Download full text - ELSA - Europa

Direct Time Integration (4)
• The final algorithm reads:
u = u +∆t⋅v
n+ 1 n n+
1/2
u M ( f B σ dV )
= −∑ ∫
n+ 1 − 1 ext( n+ 1) T e( n+
1)
e e( n+
1)
V
v = v +∆t⋅
u
n+ 3/2 n+ 1/2 n+
1
A new configuration is obtained first. On this known configuration,
equilibrium is enforced. The new mid-step velocity is obtained last.
• This scheme is explicit.
How does one
( 1)
obtain σ en+
?
(see below)
• If ∆t varies in time, the only change is in the third
equation, which becomes:
n n+
1
n n+
1 n
n+ 3/2 n+ 1/2 ∆ t +∆t
n+
1
∆t ≡t −t
v = v + ⋅ u with:
2
∆t ≡t −t
n+ 1 n+ 2 n+
1
17
Scheme start-up and marching
u 0
0
σ
u , u
0 0
step 0 1 2
0
t
0 0 0
∆t
u , σ , u
, ∆t
are given
n =−1
0 −1 ext int 0
u
= M ( f − f )
u = u +∆t⋅v
1/2
v
n+ 1 n n+
1/2
u
= M ( f − f )
n+ 1 − 1 ext int n+
1
v = v +∆t⋅
u
n+ 3/2 n+ 1/2 n+
1
u 1
1
σ
1
u
t
3/2
v
u 2
2
σ
2
u
1/2 3/2 N
fin
v = u
+ ( ∆t/2)
⋅u
n← n+
1
1 0 1/2
u = u +∆t⋅v
u
= M ( f − f )
1/2 0 0
1 −1 ext int 1
v = v +∆t⋅
u
etc.
3/2 1/2 1
t
18
t
9