Interpolating Element-free Galerkin Method Research Articles

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.

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.

A Dimension Splitting Generalized Interpolating Element-Free Galerkin Method for the Singularly Perturbed Steady Convection–Diffusion–Reaction Problems

By introducing the dimension splitting method (DSM) into the generalized element-free Galerkin (GEFG) method, a dimension splitting generalized interpolating element-free Galerkin (DS-GIEFG) method is presented for analyzing the numerical solutions of the singularly perturbed steady convection–diffusion–reaction (CDR) problems. In the DS-GIEFG method, the DSM is used to divide the two-dimensional CDR problem into a series of lower-dimensional problems. The GEFG and the improved interpolated moving least squares (IIMLS) methods are used to obtain the discrete equations on the subdivision plane. Finally, the IIMLS method is applied to assemble the discrete equations of the entire problem. Some examples are solved to verify the effectiveness of the DS-GIEFG method. The numerical results show that the numerical solution converges to the analytical solution with the decrease in node spacing, and the DS-GIEFG method has high computational efficiency and accuracy.

Magnetic Field Calculation of a Novel Electromagnetic Clutch Based on an Interpolating Element-Free Galerkin Method

In this paper, the interpolating element-free Galerkin method (IEFGM) is introduced to calculate the magnetic field distribution of a novel permanent magnet bistable electromagnetic clutch (BPMEC), which is used in the in-wheel motor drive system to flexibly connecting the wheel and motor to release the mechanical shock and electromagnetic impulsion. Firstly, the fundamental of the IEFGM is introduced, and a new weight function for magnetic field calculation is proposed. Then the nodes distribution is discussed, and the magnetic field distribution of the clutch is analyzed. Simulation results show the IEFGM can impose the boundary conditions directly and fewer nodes are employed, make the calculation efficiency higher. Finally, some experiments are carried out, the experimental results verified the effectiveness of the IEFGM in magnetic field calculation. The study results provide references for the application of the interpolating element-free Galerkin method in the magnetic field calculation.

The interpolating element-free Galerkin method for three-dimensional transient heat conduction problems

In this study, the interpolating element-free Galerkin (IEFG) method for solving three-dimensional (3D) transient heat conduction problem is presented. By using the improved interpolating moving least-squares (IIMLS) method to form the shape function, and using the weak form of 3D transient heat conduction problems to obtain the discretized equations, the formulae of the IEFG method are obtained. The shape function of the IIMLS method satisfies the property of Kronecker delta function, and then the IEFG method can apply essential boundary conditions directly, which can result in higher computational speed and accuracy. Some examples are given to discuss the convergence and advantages of the IEFG method. By analyzing the numerical results obtained by IEFG method and improved EFG method, we conclude that IEFG method has clear advantages in computational speed and accuracy.

The interpolating element-free Galerkin method for elastic large deformation problems

This paper presents an interpolating element-free Galerkin (IEFG) method for solving the two-dimensional (2D) elastic large deformation problems. By using the improved interpolating moving least-squares method to form shape function, and using the Galerkin weak form of 2D elastic large deformation problems to obtain the discrete equations, we obtain the formulae of the IEFG method for 2D elastic large deformation problems. As the displacement boundary conditions can be applied directly, the IEFG method can acquire higher computational efficiency and accuracy than the traditional element-free Galerkin (EFG) method, which is based on the moving least-squares approximation and can not apply the displacement boundary conditions directly. To analyze the influences of node distribution, scale parameter of influence domain and the loading step on the numerical solutions of the IEFG method, three numerical examples are proposed. The IEFG method has almost the same high accuracy as the EFG method, and for some 2D elastic large deformation problems the IEFG method even has higher computational accuracy.

