A unified derivation of Voronoi, power, and finite-element Lagrangian computational fluid dynamics

Abstract

Lagrangian fluid dynamics simulations often start from a definition of the volume of a particle, from which discrete versions of the spatial differential operators are derived. Recently, Gallouët and Mérigot (2018) tackled simultaneously physical dynamics and geometrical optimization, with the result that the pressure field is linked to a geometric feature: the weights of a power diagram.Inspired by this work, simultaneous geometrical and mechanical optimization is here considered, within a framework due to Arroyo and Ortiz (2006). In a systematic way, we first find a connection with the smoothed particle hydrodynamics method. In what we will call the “low-temperature limit”, the requirement of zeroth order consistency leads to the Voronoi diagram, and incompressibility leads to Gallouët and Mérigot’s method.If the requirement of first order consistency is added, the particle finite element method (pFEM) is recovered. However, it features an additional spring-like term that has been missing from previous formulations of the method. Different methods are tested on two standard inviscid single-phase cases: the rotating Gresho vortex and the Taylor–Green vortex sheet, showing the superiority of pFEM, which is slightly increased by the additional force found here.