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FE Modelling and Simulation of Gas and Fluid Supported Structures 159
St S is the size of the water surface, which can also be computed via a boundary
integral over the enclosure of the fluid volume projected onto the direction of
gravity
∗nt
St S = · g
dξdη. (26)
|g|
η
ξ
Thus the change in the water level height can be written as
∆ o u = ∆o
v 1
=
St S St S
∗nt
· ∆u dξdη (27)
and the corresponding pressure change becomes
η
ξ
∆p = ρg · ∆ o u − ρg · ∆u
= ρ |g|
∗nt
· ∆u dξdη − ρg · ∆u. (28)
St S
η
ξ
The linearized variational form of the gravity potential depending on the fluid
level is then obtained as
δ g Π ∆p
lin =
η
ξ
∆p ∗ nt · δu dξdη
= ρ |g|
δu ·
St S η ξ
∗
nt dξdη
η ξ
∗nt
· ∆u dξdη
−ρ δu · ∗ nt g · ∆u dξdη. (29)
η
ξ
Obviously the second part of this equation is a non-symmetric term. However,
combining both non-symmetric parts of δgΠ ∆n
lin and δgΠ ∆p
lin , a symmetric expression
results for the complete sum
δ g Πlin = δ g Π ∆n
lin + δ g Π ∆p
lin + δgΠt Π
= δ g Πt Π +
+ ρ |g|
δu ·
St S η ξ
∗
∗nt
nt dξdη · ∆u dξdη Term I
η ξ
− ρ
2 η ξ
⎛ ⎞ ⎛
ξ η
gpt δu 0 W W
+ ⎝ δu,ξ ⎠ · ⎝ W
η ξ 2
δu,η
ξT
0 0
W ηT
⎞ ⎛ ⎞
∆u
⎠ ⎝ ∆u,ξ ⎠ dξdη.
0 0 ∆u,η
δu · ( ∗ nt ⊗ g + g ⊗ ∗ nt)∆u dξdη Term II
Term III (30)