Multiscale Element-free Galerkin Method 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

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.

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.

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.

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.

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.

Variational multiscale element free Galerkin method for convection-diffusion-reaction equation with small diffusion

This paper presents the variational multiscale element free Galerkin method to solve convection-diffusion-reaction equation. The equation under consideration involves a small diffusivity and a large reaction coefficient, which leads to reaction-convection dominated problem. The variational multiscale element free Galerkin method is derived based on Hughes׳s variational multiscale formulation and element free Galerkin method, thus it inherits the advantages of variational multiscale and meshless methods, meanwhile, the formulation is free of any user-defined parameters owing to the stabilization parameter arises naturally. In order to investigate the presented method, both steady and unsteady 2D convection-diffusion-reaction problems are considered, and the numerical results illustrate the proposed method has the high accuracy and stability for solving convection-diffusion-reaction equation.

Numerical Modeling of Stokes Flow in a Circular Cavity by Variational Multiscale Element Free Galerkin Method

The variational multiscale element free Galerkin method is extended to simulate the Stokes flow problems in a circular cavity as an irregular geometry. The method is combined with Hughes’s variational multiscale formulation and element free Galerkin method; thus it inherits the advantages of variational multiscale and meshless methods. Meanwhile, a simple technique is adopted to impose the essential boundary conditions which makes it easy to solve problems with complex area. Finally, two examples are solved and good results are obtained as compared with solutions of analytical and numerical methods, which demonstrates that the proposed method is an attractive approach for solving incompressible fluid flow problems in terms of accuracy and stability, even for complex irregular boundaries.

Multiscale Element–Free Galerkin Method with Penalty for 2D Burgers’ Equation

In this paper, a new numerical method which is based on the coupling between multiscale method and meshless method with penalty is developed for 2D Burgers’ equation. The advantage of meshless method over the finite element method (FEM) is that remeshing process is not required. This is because the meshless method approximation is constructed entirely in terms of a set of nodes. Since the moving least squares (MLS) shape function does not satisfy the Kronecker delta property, so penalty method is adopted to enforce the essential boundary conditions in this paper. In order to obtain the fine scale approximation, the local enrichment basis is applied. The local enrichment basis may adopt the polynomial basis functions or any other analytical basis functions. Here, the polynomial basis functions are chosen as local enrichment basis. This multiscale meshless method with penalty will provide a more accurate result especially in the critical region which requires higher accuracy. It is believed that this proposed method is an attractive approach for solving more general problems which involve large deformation.

The variational multiscale element free Galerkin method for MHD flows at high Hartmann numbers

The aim of the paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics (MHD) flow problems with either fully insulating walls or partially insulating and partially conducting walls. Toward this, we first extend the influence domain of the shape function for the element free Galerkin (EFG) method to have arbitrary shape. When the influence factor approaches 1, we find that the EFG shape function almost has the Delta property at the node (i.e. the value of the EFG shape function of the node is nearly equal to 1 at the position of this node) as well as the property of slices in the influence domain of the node (i.e. the EFG shape function in the influence domain of the node is nearly constructed by different functions defined in different slices). Therefore, for MHD flow problems at high Hartmann numbers we follow the idea of the variational multiscale finite element method (VMFEM) to combine the EFG method with the variational multiscale (VM) method, namely the variational multiscale element free Galerkin (VMEFG) method is proposed. Subsequently, in order to validate the proposed method, we compare the obtained approximate solutions with the exact solutions for some problems where such exact solutions are known. Finally, several benchmark problems of MHD flows are simulated and the numerical results indicate that the VMEFG method is stable at moderate and high values of Hartmann number. Another important feature of this method is that the stabilization parameter has appeared naturally via the solution of the fine scale problem. Meanwhile, because this proposed method is a type of meshless method, it can avoid the need for meshing, a very demanding task for complicated geometry problems.

Variational multiscale element free Galerkin method for the water wave problems

A new numerical method, which is based on the coupling between variational multiscale method and meshfree methods, is developed for the water wave problems, in which the free surface capturing technique is used to capture the position of the free surface. The proposed method takes full advantage of meshfree methods, therefore, no mesh generation and mesh reconstruction are involved. Meanwhile, due to that the proposed method belongs to meshfree methods, thus it is suitable for the highly deformed free surface flow problems. Finally, two water wave problems are solved and the results have also been analyzed. The numerical results show that the proposed method can indeed obtain accurate numerical results for the water wave problems, which does not refer to the choice of a proper stabilization parameter.

Variational multiscale element-free Galerkin method for 2D Burgers’ equation

A new numerical method, which is based on the coupling between variational multiscale method and meshfree methods, is developed for 2D Burgers’ equation with various values of Re. The proposed method takes full advantage of meshfree methods, therefore, no mesh generation and mesh recreation are involved. Meanwhile, compared with the variational multiscale finite element method, a strong assumption, that is, the fine scale vanishes identically over the element boundaries although non-zero within the elements, is not needed. Subsequently two problems which have an available analytical solution of their own are solved to analyze the convergence behaviour of the proposed method. Finally a 2D Burgers’ equation having large Re is solved and the results have also been compared with the ones computed by two other methods. The numerical results show that the proposed method can indeed obtain accurate numerical results for 2D Burgers’ equation having large Re, which does not refer to the choice of a proper stabilization parameter.

On a multi-scale element-free Galerkin method for the Stokes problem

In this paper, a multi-scale element-free Galerkin method is presented for the Stokes problem. The new method is based on the Hughes’ variational multi-scale formulation, and arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. In this method, an unresolved model is obtained in which unresolved scales are incorporated analytically through the bubble functions. Modeling of the unresolved scales corrects the lack of the stability of the standard element-free Galerkin formulation and the resulting stabilized formulation possesses superior properties like that of the streamline upwind/Petrov–Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The method allows equal order basis for pressure and velocity because it violates the celebrated Babuska–Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor τ appears naturally via the solution of the fine-scale problem. Numerical results for example problems confirm that this method has some excellent properties, such as better stability and accuracy.