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6.2.3 Generalized Method for Elastodynamics
In the generalized method, we evaluate displacements and velocities at the intermediate time , where is a user-defined parameter ( ), such that where is the motion and is the displacement. In particular, it follows that the deformation gradient and its determinant are given at the intermediate time by and The material time derivative of , and the velocity gradient are normally evaluated as and In practice however, we get better numerical results when using and
According to the generalized method, we evaluate the velocity derivative at a different intermediate time , such that Since elastodynamics represent a second-order system of equations in time, the parameters and are evaluated from a single parameter using  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.
To complete the integration scheme , we evaluate then we use the Newmark integration formulas (Section 6.1↑), At the start of each time step, we initialize the variables as follows: