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32.4.3 Hilber-Hughes-Taylor

The Hilber-Hughes-Taylor method [27] (also called $ \alpha$ -method) is an extension to the Newmark method. With the Hilber-Hughes-Taylor method it is possible to introduce numerical dissipation without degrading the order of accuracy. The Hilber-Hughes-Taylor method uses the same finite difference formulas (32.41) and (32.42) as the Newmark method with fixed $ \gamma$ and $ \beta$ ( $ \gamma$ = $ {\tfrac{{1}}{{2}}}$(1 - 2$ \alpha$) , $ \beta$ = $ {\tfrac{{1}}{{4}}}$(1 - $ \alpha$)2 ). The time-discrete equation of motion is modified as follows:

\begin{displaymath}\begin{split}\mathbf{M} \: ^{{t+{\Delta t}}}\ddot{\mathbf{u}}...
...\: ^{t+(1+\alpha)\Delta t}\mathbf{f}_{\mathrm{ext}} \end{split}\end{displaymath} (32.46)

For $ \alpha$ = 0 the method reduces to the Newmark method. For - $ {\tfrac{{1}}{{3}}}$ $ \leq$ $ \alpha$ $ \leq$ 0 , $ \gamma$ = $ {\tfrac{{1}}{{2}}}$(1 - 2$ \alpha$) , and $ \beta$ = $ {\tfrac{{1}}{{4}}}$(1 - $ \alpha$)2 the scheme is second order accurate and unconditionally stable. Decreasing $ \alpha$ means increasing the numerical damping. This damping is low for low-frequency modes and high for the high-frequency modes.



DIANA-9.4.2 User's Manual - Analysis Procedures
First ed.

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