Implicit Generalized-α Method

This family of methods was initially developed by Chung and Hilbert for the resolution of dynamics in the context of computational mechanics of solids. Cardona and Géradin adapted the method to compute the dynamics of multibody systems. Many extensions have been developed in the past, such as the extension developed by O. Brüls and M. Arnold for dynamics equations formulated as an index-3 DAE. The dynamics is written at time as:

(5–52)

The acceleration-like variable a_n is defined by the recurrence relation as:

(5–53)

At the beginning of the simulation, this variable is initialized as . The following difference equations relate , and :

(5–54)

where the constants of , , , and are suitably chosen so that the scheme is stable. The algorithm is unconditionally stable if the coefficients are chosen such that for ρ_∞<1,

(5–55)

The scheme is based on a prediction step and a correction step where some Newton iterations are performed in order to solve the dynamical and the constraint residuals R_q and R_λ, defined by:

(5–56)

The Newton iterations solve the following linear system:

(5–57)

where is the inertia matrix, is the stiffness matrix, and is the damping matrix, and k denotes the number of the Newton iterations. Note that this algorithm maintains the constraints at the position level, but it can also be reformulated to write the constraints at the velocity level or the acceleration level.

When considering the constraints at the velocity level, the problem is formulated as an index-2 DAE, whose discretization is given by the first two equations of Equation 5–54, as well as:

(5–58)

In this case, the correction step is performed using Newton iterations to solve the following linear system:

(5–59)

where:

The integration error is estimated using the methodology proposed by Géradin and Cardona for the HHT schemes family. The exact value of the positions vector can be approximated using a truncated Taylor series development around t_n as follows:

(5–60)

The integration error is computed as:

(5–61)

By substituting the expression q_n+1 from Equation 5–54 into Equation 5–61, we obtain:

(5–62)

The third derivative of the position can be approximated by:

(5–63)

By substituting a_n+1 and into Equation 5–62, and by using the relations in Equation 5–54, we obtain:

(5–64)

Adaptive Time Stepping (for implicit time integration)

Much like RK4, the Implicit Generalized-α method also implements an adaptive time stepping strategy based on the energy conservation. However, the order of the Implicit Generalized-α method is lower than RK4. Consequently, the Implicit Generalized-α method may require smaller time steps than explicit method to achieve the same accuracy.

Unlike RK4, the Implicit Generalized-α method is unconditionally stable. This method remains stable even for large time steps, regardless the accuracy. As such, it is particularly suitable to address situations where the explicit RK4 method requires small time steps. The energy tolerance control can be relaxed in these situations, and the time step is driven only by the Newton-Raphson convergence. If a large number of iterations was required for the last time step, the time step decreases. Conversely, if fewer iterations were required for convergence of the previous time step, the time step increases.