Complex Variable Element-free Galerkin Research Articles

The HCVEFG method for 3D steady convection–diffusion–reaction problems

In this paper, a hybrid complex variable element-free Galerkin (HCVEFG) method is proposed for solving three-dimensional (3D) steady convection–diffusion–reaction equations. By introducing the dimension splitting method (DSM), a 3D convection–diffusion–reaction problem is changed into a series of two-dimensional (2D) ones. For each 2D form, the improved complex variable moving least-squares (ICVMLS) method is applied to obtain the required approximation shape functions, and the penalty method is used to apply the essential boundary conditions, thus we can obtain the 2D discretized equations by using the improved complex variable element-free Galerkin (ICVEFG) method. The finite difference method (FDM) is used in one-dimensional splitting direction. Thus, the final solution formula about the numerical solution for 3D convection–diffusion–reaction problem can be obtained. Three numerical examples show that the HCVEFG algorithm can cut down the CPU cost of the improved element-free Galerkin (IEFG) method greatly for solving 3D steady convection–diffusion–reaction problems.

Modeling nonlinear deformation of slender auxetic structures under follower loads with complex variable meshfree methods

The auxetic structure with negative Poisson’s ratio has broad application prospects in many engineering fields. Although it is often simplified as a linear elastic problem in theoretical models, the generally used slender structure, such as the reentrant honeycombs, will inevitably undergo large deformation resulting in significant non-conservative load effects in the service conditions. Therefore, a numerical framework for modeling nonlinear deformation of the auxetic structures under follower loads with the complex variable element-free Galerkin method is developed in this paper. The application of the complex variable meshfree method is to deal with the numerical difficulties caused by mesh distortion that may occur in large deformation problems, and at the same time, it can improve the construction efficiency of meshfree shape functions through the complex variable moving least-squares approximation. The Galerkin weak form of the incremental total Lagrangian formula for large deformation problems with the enforced essential boundary conditions using the penalty method is derived and then discretized in the complex variable meshless implementation. Five numerical examples are presented with detailed convergence study to demonstrate the accuracy of the proposed approach in dealing with non-conservative large deformation of the slender auxetic structures.

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 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.

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.

An enriched improved complex variable element-free Galerkin method for efficient fracture analysis of orthotropic materials

The fracture problem of orthotropic materials has been widely concerned by researchers in the fields of mechanics and engineering. The meshless method using nodes to discrete the problem domain is an effective way to analyze cracked structures. In this paper, an enriched improved complex variable moving least squares (EICVMLS) approximation, in which a group of intrinsic enrichments is introduced into the conventional ICVMLS adopting a well-defined weighted complex variable error norm, is proposed for capturing the crack-tip fields in orthotropic media accurately. Then, an enriched improved complex variable element-free Galerkin (EICVEFG) method is developed to simulate fracture behavior in orthotropic media. The visibility criterion is used to model the discontinuity behavior around cracks and the M−integral is considered to evaluate stress intensity factor (SIF). The stress field and the displacement field at the crack tip, the SIF for different crack inclinations, and different directions of orthotropy axes are investigated to verify the validity of the proposed method. In addition, the effects of linear basis functions and enriched basis functions on the results are compared in this study. The numerical results reveal that the established EICVEFG approach can achieve higher accuracy than the extended finite element method (XFEM) and traditional meshless methods such as the element-free Galerkin (EFG) method and the meshless finite volume (MFV) method. Similarly, the EICVEFG method has been proved to offer a higher convergence rate by parameter studies (including dimensionless factor dmax, integral size, and the number of integral cells in M−integral).

