Adaptive meshless local maximum-entropy finite element method for Navier-Stokes equations
D L Young, C-L Shih, L J Yen, C-R Chu, Adaptive meshless local maximum-entropy finite element method for Navier-Stokes equations, Journal of Mechanics, Volume 40, 2024, Pages 475–490, https://doi.org/10.1093/jom/ufae039
Based on the successful application of the adaptive meshless local maximum-entropy finite element method to solve the convection-diffusion equation, this study extends the same principle to study the 2-dimensional Navier-Stokes equations. Through extensive case studies, this work demonstrates that the present approach is a viable alternative to resolve the high Reynolds number Navier-Stokes equations. The simulation results indicate that by incorporating additional points into the elements without increasing the bandwidth or refinement via the local maximum-entropy procedure, it will enhance the accuracy and efficiency of numerical simulations. A 2-dimensional square lid-driven cavity with various Reynolds numbers will serve as the first example. In the second example, we address a more complex geometry by solving the cavity with a hole inside the cavity center. The numerical results of the model compare favorably with other numerical solutions, including the finite difference method and the finite element method. This paper provides a very powerful tool to study the boundary layer theory with irregular geometry of the Navier-Stokes equations.
When dealing with convection-dominated flow problems, such as high Peclet number flows in advection-diffusion heat transfer problems [ 1, 2] or high Reynolds number flow problems in the Navier-Stokes equation problems [ 3], we typically use upwind schemes to mitigate spurious oscillations and numerical instability. General approaches include the finite difference method (FDM) [ 4], the finite element method (FEM) [ 5], the finite volume method (FVM) [ 6] and meshless method [ 7]. The upwind meshless FEM [ 3], upwind meshless FVM [ 2], meshless GE/BC formulation [ 8] and streamline upwind Petrov-Galerkin FEM [ 9] were found in recent advancements. In this analysis, we will introduce an innovative algorithm to solve high Reynolds number flows without resorting to upwind schemes or mesh refinement procedures.
