Complex Variable Moving Least-squares Approximation Research Articles

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.

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.

On tracking arbitrary crack path with complex variable meshless methods

This study presents a numerical modelling framework based on complex variable meshless methods, which can accurately and efficiently track arbitrary crack paths in two-dimensional linear elastic solids. The key novelty of this work is that the proposed meshless modelling scheme enables a direct element-free approximation for the solutions of linear elastic fracture mechanics problems. The complex variable moving least-squares approximation with a group of simple complex polynomial basis is applied to implement this meshless model, with which the fracture problems with both stationary or progressive cracks are considered and studied. The effects of choosing different definitions of weighted complex variable error norm and different forms of complex polynomial basis on the computational accuracy of crack tip fields and crack paths prediction are analysed and discussed. Five benchmark numerical examples were studied to demonstrate the superiority of the present complex variable meshless framework over a standard element-free Galerkin method in tracking arbitrary crack paths in two-dimensional elastic solids.

A Complex Variable Boundary Element-Free Method for Potential and Helmholtz Problems in Three Dimensions

The complex variable boundary element-free method (CVBEFM) is a meshless method that takes the advantages of both boundary integral equations (BIEs) in dimension reduction and the complex variable moving least squares (CVMLS) approximation in element elimination. The CVBEFM is developed in this paper for solving 3D problems. This paper is an attempt in applying complex variable meshless methods to 3D problems. Formulations of the CVMLS approximation on 3D surfaces and the CVBEFM for 3D potential and Helmholtz problems are given. In the current implementation, the CVMLS shape function of 3D problems is formed with 1D basis functions, and the boundary conditions in the CVBEFM can be applied directly and easily. Some numerical examples are presented to demonstrate the method.

A complex variable boundary element-free method for the Helmholtz equation using regularized combined field integral equations

Based on the combined field integral equation (CFIE) and the complex variable moving least squares (CVMLS) approximation, a complex variable boundary element-free method is developed for the exterior Neumann problem for the Helmholtz equation. To eliminate both strongly singular and hypersingular integrals and to ensure the stability and efficiency of the method for all wavenumbers, a regularized CFIE is derived. To facilitate the application of the method and the approximation of the gradient of the boundary function, the CVMLS approximation is used to discretize the CFIE. Numerical results reveal that the method is pollution-free and efficient for solving the Helmholtz equation with arbitrary and large wavenumbers.

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.

The complex variable meshless local Petrov–Galerkin method for elastodynamic analysis of functionally graded materials

As an improvement of the meshless local Petrov–Galerkin (MLPG), the complex variable meshless local Petrov–Galerkin (CVMLPG) method is extended here to dynamic analysis of functionally graded materials (FGMs). In this method, the complex variable moving least-squares (CVMLS) approximation is used instead of the traditional moving least-squares (MLS) to construct the shape functions. The main advantage of the CVMLS approximation over MLS approximation is that the number of the unknown coefficients in the trial function of the CVMLS approximation is less than that of the MLS approximation, thus higher efficiency and accuracy can be achieved under the same node distributions. In implementation of the present method, the variations of the FGMs properties are computed by using material parameters at Gauss points, so it totally avoids the issue of the assumption of homogeneous in each element in the finite element method (FEM) for the FGMs. Several numerical examinations for dynamic analysis of FGMs are carried out to demonstrate the accuracy and efficiency of the CVMLPG.

Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method**Project supported by the National Natural Science Foundation of China (Grant Nos. 11171208 and U1433104).

In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin (ICVEFG) method is presented for two-dimensional (2D) elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.

Analyzing elastoplastic large deformation problems with the complex variable element-free Galerkin method

Using the complex variable moving least-squares (CVMLS) approximation, a complex variable element-free Galerkin (CVEFG) method for two-dimensional elastoplastic large deformation problems is presented. This meshless method has higher computational precision and efficiency because in the CVMLS approximation, the trial function of a two-dimensional problem is formed with a one-dimensional basis function. For two-dimensional elastoplastic large deformation problems, the Galerkin weak form is employed to obtain its equation system. The penalty method is used to impose essential boundary conditions. Then the corresponding formulae of the CVEFG method for two-dimensional elastoplastic large deformation problems are derived. In comparison with the conventional EFG method, our study shows that the CVEFG method has higher precision and efficiency. For illustration purpose, a few selected numerical examples are presented to demonstrate the advantages of the CVEFG method.

THE COMPLEX VARIABLE ELEMENT-FREE GALERKIN (CVEFG) METHOD FOR TWO-DIMENSIONAL ELASTODYNAMICS PROBLEMS

The complex variable moving least-squares (CVMLS) approximation is discussed in this paper, and the mathematical and physical meaning of the complex functional in the CVMLS approximation is presented. With the CVMLS approximation, the trial function of a two-dimensional problem is formed with a one-dimensional basis function. Then combining the CVMLS approximation and the Galerkin weak form, we investigate the complex variable element-free Galerkin (CVEFG) method for two-dimensional elastodynamics problems. The penalty method is used to apply the essential boundary conditions, and the implicit time integration method, which is the Newmark method, is used for time history analysis. Then the corresponding formulae of the CVEFG method for two-dimensional elastodynamics problems are obtained. For the purposes of demonstration, some selected numerical examples are solved using the CVEFG method. Compared with the EFG method, the CVEFG method has greater precision.

