Download full text - ELSA - Europa

Time integration
Each time increment is split into three phases:
v
w
1. Explicit Lagrangian phase: by posing
. w=
v all transport terms vanish
2. Implicit Lagrangian phase (see details
below) whereby the pressure is
iteratively refined (this stabilizes the
solution)
3. Convective flux phase whereby
the transport term contributions
are added
• This scheme is not as limpid as the purely Lagrangian one.
Second-order accuracy guaranteed only for phase 1.
Λ
v
Λ p ← f( p)
Λ
w
15
Time integration (2)
n+ 1/2 n−1/2
0. First compute v = v +∆ta n , where:
F old t
momentum transport forces at the end of
previous step
n
Fp
pressure forces
n
F b
body forces
n
F s
a
n
F + F + F + F
=
M
old n n n
t p b s
surface forces
1. Obtain new “Lagrangian” configuration:
x = x +∆t⋅v +
L n n 1/2
2. Evaluate L-volume and L-density (mass M is constant!):
L L L
L n L
V = V ( x ) ρ = M / V
3. From the internal energy eqn. without the transport term, using the
divergence theorem:
d
( ρ iV ) =− ( p + q) v ndS
dt
∫ i
S
pseudo-viscous pressure
16
8