Variational Multiscale Element Free Galerkin Research Articles

An efficient variational multiscale element free Galerkin method based on adaptive TR-AB2 scheme for convection–diffusion equations

The variational multiscale element-free Galerkin (VMEFG) method can effectively improve stability when solving convection-dominated problems. However, for time-dependent problems, the VMEFG method has a large amount of calculation. In this paper, an efficient VMEFG method for solving unsteady convection-dominated convection–diffusion equations is proposed. The trapezoidal rule (TR) with adaptive time step is used for time integration, and the explicit second-order Adams–Bashforth (AB2) method is used to control the error. The main idea is to allow the time step to change along the direction of the solution, so that the amount of calculation under the specified accuracy requirements can be minimized. We further exemplified the effectiveness and accuracy of the given method in several examples. The numerical results show that the VMEFG method combined with TR-AB2 method (VMEFG-TR-AB2) can effectively increase computing efficiency while maintaining the calculation accuracy for the convection-dominated problem.

Local maximum-entropy approximation based stabilization methods for the convection diffusion problems

Local Maximum Entropy (LME) approximation in the field of elliptic partial differential equations is more advanced compared to convection dominated flow problems. This paper presents a comparative study on the LME based meshfree stabilization methods for convection dominated convection-diffusion problems. LME approximants account for the disposal of the elements and direct imposition of boundary conditions, whereas the stabilization methods are meant to deal with the issues associated with the non-self-adjoint convective term. A new meshfree stabilization scheme, residual free bubbles based multiscale element free Galerkin method is also presented and compared with streamline upwind Petrov Galerkin meshfree method and variational multiscale element free Galerkin Method. The effect of different priors and radius of support which define the LME basis are studied to test their importance in the three methods with respect to standard benchmark problems. Cubic spline prior based LME is found to be enough to produce better stability and accuracy for all the stabilization methods when compared to quartic spline and Gaussian priors. LME based variational multiscale element free Galerkin method is observed to yield better stability and accuracy for all the priors. The convergence rate for the three stabilization methods is observed to be in good agreement with the classical FEM-SUPG.

An adaptive variational multiscale element free Galerkin method based on the residual-based a posteriori error estimators for convection–diffusion–reaction problems

As we know, the variational multiscale element free Galerkin (VMEFG) method may still suffer from non-physical oscillations near the boundary or interior layers when solving the convection–diffusion–reaction problems with strong convection-dominated. To overcome this shortcoming, we consider incorporating an adaptive algorithm based on the residual error estimations into the VMEFG, which formed the adaptive VMEFG (AVMEFG) method. With respect to the adaptive technique adopted, two residual-based a posteriori error estimators in the H1-semi norm and energy norm are respectively used to locate and remark the high-gradient numerical solution regions. Several stationary convection–diffusion–reaction problems are solved to verify the effectiveness of the proposed method. Among them, the first example makes a comparison between the proposed method and the adaptive element free Galerkin method, while the other remaining examples compare two different residual-based a posteriori error estimators for the proposed method. Numerical examples illustrate that the proposed method is effective and efficient in solving the convection-dominated problem involving various layers. Moreover, the AVMEFG methods with the two residual-based a posteriori error estimators are generally equivalent except in the case that the solution has an exponential boundary layer, where the energy norm error estimator can achieve significantly better results than the other one.

A novel stabilized Galerkin meshless method for steady incompressible Navier–Stokes equations

In the paper, a novel stabilized meshless method is presented for solving steady incompressible fluid flow problems. For this method, the standard Galerkin discretization is used to momentum equations, where the variational multiscale method is applied to mass conservation equation. Thus, the novel stabilized method can be regarded as a simplification of the variational multiscale element free Galerkin method, but it still retains the advantages of the variational multiscale element free Galerkin method. The present method allows equal linear basis approximation of both velocity and pressure and avoids the Ladyzhenskaya–Babuška–Breezi(LBB) condition. Meanwhile, it can automatically obtain the stabilization tensor. Three Stokes flow and two Navier–Stokes flow problems are applied to validate the accuracy and feasibility of the present method. It is shown that the present stabilized meshless method can guarantee the numerical stability and accuracy for incompressible fluid flow problems. Moreover, it can save computational cost evidently compared with variational multiscale element free Galerkin method.

