior of the block is analysed by solving the Galerkin weak form formulation over local support domain. The 3D moving least square (MLS) approximation is applied to build the interpolation functions of unknowns. The pre-defined number of nodes in an integration domain ranging from 10 to 60 nodes is also investigated for their effect on the studied results. The accuracy and efficiency of the studied method on 3D elastostatic responses are validated through the comparison with the solutions of standard finite element method (FEM) using linear shape functions on tetrahedral elements and the well-known commercial software, ANSYS. The results show that elastostatic responses of studied concrete block obtained by meshfree method converge faster and are more accurate than those of standard FEM. The studied meshfree method is effective in the analysis of static responses of complex geometry structures. The amount of discretised nodes within the integration domain used in building MLS shape functions should be in the range from 30 to 60 nodes and should not be less than 20 nodes.     

A Trefftz function approximation in local boundary integral equations

 In the present paper the Trefftz function as a test function is used to derive the local boundary integral equations (LBIE) for linear elasticity. Since Trefftz functions are regular, much less requirements are put on numerical integration than in the conventional boundary integral method. The moving least square (MLS) approximation is applied to the displacement field. Then, the traction vectors on the local boundaries are obtained from the gradients of the approximated displacements by using Hooke's law. Nodal points are randomly spread on the domain of the analysed body. 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. Two ways are presented to formulate the solution of boundary value problems. In the first one the local boundary integral equations are written in all nodes (interior and boundary nodes). In the second way the LBIE are written only at the interior nodes and at the nodes on the global boundary the prescribed values of displacements and/or tractions are identified with their MLS approximations. Numerical examples for a square patch test and a cantilever beam are presented to illustrate the implementation and performance of the present method. Received 6 November 2000     

Imposing boundary conditions in the meshless local Petrov-Galerkin method

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.     

Dispersion and pollution of the improved meshless weighted least-square (IMWLS) solution for the Helmholtz equation

The meshless weighted least-square (MWLS) method is a meshless method based on the moving least-square (MLS) approximation. Compared with the Galerkin based meshless methods, the MWLS avoids numerical integrations, which improves the computational efficiency significantly. The MLS may form ill-conditioned system of equations, an accurate solution of which is difficult to obtain. In this paper, by using the weighted orthogonal basis function to construct the improved moving least-square (IMLS) approximation and the Lagrange multiplier method to enforce the Dirichlet boundary condition, we derive the formulas and perform the dispersion analysis for an improved meshless weighted least-square (IMWLS) method for two-dimensional (2D) Helmholtz problems. Results demonstrated that the IMWLS is more accurate and has advantages in handling dispersion. A 2D industrial model problem illustrated that the proposed method can easily reach higher frequency without losing accuracy.     

An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials

A new meshless method for computing the dynamic stress intensity factors (SIFs) in continuously non-homogeneous solids under a transient dynamic load is presented. The method is based on the local boundary integral equation (LBIE) formulation and the moving least squares (MLS) approximation. The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by a circle centered at the collocation point. The boundary-domain integral formulation with elastostatic fundamental solutions for homogeneous solids in Laplace-transformed domain is used to obtain the weak solution for subdomains. On the boundary of the subdomains, both the displacement and the traction vectors are unknown generally. If modified elastostatic fundamental solutions vanishing on the boundary of the subdomain are employed, the traction vector is eliminated from the local boundary integral equations for all interior nodal points. The spatial variation of the displacements is approximated by the MLS scheme.     

Topology optimization of structures using meshless density variable approximants

This paper proposes a new structural topology optimization method using a dual‐level point‐wise density approximant and the meshless Galerkin weak‐forms, totally based on a set of arbitrarily scattered field nodes to discretize the design domain. The moving least squares (MLS) method is used to construct shape functions with compactly supported weight functions, to achieve meshless approximations of system state equations. The MLS shape function with the zero‐order consistency will degenerate to the well‐known ‘Shepard function’, while the MLS shape function with the first‐order consistency refers to the widely studied ‘MLS shape function’. The Shepard function is then applied to construct a physically meaningful dual‐level density approximant, because of its non‐negative and range‐restricted properties. First, in terms of the original set of nodal density variables, this study develops a nonlocal nodal density approximant with enhanced smoothness by incorporating the Shepard function into the problem formulation. The density at any node can be evaluated according to the density variables located inside the influence domain of the current node. Second, in the numerical implementation, we present a point‐wise density interpolant via the Shepard function method. The density of any computational point is determined by the surrounding nodal densities within the influence domain of the concerned point. According to a set of generic design variables scattered at field nodes, an alternative solid isotropic material with penalization model is thus established through the proposed dual‐level density approximant. The Lagrangian multiplier method is included to enforce the essential boundary conditions because of the lack of the Kronecker delta function property of MLS meshless shape functions. Two benchmark numerical examples are employed to demonstrate the effectiveness of the proposed method, in particular its applicability in eliminating numerical instabilities. Copyright © 2012 John Wiley & Sons, Ltd.     