Error analysis of interpolating element free Galerkin method to solve non-linear extended Fisher–Kolmogorov equation

Nonlinear partial differential equations (PDEs) play an important role in the modeling of the natural phenomena as they have great significance in real-world applications. This investigation proposes a new algorithm to find the numerical solution of the non-linear extended Fisher–Kolmogorov equation. Firstly, the time variable is discretized by a second-order finite difference scheme. The rate of convergence and stability of the semi-discrete formulation are studied by the energy method. The existence and uniqueness of the solution of the weak form based on the proposed technique have been proved in detail. Furthermore, the interpolating element free Galerkin approach based on the interpolation moving least-squares approximation is employed to derive a fully discrete scheme. Finally, the error estimate of the full-discrete plan is proposed and its convergence order is O(τ2+δm+1) in which τ, δ and m denote the time step, the radius of the weight function and smoothness of the exact solution of the main problem, respectively.

The interpolating element-free Galerkin method for three-dimensional elastoplasticity problems

In this paper, the interpolating element-free Galerkin (IEFG) method for solving the three-dimensional (3D) elastoplasticity problems is presented. By using the improved interpolating moving least-squares method to form the approximation function, and using the Galerkin weak form of 3D elastoplasticity problems to obtain the discretilized equations, we present the formulae of the IEFG method for the 3D elastoplasticity problems. The method can apply the displacement boundary conditions directly, which results in higher computational efficiency and accuracy. Numerical examples are given to discuss the influences of node distributions, scale parameters of influence domains and the loading steps on the computational accuracy of numerical solutions of the IEFG method. The numerical results show that, comparing with the element-free Galerkin method, the IEFG method for 3D elastoplasticity problems in this paper has higher computational efficiency and accuracy.

The interpolating element-free Galerkin (IEFG) method for three-dimensional potential problems

In this paper, an interpolating element-free Galerkin (IEFG) method for three-dimensional potential problems is presented based on the improved interpolating moving least-squares (IIMLS) method. The shape function of the IIMLS method satisfies the property of Kronecker δ function, which results in that the boundary conditions are applied directly. Compared with the improved element-free Galerkin (EFG) method, the IEFG method in this paper gives a higher computational accuracy. Moreover, this method also has greater computational efficiency than the improved EFG method under the similar accuracy. For the purposes of demonstration, four numerical examples are solved using the IEFG method to show the advantages of the IEFG method with higher computational accuracy and computational efficiency.

An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

Purpose This paper aims to present an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation, and then discuss the stability and convergence of the numerical solutions. Design/methodology/approach In the time-fractional diffusion equation, the time fractional derivatives are approximated by L1 method, and the shape functions are constructed by the interpolating moving least-squares (IMLS) method. The final system equations are obtained by using the Galerkin weak form. Because the shape functions have the interpolating property, the unknowns can be solved by the iterative method after imposing the essential boundary condition directly. Findings Both theoretical and numerical results show that the IEFG method for the time-fractional diffusion equation has high accuracy. The stability of the fully discrete scheme of the method on the time step is stable unconditionally with a high convergence rate. Originality/value This work will provide an interpolating meshless method to study the numerical solutions of the time-fractional diffusion equation using the IEFG method.

Interpolating element-free Galerkin method for viscoelasticity problems

In this paper, based on the improved interpolating moving least-square (IMLS) approximation, the interpolating element-free Galerkin (IEFG) method for two-dimensional viscoelasticity problems is presented. The shape function constructed by the IMLS approximation can overcome the shortcomings that the shape function of the moving least-squares (MLS) can-not satisfy the property of Kronecker function, so the essential boundary conditions can be directly applied to the IEFG method. Under a similar computational precision, compared with the meshless method based on the MLS approximation, the meshless method using the IMLS approximation has a high computational efficiency. Using the IMLS approximation to form the shape function and adopting the Galerkin weak form of the two-dimensional viscoelasticity problem to obtain the final discretized equation, the formulae for two-dimensional viscoelasticity problem are derived by the IEFG method. The IEFG method has some advantages over the conventional element-free Galerkin (EFG) method, such as the concise formulae and direct application of the essential boundary conditions, For the IEFG method of two-dimensional viscoelasticity problems proposed in this paper, three numerical examples and one engineering example are given. The convergence of the method is analyzed by considering the effects of the scale parameters of influence domains and the node distribution on the computational precision of the solutions. It is shown that when <i>d</i><sub>max</sub> = 1.01−2.00, the method in this paper has a good convergence. The numerical results from the IEFG method are compared with those from the EFG method and from the finite element method or analytical solution. We can see that the IEFG method in this paper is effective. The results of the examples show that the IEFG method has the advantage in improving the computational efficiency of the EFG method under a similar computational accuracy. And the engineering example shows that the IEFG method can not only has higher computational precision, but also improve the computational efficiency.

