Hermite–Cloud: a novel true meshless method

In this paper, a novel true meshless numerical technique – the Hermite–Cloud method, is developed. This method uses the Hermite interpolation theorem for the construction of the interpolation functions, and the point collocation technique for discretization of the partial differential equations. This technique is based on the classical reproducing kernel particle method except that a fixed reproducing kernel approximation is employed instead. As a true meshless technique, the present method constructs the Hermite-type interpolation functions to directly compute the approximate solutions of both the unknown functions and the first-order derivatives. The necessary auxiliary conditions are also constructed to generate a complete set of partial differential equations with mixed Dirichlet and Neumann boundary conditions. The point collocation technique is then used for discretization of the governing partial differential equations. Numerical results show that the computational accuracy of the Hermite–Cloud method at scattered discrete points in the domain is much refined not only for approximate solutions, but also for the first-order derivative of these solutions.     

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.     

A boundary meshless method with shape functions computed from the PDE

This paper presents a new meshless method using high degree polynomial shape functions. These shape functions are approximated solutions of the partial differential equation (PDE) and the discretization concerns only the boundary. If the domain is split into several subdomains, one has also to discretize the interfaces. To get a true meshless integration-free method, the boundary and interface conditions are accounted by collocation procedures. It is well known that a pure collocation technique induces numerical instabilities. That is why the collocation will be coupled with the least-squares method. The numerical technique will be applied to various second order PDE's in 2D domains. Because there is no integration and the number of shape functions does not increase very much with the degree, high degree polynomials can be considered without a huge computational cost. As for instance the p-version of finite elements or some well established meshless methods, the present method permits to get very accurate solutions.     

A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems

This paper presents a numerical framework for the highly accurate solutions of transient heat conduction problems. The numerical framework discretizes the temporal direction of the problems by introducing the Krylov deferred correction (KDC) approach, which is arbitrarily high order of accuracy while remaining the computational complexity same as in the time-marching of first-order methods. The discretization by employing the KDC method yields a boundary value problem of the inhomogeneous modified Helmholtz equation at each time step. The meshless generalized finite difference method (GFDM) or meshless finite difference method (MFDM), a meshless method, is then applied to the solution of resulting boundary value problems at each time step. Six numerical experiments in one-, two-, and three-dimensional cases show that the proposed hybrid KDC-GFDM scheme allows big time step size for a long-time dynamic simulation and has a great potential for the problems with complex boundaries. In addition, some comparisons are also presented between the present method, the COMSOL software, and the GFDM with implicit Euler method.     

The static and free vibration analysis of a nonhomogeneous moderately thick plate using the meshless local radial point interpolation method

A meshless local radial point interpolation method (LRPIM) for the bending and free vibration analysis of a nonhomogeneous moderately thick plate is presented in this paper. It uses a radial basis function coupled with a quadratic polynomial basis function as a trail function and a quartic spline function as a test function of the weighted residual method. The shape functions obtained in the trail function have the Kronecker delta function property, and the essential boundary conditions can be easily imposed. The present method is a true meshless method as it does not need any grids and all integrals can be easily evaluated over regularly shaped domains and their boundaries. In computational procedures, variations of material properties in the considered domain are modelled by adopting proper material parameters at Gauss points in integrations. Examples show that results obtained by the presented method are found to agree well with the existing solutions in the literature and with the results obtained by the finite element method, and the presented method has a number of advantages, such as high efficiency, quite good accuracy and easy implementation.     

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.     

Least‐squares collocation meshless method

A finite point method, least‐squares collocation meshless method, is proposed. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted. Unlike the direct collocation method, the equilibrium conditions are satisfied not only at the collocation points but also at the auxiliary points in a least‐squares sense. The moving least‐squares interpolant is used to construct the trial functions. The computational effort required for the present method is in the same order as that required for the direct collocation, while the present method improves the accuracy of solution significantly. The proposed method does not require any mesh so that it is a truly meshless method. Three numerical examples are studied in detail, which show that the proposed method possesses high accuracy with low computational effort. Copyright © 2001 John Wiley & Sons, Ltd.     

A hybrid radial boundary node method based on radial basis point interpolation

The hybrid boundary node method (HBNM) retains the meshless attribute of the moving least squares (MLS) approximation and the reduced dimensionality advantages of the boundary element method. However, the HBNM inherits the deficiency of the MLS approximation, in which shape functions lack the delta function property. Thus in the HBNM, boundary conditions are implemented after they are transformed into their approximations on the boundary nodes with the MLS scheme.This paper combines the hybrid displacement variational formulation and the radial basis point interpolation to develop a direct boundary-type meshless method, the hybrid radial boundary node method (HRBNM) for two-dimensional potential problems. The HRBNM is truly meshless, i.e. absolutely no elements are required either for interpolation or for integration. The radial basis point interpolation is used to construct shape functions with delta function property. So unlike the HBNM, the HRBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables, and boundary conditions can be applied directly and easily, which leads to greater computational precision. Some selected numerical tests illustrate the efficiency of the method proposed.     

偶应力理论的径向点插值无网格法研究

偶应力理论引入了位移二阶导数和相应的高阶边界条件,在有限元实施过程中,需要实现位移函数C1连续,这对传统有限元方法来说是一个苛刻的要求。无网格法能够实现位移函数的高阶导数连续,该文基于应变梯度偶应力理论下的虚功原理,导出了其无网格实施方法。无网格法的形函数一般不具有插值特性,本质边界条件施加困难,为克服这一困难,采用了带有多项式基的径向点插值法,构造的形状函数具有插值特性,可直接施加本质边界条件。数值结果表明,该方法数值结果稳定、精度高。     

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.     