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. Copyright © 2005 John Wiley & Sons, Ltd.     

Improved element-free Galerkin method for two-dimensional potential problems

Potential difficulties arise in connection with various physical and engineering problems in which the functions satisfy a given partial differential equation and particular boundary conditions. These problems are independent of time and involve only space coordinates, as in Poisson's equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, they usually cannot be solved with analytical solutions. The element-free Galerkin (EFG) method is a meshless method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by using the weighted orthogonal basis function to construct the MLS interpolants, we derive the formulae of an improved EFG (IEFG) method for two-dimensional potential problems. There are fewer coefficients in the improved MLS (IMLS) approximation than in the MLS approximation, and in the IEFG method fewer nodes are selected in the entire domain than in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed.     

IMLS方形影响域法

展示了一些最新无网格法的研究进展,给出了一种新型无网格法-IMLS方形域无网格法。该法中未知变量的近似采用IMLS技术,局部影响域形状采用方形几何形态。这些技术的具体实施展现了节点布置和数值积分的无网格特点,并自然满足Dirichlet边界条件。该方法可以容易推广到求解非线性问题以及非均匀介质的力学问题。此外,还计算了两个弹性力学平面问题的例子,所得计算结果证明:该方法是一种具有收敛快、精度高、简便有效的通用方法,在工程中具有广阔的应用前景。     

A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids

 The meshless local Petrov-Galerkin (MLPG) method is an effective truly meshless method for solving partial differential equations using moving least squares (MLS) interpolants and local weak forms. In this paper, a MLPG formulation is proposed for free and forced vibration analyses. Local weak forms are developed using weighted residual method locally from the dynamic partial differential equation. In the free vibration analysis, the essential boundary conditions are implemented through the direct interpolation form and imposed using orthogonal transformation techniques. In the forced vibration analysis, the penalty method is used in implementation essential boundary conditions. Two different time integration methods are used and compared in the forced vibration analyses using the present MLPG method. The validity and efficiency of the present MLPG method are demonstrated through a number of examples of two-dimensional solids. Received 9 October 2000     

A hybrid finite difference and moving least square method for elasticity problems

In this paper, a novel hybrid finite difference and moving least square (MLS) technique is presented for the two-dimensional elasticity problems. A new approach for an indirect evaluation of second order and higher order derivatives of the MLS shape functions at field points is developed. As derivatives are obtained from a local approximation, the proposed method is computationally economical and efficient. The classical central finite difference formulas are used at domain collocation points with finite difference grids for regular boundaries and boundary conditions are represented using a moving least square approximation. For irregular shape problems, a point collocation method (PCM) is applied at points that are close to irregular boundaries. Neither the connectivity of mesh in the domain/boundary or integrations with fundamental/particular solutions is required in this approach. The application of the hybrid method to two-dimensional elastostatic and elastodynamic problems is presented and comparisons are made with the boundary element method and analytical solutions.     

A new local meshless method for steady-state heat conduction in heterogeneous materials

In this paper a truly meshless method based on the integral form of energy equation is presented to study the steady-state heat conduction in the anisotropic and heterogeneous materials. The presented meshless method is based on the satisfaction of the integral form of energy balance equation for each sub-particle (sub-domain) inside the material. Moving least square (MLS) approximation is used for approximation of the field variable over the randomly located nodes inside the domain. In the absence of heat generation, the domain integration is eliminated from the formulation of presented method and the computational efforts are reduced substantially with respect to the conventional MLPG method. A direct method is presented for treatment of material discontinuity at the heterogeneous material in the presented meshless method. As a practical problem the heat conduction in fibrous composite material is studied and the steady-state heat conduction in unidirectional fiber–matrix composites is investigated. The solution domain includes a small area of the composite system called representative volume element (RVE). Comparison of numerical results shows that the presented meshless method is simple, effective, accurate and less costly method for micromechanical analysis of heat conduction in heterogeneous materials.