Unified discrete approach of acceleration conservation

TL;DR

This paper presents a unified discrete framework for modeling various physical phenomena, including fluid mechanics, electromagnetism, and optics, using acceleration decomposition and discrete operators for coherent numerical simulations.

Contribution

It introduces a novel discrete approach that unifies multiple physical theories through acceleration decomposition and discrete operators, enabling consistent numerical solutions.

Findings

Simulations recover classical equations like Navier-Stokes and Maxwell.

The approach demonstrates coherence across different physical domains.

Numerical results align with established theoretical models.

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-Lam{\'e} and Maxwell equations and to show the coherence of the proposed unified approach.