Velocity and displacement update

The velocity is updated with the damped acceleration, but subject to the prescribed velocity boundary conditions, that is:

$${\mathbf{u}}_a^{t+\Delta t}={\mathbf{u}}_a^t+\Delta t{\mathbf{a}}_a^{t+\Delta t}.$$

The position ${\mathbf{x}}_a$ of the node $a$ is updated by:

$${\mathbf{x}}_a^{t+\Delta t}={\mathbf{x}}_a^t+\Delta t{\mathbf{u}}_a^{t+\Delta t}.$$

Since the mesh is changed, the shape function derivates $N_a^e$ and the element volume ${\Omega }_e$ are updated every time step.