  Most approaches in Lagrangian fluid dynamics simulations proceed from the
definition of particle volumes, from which discrete versions of the spatial
differential operators are derived.
  Recently, Gallou\"et and M\'erigot [1] simultaneously tackled physical
dynamics and geometrical optimization, with the result that the pressure field
is linked to a geometric feature: the weights of a power diagram. Their
resulting dynamics, surprisingly, does not feature a pressure gradient, but
spring-like forces between each particle and the centroid of its cell.
  Inspired by this work, both geometrical and mechanical optimization are here
included within a framework due to Arroyo and Ortiz [2]. In a systematic way,
we first find a connection with the smoothed particle hydrodynamics method. In
what we will call the ``low-temperature limit'', we show that the requirement
of zeroth order consistency leads to the Voronoi diagram, and a pressure field
enforcing incompressibility leads to Gallou\"et and M\'erigot'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 \added{and the Taylor-Green vortex sheet}, showing the
superiority of pFEM, which is slightly increased by the additional force found
here.
