Stabilized and Variationally Consistent Integrated Meshfree Formulation for Advection-Dominated Problems

Meshfree methods, such as reproducing kernel particle method (RKPM), have flexibility in controlling local smoothness and approximation basis, which is suitable for modeling various fluid flow problems. However, Eulerian described partial differential equations, such as the advection-diffusion equation or Navier Stokes equation, exhibits numerical instability under conventional Bubnov-Galerkin formulation. Although it is well-known that such instability can be remedied by Petrov-Galerkin formulations such as streamline upwind Petrove Galerkin (SUPG) or variational multiscale method (VMS) [1], immature domain integration could still lead to suboptimal convergence and hourglass-like instability. Such inaccuracy and instability will be amplified under the strong advection and is commonly seen in meshfree formulations [2]. This study proposes a variationally consistent integration method based on the advection-diffusion equation to address the inaccuracy issue [3]. A gradient type of nodal stabilization based on VMS is also developed to enhance the coercivity of the system [3]. The proposed method is proven effective in various advection-dominated fluid flow problems.

Reference

1. Hughes, Thomas JR. "Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods." Computer methods in applied mechanics and engineering 127.1-4 (1995): 387-401.
2. Chen, Jiun-Shyan, Michael Hillman, and Sheng-Wei Chi. "Meshfree methods: progress made after 20 years." Journal of Engineering Mechanics 143.4 (2017): 04017001.
3. Huang, Tsung-Hui. "Stabilized and variationally consistent integrated meshfree formulation for advection-dominated problems." Computer Methods in Applied Mechanics and Engineering 403 (2023): 115698.