Moving Least Squares Shape Functions Research Articles

An improved material point method using moving least square shape functions

In this study, moving least square (MLS) shape functions are employed to reduce the cell-crossing error occurred in conventional material point method (MPM) when material particles pass through grid cell boundaries. The level of smoothness of MLS shape functions for mapping information from material particles to a background grid can be controlled by the support size of MLS weight functions. A simple method is proposed to reduce the computational cost for evaluating MLS shape functions at material particles by interpolating pre-computed MLS shape function values at sampling points in a grid cell. Numerical results show that the present method is very effective to reduce the cell-crossing error in MPM computations.

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 meshless method to solve nonlinear variable-order time fractional 2D reaction–diffusion equation involving Mittag-Leffler kernel

In this paper, an efficient and accurate meshless method based on the moving least squares (MLS) shape functions is developed to solve the generalized variable-order (V-O) time fractional nonlinear 2D reaction–diffusion equation. The V-O fractional derivative is considered in the Atangana–Baleanu–Caputo sense with Mittag-Leffler non-singular kernel. The numerical method is based on the following steps: First, the V-O fractional derivative is approximated by finite differences, and the $$\theta $$ -weighted method has been used to derive a recursive algorithm. Then, the solution of the problem is expanded by the MLS shape functions. Finally, by a substitution of this series expansion and corresponding its partial derivatives into the main equation, the problem is reduced to a linear system of algebraic equations to be solved at each time step. Several numerical examples are also given to illustrate the applicability, validity and accuracy of the presented method. The achieved numerical results reveal that the proposed method is highly accurate in solving the introduced V-O fractional model.

Topology optimization of thermal structure for isotropic and anisotropic materials using the element-free Galerkin method

ABSTRACTA numerical model of topology optimization for isotropic and anisotropic thermal structures was established by combining the element-free Galerkin (EFG) method and the rational approximation of material properties model. The relative density of EFG nodes was defined as the design variable, and the moving least-squares shape function was used to construct the relative density of calculation points. The EFG topology optimization model of anisotropic thermal structure was verified by benchmark examples based on the finite element method (FEM). The effects of the orthotropic factor and off-angle on EFG topology optimization results were studied. The results show that EFG optimal structures have better heat dissipation performance, and the boundary profiles are clearer than with FEM even without filtering techniques, enabling manufacture using 3D printing technology. The proper topology thermal structure can be obtained by adjusting the orthotropic factor and off-angle, thus improving the heat dissipation performance of the thermal structure.

An improved local boundary integral equation method implemented by the transformed MLS approximation with the delta property

The meshless local boundary integral equation (LBIE) method is a promising method for solving problems of elasticity with nonhomogeneous material properties. However, the shape functions of the LBIE method obtained by the moving least-squares (MLS) approximation, generally, do not satisfy the Kronecker delta property. To impose boundary conditions, numerical integrations of LBIE method are carried out under consideration of boundary conditions and the fictitious nodal values would be solved out. In this paper, the transformed MLS shape functions can be obtained, satisfying the Kronecker delta property and requiring no singular weight function. And then the potential and flux in 2D potential problems would be regarded as independent variables each other. The numerical integrations of LBIEs can be processed without regard to boundary conditions and the coefficient matrix of LBIE method would be irrelated to boundary conditions. Finally boundary conditions would be directly imposed and the unknown nodal potentials or the unknown fluxes on the boundary nodes would be solved out. Three numerical examples are computed to verify this feasibility. The coincidence of the numerical results obtained by the proposed method with the traditional LBIE method shows the feasibility of the proposed method.

A meshless method for solving the time fractional advection–diffusion equation with variable coefficients

In this paper, an efficient and accurate meshless method is proposed for solving the time fractional advection–diffusion equation with variable coefficients which is based on the moving least square (MLS) approximation. In the proposed method, firstly the time fractional derivative is approximated by a finite difference scheme of order O((δt)2−α),0<α≤1 and then the MLS approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the Caputo sense. Also, the validity of the proposed method is investigated in error analysis discussion. The main aim is to show that the meshless method based on the MLS shape functions is highly appropriate for solving fractional partial differential equations (FPDEs) with variable coefficients. The efficiency and accuracy of the proposed method are verified by solving several examples.

