An ALE formulation based on spatial and material settings of continuum mechanics. Part 2: Classification and applications Harm Askes a , Ellen Kuhl b , Paul Steinmann b, * a Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, NL-2600 GA, Delft, The Netherlands b Faculty of Mechanical Engineering, University of Kaiserslautern, P.O. Box 3049, D-67653 Kaiserslautern, Germany Received 16 May 2003; received in revised form 29 August 2003; accepted 4 September 2003 Abstract In the second paper of this two-part contribution, a specialisation towards Neo–Hookean material will be made of the generic hyperelastic arbitrary Lagrangian–Eulerian (ALE) formulation derived in Part 1. First, for the sake of comparison and classification, several existing ALE solution schemes are discussed, including total and updated ap- proaches, as well as monolithic and staggered algorithms. Then, implementational details are provided for the newly proposed ALE strategy. The versality and the limitations of the present formulation are shown by means of a set of one- dimensional and multi-dimensional numerical examples. In particular, it is shown that with the proposed ALE for- mulation, potential energies can be obtained that are minimum for the considered topology. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Arbitrary Lagrangian–Eulerian formulation; Spatial and material settings; Variational principle; Spatial and material forces; Finite element method 1. Introduction In the first part of this contribution [1], an arbitrary Lagrangian–Eulerian (ALE) formulation was de- rived which is based on the simultaneous solution of the spatial motion problem and the material motion problem. Whereas the notion of the spatial motion problem is well-established (normally known simply as the standard equilibrium equation or the standard equations of motion), the notion of the material motion problem is less obvious. The material motion problem is concerned with the equilibrium of material forces, whereby material forces arise as a result of material inhomogeneity, such as defects, voids, (micro-)cracks [2– 7]. However, material forces also arise as a consequence of discretisation, as has been exemplified in [8–12]. In fact, on a discretised level there is no difference whether a material force results from a material * Corresponding author. E-mail addresses: h.askes@citg.tudelft.nl (H. Askes), ekuhl@rhrk.uni-kl.de (E. Kuhl), ps@rhrk.uni-kl.de (P. Steinmann). 0045-7825/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2003.09.031 Comput. Methods Appl. Mech. Engrg. 193 (2004) 4223–4245 www.elsevier.com/locate/cma