Finite strain viscoelasticity: how to consistently couple discretizations in time and space on quadrature-point level for full order $$p \ge 2$$ and a considerable speed-up

Abstract

In computational viscoelasticity, the spatial finite element discretization for the global solution of the weak form of the balance of momentum is coupled to the temporal discretization for solving local initial value problems (IVP) of viscoelastic flow. In this contribution we show that this global-local or space-time coupling is consistent, if the total strain tensor as the coupling quantity exhibits the same approximation order $$p$$ in time as the Runge---Kutta (RK) integration algorithm. To this end we construct interpolation polynomials, based on data at $$t_{n+1}$$ , $$t_{n}$$ , $$\ldots $$ , $$t_{n+2-p}$$ , $$p\ge 2$$ , which provide consistent strain data at RK stages. This is a generalization of the idea proposed in (Eidel and Kuhn, Int J Numer Methods Eng 87(11):1046---1073, 2011). For lower-order strain interpolation, time integration exhibits order reduction and therefore low efficiency. For consistent strain interpolation, the adapted RK methods up to $$p=4$$ obtain full convergence order and thus approve the novel concept of consistency. High speed-up factors substantiate the improved efficiency compared with Backward-Euler.