A hybrid improved complex variable element-free Galerkin method for three-dimensional potential problems
Introduction
Meshless method is an important numerical method for science and engineering problems, and has developed rapidly in recent twenty years. Compared with traditional numerical methods based on mesh [1], the advantage of meshless method only need the information of the nodes in the problem domain, then it can obtain the solution with great precision for some problems, such as the large deformation [2] and dynamic crack growth [3].
The element-free Galerkin (EFG) method is one of the most important meshless methods, and it has been applied into many science and engineering problems [4], [5], [6]. The EFG method is based on the moving least-squares (MLS) approximation, which sometimes forms ill-conditional or singular matrix. In order to overcome the disadvantage of MLS approximation, Cheng et al. proposed the improved moving least-squares (IMLS) approximation by orthogonalizing the basis function [7]. Using the IMLS approximation to construct shape function, the improved element-free Galerkin (IEFG) method are presented for potential problem [8], transient heat conduction [9], wave equation [10], fracture [11] and elastodynamics [12]. The IEFG method has higher computational efficiency than the EFG method with the same accuracy.
To improve the computational accuracy and efficiency of the MLS approximation and the corresponding meshless methods, the approximation of vector function, which is the complex variable moving least-squares (CVMLS) approximation, was presented by Cheng et al. [13]. Based on the CVMLS approximation and Galerkin weak form, the complex variable element-free Galerkin (CVEFG) method was presented for elasticity [14], elastoplasticity [15], elastodynamics [16], viscoelasticity [17], and elastoplastic large deformation [18]. Comparing with the EFG method, this method has higher computational efficiency.
Based on the conjugate basis function, Bai et al. proposed the improved complex variable moving least-squares (ICVMLS) approximation to construct the shape function [19]. Bai et al. presented the improved complex invariable element-free Galerkin (ICVEFG) method by using the ICVMLS approximation. Cheng et al. presented the ICVEFG method for potential problem [20], [21], heat conduction [22], advection-diffusion [23], elasticity [19], elastoplasticity [24] elastic large deformation [25] and elastoplastic large deformation [26]. And Peng et al. studied the ICVEFG method for viscoelasticity [27]. The basis of ICVMLS approximation may yield the inherent instability, then Li et al. proposed shifted and scaled bases [28], [29], and gave the theoretical error estimation of the ICVMLS approximation and the ICVEFG method [30]. Compared with the CVEFG method, the ICVEFG method has higher accuracy under the condition of same node distribution, and has higher computational speed when the computational accuracy is similar.
The ICVEFG method cannot be applied into three-dimensional problems because that the complex theory is used. Up to now the IEFG method is the best method for three-dimensional problems because it has higher computational precision and efficiency than the EFG method [8−10]. However, the shape functions used in the IEFG and EFG methods are more complicated than the ones in finite element method (FEM). And the shape functions in the IEFG and EFG methods must be computed at each node, then the computational efficiency of the IEFG and EFG methods are very low. Therefore, it is necessary to study the meshless methods for three-dimensional problems with great computational efficiency.
The dimension splitting method was proposed by Li et al. firstly to solve the compressible Navier-Stokes equation [31]. Then Li et al. [32], [33] solved three-dimensional compressible and incompressible Navier-Stokes equations by using the dimension splitting method. Hou et al. studied a finite element dimension splitting algorithm for a three-dimensional elliptic equation [34]. The dimension splitting method can improve the computational efficiency of numerical methods for three-dimensional problems [34].
In this paper, we introduce the dimension splitting method into the ICVEFG method, and a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional potential problems. The main idea of this method is that a three-dimensional potential problem is transformed into a series of two-dimensional ones which can be solved with the ICVEFG method. In the ICVEFG method for each two-dimensional problem, the 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.
For the H-ICVEFG method presented in this paper, the effects of the scale parameter, the penalty factor, the number of nodes, the step number and the weight functions on the computational precision and efficiency are discussed respectively. Some numerical examples are given, and the numerical results are compared with the ones of the IEFG method, which shows that the new method in this paper can not only achieve higher computational accuracy but also improve the computational efficiency greatly.
Section snippets
The improved complex variable moving least-squares approximation
The approximation of a function u(z) in the ICVMLS approximation is where m is the number of basis function, is the basis function vector which equals the conjugate of the basis function vector pT(z), and is the corresponding coefficient vector.
In general, the linear and the quadratic basis function vectors in the plane domain are given respectively by
The H-ICVEFG method for three-dimensional potential problems
The Poisson's equation of three-dimensional problem is with the following essential and natural boundary conditions where Ω is the domain of problem, Γ is the boundary of Ω; u(x) is the potential function, b(x) is a given source function; is the given potential on essential boundary Γu, is the given gradient on natural boundary Γq, Γ = Γu∪Γq, Γu∩Γq = ∅; and n(x) is the unit outward normal to
Numerical examples
In order to verify the advantage of the H-ICVEFG method for three-dimensional potential problems, we present four numerical examples in this section, and compared the accuracy and efficiency of the H-ICVEFG method with the ones of the IEFG method for three-dimensional problems.
In order to compare the accuracy of the H-ICVEFG method with the IEFG method, we define the relative error as where is the L2-norm of the error.
In this
Conclusions
Combining the dimension splitting method and the ICVEFG method, the H-ICVEFG method for three-dimensional potential problems is presented in this paper.
The influences of the scale parameter, the penalty factor, the node distribution, the step number and the weight functions on the computational precision of the H-ICVEFG method are discussed respectively, and relevant conclusions are given.
Four numerical examples are given, and the numerical results of the H-ICVEFG method are compared with the
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 11571223 and U1433104).
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