A meshless local natural neighbour interpolation method for analysis of two-dimensional piezoelectric structures

A novel meshless method applied to solve two-dimensional piezoelectric structures is presented and discussed in this paper. It is called meshless local natural neighbour interpolation (MLNNI) method, which is derived from the generalized meshless local Petrov–Galerkin (MLPG) method as a special case. In the present method, nodal points are spread on the analysed domain and each node is surrounded by a polygonal sub-domain, which can be conveniently constructed with Delaunay tessellations. The spatial variation of the displacements and the electric potential are interpolated by the natural neighbour interpolation. As the shape functions so constructed possess the delta function property, the essential boundary conditions can be imposed by directly substituting the corresponding terms in the system of equations. Furthermore, the usage of three-node triangular FEM shape functions as test functions reduces the order of integrands involved in domain integrals. Numerical examples are presented at the end to demonstrate the applicability and accuracy of the present approach in analysing two-dimensional piezoelectric structures.     

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.     

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.     

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.     

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 meshless method for axi‐symmetric poroelastic simulations: numerical study

A meshless model, based on the meshless local Petrov–Galerkin (MLPG) approach, is developed and implemented for the solution of axi‐symmetric poroelastic problems. The solution accuracy and the code performance are investigated on a realistic application concerning the prediction of land subsidence above a deep compacting reservoir. The analysis addresses several numerical issues, including the parametric selection of the optimal size of the local sub‐domains for the weak form and the nodal supports, the appropriate integration rule, and the linear system solver. The results show that MLPG can be more accurate than the standard finite element (FE) method on coarse discretizations, with its superiority decreasing as the nodal resolution increases. This is due to both a slower convergence rate and a progressively higher computational cost compared to FE. These drawbacks can be partially mitigated by improving the efficiency of the numerical integration and the system solver with the aid of projection techniques based on Krylov subspace methods. The outcome of the present analysis supports the development of coupled methods where a limited number of MLPG nodes are used to locally improve a FE solution. Copyright © 2006 John Wiley & Sons, Ltd.     

Non‐linear dynamic analyses by meshless local Petrov–Galerkin formulations

In this work, meshless methods based on the local Petrov–Galerkin approach are proposed for the solution of dynamic problems considering elastic and elastoplastic materials. Formulations adopting the Heaviside step function and the Gaussian weight function as the test functions in the local weak form are considered. The moving least‐square method is used for the approximation of physical quantities in the local integral equations. After spatial discretization is carried out, a non‐linear system of ordinary differential equations of second order is obtained. This system is solved by Newmark/Newton–Raphson techniques. At the end of the paper numerical results are presented, illustrating the potentialities of the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

A pseudo-elastic local meshless method for analysis of material nonlinear problems in solids

This paper aims to develop an effective meshless technique for the analysis of elasto-plastic problems. The material nonlinearity will be studied by a new pseudo-elastic local radial point interpolation formulation which is based on the local Petrov–Galerkin form and the radial basis function (RBF) interpolation. Hencky's total deformation theory is used to define the effective Young's modulus and Poisson's ratio, which are treated as spatial field variables, and considered as functions of the final stress state and material properties. These effective material parameters are obtained in an iterative manner using the strain controlled projection method. Several numerical examples are presented to illustrate the effectivity of the newly developed formulation, and the numerical results obtained by the present method closely agree with the results obtained by other methods. It has proven that the present pseudo-elastic local meshless method is effective and easy to apply to the analysis of elasto-plastic materials subjected to proportional loading.     

A 3D shell-like approach using a natural neighbour meshless method: Isotropic and orthotropic thin structures

A three-dimensional shell-like approach for the analysis of composite thin plates and shells using a meshless method, the natural neighbour radial point interpolation method (NNRPIM), is presented. In the NNRPIM the nodal connectivity is enforced using the natural neighbour concept. The node-depending background mesh used in the numerical integration of the NNRPIM interpolation functions is entirely created from the unstructured nodal arrangement. The radial point interpolators are used to construct the NNRPIM interpolation functions, which possesses the delta Kronecker property, used in the Galerkin weak form. The novelty of this work lays on the development of a unique NNRPIM approach when 3D thin structures are considered. This new approach leads to remarkable results and it is extremely suitable to the composite structure problem. In order to demonstrate the effectiveness of the method the 3D shell-like NNRPIM analysis is used to solve several isotropic and orthotropic thin plates and shells problems.     

