Computational Mechanics

, Volume 52, Issue 6, pp 1281–1299 | Cite as

Lie-group interpolation and variational recovery for internal variables

  • Alejandro MotaEmail author
  • WaiChing Sun
  • Jakob T. Ostien
  • James W. FoulkIII
  • Kevin N. Long
Original Paper


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 \(L_2\) projection that minimizes the distance between the source and target internal variables as measured in the \(L_2\) 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.


Variational methods Internal variables Recovery  Interpolation Lie groups 



Support for this work was received through the U.S. Department of Energy’s (DOE) Advanced Simulation and Computing (ASC) Program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. This work was also supported in part by the Joint U.S. Department of Defense (DoD)/DOE Munitions Technology Development Program.


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Copyright information

© Springer-Verlag Berlin Heidelberg (outside the USA) 2013

Authors and Affiliations

  • Alejandro Mota
    • 1
    Email author
  • WaiChing Sun
    • 1
  • Jakob T. Ostien
    • 1
  • James W. FoulkIII
    • 1
  • Kevin N. Long
    • 2
  1. 1.Mechanics of Materials DepartmentSandia National LaboratoriesLivermoreUSA
  2. 2.Solid Mechanics DepartmentSandia National LaboratoriesAlbuquerqueUSA

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