Error analysis and numerical simulation of magnetohydrodynamics (MHD) equation based on the interpolating element free Galerkin (IEFG) method

The MHD equation has some applications in physics and engineering. The main aim of the current paper is to propose a new numerical algorithm for solving the MHD equation. At first, the temporal direction has been discretized by the Crank–Nicolson scheme. Also, the unconditional stability and convergence of the time-discrete scheme have been investigated by using the energy method. Then, an improvement of element free Galerkin (EFG) i.e. the interpolating element free Galerkin method has been employed to discrete the spatial direction. Furthermore, an error estimate is presented for the full discrete scheme based on the Crank–Nicolson scheme by using the energy method. We prove that convergence order of the numerical scheme based on the new numerical scheme is O(τ2+δm). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. Numerical examples confirm the efficiency and accuracy of the proposed scheme.

Interpolating element-free Galerkin method for the regularized long wave equation and its error analysis

Combining the simplified interpolating moving least-squares (IMLS) method with the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method is presented for the numerical study of the regularized long wave (RLW) equation. The trial function in the IMLS method has one less unknown coefficient than in the traditional MLS approximation, and the shape functions satisfy the interpolating property at nodes. Then the IEFG method can apply the essential boundary conditions naturally and directly, and has high computational accuracy. And then this paper studies the error estimates of the IEFG method for RLW equation. The error estimate of the semi-discrete IEFG method for the RLW equation on the influence domain radius is first studied, and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied. The theoretical results show that the IEFG method for RLW equation has good convergence rate. For the purpose of demonstration, three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper.

Accuracy Analysis of Interpolating Element-Free Galerkin (IEFG) Method in Solving Transient Heat Conduction Problems

For the interpolating element-free Galerkin (IEFG) method set up by combining the shape function constructed with interpolating moving least-squares (IMLS) approximation and Galerkin integral weak form of partial differential equation, we show that the task of accuracy analysis of the IEFG method in solving transient heat conduction problems can be reduced to the estimation of errors in the procedure of IMLS approximation and the integral weak form of heat conduction equation. Following this practice, accuracy analysis of IEFG method is derived. To illustrate our results, numerical tests are discussed by two examples and give an optimal convergence rate with respect to two types of norm.

An Improved Interpolating Element-Free Galerkin Method for Elastoplasticity via Nonsingular Weight Functions

An improved interpolating element-free Galerkin (IIEFG) method for elastoplasticity is proposed in this paper. In the IIEFG method, the shape functions are constructed by the improved interpolating moving least-squares (IIMLS) method, and the final system equations are obtained by using the Galerkin weak form of elastoplasticity. Compared with the interpolating moving least-squares (IMLS) method, the weight functions are not singular in the IIMLS method, in which the shape functions have the interpolating property. The IIMLS method has fewer unknown coefficients to be solved in the trial functions than the moving least-squares (MLS) approximation. Hence, the IIEFG method is able to directly enforce the displacement boundary condition and obtain numerical solutions with high computational accuracy and efficiency. To show advantages of the IIEFG method, some selected elastoplastic examples are given.

Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: Meshless interpolating element free Galerkin (IEFG) and finite element methods

In this paper we apply a finite element scheme and interpolating element free Galerkin technique for the numerical solution of the two-dimensional time fractional diffusion-wave equation on the irregular domains. The time fractional derivative which has been described in the Caputo׳s sense is approximated by a scheme of order O(τ3−α), 1<α<2, and the space derivatives are discretized with finite element and interpolating element free Galerkin techniques. We prove the unconditional stability and obtain an error bound for the two new schemes using the energy method. However we would like to emphasize that the main aim of the current paper is to implement the Galerkin finite element method and interpolating element free Galerkin method on complex domains. Also we present error estimate for both schemes proposed for solving the time fractional diffusion-wave equation. Numerical examples demonstrate the theoretical results and the efficiency of the proposed scheme.

Error estimates for the interpolating moving least-squares method in n-dimensional space

In this paper, the interpolating moving least-squares (IMLS) method is discussed in details. A simpler expression of the approximation function of the IMLS method is obtained. Compared with the moving least-squares (MLS) approximation, the shape function of the IMLS method satisfies the property of Kronecker δ function. Then the meshless method based on the IMLS method can overcome the difficulties of applying the essential boundary conditions. The error estimates of the approximation function and its first and second order derivatives of the IMLS method are presented in n-dimensional space. The theoretical results show that if the weight function is sufficiently smooth and the order of the polynomial basis functions is big enough, the approximation function and its partial derivatives are convergent to the exact values in terms of the maximum radius of the domains of influence of nodes. Then the interpolating element-free Galerkin (IEFG) method based on the IMLS method is presented for potential problems. The advantage of the IEFG method is that the essential boundary conditions can be applied directly and easily. For the purpose of demonstration, some selected numerical examples are given to prove the theories in this paper.

AN INTERPOLATING ELEMENT-FREE GALERKIN METHOD FOR STEADY-STATE HEAT CONDUCTION PROBLEMS

In this paper, an interpolating element-free Galerkin (IEFG) method is presented for steady-state heat conduction problems with heat generation and spatially varying conductivity. The shape function in the moving least-squares (MLS) approximation does not satisfy the property of Kronecker delta function, so an interpolating moving least-squares (IMLS) method is discussed, then combining the shape function constructed by the IMLS method and Galerkin weak form of the two-dimensional steady-state heat conduction problems, the IEFG method for heat conduction problems is presented, and the corresponding formulae are obtained. The main advantage of this approach over the conventional meshless methods is that essential boundary conditions can be applied directly. Numerical results show that the IEFG method not only has high computational accuracy, but also enhances computational efficiency greatly.

A novel interpolating element-free Galerkin (IEFG) method for two-dimensional elastoplasticity

Using the interpolating moving least-squares (IMLS) method to obtain the shape function, we present a novel interpolating element-free Galerkin (IEFG) method to solve two-dimensional elastoplasticity problems. The shape function of the IMLS method satisfies the property of Kronecker δ function, then in the meshless methods based on the IMLS method, the essential boundary conditions can applied directly. Based on the Galerkin weak form, we obtain the formulae of the IEFG method for solving two-dimensional elastoplasticity problems. The IEFG method has some advantages, such as simpler formulae and directly applying the essential boundary conditions, over the conventional element-free Galerkin (EFG) method. The results of three numerical examples show that the computational precision of the IEFG method is higher than that of the EFG method.

The Error Estimates of the Interpolating Element-Free Galerkin Method for Two-Point Boundary Value Problems

The interpolating moving least-squares (IMLS) method is discussed in detail, and a simpler formula of the shape function of the IMLS method is obtained. Then, based on the IMLS method and the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method for two-point boundary value problems is presented. The IEFG method has high computing speed and precision. Then error analysis of the IEFG method for two-point boundary value problems is presented. The convergence rates of the numerical solution and its derivatives of the IEFG method are presented. The theories show that, if the original solution is sufficiently smooth and the order of the basis functions is big enough, the solution of the IEFG method and its derivatives are convergent to the exact solutions in terms of the maximum radius of the domains of influence of nodes. For the purpose of demonstration, two selected numerical examples are given to confirm the theories.