A meshfree radial basis function method for simulation of multi-dimensional conservation problems

Many computational fluid dynamics problems utilize finite volume frameworks for simulation, due to the simplifications provided by conservative formulation of the driving partial differential equations (PDEs). However, fluid dynamics applications can often involve temporal shifts in the domain structure-such as moving boundaries, or pore structure changes-requiring mesh adaptation throughout computation. These mesh adaptations often render classical numerical methods such as the finite volume method infeasible, due to their reliance on a well-defined static mesh structure. This limitation has led to the development of a wide variety of meshless methods-techniques that can simulate PDEs without requiring a rigid connective structure between nodes. However, most meshless methods are typically based on finite element or finite difference formulations, and the limited number of meshless finite volume methods (MFVMs) either introduce a weak background mesh, or use weak-form approximations that do not take full advantage of the strong conservative form of the driving equations. Addressing this gap within this study we outline a meshfree numerical scheme for simulation of partial different equations, based on strong-form finite volume style formulations. Building upon the previously developed MFVM, this technique uses radial basis functions to interpolate the problem domain, and approximate fluxes in a disjoint finite volume scheme, removing reliance on a mesh structure. We present method derivation, including promising new techniques for enforcing boundary conditions in a meshless environment. Following this we discuss method accuracy and computational performance across a variety of problems in two and three dimensions. We then illustrate how this method may prove beneficial for applications in porous media modeling, and computational fluid dynamics. For completeness, we provide a sensitivity analysis of the method hyper-parameters and investigate the conservative properties of the method. We also illustrate similarities of this approach to the widely used meshless point collocation methods. We close with a discussion of the strengths, limitations, and broader applicability of the technique.

Automated summary

This study proposes a meshfree numerical scheme based on strong-form finite volume style formulations. The technique uses radial basis functions to interpolate the problem domain and approximate fluxes in a disjoint finite volume scheme, eliminating the reliance on a mesh structure. The method shows potential for applications in porous media modeling and computational fluid dynamics.