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Multiple scales in time
• Use appropriate time scale in each S/D
• Simplest case: constant
steps, hierarchic case
(exact multiples: ∆ t1 = m∆t2)
• Enforce continuity on
finest time scale:
CU
CU
j j
1 1
+
2 2
= 0
at generic instant t j = t n + j∆t
• By decomposing the velocities this becomes:
CU + CU =− ( CU + CU
)
j j j j
1 1,link 2 2,link 1 1,free 2 2,free
2
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Multiple scales in time (2)
• Solve free problem for both S/Ds:
M U
j
= F M U
= F
j
1 1,free 1
2 2,free 2
1
t + t
for S/D 1 at time n
for S/D 2 at time j
⇒ U
1,free
and U
j
2,free
j
• Estimate free velocity of S/D 1 at t by linear interpolation:
U = (1 − α ) U + α U with α j / m
j
n
1,free j 1,free j 1,free
j
• Same thing may be done for link velocity:
U = (1 − α ) U + α U
j
n
1,link j 1,link j 1,link
120
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