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Multiple scales in time (3)
• Express link velocities via link accelerations (multipliers).
For S/D 2:
j ∆t2
1 T j
U
2,link
= M −
2
C2
Λ
2
• For S/D 1:
U
∆t
= M C Λ
2
∆t1
1 T
= M − C Λ
2
n 1 − 1 T n
1,link 1 1
at time t
n
1
1,link 1 1 at time t n+
U
• This, replaced into previous linear interpolation, yields:
j t1
1 T [(1 ) n
U
∆ −
]
1,link
= M1 C1 −α j
Λ + α
jΛ
2
Λ
j
121
Multiple scales in time (4)
• This corresponds to assuming that the interface stresses vary
n n 1
linearly between any two “coarse” instants and .
• Finally we obtain the linear system:
H
j
Λ =
B
j
∆t2
−1 T
−1
T
H ( mC1M 1
C1 + C2M 2
C2
)
2
B − ( CU
+ CU
)
j j j
1 1,free 2 2,free
t
t +
122
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