Interpolating Element-free Galerkin 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.

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.

A reduced-order variational multiscale interpolating element free Galerkin technique based on proper orthogonal decomposition for solving Navier–Stokes equations coupled with a heat transfer equation: Nonstationary incompressible Boussinesq equations

In the recent decade, meshless methods have been handled for solving some PDEs due to their easiness. One of the most efficient meshless methods is the element free Galerkin (EFG) method. The test and trial functions of the EFG are based upon the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG (VMEFG) procedure. In the current article, the shape functions of interpolating moving least squares (IMLS) approximation are applied to the variational multiscale EFG technique to numerical study the Navier–Stokes equations coupled with a heat transfer equation such that this model is well-known as two-dimensional nonstationary Boussinesq equations. In order to reduce the computational time of simulation, we employ a reduced order model (ROM) based on the proper orthogonal decomposition (POD) technique. In the current paper, we developed a new reduced order model based on the meshless numerical procedure for solving an important model in fluid mechanics. To illustrate the reduction in CPU time as well as the efficiency of the proposed method, we investigate two-dimensional cases.

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.

Numerical and analytical investigations for solving the inverse tempered fractional diffusion equation via interpolating element-free Galerkin (IEFG) method

This manuscript is devoted to analysis of a novel meshless numerical procedure for solving the inverse tempered fractional diffusion equation. The employed numerical technique is based on a modification of element-free Galerkin (EFG) method, and the shape functions of interpolating moving least squares approximation are utilized for ingredients of the test and trial functions. At the first stage, the time derivative is discretized by a Crank–Nicolson idea to derive a semi-discrete scheme. In the next stage, the space variable is approximated by the EFG procedure. The convergence rate and stability of the time-discrete formulation are analyzed. Furthermore, the error estimate of the full-discrete plan is discussed in detail. In the end, some numerical experiments are investigated to check the theoretical results and the efficiency of the developed technique.

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.

Analysis and application of the interpolating element free Galerkin (IEFG) method to simulate the prevention of groundwater contamination with application in fluid flow

We develop a meshless numerical procedure to simulate the groundwater equation (GWE). The used technique is based on the interpolating element free Galerkin (IEFG) method. The interpolating moving least squares (IMLS) approximation produces a set of functions such that they are well-known as “shape functions”. The IEFG technique employs the shape functions of IMLS approximation. The shape functions of IMLS approximation vanish on the boundary and also they satisfy the property of the Kronecker Delta function. Thus, Dirichlet boundary conditions can be exactly imposed. In this paper, we check the unconditional stability and convergence of the proposed numerical scheme based on the energy method. The numerical results confirm the theoretical analysis.

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.

Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method

A delay PDE is different from a PDE in which it depends not only on the solution at a present stage but also on the solution at some past stage(s). In the current paper, we develop interpolating stabilized EFG method for a neutral delay PDE with fractional derivative in terms of Caputo fractional derivatives. The considered model in this paper is a generalized form of the other equations such that it contains a delay term. Thus, the existence and uniqueness of its analytic solution have been proven by the separation variable technique. At first, the temporal direction is discretized with a finite difference scheme. Then the EFG method has been employed to discrete the spatial direction. The stability of time-discrete scheme is studied by using the energy method. Also, we show the convergence order of the developed technique is O(τ3−α) in the temporal direction and the convergence order in spatial direction is O(rp+1). Finally, two test problems have been considered to demonstrate the capability and efficiency of the proposed numerical technique.

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.

The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau-regularized long wave (RRLW) equation with error analysis

ABSTRACTThe two-grid method is a technique to solve the linear system of algebraic equations for reducing the computational cost. In this study, the two-grid procedure has been combined with the EFG method for solving nonlinear partial differential equations. The two-grid FEM has been introduced in various forms. The well-known two-grid FEM is a three-step method that has been proposed by Bajpai and Nataraj (Comput. Math. Appl. 2014;68:2277–2291) that the new proposed scheme is an ecient procedure for solving important nonlinear partial differential equations such as Navier–Stokes equation. By applying shape functions of IMLS approximation in the EFG method, a new technique that is called interpolating EFG (IEFG) can be obtained. In the current investigation, we combine the two-grid algorithm with the IEFG method for solving the nonlinear Rosenau-regularized long-wave (RRLW) equation. In other hand, we demonstrate that solutions of steps 1, 2, and 3 exist and are unique and also we achieve an error estimate for them. Moreover, three test problems in one- and two-dimensional cases are given which support accuracy and efficiency of the proposed scheme.

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.

A meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) method

A new numerical algorithm has been investigated for solving time fractional reaction–subdiffusion equation. The fractional derivative of the considered equation is described in the Riemann–Liouville sense. Firstly, we discrete the temporal dimension of the considered model using a finite difference scheme. A central difference scheme has been applied to discrete the first time derivative and then for discretizing the fractional integral term a difference scheme has been employed with convergence order O(τ1+γ). Moreover, to achieve a full discretization scheme a type of meshless method has been improved that is known as element free Galerkin (EFG) method. The EFG method for integration uses a background mesh. This method is based on the Galerkin weak form in which the test and trial functions are shape functions of moving least squares (MLS) approximation. Since the shape functions of traditional MLS lack the Kronecker δ property, essential boundary conditions of a boundary value problem can not be directly computed and other methods must be employed for this issue. To this end, a new class of MLS shape functions has been applied that is called shape functions of interpolating MLS. The new shape function has the mentioned property. In the EFG method, calculating the appeared two-dimensional integrals is a basic issue. In this research work, the alternating direction implicit approach is combined with the element free Galerkin method. Then, using the new proposed method, the two-dimensional integrals on rectangular domain will be changed to simple one-dimensional integrals. We prove that the new numerical algorithm is unconditionally stable and also we obtain an error bound for the new procedure using the energy method. Numerical examples are reported which demonstrate the theoretical results and the efficiency of proposed scheme.