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<H3><A NAME="SECTION00079700000000000000">
Direct integration dynamic analysis</A>
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In a direct integration dynamic analysis, activated by the <A HREF="node229.html#dynamic">*DYNAMIC</A> key word, the equation of motion is integrated in time using the <B><IMG
WIDTH="15" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img110.png"
ALT="$ \alpha$"></B>-method developed by Hilber, Hughes and Taylor [<A
HREF="node378.html#Miranda">53</A>]. The parameter <B><IMG
WIDTH="15" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img110.png"
ALT="$ \alpha$"></B> lies in the interval [-1/3,0] and controls the high frequency dissipation: <B><IMG
WIDTH="15" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img110.png"
ALT="$ \alpha$"></B>=0 corresponds to the classical Newmark method inducing no dissipation at all, while <B><IMG
WIDTH="15" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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ALT="$ \alpha$"></B>=-1/3 corresponds to maximum dissipation. The user can choose between an implicit and explicit version of the algorithm. The implicit version (default) is unconditionally stable.
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In the explicit version, triggered by the parameter EXPLICIT in the *DYNAMIC keyword card, the mass matrix is lumped, and a forward integration scheme is used so that the solution can be calculated without solving a system of equations. Each iteration is much faster than with the implicit scheme. However, the explicit scheme is only conditionally stable: the maximum time step size is proportional to the time a mechanical wave needs to cross the smallest element in the mesh. For linear elements the proportionality factor is 1., for quadratic elements it is <!-- MATH
$1/\sqrt{6}$
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<B><IMG
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SRC="img1022.png"
ALT="$ 1/\sqrt{6}$"></B>. For elastic materials, the wave speed in a rod is <!-- MATH
$\sqrt{E/\rho}$
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<B><IMG
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<ADDRESS>
guido dhondt
2016-07-31
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