Explicit Time Integration

The explicit time integration scheme can be written as

$$a_{i,t}=\frac{f_{i,t}}{m_i}$$ $${v_{i,t + \frac{\Delta t}{2}} = v_{i,t - \frac{\Delta t}{2}} + a_{i,t}\Delta t }$$ $${u_{i,t + \Delta t} = u_{i,t} + v_{i,t + \frac{\Delta t}{2}}\Delta t}$$

The force vector at time t is the sum of all the forces acting on the nodes (degrees of freedom).

$$f_{i} = f_{i,app} + f_{i,int}$$

For an element the internal force vector for linear and geometrically nonlinear problems is calculated from

$$\left\{ f_{int} \right\} = \lbrack K\rbrack\left\{ u_{t} \right\}$$

and if Rayleigh damping is included

$$\left\{ f_{int} \right\} = \lbrack K\rbrack\left( \left\{ u_{t} \right\} + \beta\left\{ v_{t} \right\} \right) + \alpha\lbrack M\rbrack\left\{ v_{t} \right\}$$