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3.6.2 Time Integration
In the generalized method we evaluate displacements and velocities at the intermediate time , such that In particular, it follows that In practice however, we get better numerical results when using Similarly, we evaluate velocity derivatives at the intermediate time , such that The parameters and are evaluated from a single parameter using for first-order systems, or for second-order systems, where . This parameter is the spectral radius for an infinite time step, which controls the amount of damping of high frequencies; a value of zero produces the greatest amount of damping, anihilating the highest frequency in one step, whereas a value of one preserves the highest frequency. Since the solid motion is governed by a second-order differential equation in time, we adopt the formulas for second-order systems.
To complete the integration scheme, we evaluate then Thus, According to this method , the virtual work is evaluated using intermediate time step values, at for all parameters except , , and , which are evaluated at .