Element-free Galerkin Method Research Articles

Analyzing 3D Helmholtz equations by using the hybrid complex variable element-free Galerkin method

In this study, we present the hybrid complex variable element-free Galerkin (HCVEFG) method for solving 3D Helmholtz equations. The dimension splitting method (DSM) will be introduced into the corresponding governing equation, thus a series of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation. For every 2D problem, the shape function can be obtained by using the improved complex variable moving least-squares (ICVMLS) approximation, and the essential boundary condition can be imposed by using the penalty method, thus the discretized equations of 2D problems can be derived by using the corresponding Galerkin weak form. These equations can be coupled by using the finite difference method (FDM) in the dimension splitting direction, thus final formulae of the numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors are given, and the convergence is analyzed numerically. The numerical result of these examples illustrates that the calculation speed can be improved greatly when the HCVEFG method is used rather than the improved element-free Galerkin (IEFG) method.

A pure complex variable enrichment method for modeling progressive fracture of orthotropic functionally gradient materials

Reasonably reducing degrees of freedom (DOFs) to achieve higher numerical efficiency and robustness is a long-standing challenge in all aspects of computational mechanics and this is especially true in the case of computational fracture analysis. In this work, a novel technique pursuing the further reduction of the number of the general singular enriched terms for fracture modelling is developed by means of the complex variable basis system inspirited by the Euler’s identity. By introducing the complex solution, the desirable enrichment to capture the crack-tip field can be constructed as a complex variable form with fewer terms. Thereby, the necessary number of nodes, as well as the DOFs, can be reasonably reduced for crack-tip modelling. The proposed novel pure complex variable enriched basis system, which is jointed with a standard complex variable meshless scheme, is applied to analyze the crack propagation problems in orthotropic functional gradient materials (FGM). The numerical results, of both the stress fields and the crack path, demonstrated that the proposed novel pure complex variable enrichment can effectively model fractures in orthotropic FGM with fewer DOFs compared with the element-free Galerkin method and the extended finite element method.

A comparison study of the efficiency and accuracy of IEFG in solving elasticity problems using different nodal integration schemes

The interpolation property (Kronecker delta property) is a favorable feature for the shape functions of meshless methods. The meshless methods with interpolation property can directly impose the essential boundary conditions of problems, which saves computing resources. The interpolating element-free Galerkin method (IEFG) based on the improved interpolating moving least-squares (IMLS) method is an efficient meshless method with high accuracy, which changes the disadvantage that the traditional element-free Galerkin method (EFG) does not have interpolation property. To achieve higher efficiency, direct nodal integration (DNI) is introduced into IEFG in this paper. DNI is simple in execution and cheaper in computing resources, compared with Gauss integration (GI), but it will produce stability issues and sub-optimal convergence. To eliminate the instability of DNI, this paper adopts three improved direct nodal integration schemes, residual balance direct nodal integration (RDNI), naturally stabilized direct nodal integration (NDNI), and modified direct nodal integration (MDNI). Through three 2D elasticity numerical examples, the meshless method combining IEFG with improved direct nodal integration schemes is verified and proved to be efficient, and the effectiveness of several nodal integration schemes in improving integration efficiency and retaining integration stability is compared.

Shear fracture propagation in quasi-brittle materials by an element-free Galerkin method

In this study, an element-free Galerkin (EFG) method for analysis of shear crack growth in brittle and quasi-brittle materials using a cohesive crack model is developed. The Mohr-Coulomb criterion is utilized as a damage threshold to check the failure of the material and predict the shear failure directions. The energy release rate criterion is adopted for defining the crack propagation from two possible shear planes. Linear cohesion softening is chosen to describe the strain softening, and residual strength of the material. Numerical experiments and physical model tests of two-dimensional crack propagation were carried out to verify the present approach. Furthermore, the developed EFG method is adopted to investigate the influence of the fracture energy and incremental crack lengths on predicted failure paths. A large incremental crack length can be utilized in the analysis of materials with high fracture energy, to reduce the computational cost. For materials with low fracture energy, small incremental crack lengths are required for accurate results.