An Efficient Hybrid Implementation of MLPG Method

The computational implementation of moving least squares (MLS) shape functions is an important step to consider in some versions of the meshless local Petrov–Galerkin (MLPG) method for a variety of two-dimensional engineering problems. Here, the usage of conventional Gaussian quadrature in the MLPG may require an excessive number integration points to achieve acceptable accuracy. In addition, since for each integration point a search for nearby points contributing to the construction of the MLS shape functions is required, considerable increase in computational cost is often observed. Herein, an efficient hybrid implementation of CPU and GPU is proposed to accelerate the construction of MLS shape functions for MLPG. To this end, a new K-d-Tree (K-dimensional tree)-based data structure is introduced in order to accelerate the calculations involving computational geometry formulas such as MLS. The results are compared for implementations in CPU and [Formula: see text] using K-d-Tree with the traditional algorithm of brute force search of the neighboring points when dealing with two-dimensional linear elasticity problems. Finally, the performance gain in the numerical approximation is obtained when using the highlighted K-d-Tree.

Using micro-planes concrete damage model in discrete least squares meshless method for predicting the crack growth

In the present study, the author’s previously developed micro-planes concrete damage model has been implemented in the Discrete Least Squares Meshless method (DLSM) for assessment of the crack performance in concrete domains. DLSM is a true meshless method needing no partition in the local approximation as well as in the assembling of the local approximations in order to obtain a global solution. Instead it uses a collection of the un-structured nodal points in place of the elements and discretizes the governing differential equations with the discrete least squares approach. This can reduce considerably the pre-processing efforts of the calculations. However, DLSM like other meshless methods loses its efficiency when it is applied for the analysis of the solid domains dealing with the growing cracks. In recent years, some researchers have utilized some techniques to overcome this problem such as visibility, transparency and diffraction for precisely detecting the crack discontinuity in the case of a static non-growing crack, but these methods require some reciprocal corrective operations within each step of the calculation process extending the time of processing. In this study, however, a new, straightforward and general applicable method has been used for accommodating this issue using the advantages of the Moving Least Squares (MLS) shape functions constructed based on the Voronoi tessellation algorithm and micro-planes concrete damage constitutive model. Voronoi based MLS shape functions have been implemented for evaluating the crack effects accurately as a discontinuity during the analysis processing and micro-planes concrete damage approach used for proper detecting the crack growth trajectory. In the proposed method, the domain of interest is divided into the Voronoi cells based on the classical Voronoi tessellation approach, a cell for each node, and then the support domain of each node located inside the domain as well as on or near the crack faces is determined based on a neighboring criterion. During each step of the analysis for all the integration points, damages are computed according to the micro-planes concrete damage model and the crack path for the next step will be defined. After the end of the iterations in each step, both the stiffness tensor and internal load vector are modified based on the final stress fields obtained on and near the crack. The accuracy and efficiency of the proposed method have been investigated by solving some benchmark examples and comparing the obtained results with the analytical or valid finite element analyses’ results.

A meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) method

A new numerical algorithm has been investigated for solving time fractional reaction–subdiffusion equation. The fractional derivative of the considered equation is described in the Riemann–Liouville sense. Firstly, we discrete the temporal dimension of the considered model using a finite difference scheme. A central difference scheme has been applied to discrete the first time derivative and then for discretizing the fractional integral term a difference scheme has been employed with convergence order O(τ1+γ). Moreover, to achieve a full discretization scheme a type of meshless method has been improved that is known as element free Galerkin (EFG) method. The EFG method for integration uses a background mesh. This method is based on the Galerkin weak form in which the test and trial functions are shape functions of moving least squares (MLS) approximation. Since the shape functions of traditional MLS lack the Kronecker δ property, essential boundary conditions of a boundary value problem can not be directly computed and other methods must be employed for this issue. To this end, a new class of MLS shape functions has been applied that is called shape functions of interpolating MLS. The new shape function has the mentioned property. In the EFG method, calculating the appeared two-dimensional integrals is a basic issue. In this research work, the alternating direction implicit approach is combined with the element free Galerkin method. Then, using the new proposed method, the two-dimensional integrals on rectangular domain will be changed to simple one-dimensional integrals. We prove that the new numerical algorithm is unconditionally stable and also we obtain an error bound for the new procedure using the energy method. Numerical examples are reported which demonstrate the theoretical results and the efficiency of proposed scheme.

