An ALE formulation based on spatial and material settings of continuum mechanics. Part 1: Generic hyperelastic formulation Original Research Article

The present contribution deals with the derivation of a generic hyperelastic Arbitrary Lagrangian–Eulerian formulation on the basis of a consistent variational framework. The governing equations follow straightforwardly from the Dirichlet principle for conservative mechanical systems. Thereby, the key idea is the reformulation of the total variation of the potential energy at fixed referential coordinates in terms of its variation at fixed material and at fixed spatial coordinates. The corresponding Euler–Lagrange equations define the spatial and the material motion version of the balance of linear momentum, i.e. the balance of spatial and material forces, in a consistent dual format. In the discretised setting, the governing equations are solved simultaneously rendering the spatial and the material configuration which minimise the overall potential energy of the system. The remeshing strategy of the ALE formulation is thus no longer user-defined but objective in the sense of energy minimisation. If the governing equations are derived from a potential, i.e. either from an incremental potential or from a total potential as in the present case, they are inherently symmetric, both in the continuous case and in the discrete case. This symmetry property is particularly appealing since it ensures symmetric system matrices upon discretisation.