Theory Manual Version 3.4
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Subsection 3.8.3: Line Search Method Up Chapter 3: The Nonlinear FE Method Chapter 4: Element Library

## 3.9 Generalized Method

The generalized method is used for temporal discretization of governing equations in fluid mechanics. For this method we combine the degrees of freedom into , where the subscript denotes time ; similarly, we let . According to this method [42], the virtual work is evaluated at , where and . Here, 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.
The linearization of reported in Section 3.5.2↑ is effectively performed along an increment of so that the solution to produces . Based on Newmark integration, we have where, according to the generalized method, Therefore, in this scheme, is evaluated from Using (3.9-1) and (3.9-5), we find that Given the solution , the solution at is evaluated from
Four different options are presented in [42] for initializing and at the beginning of time step ; the first three of these have been implemented in FEBio. For steady flows these authors recommend disregarding and setting to recover the backward Euler scheme.
Subsection 3.8.3: Line Search Method Up Chapter 3: The Nonlinear FE Method Chapter 4: Element Library