Topology optimization of compliant mechanisms using element-free Galerkin method

This paper will propose a topology optimization approach for the design of large displacement compliant mechanisms with geometrical non-linearity by using the element-free Galerkin (EFG) method. In this method, the Shepard function is applied to construct a physically meaningful density approximant, to account for its non-negative and range-bounded property. Firstly, in terms of the original nodal density field, the Shepard function method functionally similar to a density filter is used to generate a non-local nodal density field with enriched smoothness over the design domain. The density of any node can be evaluated according to the nodal density variables located inside the influence domain of the interested node. Secondly, in the numerical implementation the Shepard function method is again employed to construct a point-wise density interpolant. Gauss quadrature is used to calculate the integration of background cells numerically, and the artificial densities over all Gauss points can be determined by the surrounding nodal densities within the influence domain of the concerned computational point. Finally, the moving least squares (MLS) method is applied to construct the shape functions using the weight functions with compact support for assembling the meshless approximations of state equations. Since MLS shape functions are lack of the Kronecker delta function property, the penalty method is applied to enforce the essential boundary conditions. A typical large-deformation compliant mechanism is used as the numerical example to demonstrate the effectiveness of the proposed method.

Global interpolating MLS shape functions for structural problems with discrete nodal essential boundary conditions

The moving least squares (MLS)-based element-free Galerkin method (EFGM) has been widely studied, yielding valuable results and becoming a robust mesh-free technique for structural analysis. Due to the non-interpolating nature of MLS shape functions, the EFGM procedure leads to stiffness and mass matrices based on nodal parameters instead of true nodal displacements. The interpolating moving least squares (IMLS) methods with singular weight functions or the weighted nodal least squares (WNLS) methods were proposed in order to arrange the formulation in terms of actual displacements at the nodes. In this work, a later transformation matrix to retrieve the classic stiffness and mass matrices is discussed and numerically tested in problems with discrete prescribed displacements at boundaries, for which a straightforward finite element method (FEM)-type imposition of boundary conditions is possible.

Meshless boundary node methods for Stokes problems

In this study, we present the first meshless boundary node method (BNM) for two-dimensional and three-dimensional Stokes problems. The BNM exploits the advantages of the reduced dimensionality of boundary integral equations (BIEs) and the meshless attribute of moving least squares (MLS) approximations. However, because MLS shape functions lack the property of a delta function, the direct collocation scheme used in the BNM to impose boundary conditions doubles the numbers of unknowns and system equations. To overcome this drawback, we propose a dual boundary node method (DBNM), which uses the velocity BIE on the velocity boundary and the traction BIE on the traction boundary. In the DBNM, the boundary conditions are incorporated directly into the BIEs and they can be imposed easily, while the numbers of unknowns and system equations are only half of those in the BNM, thereby leading to higher computational precision and speed. Selected numerical tests illustrate the efficiency of the BNM and the DBNM.

Error analysis for moving least squares approximation in 2D space

In this paper, we provide a theoretical analysis of the moving least squares (MLS) approximation, which belongs to the family of meshless methods. First the non matrix form of the MLS shape function in two-dimensional space is obtained by using consistency conditions. The error estimates for MLS approximation in Sobolev space are presented when u(x,y)∈Cm+1(Ω), and u(x,y)∈Wm+1,q(Ω), respectively. We establish the error estimates for interpolating element-free Galerkin (IEFG) method when it is used for solving Poisson’s equation. The error bound is related to the radii of the weight functions and the bound of the norm of derivatives of shape functions. Three numerical examples are selected to confirm our analysis.

