Explicit time integration

The accelerations at the beginning of the current increment (time $t$) are calculated as

Since the explicit procedure always uses a diagonal, or lumped, mass matrix, solving for the accelerations is trivial; there are no simultaneous equations to solve. The acceleration of any node is determined completely by its mass and the net force acting on it, making the nodal calculations very inexpensive.

The accelerations are integrated through time using the central difference rule, which calculates the change in velocity assuming that the acceleration is constant. This change in velocity is added to the velocity from the middle of the previous increment to determine the velocities at the middle of the current increment:

The velocities are integrated through time and added to the displacements at the beginning of the increment to determine the displacements at the end of the increment:

Thus, satisfying dynamic equilibrium at the beginning of the increment provides the accelerations. Knowing the accelerations, the velocities and displacements are advanced explicitly through time. The term explicit refers to the fact that the state at the end of the increment is based solely on the displacements, velocities, and accelerations at the beginning of the increment. This method integrates constant accelerations exactly. For the method to produce accurate results, the time increments must be quite small so that the accelerations are nearly constant during an increment. Since the time increments must be small, analyses typically require many thousands of increments. Fortunately, each increment is inexpensive because there are no simultaneous equations to solve. Most of the computational expense lies in the element calculations to determine the internal forces of the elements acting on the nodes. The element calculations include determining element strains and applying material constitutive relationships (the element stiffness) to determine element stresses and, consequently, internal forces.

Here is a summary of the explicit dynamics algorithm:

1. Nodal calculations.

   1. Dynamic equilibrium.

      $${\ddot{\mathbf{u}}}_{\left(t\right)}={\mathbf{M}}^{-1}\left({\mathbf{P}}_{\left(t\right)}-{\mathbf{I}}_{\left(t\right)}\right)$$

   2. Integrate explicitly through time.

      $${\dot{\mathbf{u}}}_{\left(t+\frac{\mathrm{\Delta }t}{2}\right)}={\dot{\mathbf{u}}}_{\left(t-\frac{\mathrm{\Delta }t}{2}\right)}+\frac{\left(\mathrm{\Delta }{t}_{\left(t+\mathrm{\Delta }t\right)}+\mathrm{\Delta }{t}_{\left(t\right)}\right)}{2} {\ddot{\mathbf{u}}}_t$$
      $${\mathbf{u}}_{\left(t+\mathrm{\Delta }t\right)}={\mathbf{u}}_{\left(t\right)}+\mathrm{\Delta }{t}_{\left(t+\mathrm{\Delta }t\right)}{\dot{\mathbf{u}}}_{\left(t+\frac{\mathrm{\Delta }t}{2}\right)}.$$

2. Element calculations.

   1. Compute element strain increments, $\mathrm{\Delta }\mathit{\epsilon }$, from the strain rate, $\dot{\mathit{\epsilon }}$.

   2. Compute stresses, $\mathit{\sigma }$, from constitutive equations.

      $${\mathit{\sigma }}_{\left(t+\mathrm{\Delta }t\right)}=f\left({\mathit{\sigma }}_{\left(t\right)},\mathrm{\Delta }\mathit{\epsilon }\right)$$

   3. Assemble nodal internal forces, ${\mathbf{I}}_{\left(t+\mathrm{\Delta }t\right)}$.

3. Set $t+\mathrm{\Delta }t$ to $t$ and return to Step 1.