An adaptive variational multiscale element free Galerkin method for convection–diffusion equations

An adaptive variational multiscale element free Galerkin method for convection–diffusion equations

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.

Maximum entropy-based variational multiscale element-free Galerkin methods for scalar advection–diffusion problems

This paper presents a new parameter-free stability term via Galerkin mesh-free scheme to solve the multiscale behavior of steady-state advection-dominated advection–diffusion problems where the classical Galerkin method is shown to be unstable. A variational multiscale element-free Galerkin method (VMEFG) is proposed based on first-order local maximum entropy (LME) basis functions, which are owning weak Kronecker delta property that enables the imposition of essential boundary conditions with ease. In the present work, the finite space of the problem is enlarged by the infinite bubble space in the background integration cell. These bubbles constitute the fine scale solve locally (captures) the thin layers associated with the flow. Then, this bubble space solution is substituted into the coarse scales to recover the VMEFG method globally, which determines the parameter-free stability term naturally. The developed LME-based VMEFG is studied against three standard benchmark problems exhibiting boundary/internal layers and observed to be stable and robust in resolving all the layers. The effect of various values of locality parameter in defining the LME approximants is carried out between the values 1.2 and 2. It is found that the suboptimal solution is arrived at the locality parameter value, 1.5 and above 1.8, is not recommended. Convergence test is also carried out with two different mass functions and validated with the analytical solution and found that the computational efficiency of the present method is in good agreement with analytical solution.

A hybrid generalized interpolated element-free Galerkin method for Stokes problems

By coupling the ideas of dimension splitting method (DSM) and the generalized element-free Galerkin (GEFG) method, a hybrid generalized interpolated element-free Galerkin (HGIEFG) method is developed for Stokes problems. In the HGIEFG method, the two-dimensional problem splits into a battery of one-dimensional problems by the DSM. Combining the improved interpolating moving least-squares (IIMLS) method, the GEFG method is applied to obtain the discrete equations of the one-dimensional problems on the splitting plane. And then final discretized equations of the whole Stokes problems are assembled by the IIMLS method. Compared with the EFG method, the HGIEFG method can obtain non-oscillating solutions of pressure for Stokes problems. Compared with the variational multiscale element-free Galerkin (VMEFG) method, the stabilization technique in HGIEFG method is only related to nodes but not to integral grids. Compared with the EFG, VMEFG and GEFG methods, the HGIEFG method has higher computational efficiency. Numerical examples show that the HGIEFG method can obtain non-oscillating solutions of pressure and velocity, and has higher efficiency and accuracy for Stokes problems.

A Hybrid Variational Multiscale Element-Free Galerkin Method for Convection-Diffusion Problems

By coupling the dimension splitting method (DSM) and the variational multiscale element-free Galerkin (VMEFG) method, a hybrid variational multiscale element-free Galerkin (HVMEFG) method is developed for the two-dimensional convection-diffusion problems. In the HVMEFG method, the two-dimensional problem is converted into a battery of one-dimensional problems by the DSM. Combining the non-singular improved interpolating moving least-squares (IIMLS) method, the VMEFG method is used to obtain the discrete equations of the one-dimensional problems on the splitting plane. Then, final discretized equations of the entire convection-diffusion problems are assembled by the IIMLS method. The HVMEFG method has high accuracy and efficiency. Numerical examples show that the HVMEFG method can obtain non-oscillating solutions and has higher efficiency and accuracy than the EFG and VMEFG methods for convection-diffusion problems.

A generalized element-free Galerkin method for Stokes problem

A generalized element-free Galerkin (GEFG) method is developed in this paper for solving Stokes problem in primitive variable form. To obtain stable numerical results for both velocity and pressure, extended terms are only introduced into the approximate space of velocity in a special way as that in the generalized finite element method. Theoretical analysis shows that the GEFG method implies a stabilized formulation similar to that in the variational multiscale element-free Galerkin (VMEFG) method. Numerical results show the efficiency of the present method and reveal that both computational errors and CPU times of the present method are less than those of the VMEFG and the finite element methods.

Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions

Solving partial differential equations with discontinuous solutions is an important challenging problem in numerical analysis. To this end, there are some methods such as finite volume method, discontinuous Galerkin approach and particle technique that are able to solve these problems. In the current paper, the moving Kriging element-free Galerkin method has been combined with the variational multiscale algorithm to obtain acceptable and high-resolution solutions. For testing this technique, we select some PDEs with discontinuous solution such as Burgers’, Sod’s shock tube, advection–reaction–diffusion, Kuramoto–Sivashinsky, Boussinesq and shallow water equations. First, we obtain a time-discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and also obtain their shape functions. We use the element-free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element-free Galerkin method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. The element-free Galerkin method is a global method while finite element method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.

