Convergence of different time integration methods in viscoelasticity. E: Backward Euler, R2l: Radau IIa with linear interpolation, R2q: Radau IIa with quadratic interpolation. Note, that only the novel R2q method shows order 3, i.e. full order of convergence, whereas R2l shows order reduction.

Associated people

B. Eidel, C. Kuhn (LTM, Uni Kaiserslautern), J. Schröder

Abstract

Time integration is the numerical kernel of inelastic finite element calculations, which largely determines their accuracy and efficiency. If higher order Runge–Kutta (RK) methods, p>2, are used for integration in a standard manner, they do not achieve full convergence order but fall back to second-order convergence. This deficiency called order reduction is a longstanding problem in computational inelasticity. We analyze it for viscoelasticity, where the evolution equations follow ordinary differential equations (ODE). We focus on RK methods of third order. We prove that the reason for order reduction is the (standard) linear interpolation of strain to construct data at the RK-stages within the considered time interval. We prove that quadratic interpolation of strain based on t_(n), t_(n+1) and, additionally, t_(n-1) data implies consistency order three for total strain, viscoelastic strain and stress. Simulations applying the novel interpolation technique are in perfect agreement with the theoretical predictions. The present methodology is advantageous, since it preserves the common, staggered structure of finite element codes for inelastic stress calculation. Furthermore, it is easy to implement, the overhead of additional history data is small and the computation time to obtain a defined accuracy is considerably reduced compared with backward Euler. 

Ongoing research work has been focused on the more delicate problem of order reduction in elasto-plasticity. This broad class of inelastic continuum constitutive laws is usually described by ordinary differential equations for the evolution of plastic flow, which is subject to the yield condition as an algebraic constraint, thus forming altogether a set of differential algebraic equations (DAE).

References

Eidel, B. & Kuhn, C. (2011), "Order reduction in computational inelasticity: why it happens and how to overcome it - the ODE-case of viscoelasticity", International Journal for Numerical Methods in Engineering. Vol. accepted for publication.