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n+ 1 n ∆t
n n+
1
u = u + ( u + u
)
2
n+
1 n n ∆t
n
u = u +∆ t( u
+ u
)
2
Direct Time Integration (2)
• These formulas are a particularization of the well-known
Newmark integration formulas:
n+ 1 n n n+
1
u = u +∆t[(1 − γ) u + γu
]
2
n+ 1 n n ∆t
n n+
1
u = u +∆ tu + [(1 − 2 β) u + 2 βu
]
2
written for γ = 1/2 and β = 0 .
• These two equations, plus the equilibrium, may be solved
for u, u,
u
upon step-by-step marching in time.
• This particular choice for β renders the scheme explicit,
while the chosen γ ensures no numerical damping.
15
n+ 1 n ∆t
n n+
1
u = u + ( u + u
)
2
n+
1 n n ∆t
n
u = u +∆ t( u
+ u
)
2
Direct Time Integration (3)
How is the scheme used in practice?
1/ 2 t
v
• Introduce a mid-step velocity:
+ u
+
∆ u
2
which transforms configuration n into n+1 over the step.
• The second equation becomes:
u = u +∆t⋅v
n+ 1 n n+
1/2
n n n
• Carry on mid-step velocities rather than full-step ones. The
first equation becomes:
n+ 3/2 n+ 1/2 n+
1
v = v +∆t⋅
u
∆t ∆t ∆t ∆t
v u + u = u + u + u + u = v +∆t⋅u
2 2 2 2
n + 3/2 n + 1 n + 1 n n n + 1 n + 1 n + 1/2 n + 1
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