Adaptive variational multiscale element free Galerkin method for elliptical ring solitons

In this work, a coupling of adaptive element free Galerkin method with variational multiscale element free Galerkin method has been proposed in solving two-dimensional elliptical ring solitons. Meshless method is suitable in incorporating into adaptive refinement as the additional nodes can be freely inserted at the steep gradient regions without the reconstruction of geometry as in finite element method. Hence, the advantageous of adaptive refinement technique and variational multiscale method are inherited in this proposed method in order to obtain higher accuracy solution.The performance of the proposed method is first validated with solving Burgers' equation. The obtained numerical solutions are more accurate than the standard element free Galerkin method. Besides, this coupling method is then extended to solve two-dimensional elliptical ring solitons. The numerical results are found to be in good agreement with the numerical solutions that exist in literature. Therefore, this confirms the ability of the proposed method to deal with the large gradient problems and sine-Gordon equation.

A variational multiscale interpolating element-free Galerkin method for convection-diffusion and Stokes problems

By combining the interpolating moving least squares (IMLS) method with the variational multiscale method, a variational multiscale interpolating element-free Galerkin (VMIEFG) method is presented in this paper for the numerical solutions of convection-diffusion and Stokes problems. In the VMIEFG, the IMLS is used to construct shape functions based on using shifted and scaled polynomial bases. Compared with the variational multiscale element-free Galerkin (VMEFG) method, the VMIEFG method can directly apply the essential boundary conditions. The VMIEFG method is an effective meshless method, especially, for convection-dominated problems. Numerical examples show that the VMIEFG method avoids the oscillation in the element-free Galerkin (EFG) method, and the computational precision of the VMIEFG method is higher than that of the EFG, the VMEFG and the finite element methods.

Variational multiscale element free Galerkin method for natural convection with porous medium flow problems

In this paper, the variational multiscale element free Galerkin (VMEFG) method is extended to simulate the two-dimensional natural convection with porous medium flow problems. The Darcy-Brinkman-Forchheimer extended model is used for the non-Darcy porous medium flow. The VMEFG method is a meshless method which had successfully applied to simulate incompressible flow problems with equal order basis approximation for the velocity and pressure fields, meanwhile, this method can easily deal with complex geometry problems. Three numerical examples, including the natural convection in a square cavity, a cavity with cosine vertical wavy wall and a triangular cavity with zig-zag shaped bottom are considered to investigate the validation and accuracy of the proposed method. The numerical results are presented in the form of streamlines, isotherms and average Nusselt numbers. A good agreement is observed between the present results and those available in the previous studies. Meanwhile, effects of Rayleigh number, porosity of the porous medium, amplitude of wavy surface and number of undulation on the flow structure and heat transfer are also investigated.

Proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier–Stokes equation

Recently, because of simplicity of meshless methods, they have been employed for solving many partial differential equations. One of the meshless methods is element free Galerkin technique. The element free Galerkin is very similar to the finite element method with this difference that the test and trial spaces of EFG procedure are shape functions of moving least squares approximation. Using the mentioned shape functions, solving a problem on a complex domain is very simple. Some researchers proposed several modifications and enriched approaches for improving the element free Galerkin method that one of them is variational multiscale element free Galerkin procedure. Up to the best of authors’ knowledge, the element free Galerkin method based on the shape functions of moving least squares approximation needs more CPU time than the element free Galerkin method based on the shape functions of moving Kriging interpolation. Thus, in the current paper, we employ the variational multiscale element free Galerkin based on the shape functions of moving Kriging interpolation. Also, for reducing the CPU time of the presented numerical scheme, we use the proper orthogonal decomposition (POD) approach. Therefore, in the current paper, we propose the proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) method for solving time-dependent incompressible Navier–Stokes equation. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme.