Speed-Up Technique of Extended Boundary Node Method for Large-Scale Simulation

The eXtended Boundary Node Method (X-BNM) with the periodic Radial Point Interpolation Method (RPIM) shape function is proposed and its performance is investigated numerically. The results of computations show that the accuracy of the X-BNM with the periodic RPIM shape function is almost equal to that with the Moving Least-Squares (MLS) shape function. In addition, the speed of the X-BNM with the periodic RPIM shape function is extremely faster than that with the MLS shape function. Therefore, the periodic RPIM shape function is useful for improving the performance of the X-BNM.

Local Kronecker delta property of the MLS approximation and feasibility of directly imposing the essential boundary conditions for the EFG method

The element-free Galerkin (EFG) method is a promising method for solving many engineering problems. Because the shape functions of the EFG method obtained by the moving least-squares (MLS) approximation, generally, do not satisfy the Kronecker delta property, special techniques are required to impose the essential boundary conditions. In this paper, it is proved that the MLS shape functions satisfy the Kronecker delta property when the number of nodes in the support domain is equal to the number of the basis functions. According to this, a local Kronecker delta property, which is satisfying the Kronecker delta property only at boundary nodes, can be obtained in one- and two-dimension. This local Kronecker delta property is an inherent property of the one-dimensional MLS shape functions and can be obtained for the two-dimensional MLS shape functions by reducing the influence domain of each boundary node to weaken the influence between them. The local Kronecker delta property provides the feasibility of directly imposing the essential boundary conditions for the EFG method. Four numerical examples are computed to verify this feasibility. The coincidence of the numerical results obtained by the direct method and Lagrange multiplier method shows the feasibility of the direct method.

Solving heat conduction problems by the Direct Meshless Local Petrov-Galerkin (DMLPG) method

As an improvement of the Meshless Local Petrov---Galerkin (MLPG), the Direct Meshless Local Petrov---Galerkin (DMLPG) method is applied here to the numerical solution of transient heat conduction problem. The new technique is based on direct recoveries of test functionals (local weak forms) from values at nodes without any detour via classical moving least squares (MLS) shape functions. This leads to an absolutely cheaper scheme where the numerical integrations will be done over low---degree polynomials rather than complicated MLS shape functions. This eliminates the main disadvantage of MLS based methods in comparison with finite element methods (FEM), namely the costs of numerical integration.

THE MESHLESS GALERKIN BOUNDARY NODE METHOD FOR TWO-DIMENSIONAL SOLIDS

The Galerkin boundary node method (GBNM) is developed for two-dimensional solid mechanics problems. The GBNM is a boundary only meshless method that combines an equivalent variational form of boundary integral formulations for governing equations with the moving least-squares (MLS) approximations for construction of the trial and test functions. In this method, boundary conditions can be implemented directly and easily despite the MLS shape functions lack the delta function property, and the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The optimal asymptotic error estimates of this approach for displacements and stresses are derived in detail in Sobolev spaces. Numerical tests are also given to demonstrate the developed algorithms.

Topology optimization of structures using meshless density variable approximants

SUMMARYThis 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 &amp; Sons, Ltd.

Corrected discrete least-squares meshless method for simulating free surface flows

Abstract In this paper, a modified version of discrete least-squares meshless (DLSM) method is used to simulate free surface flows with moving boundaries. DLSM is a newly developed meshless approach in which a least-squares functional of the residuals of the governing differential equations and its boundary conditions at the nodal points is minimized with respect to the unknown nodal parameters. The meshless shape functions are also derived using the Moving Least Squares (MLS) method of function approximation. The method is, therefore, a truly meshless method in which no integration is required in the computations. Since the second order derivative of the MLS shape function are known to contain higher errors compared to the first derivative, a modified version of DLSM method referred to as corrected discrete least-squares meshless (corrected DLSM) is proposed in which the second order derivatives are evaluated more accurately and efficiently by combining the first order derivatives of MLS shape functions with a finite difference approximation of the second derivatives. The governing equations of fluid flow (Navier–Stokes) are solved by the proposed method using a two-step pressure projection method in a Lagrangian form. Three benchmark problems namely; dam break, underwater rigid landslide and Scott Russell wave generator problems are used to test the accuracy of the proposed approach. The results show that proposed corrected DLSM can be employed to simulate complex free surface flows more accurately.

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.