Discrete Hamilton's equations for arbitrary Lagrangian–Eulerian dynamics of viscous compressible flow

Abstract

A number of different arbitrary Lagrangian–Eulerian (ALE) formulations of continuum fluid and solid dynamics problems have been developed, to address applications where more conventional Lagrangian or Eulerian modeling techniques are difficult to apply. In general these ALE formulations are based on finite difference or weighted residual finite element solutions of the partial differential equations for the system. An alternative, energy based ALE model for fluid dynamics simulations may be obtained, by direct application of Hamilton's canonical equations to a finite element discretization of an open, deforming control volume. Formulated in terms of convected coordinates and incorporating an adaptive mesh scheme, this modeling approach yields a simple but general description of viscous compressible flows. Numerical application of the method demonstrates accurate results in the solution of several shock problems, whether the calculations are performed using a Lagrangian, an Eulerian, or an ALE mesh.