A fast and stabilized meshless method for the convection-dominated convection-diffusion problems

ABSTRACTThis paper presents a fast and stabilized meshless method that combines variational multi-scale element free Galerkin (VMEFG) method and proper orthogonal decomposition (POD) method, namely VMEFG&POD, to solve convection-diffusion problems. Variational multi-scale method is applied to overcome the numerical oscillation for the convection-dominated problems and the POD method is used to improve the computational efficiency of the VMEFG method. This method is validated by considering the simulation of two-dimensional convection-diffusion problems with a small diffusion coefficient. It is demonstrated that the proposed method can largely improve the computational efficiency without a significant loss in accuracy.

A numerical study of natural convection in an inclined square enclosure with an elliptic cylinder using variational multiscale element free Galerkin method

In this paper, a numerical study for steady-state natural convection in a cold outer square enclosure containing a hot inner elliptic cylinder is investigated using the variational multiscale element free Galerkin method (VMEFG). The VMEFG method is a meshless method which coupled element free Galerkin method and variational multiscale method, it allows equal order basis for the velocity and the pressure field. The numerical calculations are performed for various Rayleigh numbers ranging from 103 to 106, the inclination angle of the outer square enclosure from 0° to 45° and major axis of the inner cylinder from 0.2L to 0.4L. The results show that the Rayleigh number, the size of the inner cylinder and the inclination angle of the outer square enclosure have remarkable effects on the streamlines and temperature contours in the enclosure, but the inclination angle of the enclosure has a small effect on average Nusselt number on the inner elliptic cylinder.

Variational Multiscale Element Free Galerkin Method Coupled with Low-Pass Filter for Burgers’ Equation with Small Diffusion

Variational multiscale element free Galerkin (VMEFG) method is applied to Burgers’ equation. It can be found that, for the very small diffusivity coefficients, VMEFG method still suffers from instability in the presence of boundary or interior layers. In order to overcome this problem, the high order low-pass filter is used to smooth the solution. Three test examples with very small diffusion are presented and the solutions obtained are compared with exact solutions and some other numerical methods. The numerical results are found in which the VMEFG coupled with low-pass filter works very well for Burgers’ equation with very small diffusivity coefficients.

Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction–diffusion system with and without cross-diffusion

The finite element method (FEM) is one of the basic methods for solving deterministic and stochastic partial differential equations. This method is proposed in the 19 decade and after years several modifications for this well-known technique such as mortal FEM, discontinuous Galerkin FEM, extended FEM, least squares FEM, spectral FEM, mixed FEM, immersed FEM, adaptive FEM, etc. have been proposed. Also, for improving the accuracy and for considering complex domain, shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique. The name of this improved method is element free Galerkin (EFG) method which is classified in the category of meshless methods. The EFG method has been applied for solving many problems in engineering. In the current paper, we select two numerical procedures that are extracted from FEM. The discontinuous Galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity. It should be noted that many enriched ideas have been explained for improving accuracy of EFG method. In the current paper, the local discontinuous Galerkin and variational multiscale EFG methods are applied for obtaining numerical solution of two-dimensional Brusselator system with and without cross-diffusion. The Brusselator system is a theoretical model for a type of autocatalytic reaction. Up to best of authors’ knowledge, the accuracy of the obtained numerical solutions using local discontinuous Galerkin method is very related to the used time-discrete scheme. Therefore, fourth-order exponential time differencing method has been employed for discretizing the time variable. Also, to achieve a full discretization scheme, the local discontinuous Galerkin finite element method is used. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.

Meshless modeling of natural convection problems in non-rectangular cavity using the variational multiscale element free Galerkin method

In this paper, the two-dimensional natural convection problems in complex geometries were solved by using the variational multiscale element free Galerkin (VMEFG) method. The VMEFG method is a meshless method which coupled element free Galerkin method and variational multiscale method, thus it inherits the advantages of variational multiscale and meshless methods. In this method, the field variables are decomposed into coarse and fine scales first, then solved fine scale problem analytically by using bubble functions, in the process, the stabilization parameters had appeared naturally. Moreover, it ensures that the resultant formulations yield a consistent stabilized method. From the viewpoint of application, the presented method can employ equal order basis for pressure and velocity, which is not only easy to implement but also avoid the restriction of the Babuŝka–Brezzi condition and eliminate non-physical oscillations. Several test problems, including natural convection in the semicircular cavity, triangular cavity and triangular cavity with zig-zag shaped bottom wall are considered to investigate the accuracy of the proposed method. The numerical results obtained using VMEFG showed very good agreement with those available in the literature.