Numerical investigation of the element-free Galerkin method (EFGM) with appropriate temporal discretization techniques for transient wave propagation problems

In the earlier paper, it is known that the solution accuracy usually can not be monotonically increased as the temporal discretization interval decreases when the standard finite element method (FEM) with the conventional temporal discretization approach is exploited for elastodynamics, hence the time integration step should be carefully determined for sufficiently fine solutions. The present work aims to investigate the behaviors of the classical element-free Galerkin method (EFGM), which is a typical meshless approach, with the Bathe temporal discretization scheme for elastodynamics. The main insights are that we explicitly show the total numerical dispersion errors in the computed numerical results actually consists of two different parts corresponding to the spatial and temporal discretization, respectively; both of them are responsible for solution accuracy. By performing the dispersion analysis, how the solution accuracy is affected by temporal and spatial discretization is shown, it is seen that we can improve the solution accuracy monotonically as the temporal step size decreases as long as the spatial dispersion error is sufficiently small and the related mathematical proofs are also given. From several supporting numerical examples, we can clearly see that the EFGM with the Bathe time integration scheme can basically provide monotonically convergent solutions as long as the reasonable node arrangement pattern and sufficiently large shape function supports are employed, namely the monotonic convergence property with respect to the temporal discretization interval can be achieved. This attractive and important feature makes the EFGM more competitive than the FE approach for elastodynamics.

The interpolating element-free Galerkin method for the p-Laplace double obstacle mixed complementarity problem

The interpolating element-free Galerkin method for the p-Laplace double obstacle mixed complementarity problem

The Dimension Splitting Interpolating Element-Free Galerkin Method for Solving 3D Wave Equations

Based on the idea of dimension splitting method, a dimension splitting interpolating element-free Galerkin (DSIEFG) method for 3D wave propagation problems is proposed. In the splitting direction, the 3D domain of a problem is divided into a series of related 2D subdomains. Using the improved interpolating moving least-squares (IMLS) method to acquire the shape function in the 2D subdomains, the discretized equations are formed based on Galerkin weak form of 2D problem. The discretized equations are coupled by using the difference method in the splitting direction. Then, the final equations of the DSIEFG method for 3D wave propagation problems are obtained. Numerical examples are given to study the effects of node distribution, number of split layers, influence domain parameters, splitting direction, and time step on the computational accuracy of the DSIEFG method. The results of numerical examples show that the DSIEFG method is more efficient and accurate compared with the improved EFG method for 3D wave equations.

An Improved Element-Free Galerkin Method Based on the Dimension Splitting Moving Least-Squares Method for 2D Potential Problems in Irregular Domains

By introducing the dimension splitting (DS) method into the moving least-squares (MLS) approximation, a dimension splitting moving least-squares (DS-MLS) method is proposed in this paper. In the DS-MLS method, the operator splitting and independent variable splitting of the DS method are used to reduce the dimension, thereby reducing the computational complexity of the matrix. The shape function of the DS-MLS method has the advantages of simple derivation and high computational efficiency. Then, by coupling DS-MLS method and Galerkin weak form, and performing the coordinate transformation, an improved element-free Galerkin method (IEFGM) based on the DS-MLS method is proposed for two-dimensional (2D) potential problems on irregular domains. The effectiveness of the method in this paper is verified by some numerical examples. The numerical results show that, compared with the element-free Galerkin (EFG) method, the IEFGM based on the DS-MLS method in this paper consumes less CPU time and has higher computational accuracy for some 2D potential problems on irregular domains.

Local maximum-entropy approximation based stabilization methods for the convection diffusion problems

Local Maximum Entropy (LME) approximation in the field of elliptic partial differential equations is more advanced compared to convection dominated flow problems. This paper presents a comparative study on the LME based meshfree stabilization methods for convection dominated convection-diffusion problems. LME approximants account for the disposal of the elements and direct imposition of boundary conditions, whereas the stabilization methods are meant to deal with the issues associated with the non-self-adjoint convective term. A new meshfree stabilization scheme, residual free bubbles based multiscale element free Galerkin method is also presented and compared with streamline upwind Petrov Galerkin meshfree method and variational multiscale element free Galerkin Method. The effect of different priors and radius of support which define the LME basis are studied to test their importance in the three methods with respect to standard benchmark problems. Cubic spline prior based LME is found to be enough to produce better stability and accuracy for all the stabilization methods when compared to quartic spline and Gaussian priors. LME based variational multiscale element free Galerkin method is observed to yield better stability and accuracy for all the priors. The convergence rate for the three stabilization methods is observed to be in good agreement with the classical FEM-SUPG.

Natural convection flow on a bent wavy vertical enclosure filled with power-law nanofluid simulated by Element Free Galerkin method

The present paper contains a study of heat transfer and flow of a power-law nanofluid in bent vertical wavy walls enclosure. The system of governing equations has been solved by mesh free method (EFGM) with a parallel scheme. The results are found in terms of heat transfer rate at the hot wall, isotherms and streamlines under the influence of Aspect ratio, Rayleigh number, Prandtl number, power-law index and wave amplitude. It is observed that the average Nusselt number improves with enhancing the Prandtl number, Rayleigh number and aspect ratio but it decrements with enhancing the power-law index and surface waviness. A parallel implementation scheme gives better efficiency as compared to the sequential scheme. Such problems have direct application in electronic devices, food processing industries, and biomedical applications.

