A Novel Meshless Method Based on the Virtual Construction of Node Control Domains for Porous Flow Problems

Abstract: In this paper, a novel meshless method that can handle porous flow problems with singular source terms is developed by virtually constructing the node control domains. By defining the connectable node cloud, this novel meshless method uses the integral of the diffusion term and generalized difference operators to derive overdetermined equations of the node control volumes. An empirical method of calculating reliable node control volumes and a triangulation-based method to determine the connectable point cloud are developed. NCDMM only focuses on the volume of the node control domain rather than the specific shape, so the construction of node control domains is called virtual, which will not increase the computational cost. To our knowledge, this is the first time to construct node control volumes in the meshless framework, so this novel method is named a node control domains-based meshless method, abbreviated as NCDMM, which can also be regarded as an extended finite volume method (EFVM). Taking two-phase porous flow problems as an example, the NCDMM discrete schemes meeting local mass conservation are derived by integrating the generalized finite difference schemes of governing equations on each node control domain. Finally, existing commonly used low-order finite volume method (FVM) based nonlinear solvers for various porous flow models can be directly employed in the proposed NCDMM, significantly facilitating the general-purpose applications of the NCDMM. Four numerical cases are implemented to test the computational accuracy, efficiency, convergence, and good adaptability to the calculation domain with complex geometry and various boundary conditions.