The Improved Interpolating Complex Variable Element-Free Galerkin Method for Two-Dimensional Elastic Problems

The Improved Interpolating Complex Variable Element-Free Galerkin Method for Two-Dimensional Elastic Problems

Analysis of Elastic–Plastic Problems Using the Improved Interpolating Complex Variable Element Free Galerkin Method

A numerical model for the two-dimensional nonlinear elastic–plastic problem is proposed based on the improved interpolating complex variable element free Galerkin (IICVEFG) method and the incremental tangent stiffness matrix method. The viability of the proposed model is verified through three elastic–plastic examples. The numerical analyses show that the IICVEFG method has good convergence. The solutions using the IICVEFG method are consistent with the solutions obtained from the finite element method using the ABAQUS program. Moreover, the IICVEFG method shows greater computing precision and efficiency than the non-interpolating meshless methods.

The hybrid complex variable element-free Galerkin method for 3D elasticity problems

Based on the dimension splitting method (DSM) and the improved complex variable element-free Galerkin (ICVEFG) method, the hybrid complex variable element-free Galerkin (HCVEFG) method for three-dimensional (3D) elasticity is proposed in this paper. The equilibrium equation of a 3D elasticity is written as three sets of equations, which contains the equilibrium equations of any two directions. By introducing the DSM, a set of the equilibrium equation of a 3D elasticity problem is transformed into a series of two-dimensional (2D) problems which are solved by using the ICVEFG method. In splitting direction, the finite difference method is used for coupling those 2D discretized equations. Similarly, the discretized equation of another set of the equilibrium equation can be derived. Then by coupling those two discretized system equations, the numerical solutions of the HCVEFG method for 3D elasticity problems can be obtained. The error and the convergence are analyzed by numerical examples, and the results show that the HCVEFG method has greater computational speed.

An improved interpolating complex variable element free Galerkin method for the pattern transformation of hydrogel

In this paper, an effective interpolating complex variable element free Galerkin method is proposed for the steady large deformation analysis of hydrogels. In this meshless method, the essential boundary conditions can be applied directly without using other special methods, which leads to less unknown coefficients. Through computing the swelling large deformation of a square hydrogel with three constrained boundaries, the presented meshless method is validated with higher accuracy and efficiency than other non-interpolating meshless methods. Based on the developed meshless method, different samples with elliptical, circular and square holes in a square hydrogel are simulated for analyzing the pattern transformation. The numerical results indicate that the shapes and initial sizes of holes in hydrogels lead to different pattern transformation rules, actuating behaviors and area reductions of holes. Finally, according to the new findings, the potential applications of a square hydrogel are discussed by designing different shapes and initial sizes of holes.

The Improved Interpolating Complex Variable Element Free Galerkin Method for Two-Dimensional Potential Problems

In this paper, the improved interpolating complex variable moving least squares (IICVMLS) method is applied, in which the complete basis function is introduced and combined with the singular weight function to achieve the orthometric basis function. Then, the interpolating shape function is achieved to construct the interpolating trial function. Incorporating the IICVMLS method and the Galerkin integral weak form, an improved interpolating complex variable element free Galerkin (IICVEFG) method is proposed to solve the 2D potential problem. Because the essential boundary conditions can be straightaway imposed in the above method, the expressions of final dispersed matrices are more concise in contrast to the non-interpolating complex variable meshless methods. Through analyzing four specific potential problems, the IICVEFG method is validated with greater computing precision and efficiency.

The Improved Complex Variable Element-Free Galerkin Method for Bending Problem of Thin Plate on Elastic Foundations

In this paper, the improved complex variable element-free Galerkin (ICVEFG) method is proposed for solving the bending problem of thin plate on elastic foundations. In the ICVEFG method, the approximation function regarding the deflection of thin plate is formed with the improved complex moving least-squares (ICVMLS) approximation, the discrete equation is obtained from Galerkin weak form of bending problem of thin plate on different elastic foundations, and essential boundary conditions are considered based on penalty method. As the ICVMLS approximation is based on the complex variable theory, it can obtain the shape function quickly with high precision. Three sample problems are used to discuss the advantages of the ICVEFG method, and the numerical results show that the ICVEFG method presented in this paper has a fast convergence speed and great computational accuracy.

