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Multiple scales in frequency
• Some large (CPU!) problems contain multiple frequency scales.
• Impact raises high frequencies
and non-linear effects locally.
• However, the bulk structure has
low-frequency vibratory
behaviour (linear).
• Fine FE discretization (to fit
complex geometry) is too detailed
to describe linear vibrations.
• Use modal projection: replace FE unknowns (dofs) by less
numerous ones, specific to the chosen frequency spectrum.
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Multiple scales in frequency (2)
• Domain decomposition allows to separate the frequency scales.
• How can one suitably couple a linear, “modal” S/D with a nonlinear,
“direct” one? One arrives to a form (S/D 1 is modal):
⎡ ∆t ˆ
∆t ˆ T ⎤ ⎡ ∆t
ˆ ⎤
⎢
M1 0 − C1 1 1
2 2 ⎥ ⎢
Kα
2 ⎥
⎢ ⎥⎡
α1
⎤ ⎢ ⎥
⎢ ∆t ∆t T
∆t
0 M2 − C ⎥⎢
2
U
⎥
2
= ⎢ F ⎥
2
⎢ 2 2 ⎥⎢
⎥ ⎢ 2 ⎥
⎢
t ˆ t
⎥⎢⎣Λ
⎥⎦
⎢
ˆ p n p n⎥
⎢
∆ ∆
− C
1 1 2 2
1
− C2
0 ⎥ ⎢C α + C U
⎥
⎢⎣ 2 2
⎥⎦ ⎢⎣ ⎥⎦
• This coupling is totally independent
from the nature of the base used for
modal reduction (free choice!).
• This is possible thanks to dual approach:
interface dofs don’t need to remain “visible”
after reduction.
α1 generalized unknowns:
⎧U
1
=Φ1α
1
⎪
⎨U
1
=Φ
1α
1
⎪
U
⎩ 1
=Φ
1α
1
linear behaviour:
F = KU = KΦα
1 1 1 1 1 1
"projected" operators:
ˆ T
⎧M1 Φ1M1Φ1
⎪ ˆ T
⎨K1 Φ1K1Φ1
⎪
Cˆ
⎪⎩ 1
C1Φ
1
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