A monolithic finite element approach using high-order schemes in time and space applied to finite strain thermo-viscoelasticity

This article addresses a thermo-mechanically coupled problem of thermo-viscoelasticity at finite strains using a monolithic approach. The underlying equations are based on the non-linear transient heat equation, the local equilibrium conditions and the evolution equations of the internal variables. The latter describe the hardening behavior of the material. If the method of vertical lines is applied, its first step – namely the spatial discretization – yields a system of differential-algebraic equations (DAE-system). Here, we employ the p-version of the finite element method based on integrated Legendre polynomials. This can lead to very precise solutions in the spatial domain. In order to be accurate in the time-domain as well, stiffly accurate, diagonally-implicit Runge-Kutta methods are applied to solve the DAE-system yielding a coupled system of non-linear algebraic equations. In this article, the system is solved monolithically by employing the Multilevel-Newton algorithm. Accordingly, a high-order result is obtained in the space and the time domain. The numerical concept is applied to a constitutive model of finite strain thermo-viscoelasticity. Several examples are applied to demonstrate the efficiency and applicability of the numerical scheme. It is especially the transient problems that call for time-adaptive schemes which are naturally embedded in the concept.