The local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity

 The meshless method based on the local boundary integral equation (LBIE) is a promising method for solving boundary value problems, using an local unsymmetric weak form and shape functions from the moving least squares approximation. In the present paper, the meshless method based on the LBIE for solving problems in linear elasticity 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 can be easily imposed even when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present method. The numerical examples show 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.     

Sensitivity analysis and shape optimization based on FE–EFG coupled method

By use of 4-node isoparametric quadrangle interface element between finite element (FE) and meshless regions, a collocation approach is introduced to couple firstly FE and element-free Galerkin (EFG) method in this paper. By taking derivative of discreteness equilibrium equation at interface element with respect to design variable, a numerical method for discreteness-based shape design sensitivity analysis in interface element is obtained. The design sensitivity analysis (DSA) of coupled FE–EFG method is achieved by employing the DSA of nodal displacement at the interface element. The numerical method presented is testified by examples. It can be observed excellent agreement between the numerical results and the analytical solution. Finally the shape optimization of fillet is achieved by using coupled FE–EFG method. The result obtained show that imposing of the essential boundary condition is easy to implement, the computational time is reduced and the distortion of mesh is avoided.     

A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems

A meshless Local Petrov-Galerkin approach based on the moving Kriging interpolation (Local Kriging method; LoKriging hereafter) is employed for solving partial different equations that govern the heat flow in two- and three-dimensional spaces. The method is developed based on the moving Kriging interpolation for constructing shape functions at scattered points, and the Heaviside step function is used as a test function in each sub-domain to avoid the need for domain integral in symmetric weak form. As the shape functions possess the Kronecker delta function property, essential boundary conditions can be implemented without any difficulties. The traditional two-point difference method is selected for the time discretization scheme. For computation of 3D problems, a novel local sub-domain from the polyhedrons is used for evaluating the integrals. Several selected numerical examples are presented to illustrate the performance of the LoKriging method.     

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.     

Kinematically admissible meshless approximation using modified weight function

In meshless methods, in general, the shape functions do not satisfy Kronecker delta properties at nodal points. Therefore, imposing essential boundary conditions is not a trivial task as in FEM. In this regard, there has been a great deal of endeavor to find ways to impose essential boundary conditions. In this study, a new scheme for imposing essential boundary conditions is developed. Weight functions are modified by multiplying with auxiliary weight functions and the resulting shape functions satisfy Kronecker delta properties on the boundary nodes. In addition, the resulting shape functions possess linear interpolation features on the boundary segments where essential boundary conditions are prescribed. Therefore, the essential boundary conditions can be exactly satisfied with the new method. More importantly, the imposition of essential boundary conditions using the present method is infinitely easy as in the finite element method. Numerical examples show that the method also retains high convergence rate comparable to the Lagrange multiplier method. Copyright © 2001 John Wiley & Sons, Ltd.     

Development of a new meshless — point weighted least-squares (PWLS) method for computational mechanics

A truly meshless approach, point weighted least-squares (PWLS) method, is developed in this paper. In the present PWLS method, two sets of distributed points are adopted, i.e. fields node and collocation point. The field nodes are used to construct the trial functions. In the construction of the trial functions, the radial point interpolation based on local supported radial base function are employed. The collocation points are independent of the field nodes and adopted to form the total residuals of the problem. The weighted least-squares technique is used to obtain the solution of the problem by minimizing the functional of the summation of residuals. The present PWLS method possesses more advantages compared with the conventional collocation methods, e.g. it is very stable; the boundary conditions can be easily enforced; and the final coefficient matrix is symmetric. Several numerical examples of one- and two-dimensional ordinary and partial differential equations (ODEs and PDEs) are presented to illustrate the performance of the present PWLS method. They show that the developed PWLS method is accurate and efficient for the implementation.     

An improved meshless method with almost interpolation property for isotropic heat conduction problems

In the paper an improved element free Galerkin method is presented for heat conduction problems with heat generation and spatially varying conductivity. In order to improve computational efficiency of meshless method based on Galerkin weak form, the nodal influence domain of meshless method is extended to have arbitrary polygon shape. When the dimensionless size of the nodal influence domain approaches 1, the Gauss quadrature point only contributes to those nodes in whose background cell the Gauss quadrature point is located. Thus, the bandwidth of global stiff matrix decreases obviously and the node search procedure is also avoided. Moreover, the shape functions almost possess the Kronecker delta function property, and essential boundary conditions can be implemented without any difficulties. Numerical results show that arbitrary polygon shape nodal influence domain not only has high computational accuracy, but also enhances computational efficiency of meshless method greatly.     

A meshless local Petrov-Galerkin scaled boundary method

The scaled boundary finite-element method is a new semi-analytical approach to computational mechanics developed by Wolf and Song. The method weakens the governing differential equations by introducing shape functions along the circumferential coordinate direction(s). The weakened set of ordinary differential equations is then solved analytically in the radial direction. The resulting approximation satisfies the governing differential equations very closely in the radial direction, and in a finite-element sense in the circumferential direction. This paper develops a meshless method for determining the shape functions in the circumferential direction based on the local Petrov-Galerkin approach. Increased smoothness and continuity of the shape functions is obtained, and the solution is shown to converge significantly faster than conventional scaled boundary finite elements when a comparable number of degrees of freedom are used. No stress recovery process is necessary, as sufficiently accurate stresses are obtained directly from the derivatives of the displacement field.