Remarks on the interpretation of current nonlinear finite element analyses as differential-algebraic equations (Q2745448)

For the numerical solution of materially nonlinear problems like in computational plasticity or viscoplasticity, the finite element (FE) discretization in space is usually coupled with pointwise defined evolution equations characterizing the material behaviour. The interpretation of such systems as differential-algebraic equations (DAE) allows modern-day integration algorithms from numerical mathematics to be efficiently applied. Especially, the application of diagonally implicit Runge-Kutta methods together with a multilevel-Newton (MN) method preserves the algorithmic structure of current FE implementations which are based on the principle of virtual displacements and on backward Euler schemes for local time integration. The quadratical order of convergence of the MN algorithm is usually validated by numerical studies. Here the authors show that an analytical proof can be applied in the current context, based on the DAE interpretation mentioned above. The article concludes with some examples showing the performance of step-size control techniques as well as different integration procedures.