This contribution aims at providing the formulation and implementation details of arbitrary Lagrangian Eulerian hyperelastodynamic problem classes. This ALE formulation is based on the dual balance of momentum in terms of spatial forces (the well-known Newtonian forces) as well as material forces (also known as configurational forces). The balance of spatial momentum results in the usual equation of motion, whereas the balance of the material momentum indicates deficiencies in the nodal positions, hence providing an objective criterion to optimize the finite element mesh. The main difference with traditional ALE approaches is that the combination of the Lagrangian and Eulerian description is no longer arbitrary, in other words the mesh motion is no longer user defined but completely embedded within the formulation.
The present work aims at developing spatial and material variational equations based on the Hamiltonian principle. These equations will be discretised to obtain the weak form of the momentum and continuity equations. The discretized ALE Hamiltonian equations of the spatial motion problem introduces the balance of the discretised spatial momentum and the discretised spatial continuity equation while the corresponding material motion problem defines the balance of the discretised material (or configurational) momentum and the discretised material continuity equation. We will deal with two systems of partial differential equations: the scalar continuity equation and the vector momentum equation. The momentum and continuity equations will then be linearised. The time integration of both the spatial and the material equations is performed with the Newmark scheme. A monolithic solution strategy solving the spatial and the material momentum equations simultaneously has been carried out while updating of the spatial and the material densities was attained through solving the spatial and material continuity equations (mass conservation). The solution defines the optimal spatial and material configuration in the context of energy minimization.