A hybrid improved complex variable element-free Galerkin method for three-dimensional advection-diffusion problems

In this paper, combining the dimension splitting method with the improved complex variable element-free Galerkin method, a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional advection-diffusion problems. Using the dimension splitting method, a three-dimensional advection-diffusion problem is transformed into a series of two-dimensional ones which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method, the improved complex variable moving least-squares (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the one-dimensional direction. And Galerkin weak form of three-dimensional advection-diffusion problems is used to obtain the final discretized equations. Then the H-ICVEFG method for three-dimensional advection-diffusion problems is presented. Numerical examples are provided to discuss the influences of the weight functions, the effects of the scale parameter, the penalty factor, the number of nodes, the step number and the time step on the numerical solutions. And the advantages of the H-ICVEFG method with higher computational accuracy and efficiency are shown.

Three-dimensional complex variable element-free Galerkin method

• The complex variable moving least squares (CVMLS) approximation is developed for 3D functions with 2D basis functions. • The complex variable element-free Galerkin (CVEFG) method is developed for 3D elliptic problems and 3D wave equations. • Error and convergence of the CVMLS approximation and the CVEFG method are analyzed theoretically. • Numerical results reveal that the present method has better accuracy and computational efficiency than other methods. The complex variable element-free Galerkin (CVEFG) method is an efficient meshless Galerkin method that uses the complex variable moving least squares (CVMLS) approximation to form shape functions. In the past, applications of the CVMLS approximation and the CVEFG method are confined to 2D problems. This paper is devoted to 3D problems. Computational formulas and theoretical analysis of the CVMLS approximation on 3D domains are developed. The approximation of a 3D function is formed with 2D basis functions. Compared with the moving least squares approximation , the CVMLS approximation involves fewer coefficients and thus consumes less computing times. Formulations and error analysis of the CVEFG method to 3D elliptic problems and 3D wave equations are provided. Numerical examples are given to verify the convergence and accuracy of the method. Numerical results reveal that the CVEFG method has better accuracy and higher computational efficiency than other methods such as the element-free Galerkin method.

Large deformation analysis of a gel using the complex variable element-free Galerkin method

Deformation of a gel is a complicated multiphysical process involving the diffusion of solvent and the mechanical stretching of polymeric networks. This process leads to a mechanical and diffusional equilibrium after a certain period of evolution. Here we present a large deformation analysis of a gel in the equilibrium state under environmental triggers using the complex variable element-free Galerkin (CVEFG) method. The work mainly addresses the numerical challenges encountered while a gel in contact with a solvent deforms only with the geometric constraints but without any force boundary conditions (implying that the force column in the final standard algebraic equations equals zero) and emphasizes the implementation of a different CVEFG approach. The material model is based on the work of Hong et al. (2008) and incorporates our previous efforts in the numerical implementation. The discretized equation system is derived from the complex variable moving least squares approximation and the Galerkin method. The essential boundary conditions are imposed through the penalty method. The proposed approach is verified by the simulation of the swelling-induced large deformation and surface instability of a confined single gel layer.

Analyzing wave propagation problems with the improved complex variable element-free Galerkin method

Two-dimensional wave propagation problems in this paper are analyzed with improved complex variable element-free Galerkin (ICVEFG) method. Improved complex variable moving least-squares (ICVMLS) approximation is applied to construct the shape function, and modified Galerkin weak form of wave propagation problems is used for obtaining the final system equations, then all equations of ICVEFG method for wave propagation problems are obtained. The convergences of ICVEFG method about node distribution and time step are analyzed, and the influences of weight functions, scale parameter of the influence domain and penalty factor on the numerical solutions are discussed. Example problems show that ICVEFG method has higher computational speed with higher precision.

