Theory |
 
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Theory |
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The basic finite element equation of motion is:
(1)
where 
, 
, and 
are the mass, damping, and stiffness matrices, respectively, 
is the vector of applied loads, and 
is the vector of displacement unknowns.
Time integration algorithms such as Generalised-α and Hilber-Hughes-Taylor have the following common form. The vectors 
, 
, and 
are given approximations to 
, 
and 
, respectively. Expressions for 
and 
are specified as linear combinations of 
, 
, 
and 
. Algorithms having this form may be classified as one-step, three-stage (or three-level) time integration methods. The algorithms are one-step methods because the solution at time 
depends only on the solution history at time 
. The three-stage designation refers to the solution being described by the three solution vectors: 
, 
, and 
.
The basic form of the generalized-a method is given by:
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
The structure of the displacement (2) and velocity (3) update equations above is obtained by restricting the sum of the coefficients of their acceleration terms to equal the coefficient of the acceleration term in a Taylor series expansion of 
and 
about 
. Simple numerical experiments have shown that this update equation structure results in a monotone increase per period in the peak displacement and velocity errors. The modified balance equation, (4), is effectively a combination of the Hilber-Hughes-Taylor and Wood et al.(1981) balance equations. With appropriate expressions for γ and β, if αm = 0, the algorithm reduces to the Hilber-Hughes-Taylor method. The generalized-α method is second-order accurate, provided:
(10)
High-frequency dissipation is maximized when:
(11)
You are referred to publications on the Generalised-α method by Chung, and Hulbert (1993), and the Hilber-Hughes-Taylor method by Hilber et al. (1977), for further discussion of these step-by-step time integration algorithms.