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 :