A novel meshless method based on the virtual construction of node control domains for porous flow problems

In this paper, a novel meshless method that can handle porous flow problems with singular source terms is developed by virtually constructing node control domains. By defining a connectable node cloud, this method uses the integral of the diffusion term and generalized finite-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. The method focuses only on the volume of the node control domain rather than the specific shape, so the construction of node control domains is virtual and does not increase the computational cost. To our knowledge, this is the first study to construct node control volumes in the meshless framework, termed the node control domain-based meshless method (NCDMM), which can also be regarded as an extended finite-volume method (EFVM). Taking two-phase porous flow problems as an example, discrete NCDMM schemes satisfying local mass conservation are derived by integrating the generalized finite-difference schemes of governing equations on each node control domain. Finally, 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 general-purpose application. Theoretically, the proposed NCDMM has the advantages of previous meshless methods for discretizing computational domains with complex geometries, as well as the advantages of traditional low-order FVMs for stably handling a variety of porous flow problems with local mass conservation. 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.