The Hybrid Element-Free Galerkin Method for 3D Helmholtz Equations

In this study, the hybrid element-free Galerkin (HEFG) method is studied to solve the 3D Helmholtz equations. The idea of the dimension splitting method (DSM) is introduced into the improved element-free Galerkin (IEFG) method, thus a sequence of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation, each 2D form can be discretized by using the IEFG method, thus the corresponding 2D discretized equations can be obtained, and these equations can be coupled by employing the finite difference method (FDM) in the dimension splitting direction, thus final formulae of numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors and the convergence are analyzed, respectively, and the numerical results show that the calculation resources can be saved greatly without losing the computational accuracy when using the HEFG method.

A Hermite interpolation element-free Galerkin method for elasticity problems

A Hermite interpolation element-free Galerkin method for elasticity problems

A fast interpolating meshless method for 3D heat conduction equations

A fast interpolating meshless (FIM) method for three-dimensional (3D) heat conduction equations is presented. Transforming a 3D problem into the relevant two-dimensional (2D) problems using the dimension splitting method (DSM) is the main idea of FIM method. The improved interpolating moving least-squares (IIMLS) method is applied in 2D problems to obtain required approximation function with interpolation property. Finite difference method (FDM) is utilized in time domain and the direction of splitting. Take the improved element-free Galerkin (IEFG) method as a comparison, difficulties created by the singularity of weight functions, such as truncation error and calculation inconvenience, are overcome by the FIM method. And it can directly implement the Dirichlet boundary conditions. To prove the advantages of the new method, three examples are selected and solved by the FIM method. Comparing and analyzing the calculation results, it can be shown that the FIM method effectively improves computation speed and precision.

A complex variable EFG method for hyperelastic large deformation analysis under non-conservative loads

The finite element method is extremely popular in almost all fields of engineering and scientific research due to its powerful and accessible industrial software for computational modelling. However, alternatives including the meshless methods, have received increasing attention on the large deformation problems of hyperelastic materials that need to avoid mesh distortion. In this paper, based on the complex variable moving least-squares approximation, the complex variable element-free Galerkin (CVEFG) method for the large deformation problem of hyperelastic materials under non-conservative loads is established. In the present method, the Newton-Raphson method is used for solving the nonlinear equations. The algorithm for hyperelastic large deformation problems under non-conservative loads are established. Three numerical examples are used to verify the effectiveness of the CVEFG method in this paper. Numerical results show that the solution obtained using the CVEFG method is closer to that obtained by ABAQUS and is more efficient than the element-free Galerkin method.

Interpolating Modified Moving Least Squares based element free Galerkin method for fracture mechanics problems

Interpolating Modified Moving Least Squares (IMMLS) based element-free Galerkin method (EFGM) is a simple, robust technique which addresses the limitations of classical Moving Least Squares Approximation (MLSA) based EFGM and is well-demonstrated for tissue deformation. However, its application to fracture mechanics featuring strong discontinuities has not been explored. To address this gap, the present work explores isotropic and orthotropic thermoelastic Linear Elastic Fracture Mechanics (LEFM) cases — with initial isotropic and orthotropic LEFM formulations extended to respective thermoelastic cases — and an isotropic Elastoplastic Fracture Mechanics (EPFM) case employing isotropic bilinear stress–strain curve. The results including field-variables and stress intensity factors (SIFs) are corroborated with converged finite element method (FEM) results; in addition, the results also include classical-MLSA results for a comparative study. L2 norms with respect to FEM indicate an overall close correspondence with a better match for primary variables than secondary variables — indicating the efficacy of IMMLS technique for fracture mechanics. In comparison with classical MLSA-based EFGM results, IMMLS-based EFGM (IN-EFGM) results are nearly similar. However, the advantages demonstrated by IMMLS technique over the classical MLSA-based EFGM in enforcing essential boundary conditions, a larger set of admissible nodal configuration and lesser computational time render IN-EFGM as a potential tool for fracture mechanics.

Interpolating Meshless Methods for 3D Elastic Problems

Interpolating meshless methods can directly impose boundary conditions because of the interpolation property which shows advantages in dealing with problems with boundary conditions. The interpolating element-free Galerkin method (IEFGM), the improved interpolating element-free Galerkin method (IIEFGM), and the radial point interpolation method (RPIM) are applied in this paper to solve the two-dimensional and three-dimensional elastic problems. IEFGM and IIEFGM are two different ways to change the status that the traditional element-free Galerkin method (EFG) does not have the interpolation property. IEFGM uses an improved interpolating moving least-squares (IMLS) method that employed singular weight functions while IIEFGM takes the improved interpolating moving least-squares method based on non-singular weight function. RPIM, one of the most widely used interpolating meshless methods, is compared with IEFGM and IIEFGM in this paper. The numerical results of two-dimensional and three-dimensional elastic problems show that the three types of interpolating meshless methods obtain high precision displacement solutions and stress solutions.

