A two-level variational multiscale meshless local Petrov-Galerkin (VMS-MLPG) method is presented for incompressible Navier-Stokes equations based on two local Gauss integrations which effectively replace a unit operator (first level) and an orthogonal project operator (second level). The present VMS-MLPG method allows arbitrary combinations of interpolation functions for the velocity and pressure fields, specifically the equal-order interpolations that are easy to implement and satisfy the Babuska-Breezi (B-B) condition. The prediction accuracy and the numerical stability of the proposed method for the lid-driven cavity flow and the backward facing step flow problems are analyzed and validated by comparing with the SUMLPG method and the benchmark solutions. It is shown that the present VMS-MLPG method can guarantee the numerical stability and obtain the reasonable solutions for incompressible Navier-Stokes equation.

The two-dimensional steady-state heat conduction in irregular geometry is solved by a MLPG–FVM coupled method. The meshless local Petrove–Galerkin (MLPG) method is applied to the sub-region with skewed wall surface while the finite volume method (FVM) is used in the rest of the domain. The Dirichlet–Dirichlet method is adopted to couple the temperature between MLPG and FVM methods. In MLPG method, the Dirac's Delta function is taken as the test function to avoid the local domain integration which does not need the numerical integration and the solution is independent of the size of the test function. The proposed MLPG–FVM method is validated and proved to be an efficient numerical method for 2-D heat conduction in irregular geometry, which can exert their own advantages of MLPG and FVM.

A meshless local Petrov-Galerkin (MLPG) approach based on the streamline upwind (SU) idea and the variational multiscale (VMS) concept, called as VMS-SUMLPG method, is herein proposed to solve the convection-dominated problems. In the present VMS-SUMLPG method, the streamline upwind is constructed in the test function to solve the non-self-adjoint matrix. Meanwhile, the VMS concept as a stability term is adopted to alleviate the numerical instability such as spurious oscillations, overshoots, and undershoots. Its numerical accuracy and stability are validated by comparing with the streamline upwind Petrov-Galerkin (SUMLPG) method and the finite volume method with high-order difference schemes for two classical convection-dominated problems at the Peclet number ranging from 10(6) to 10(8). It is shown that the numerical solutions of the present VMS-SUMLPG method are accuracy, smoothness, and stability.

The numerical oscillation will occur in convection-diffusion equations as special linear problems at large Peclet number (Pe) in the numerical calculation process. In this article, we propose a new definition of the stability parameter in streamline upwind meshless Petrov-Galerkin (SUMLPG) method. The most important feature of the proposed method is that the test function in the stabilization term is taken into the differential operator-like form v* = v + tau L(adv)v. The stability parameter tau is designed to adjust the convection strength to achieve accurate and stable numerical solutions. Several classical examples are adopted to assessment the accuracy and stability of the proposed stability parameter. It is proven that the proposed method is especially suitable for convection-diffusion problems with large Pe.

It is challengeable to obtain the stable and accurate solutions of convection-diffusion problems with large Peclet number (Pe) since the convection term may cause oscillation solutions at large Pe. In this paper, a unit operator (first level) and an orthogonal project operator (second level) are constructed to act as the stability terms for meshless local Petrov-Galerkin (MLPG) method, which is called a two-level variational multiscale MLPG (VMS-MLPG) method. The VMS-MLPG method is applied to eliminate oscillation, overshoots and undershoots of MLPG method at large Pe. The prediction accuracy and the numerical stability of the proposed method for the Smith-Hutton and the Brezzi problems are analyzed and validated by comparing with the MLPG method and the finite volume method (FVM) with various difference schemes. It is showed that the present VMS-MLPG method can guarantee the stable and reasonable solutions of convection-diffusion problems with large Peclet number.

It is a difficult to solve convection-diffusion equations with large Pelect number(Pe).The stiff ratio is too large for the stiff matrix to be self-adjoint,which causes streamwise false diffusion or serious overshoots and undershoots in numerical results.In this paper,we introduce an upwind scheme and a variational multiscale method to improve the stiff matrix,and validate the reliability through the classical rotation field problem.

In this paper, the Diffusion Limited Cluster Aggregation (DLCA) method is employed to reconstruct the three-dimensional network of silica aerogel. Then, simulation of nitrogen adsorption at 77 K in silica aerogel is conducted by the Grand Canonical Monte Carlo (GCMC) method. To reduce the computational cost and guarantee accuracy, a continuous-discrete hybrid potential model, as well as an adsorbed layer thickness estimation method, is employed. Four different structures are generated to investigate impacts of specific surface area and porosity on adsorptive capacity. Good agreement with experimental results is found over a wide range of relative pressures, which proves the validity of the model. Specific surface area and porosity mainly affect nitrogen uptake under low pressure and high pressure, respectively.