Time step size
Wave Propagation in 3D - Continuum Wave propagation velocity in 3D-continuum:
comparison to rod :
Critical time step :
comparison of critical time steps
materials (ν = 0.5): α --> 0
Wave Propagation in Plane Media
Wave propagation velocity in 2D-continuum:
(twodimensional stress state)
comparison to rod :
- Solid elements : c 3D-continuum
- Shell elements : c 2D-continuum
- Beams & trusses : c rod
Remarks:
- The wave propagation velocity of the rod crod has the smallest value in comparison to the 2D - and 3D-continuum.
- The wave propagation velocity for membrane deformations determines the critical time step for shell and beam elements.
Time Step Control for Beam and Truss Elements
For the Hughes-Liu beam and truss elements, the time step size is given by:
where L is the length of the element and c is the sound speed:
For the Belytschko beam the time step size given by the longitudinal sound speed is used, unless the bending-related time step size given by [Belytschko and Tsay 1982] governs
is smaller, where I and A are the maximum value of the moment of inertia and area of the cross section, respectively.
Characteristic length lc for Time Step
warped elements :
several alternatives can be selected via *CONTROL_TIMESTEP variable ISDO (Control Card 9, Columns 21-30), e.g.:
where β = 0 for quadrilateral and β = 1 for triangular shell elements.
Time Step Control for Solid Shell Elements
A critical time step size, Δ te is computed for solid shell elements from
where Ve is the element volume, Aemax ist the area of the largest side, and c is the plane stress sound speed
Critical Time Step for Spring Elements
Problem : There is no wave propagation velocity c to calculate critical time step size.
Motivation : Consider free vibration of spring with nodal mass m1 and m2
Recall critical time step of rod :