Efficient implementation of energy conservation for higher order finite elements with variational integrators

In this paper, we introduce a variational integrator for higher order finite element models. The solution quality of finite element simulation strongly depends on the approximation in time and space. A variational integrator is robust due to conservation of both the balance of total linear momentum and the balance of total angular momentum. It also binds the error in the balance of total energy. We extend the variational integrator to an integrator that conserves the balance of total energy as well. However, the approximation in time cannot mend the approximation error in space. Therefore, we use a higher order approximation in space to obtain a better solution compared to the real live model that should be simulated. We show different numerical examples with Dirichlet and Neumann boundaries. For the Dirichlet boundaries we use the Lagrange multiplier method and for the Neumann boundaries we introduce the forces on the nodes by constant pressure.