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SUMMARYFor the numerical solution of materially non-linear problems like in computational plasticity or viscoplasticity the ÿnite element discretization in space is usually coupled with point-wise deÿned evolution equations characterizing the material behaviour. The interpretation of such systems as di erentialalgebraic equations (DAE) allows modern-day integration algorithms from Numerical Mathematics to be e ciently applied. Especially, the application of diagonally implicit Runge-Kutta methods (DIRK) together with a Multilevel-Newton method preserves the algorithmic structure of current ÿnite 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 Multilevel-Newton algorithm is usually validated by numerical studies. However, an analytical proof of this second order convergence has already been given by authors in the ÿeld of non-linear electrical networks. We show that this proof can be applied in the current context based on the DAE interpretation mentioned above. We ÿnally compare the proposed procedure to several well-known stress algorithms and show that the inclusion of a step-size control based on local error estimations merely requires a small extra time-investment.

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Usually the notion ''Newton-Raphson method'' is used in the context of non-linear finite element analysis based on quasi-static problems in solid mechanics. It is pointed out that this is only true in the case of non-linear elasticity. In the case of constitutive equations of evolutionary-type, like in viscoelasticity, viscoplasticity or elastoplasticity, the ''Multilevel-Newton algorithm'' is usually applied yielding the notions of global and local level (iteration), as well as the consistent tangent operator. In this paper, we investigate the effects of a consistent application of the classical NewtonRaphson method in connection with the finite element method, and compare it with the classical MultilevelNewton algorithm. Furthermore, an improved version of the Multilevel-Newton method is applied.