To relax this constraint, we present asynchronous multi-domain variational time integrators together with a dual domain decomposition method for a hyperbolic initial-boundary value problem in hyperelasticity. We construct variational time integration schemes based on the principle of least action within a Lagrangian framework for the equation of motion, implemented in a variational finite element framework systematically derived from the three-field de Veubeke-Hu-Washizu variational principle to accommodate nearly-incompressible materials such as solid propellants. For efficient parallel computing, we use a dual domain decomposition method with local Lagrange multipliers to ensure continuity of the displacement field at the interface between subdomains. The α-method for time discretization and multi-domain spatial decomposition enable us to use different types of integrators (explicit or implicit) and different time steps for different parts of the computational domain, and thus efficiently capture the underlying physics with less computational effort. We outline preliminary findings on the stability and accuracy of our nonlinear system, investigated by means of Lyapunov stability.
We illustrate the performance of our variational multi-domain time integrators by means of two examples. First, the method of manufactured solutions is used to examine the consistency of the formulation. In the second example, we apply our method to the motion of a heterogeneous plane domain, where different integrators and time discretization steps are used accordingly, with disparate material data for individual parts. Finally, we discuss recent progress and future prospects.