Complex Variable Moving Least-squares Research Articles

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.

A meshless complex variable Galerkin boundary node method for potential and Stokes problems

In this study, combining the boundary integral equations (BIEs) with the complex variable moving least squares (CVMLS) approximation, a symmetric and boundary-only meshless method, the complex variable Galerkin boundary node method (CVGBNM), is developed. Numerical applications and theoretical error estimates of the CVGBNM are derived for BIEs, potential problems and Stokes problems. Finally, numerical examples are given to demonstrate the efficacy of the method.

Complex Variable Meshless Manifold Method for Transient Heat Conduction Problems

On the basis of the complex variable meshless manifold method (CVMMM), the time domain is discretized using the traditional two-point difference method, and the complex variable moving least-squares (CVMLS) method is used in the spatial domain. According to the equivalent integral weak form of the governing equation, the CVMMM for transient heat conduction problem is established, and the corresponding discreted equations are derived by introducing the penalty function method to the essential boundary conditions. The numerical examples show that compared with the meshless manifold method (MMM), the CVMMM for transient heat conduction problem has the advantages of high precision and less calculation. This proposed method is a new numerical calculation method which has great development potential.

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.

The complex variable meshless local Petrov–Galerkin method for elasticity problems of functionally graded materials

This paper proposed the complex variable meshless local Petrov–Galerkin (CVMLPG) method for the static analysis of functionally graded materials (FGMs). In the presented method, the complex variable moving least-square (CVMLS) approximation, which is established based on the moving least-square (MLS) approximation by introducing the complex variable theory, is adopted for construction of the field approximation function. Compared with the conventional MLS method, the number of the unknown coefficients in the trial function of the CVMLS method is less than that of the MLS approximation, thus higher efficiency and accuracy can be achieved under the same node distributions. One advantage of the CVMLPG method for the FGMs is that the variations of the functionally graded material properties are simulated 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. Some of selected benchmark examples are considered to confirm the validity and accuracy of the proposed method.

The complex variable meshless local Petrov–Galerkin method for elastodynamic problems

The complex variable meshless local Petrov–Galerkin (CVMLPG) method is further developed for structural dynamic analyses of two-dimensional (2D) solids. The complex variable moving least-square (CVMLS) approximation is used to construct the shape function and the Heaviside step function is chosen for representing the test function. In the construction of the well-performed shape function, the trial function of 2D problem is formed with a one-dimensional (1D) basis function, thus improving computational efficiency. The governing elastodynamic equations are transformed into a standard local weak formulation, and then it is discretized into a meshfree system of time-dependent equations, which are solved by the standard implicit Newmark time integration scheme. Numerical results obtained from the proposed CVMLPG method are compared with the exact solutions and those of the conventional MLPG method and excellent agreement is achieved.

Stress Intensity Factor for Interface Cracks in Bimaterials Using Complex Variable Meshless Manifold Method

This paper describes the application of the complex variable meshless manifold method (CVMMM) to stress intensity factor analyses of structures containing interface cracks between dissimilar materials. A discontinuous function and the near-tip asymptotic displacement functions are added to the CVMMM approximation using the framework of complex variable moving least-squares (CVMLS) approximation. This enables the domain to be modeled by CVMMM without explicitly meshing the crack surfaces. The enriched crack-tip functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The complex stress intensity factors for bimaterial interfacial cracks were numerically evaluated using the method. Good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized.

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.

An improved complex variable element-free Galerkin method for two-dimensional large deformation elastoplasticity problems

This paper presents the formulation and numerical implementation of the improved complex variable element-free Galerkin (ICVEFG) method for two-dimensional large deformation problems of elastoplasticity in total Lagrangian description. The ICVEFG method is a novel element free Galerkin (EFG) method based on the improved complex variable moving least-squares (ICVMLS) approximation. The ICVMLS approximation has all the advantages inherited from the complex variable moving least-squares (CVMLS) approximation. The function J in the ICVMLS approximation has an explicit physical meaning, compared with the former. The Galerkin weak form is employed to obtain the equations system and the penalty method is used to apply essential boundary conditions. Several numerical examples presented show that the ICVEFG method has greater precision and efficiency compared to the EFG and CVEFG methods.