Unified Complex Variable Element-Free Methods: A Comprehensive Framework of Complex Variable Meshless Approaches

There are long-term traditions and efforts to apply the complex variable methods to the theoretical and numerical solutions of applied mechanics. Based on the aim of establishing a unified format of the complex variable meshless approximation, this paper proposes a unified complex variable moving least-squares (UCVMLS) approximation for any pair of independent real field variables. In the UCVMLS approximation, based on the complex variable form of real variable pair and the adopted complex variable basis, four independent approximation functions can be constructed, in which two different improved complex variable moving least-squares (ICVMLS) approximations developed before are included. The four trial functions in UCVMLS are proved to be able to construct two different approximations to the same real variable pair. The unified complex variable element-free Galerkin (UCVEFG) method for elasticity is established. Numerical examples are provided by applying UCVEFG method in three elastic problems to show its accuracy.

The element-free Galerkin method for the variational–hemivariational inequality of the dynamic Signorini–Tresca contact problems with friction in elastic materials

The element-free Galerkin method is presented for the variational–hemivariational inequality of the dynamic Signorini–Tresca contact problems with friction in elastic materials. The Dirichlet boundary conditions and the constrained conditions are imposed by the penalty method. The error estimates of the element-free Galerkin method indicate that the convergence order depends on the spatial step, the time step, the largest degree of basis functions in the moving least-squares approximation and the penalty factor. Numerical examples verify our theoretical results.

Application of the CUF–EFG method for buckling analysis of the multilayer GPLs–CNTs-reinforced FG plates with cutout

In this article, the element free Galerkin (EFG) method and Carrera’s Unified Formulation (CUF) are employed as the CUF–EFG method for the buckling analysis of multilayer GPLs–CNTs-reinforced functionally graded (FG) plates with cutout. The radial basis shape functions which possess the Kronecker delta function property are used as the shape functions. In CUF, a Taylor expansion with first to 4th order is employed as the thickness function. In the proposed method, the stretching effect (effect of thickness extension ‘ ε zz ≠ 0 ’) is taken into account. The plate is under axial, biaxial, shear and combined loadings and composed of a number of layers stacked up in the thickness direction. The carbon nanotubes (CNTs) and graphene platelets (GPLs) uniformly distributed in each individual layer but their weight fractions gradually vary in each layer to achieve functionally graded properties. Four types of distribution patterns with the same average GPLs and CNTs weight fraction are considered in the problem. The mechanical properties of plates with various grading patterns of GPLs and CNTs distributions are estimated using the modified Halpin-Tsai method and the rule of mixture. A comparison between the obtained results with both analytical solution and other shear deformation theories given in the published literatures is performed to show the accuracy and efficiency of the proposed CUF–EFG method. A parametric study is carried out to evaluate the influences of the boundary conditions, aspect ratio, CNT and GPLs weight fraction, number of layers, stacking sequences of layers, dimension and location of circular cutout on the buckling capacity of plates.

Thermoelastic fracture analysis in orthotropic media using optimized element free Galerkin algorithm

In this work the effectuality of element free Galerkin method (EFGM) have been enhanced by proposing a novel algorithm for modeling discrete cracks in two-dimensional orthotropic media that are subjected to thermoelastic loading. The standard element-free Galerkin method (EFGM) is amended at the approximation level in conjunction with parametric optimization at computational level. The proposed algorithm not only extends the capability of element free Galerkin method for simulating thermoelastic fracture in orthotropic media but additionally it demonstrates a higher degree of computational efficiency and solution accuracy. The proposed algorithm utilizes variable quadrature points from the problem domain for the purpose of numerical integration. A parametric optimization has also been incorporated in the proposed algorithm for minimizing the computational time of simulation. The proposed algorithm is proficient to simulate near tip stress field in both convex and non-convex problem domains. A modified conservative M-integral technique has been used in order to extract the thermal stress intensity factors (SIFs) for the simulated problems. Few static and quasi-static crack problems have been modeled and simulated to illustrate and establish the accuracy and effectiveness of the proposed algorithm. Moreover, two component level problems, having non-convex domains, have also been modeled in order to establish the versatility of proposed algorithm. A good agreement between the obtained results with those of the available reference results establishes the capability and robustness of proposed algorithm for simulating fracture in orthotropic media subjected to thermoelastic loads. Moreover, the proposed EFGM algorithm significantly reduces the computational time which further enhances and augments its modeling prowess.