Unified discrete approach of acceleration conservation Jean-Paul Caltagirone Université de Bordeaux Institut de Mécanique et d’Ingéniérie Département TREFLE, UMR CNRS nř 5295 16 Avenue Pey-Berland, 33607 Pessac Cedex
[email protected] Abstract Discrete mechanics is used to present fluid mechanics, fluid-structure interactions, electromagnetism and optical physics in a coherent theoretical and numerical approach. Acceleration considered as an absolute quantity is written as a sum of two terms, i.e. an irrotational and a divergence-free component corresponding to a formal Hodge-Helmholtz decomposition. The variables of this equation of discrete motion are only the scalar and vector potential of the acceleration, whatever the physical field. These, like the physical properties, are only expressed as a function of two fundamental units, namely a length and a time. The numerical methodology associated with this equation of motion is based on discrete operators, gradient, divergence, primal and dual curl applied to the velocity components of the primal geometric topology. Some solutions resulting from simulations carried out in each domain make it possible to find the results obtained from the Navier-Stokes, Navier-Lamé and Maxwell equations and to show the coherence of the proposed unified approach. Keywords: Discrete Mechanics; Weak Equivalence Principle; Hodge-Helmholtz Decomposition; Navier-Stokes equations; Maxwell equations 1 Introduction The Navier-Stokes, Navier-Lamé and Maxwell equations are fully representative of physical phenomena in the fields of fluid mechanics, solid mechanics and electromagnetism. Continuum mechanics [25], [26] was supposed to unify mechanics but two centuries after their derivation, it turns out that its (Eulerian or Lagrangian) formulations are different; velocity is the variable in fluid media whereas it is displacement in solid media.