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Explicit Time Integration

The explicit time integration scheme can be written as

vi,t+Δt2=vi,tΔt2+ai,tΔt{v_{i,t + \frac{\Delta t}{2}} = v_{i,t - \frac{\Delta t}{2}} + a_{i,t}\Delta t }
ui,t+Δt=ui,t+vi,t+Δt2Δt{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).

fi=fi,app+fi,intf_{i} = f_{i,app} + f_{i,int}

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

{fint}=[K]{ut}\left\{ f_{int} \right\} = \lbrack K\rbrack\left\{ u_{t} \right\}

and if Rayleigh damping is included

{fint}=[K]({ut}+β{vt})+α[M]{vt}\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\}