Variational h-adaption in finite deformation elasticity and plasticity
We propose a variational h-adaption strategy in which the evolution of the mesh is driven directly by the governing minimum principle. This minimum principle is the principle of minimum potential energy in the case of elastostatics; and a minimum principle for the incremental static problem of elasto-viscoplasticity. In particular, the mesh is refined locally when the resulting energy or incremental pseudo-energy released exceeds a certain threshold value. In order to avoid global recomputes, we estimate the local energy released by mesh refinement by means of a lower bound obtained by relaxing a local patch of elements. This bound can be computed locally, which reduces the complexity of the refinement algorithm to O(N). We also demonstrate how variational h-refinement can be combined with variational r-refinement to obtain a variational hr-refinement algorithm. Because of the strict variational nature of the h-refinement algorithm, the resulting meshes are anisotropic and outperform other refinement strategies based on aspect ratio or other purely geometrical measures of mesh quality. The versatility and rate of convergence of the resulting approach are illustrated by means of selected numerical examples.
© 2007 John Wiley & Sons. Received 20 February 2006; Revised 8 January 2007; Accepted 8 January 2007. Support from the DoE through Caltech's ASC/ASAP Center for the Simulation of the Dynamic Response of Solids is gratefully acknowledged. J. M. is also grateful for support from the Deutsche Forschungsgemeinschaft (DFG) under contract/grant number: Mo 1389/1-1.