Enriching the finite element method with meshfree particles in structural mechanics

The automatic generation of meshes for the finite element (FE) method can be an expensive computational burden, especially in structural problems with localized stress peaks. The use of meshless methods can address such an issue, as these techniques do not require the existence of an underlying connection among the particles selected in a general domain. This study advances a numerical strategy that blends the FE method with the meshless local Petrov–Galerkin technique in structural mechanics, with the aim at exploiting the most attractive features of each procedure. The idea relies on the use of FEs to compute a background solution that is locally improved by enriching the approximation space with the basis functions associated to a few meshless points, thus taking advantage of the flexibility ensured by the use of particles disconnected from an underlying grid. Adding the meshless particles only where needed avoids the cost of mesh refining, or even of remeshing, without the prohibitive computational cost of a thoroughly meshfree approach. In the present implementation, an efficient integration strategy for the computation of the coefficients taking into account the mutual FE–meshless local Petrov–Galerkin interactions is introduced. Moreover, essential boundary conditions are enforced separately on both FEs and meshless particles, thus allowing for an overall accuracy improvement also when the enriched region is close to the domain boundary. Numerical examples in structural problems show that the proposed approach can significantly improve the solution accuracy at a local level, with no remeshing effort, and at a low computational cost. Copyright © 2016 John Wiley & Sons, Ltd.     

A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions

 A meshless method is developed for the stress analysis of two-dimensional solids, based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain. Trial functions are constructed using radial basis functions (RBF). The present method is a truly meshless method based only on a number of randomly located nodes. No domain integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behaviour of shape parameters of multiquadrics (MQ) has been systematically studied. Example problems in elastostatics are presented and compared with closed-form solutions and show that the proposed method is highly accurate and possesses no numerical difficulties. Received: 10 November 2002 / Accepted: 5 March 2003     

A new meshless regular local boundary integral equation (MRLBIE) approach

A new meshless method based on a regular local integral equation and the moving least‐squares approximation is developed. The present method is a truly meshless one as it does not need a ‘finite element or boundary element mesh’, either for purposes of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three‐dimensional problems) and their boundaries. No derivatives of the shape functions are needed in constructing the system stiffness matrix for the internal nodes, as well as for those boundary nodes with no essential‐boundary‐condition‐prescribed sections on their local boundaries. Numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable, and the computational results for the unknown variable and its derivatives are very accurate. No special post‐processing procedure is required to compute the derivatives of the unknown variable, as the original result, from the moving least‐squares approximation, is smooth enough. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

A meshless method with conforming and nonconforming sub‐domains

An accurate and easy integration technique is desired for the meshless methods of weak form. As is well known, a sub‐domain method is often used in computational mechanics. The conforming sub‐domains, where the sub‐domains are not separated nor overlapped each other, are often used, while the nonconforming sub‐domains could be employed if needed. In the latter cases, the integrations of the sub‐domains may be performed easily by choosing a simple configuration. And then, the meshless method with nonconforming sub‐domains is considered one of the reasonable choices for computational mechanics without the troublesome integration. In this paper, we propose a new sub‐domain meshless method. It is noted that, because the method can employ both the conforming and the nonconforming sub‐domains, the integration for the weak form is necessarily accurate and easy by selecting the nonconforming sub‐domains with simple configuration. The boundary value problems including the Poisson's equation and the Helmholtz's equation are analyzed by using the proposed method. The numerical solutions are compared with the exact solutions and the solutions of the collocation method, showing that the relative errors by using the proposed method are smaller than those by using the collocation method and that the proposed method possesses a good convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