An Improved Interpolating Complex Variable Meshless Method for Bending Problem of Kirchhoff Plates

An improved interpolating complex variable moving least squares (IICVMLS) method is proposed for numerical simulations of structures, in which a complete basis function and singular weight function are used to form a new basis function through the orthogonalization process. In this method, a new shape function which has the property of Kronecker [Formula: see text] function is derived to build the interpolating function. Based on the IICVMLS method, an improved interpolating complex variable element free Galerkin (IICVEFG) method is obtained for bending problem of Kirchhoff plates. In the IICVEFG method, the essential boundary conditions can be satisfied directly, and thus the final discrete matrix equation is more concise than that in the non-interpolating complex variable element free Galerkin methods. Hence, the proposed meshless method is more accurate and efficient than conventional complex variable meshless methods. Numerical examples of bending problem of Kirchhoff plates are presented to validate the advantages of the IICVEFG method.

A Fast Complex Variable Element-Free Galerkin Method for Three-Dimensional Wave Propagation Problems

In this paper, combining the dimension splitting method with the improved complex variable element-free Galerkin (ICVEFG) method, we present a fast ICVEFG method for three-dimensional wave propagation problems. Using the dimension splitting method, the equations of three-dimensional wave propagation problems are translated into a series of two-dimensional ones in another one-dimensional direction. The new Galerkin weak form of the dimension splitting method for three-dimensional wave propagation problems is obtained. The improved complex variable moving least-square (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions, finite difference method is used in the one-dimensional direction, then the formulae of the ICVEFG method for three-dimensional wave propagation problems are obtained. The convergence and the corresponding parameters in the ICVEFG method are discussed. Some numerical examples are given to show that the new method has higher computational precision, and can improve the computational efficiency of the conventional meshless methods for three-dimensional problems greatly.

A hybrid improved complex variable element-free Galerkin method for three-dimensional potential problems

Combining the dimension splitting method with the improved complex variable element-free Galerkin method, a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional potential problems. Using the dimension splitting method, a three-dimensional potential problem is transformed into a series of two-dimensional ones which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method for each two-dimensional problem, the improved complex variable moving least-square (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the one-dimensional direction. And Galerkin weak form of three-dimensional potential problem is used to obtain the final discretized equations. Then the H-ICVEFG method for three-dimensional potential problems is presented. Four numerical examples are given to show that the new method has higher computational efficiency.

The improved complex variable element-free Galerkin method for the analysis of Kirchhoff plates

Based on the improved complex variable moving least-squares (ICVMLS) approximation, the improved complex variable element-free Galerkin (ICVEFG) method for the bending problem of Kirchhoff plate is presented. Compared with the moving least-squares (MLS) approximation, in the ICVMLS approximation, the approximation function of two-dimensional problems can be obtained with one-dimensional basis function, then the computational efficiency of the shape functions is higher. Compared with the meshless methods based on the MLS approximation, under the same node distributions, the ones using the ICVMLS approximation can obtain the solutions with higher computational accuracy; and under the similar computational accuracy, the ones using the ICVMLS approximation have higher computational efficiency. The ICVMLS approximation is used to form the approximation function of the deflection of a Kirchhoff plate, the Galerkin weak form of the bending problem of Kirchhoff plates is adopted to obtain the discretized system equations, and the penalty method is employed to enforce the essential boundary conditions, then the corresponding formulae of the ICVEFG method for the bending problem of Kirchhoff plates are presented. By computing and analyzing four typical examples, it is shown that the ICVEFG method of Kirchhoff plates in this paper is efficient. And the computational precision of the numerical solutions is analyzed to select the basis function, weight function, scaling factor, node distribution and penalty factor in the ICVEFG method. Numerical examples show that the method in this paper has better convergence and higher accuracy. When the quadratic polynomial basis function and the cubic or quartic spline weight function are used, and d m a x = 4.2−4.4 , the ICVEFG method of Kirchhoff plates in this paper can obtain the solutions with high computational accuracy. And more nodes are distributed in the problem domain, higher computational accuracy of the solution will obtained, which shows that the method in this paper has better convergence.