No abstract

No abstract

Implicit Runge-Kutta methods for the dual problem of elastoplasticity are analyzed and classified. The choice of Runge-Kutta time integration is inspired by the problem structure, which consists of a coupled system of balance equations and unilaterally constrained evolution equations and which can be viewed as an infinite-dimensional differential-algebraic equation. Focussing on the time axis and leaving the space variables continuous, a grid-independent existence and uniqueness result is given for the class of coercive Runge-Kutta methods. Moreover, contractivity preservation and convergence are shown for methods that are also algebraically stable. Introduction.We analyze implicit Runge-Kutta methods for an infinitedimensional system of constrained evolution equations, the so-called dual problem of elastoplasticity. Our approach discretizes only the time axis and leaves the space variables continuous. In this way, we obtain a grid-independent formulation and lay the foundation for an implementation in the fashion of Rothe's method. The main results are first existence and uniqueness of the numerical solution for coercive RungeKutta methods, second contractivity preservation for algebraically stable methods, and third convergence for methods that feature both properties as well as a certain stage order.Elastoplastic models are used for materials where the deformation process shows a time-dependent and irreversible behavior. Applications comprise, e.g., the stretch formation of metal sheets, wear effects in turbine blades, and the behavior of micromechanical devices. The mathematical model consists of a coupled system of balance equations and unilaterally constrained evolution equations, where the first set of equations stands for the balance of momentum and the second for the properties of the material under consideration [16]. We consider here materials that satisfy the principle of maximum plastic dissipation and possess a quadratic internal free energy.In elastoplasticity, numerical simulation is mostly based on the method of lines [15,16]. More precisely, one discretizes the balance equations by the finite element method (FEM) and reduces at the same time the evolution equations to the quadrature nodes of the grid. In these nodes, the evolution is integrated in time, with the implicit Euler as standard method and the return-mapping scheme for the handling of the unilateral constraint. Due to this constraint, the so-called yield condition, one deals actually with a differential-algebraic equation (DAE) of index 2 [8,12].Numerical experiments [4,7,18] indicate that at least order 2 is possible for

The equations of rate-independent elastoplasticity form a differential-algebraic equation (DAE) with discontinuities. For the numerical solution, implicit Runge-Kutta methods are applied and combined with the return mapping strategy of computational plasticity. It turns out that the convergence order depends crucially on the switching point detection. Further, it is shown that algebraically stable Runge-Kutta methods preserve the contractivity of the elastoplastic flow.