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FE Modelling and Simulation of Gasand Fluid Supported Structures 163
phg ρt
K = − N
2
e η ξ
T ( ∗ ng T + g ∗ n T )N dξdη
+ 1
⎛ ⎞ ⎞T ⎛
ξ η
N 0 W W
phgpt ⎝ N,ξ ⎠ ⎝ W
2
e η ξ N,η
ξT
0 0
W ηT
⎞ ⎛ ⎞
N
⎠ ⎝ N,ξ ⎠
0 0 N,η
dξdη,
a =
e
η
(52)
N
ξ
T ∗ nt dξdη, (53)
b =
phgαtNT ∗ nt dξdη, (54)
e
η
ξ
phg
f =
phgptNT ∗ nt dξdη, (55)
e
η
ξ
phg
αt = K
2V −
cpt −x pt
,
vt
(56)
phg
pt = c p − x p − h p. (57)
The nodal displacement vector for one iteration step is then given by a linear
combination of the three auxiliary solution vectors: d = d(d1, d2, d3). For
further details and the combination with arc-length schemes, we refer to [9]
and [5]. For the cases pneumatics, hydraulics and incompressible fluids with
open surfaces only a rank-one update is found. For pneumatics and hydraulics
the pressure terms due to gravity disappear with c p = x p =0, as well as
the first term in phg K,asg is not existent then. Depending on the pressure
equation for gas or fluid
k
h pt
αt = κ
vt
or
h αt = K
V
= const. (58)
and only a constant pressure term remains phg pt = − h pt.
For incompressible fluids phg K has the same structure as for compressible
fluids, but the pressure is dependent on the current coordinate of the fluid
only hg pt = x p = ρg(−xt). However, we have to note that incompressibility
can only be considered for fluids with open surfaces. For compressible fluids
with open surface the pressure term h p due to overpressure is not present.
Thenthevolumechange due to the fluid weight is included in the c p-term.
However, in this case we have to keep also track of the current height of the
fluid level, which is discussed in a forthcoming paper.
5 Numerical Examples
5.1 Pneumatic Multi-Chamber Structure Under Torsional Loading
A pneumatic multi-chamber structure see Fig. 1, two chambers with different
initial internal pressure (left pl0 = 0.01 bar,right pr0 =1bar, with material