Discrete Lagrange equations for reacting thermofluid dynamics in arbitrary Lagrangian–Eulerian frames

Abstract

Interest in the development of multiscale methods has focused attention on the marked differences in the numerical modeling techniques typically applied at different scales. Most continuum dynamics models construct approximate solutions to partial differential equations, while most nanoscale models employ a discrete Hamiltonian approach. Previous research has demonstrated that the introduction of entropies or internal energies as generalized coordinates, along with separation of the discretization and model formulation steps, allows general thermomechanical models to be developed, at the continuum scale, using a nonholonomic Hamiltonian or Lagrange equation formulation. With the introduction of additional state variables and nonholonomic constraints, the latter work may be further extended in order to model reacting systems. Employing a finite element interpolation and a Lagrangian, an Eulerian, or an ALE mesh, the new formulation has been validated by solving several one-dimensional reacting shock physics problems. The result is a continuum dynamics modeling approach highly compatible with the discrete energy methods normally used at the nanoscale.