Variational-based locking-free energy–momentum schemes of higher-order for thermo-viscoelastic fiber-reinforced continua

Abstract

Locking-free finite elements and energy–momentum time integration schemes are two of the best-known algorithmic improvements of finite element methods. Both are developed since the middle of the eighties of the last century, but usually independently from each other. Therefore, a smart interface between both algorithms is rarely a development goal of the researcher. In this paper, we present such a smart interface, namely the Hu–Washizuprocedure applied to the principle of virtual power. By using of the resulting mixed variational principle, we avoid locking in a spatial finite element discretization of non-isothermal inelastic fiber-reinforced materials, and obtain a family of corresponding higher-order accurate energy–momentum schemes. Thereby, we consider volumetric locking in the matrix material and line locking in the fibers. We show that this reduction of locking in the energy–momentum schemes leads to an increase of the maximum time step size, such that the efficiency of the time integration is improved in the sense that less CPU time is required. This could be shown by using an automatic time step size control with the iteration number of the applied Newton–Raphson scheme as target function. As numerical examples, we consider slender fiber-reinforced structures as a turbine rotor and a lightweight beam consisting of fiber-reinforced trusses. Here, we simulate different combinations of mechanical and thermal Dirichlet and Neumann boundary conditions.