2013 Articles

Lie-group interpolation and variational recovery for internal variables

Mota, Alejandro; Sun, WaiChing; Ostien, Jakob T.; Long, Kevin N.; Foulk, James W.

We propose a variational procedure for the recovery of internal variables, in effect extending them from integration points to the entire domain. The objective is to perform the recovery with minimum error and at the same time guarantee that the internal variables remain in their admissible spaces. The minimization of the error is achieved by a three-field finite element formulation. The fields in the formulation are the deformation mapping, the target or mapped internal variables and a Lagrange multiplier that enforces the equality between the source and target internal variables. This formulation leads to an L2 projection that minimizes the distance between the source and target internal variables as measured in the L2 norm of the internal variable space. To ensure that the target internal variables remain in their original space, their interpolation is performed by recourse to Lie groups, which allows for direct polynomial interpolation of the corresponding Lie algebras by means of the logarithmic map. Once the Lie algebras are interpolated, the mapped variables are recovered by the exponential map, thus guaranteeing that they remain in the appropriate space.