FEBio can perform a nonlinear dynamic analysis by iteratively solving the following nonlinear semi-discrete finite element equations . Here, is the mass matrix, the stiffness matrix, the internal force (stress) vector and the externally applied loads. The upperscript index refers to the iteration number, the subscript refers to the time increment. The trapezoidal (or midpoint) rule is used to perform the time integration. This results in the following approximations for the displacement and velocity updates. Using (6-2) we can solve for , Substituting this into equation (6-1) results in the following linear system of equations. Solving this equation for and using (6-1) gives the new displacement vector . The acceleration vector can then be found from (6-3) and the velocity vector from(6-3). This algorithm is repeated until convergence is reached.