A complex variable meshless local Petrov—Galerkin method for transient heat conduction problems

On the basis of the complex variable moving least-square (CVMLS) approximation, a complex variable meshless local Petrov—Galerkin (CVMLPG) method is presented for transient heat conduction problems. The method is developed based on the CVMLS approximation for constructing shape functions at scattered points, and the Heaviside step function is used as a test function in each sub-domain to avoid the need for a domain integral in symmetric weak form. In the construction of the well-performed shape function, the trial function of a two-dimensional (2D) problem is formed with a one-dimensional (1D) basis function, thus improving computational efficiency. The numerical results are compared with the exact solutions of the problems and the finite element method (FEM). This comparison illustrates the accuracy as well as the capability of the CVMLPG 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 meshless local Petrov—Galerkin method of solving two-dimensional potential problems

Based on the complex variable moving least-square (CVMLS) approximation and a local symmetric weak form, the complex variable meshless local Petrov—Galerkin (CVMLPG) method of solving two-dimensional potential problems is presented in this paper. In the present formulation, 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-square (MLS) approximation. The essential boundary conditions are imposed by the penalty method. The main advantage of this approach over the conventional meshless local Petrov—Galerkin (MLPG) method is its computational efficiency. Several numerical examples are presented to illustrate the implementation and performance of the present CVMLPG method.

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.

A new complex variable element-free Galerkin method for two-dimensional potential problems

In this paper, based on the element-free Galerkin (EFG) method and the improved complex variable moving least-square (ICVMLS) approximation, a new meshless method, which is the improved complex variable element-free Galerkin (ICVEFG) method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square (CVMLS) approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.

Complex variable element-free Galerkin method for viscoelasticity problems

Based on the complex variable moving least-square (CVMLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional viscoelasticity problems under the creep condition is presented in this paper. The Galerkin weak form is employed to obtain the equation system, and the penalty method is used to apply the essential boundary conditions, then the corresponding formulae of the CVEFG method for two-dimensional viscoelasticity problems under the creep condition are obtained. Compared with the element-free Galerkin (EFG) method, with the same node distribution, the CVEFG method has higher precision, and to obtain the similar precision, the CVEFG method has greater computational efficiency. Some numerical examples are given to demonstrate the validity and the efficiency of the method.

A novel complex variable element-free Galerkin method for two-dimensional large deformation problems

Based on complex variable theory and moving least-squares (MLS) approximation, the improved complex variable moving least-squares (ICVMLS) approximation is discussed in this paper. Compared with complex variable moving least-squares (CVMLS) approximation, the function in the ICVMLS approximation has an explicit physics meaning. By using a new basis function, the ICVMLS approximation can obtain greater precision and computational efficiency. Based on the ICVMLS approximation, an improved complex variable element-free Galerkin (ICVEFG) method, which belongs to a novel element free Galerkin (EFG) method, is presented for two-dimensional large deformation problems. The Galerkin weak form is employed to obtain the equations, while the penalty method is used to apply the essential boundary conditions. Then the corresponding formulae of the ICVEFG method for two-dimensional large deformation problems are obtained. Compared with the EFG method, the ICVEFG method has greater precision and efficiency.

An improved complex variable element-free Galerkin method for two-dimensional elasticity problems

In this paper, the improved complex variable moving least-squares (ICVMLS) approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squares (CVMLS) approximations presented by Cheng and Ren, the ICVMLS approximation has a great computational precision and efficiency. Based on the element-free Galerkin (EFG) method and the ICVMLS approximation, the improved complex variable element-free Galerkin (ICVEFG) method is presented for two-dimensional elasticity problems, and the corresponding formulae are obtained. Compared with the conventional EFG method, the ICVEFG method has a great computational accuracy and efficiency. For the purpose of demonstration, three selected numerical examples are solved using the ICVEFG method.

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.