A hybridized discontinuous Petrov–Galerkin scheme for scalar conservation laws

We present a hybridized discontinuous Petrov–Galerkin (HDPG) method for the numerical solution of steady and time‐dependent scalar conservation laws. The method combines a hybridization technique with a local Petrov–Galerkin approach in which the test functions are computed to maximize the inf‐sup condition. Since the Petrov–Galerkin approach does not guarantee a conservative solution, we propose to enforce this explicitly by introducing a constraint into the local Petrov–Galerkin problem. When the resulting nonlinear system is solved using the Newton–Raphson procedure, the solution inside each element can be locally condensed to yield a global linear system involving only the degrees of freedom of the numerical trace. This results in a significant reduction in memory storage and computation time for the solution of the matrix system, albeit at the cost of solving the local Petrov–Galerkin problems. However, these local problems are independent of each other and thus perfectly scalable. We present several numerical examples to assess the performance of the proposed method. The results show that the HDPG method outperforms the hybridizable discontinuous Galerkin method for problems involving discontinuities. Moreover, for the test case proposed by Peterson, the HDPG method provides optimal convergence of order k + 1. Copyright © 2012 John Wiley & Sons, Ltd.     

轴对称动力学问题的无网格自然邻接点Petrov-Galerkin法

基于无网格自然邻接点Petrov-Galerkin法,提出了复杂轴对称动力学问题求解的一条新途径。几何形状和边界条件的轴对称特点,将原来的空间问题转化为平面问题求解。计算时仅仅需要横截面上离散节点的信息,无论积分还是插值都不需要网格。自然邻接点插值构造的试函数具有Kronecker delta函数性质,因此能够直接准确地施加本质边界条件。有限元三节点三角形单元的形函数作为权函数,可以减少域积分中被积函数的阶次,提高了计算效率。数值算例结果表明,本文提出的方法对求解轴对称动力学问题是行之有效的。     

The natural radial element method

In this work an innovative numerical approach is proposed, which combines the simplicity of low‐order finite elements connectivity with the geometric flexibility of meshless methods. The natural neighbour concept is applied to enforce the nodal connectivity. Resorting to the Delaunay triangulation a background integration mesh is constructed, completely dependent on the nodal mesh. The nodal connectivity is imposed through nodal sets with reduce size, reducing significantly the test function construction cost. The interpolations functions, constructed using Euclidian norms, are easily obtained. To prove the good behaviour of the proposed interpolation function several data‐fitting examples and first‐order partial differential equations are solved. The proposed numerical method is also extended to the elastostatic analysis, where classic solid mechanics benchmark examples are solved. Copyright © 2013 John Wiley & Sons, Ltd.     

A meshless method for two-dimensional diffusion equation with an integral condition

This paper presents a new approach based on the meshless local Petrov–Galerkin (MLPG) and collocation methods to treat the parabolic partial differential equations with non-classical boundary conditions. In the presented method, the MLPG method is applied to the interior nodes while the meshless collocation method is applied to the nodes on the boundaries, and so the Dirichlet boundary condition is imposed directly. To treat the complicated integral boundary condition appearing in the problem, Simpson's composite numerical integration rule is applied. A time stepping scheme is employed to approximate the time derivative. Finally, two numerical examples are presented showing the behavior of the solution and the efficiency of the proposed method.     

A local boundary integral-based meshless method for Biot's consolidation problem

Traditional numerical techniques such as FEM and BEM have been successfully applied to the solutions of Biot's consolidation problems. However, these techniques confront some difficulties in dealing with moving boundaries. In addition, pre-designing node connectivity or element is not an easy task. Recently, developed meshless methods may overcome these difficulties. In this paper, a meshless model, based on the local Petrov–Galerkin approach with Heaviside step function as well as radial basis functions, is developed and implemented for the numerical solution of plane strain poroelastic problems. Although the proposed method is based on local boundary integral equation, it does not require any fundamental solution, thus avoiding the singularity integral. It also has no domain integral over local domain, thus largely reducing the computational cost in formulation of system stiffness. This is a truly meshless method. The solution accuracy and the code performance are evaluated through one-dimensional and two-dimensional consolidation problems. Numerical examples indicate that this meshless method is suitable for either regular or irregular node distributions with little loss of accuracy, thus being a promising numerical technique for poroelastic problems.