The complex variable interpolating moving least-squares method

In this paper, a new method for deriving the moving least-squares (MLS) approximation is presented first. Considering the unclear physical meaning of the functional in the existing complex variable moving least-squares (CVMLS) approximation, an improved CVMLS approximation is presented by constructing a new functional with an explicit physical meaning. Based on the improved CVMLS approximation, the complex variable interpolating moving least-squares (CVIMLS) method is presented, and the interpolating property of the corresponding shape function of the CVIMLS method is proved.

The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems

Based on the complex variable moving least-squares (CVMLS) approximation and element-free Galerkin (EFG) method, the complex variable element-free Galerkin (CVEFG) method for two-dimensional elasto-plasticity problems is presented in this paper. The CVMLS approximation is an approximation method for a vector function. Under the same node distribution the meshless method based on the CVMLS approximation has higher precision than the one based on the moving least-squares (MLS) approximation. For two-dimensional elasto-plasticity problems, the Galerkin weak form is employed to obtain the equations system, and the penalty method is used to apply the essential boundary conditions, then the corresponding formulae of the CVEFG method for two-dimensional elasto-plasticity problems are obtained. Compared with the EFG method, the CVEFG method can obtain greater precision. For the purposes of demonstration, some selected numerical examples are solved using the CVEFG method.

A COMPLEX VARIABLE MESHLESS MANIFOLD METHOD FOR FRACTURE PROBLEMS

Based on the complex variable moving least-squares (CVMLS) approximation and the finite cover theory, the complex variable meshless manifold method (CVMMM) for fracture problems is presented in this paper. The CVMMM employs two cover systems which are the mathematical cover system and the physical cover system. The shape function in the CVMMM is derived with the CVMLS approximation and the finite cover theory. The finite cover theory is used to model cracks which lead to interior discontinuous displacements. At the tip of a crack of a problem, we use the analytical solution near the tip of a crack to extend the trial function of the CVMMM, then the corresponding approximation function is obtained. From the minimum potential energy principle, the corresponding formulae of the CVMMM for fracture problems are presented. Some numerical examples are presented to demonstrate the method in this paper.

Complex variable boundary element-free method for two-dimensional elastodynamic problems

We proposed a new direct meshless boundary integral equation technique – the complex variable boundary element-free method (CVBEFM) based on the complex variable moving least-squares (CVMLS) approximation and the boundary element-free method (BEFM), to study the two-dimensional elastodynamic problems. With the CVMLS approximation, the trial function of a two-dimensional problem is formed with a one-dimensional basis function. The number of unknown coefficients in the trial function of the CVMLS approximation is less than that in the trial function of the moving least-squares (MLS) approximation. Therefore it requires fewer nodes in the meshless method which formed from the CVMLS approximation than that formed from the MLS approximation with no lose of precision. The Laplace transform is used to formulate the boundary integral equations of the two-dimensional elastodynamics and then the formulae of the CVBEFM for two-dimensional elastodynamic problems are derived. The CVBEFM is a direct numerical method in which the basic unknown quantities are the real solutions of the nodal variables. Moreover in the CVBEFM, the boundary conditions can be applied directly and easily that leads to a greater computational precision. In this paper, we selected a few numerical examples to illustrate the applicability of the CVBEFM.

THE COMPLEX VARIABLE ELEMENT-FREE GALERKIN (CVEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS

Based on element-free Galerkin (EFG) method and the complex variable moving least-squares (CVMLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems is presented in this paper. With the CVMLS approximation, the trial function of a two-dimensional problem is formed with a one-dimensional basis function. The number of unknown coefficients in the trial function of the CVMLS approximation is less than in the trial function of moving least-squares (MLS) approximation, and we can thus select fewer nodes in the meshless method that is formed from the CVMLS approximation than are required in the meshless method of the MLS approximation with no loss of precision. The formulae of the CVEFG method for two-dimensional elasticity problems is obtained. Compared with the conventional meshless method, the CVEFG method has a greater precision and computational efficiency. For the purposes of demonstration, some selected numerical examples are solved using the CVEFG method.

Complex variable moving least‐squares method: a meshless approximation technique

AbstractBased on the moving least‐squares (MLS) approximation, we propose a new approximation method—the complex variable moving least‐squares (CVMLS) approximation. With the CVMLS approximation, the trial function of a two‐dimensional problem is formed with a one‐dimensional basis function. The number of unknown coefficients in the trial function of the CVMLS approximation is less than in the trial function of the MLS approximation, and we can thus select fewer nodes in the meshless method that is formed from the CVMLS approximation than are required in the meshless method of the MLS approximation with no loss of precision. The meshless method that is derived from the CVMLS approximation also has a greater computational efficiency. From the CVMLS approximation, we propose a new meshless method for two‐dimensional elasticity problems—the complex variable meshless method (CVMM)—and the formulae of the CVMM for two‐dimensional elasticity problems are obtained. Compared with the conventional meshless method, the CVMM has a greater precision and computational efficiency. For the purposes of demonstration, some selected numerical examples are solved using the CVMM. Copyright © 2006 John Wiley & Sons, Ltd.