Journal article
Remarks on the Interpretation of Current Nonlinear Finite Element Analyses as DifferentialAlgebraic Equations
Publication Details
Authors:  Ellsiepen, P.; Hartmann, S.

Journal:  International Journal for Numerical Methods in Engineering 
Abstract
For the numerical solution of materially nonlinear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with pointwise defined evolution equations characterizing the material behaviour. The interpretation of such systems as differentialalgebraic equations (DAE) allows modemday integration algorithms from Numerical Mathematics to be efficiently applied. Especially, the application of diagonally implicit RungeKutta methods (DIRK) together with a MultilevelNewton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward Euler schemes for the local time integration. Moreover, the notion of the consistent tangent operator becomes more obvious in this context. The quadratical order of convergence of the MultilevelNewton algorithm is usually validated by numerical studies. However, an analytical proof of this second order convergence has already been given by authors in the field of nonlinear electrical networks. We show that this proof can be applied in the current context based on the DAE interpretation mentioned above. We finally compare the proposed procedure to several wellknown stress algorithms and show that the inclusion of a stepsize control based on local error estimations merely requires a small extra timeinvestment. Copyright (C) 2001 John Wiley & Sons, Ltd.