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Finding the Lagrange Multipliers (2)
• The CD scheme for the velocity and constant ∆t is:
v = v +∆t⋅a
n+ 3/2 n+ 1/2 n+
1
• Substituting this into the constraint Cv = b gives:
n+ 3/2 n+ 1/2 n+
1
Cv = Cv +∆t⋅ Ca = b
• From this we obtain:
1 n+ 1/2 1
n+
1/2
Ca = ( b − Cv ) = ( b −Cv
)
∆t
γ
• For a variable ∆t in time, one has simply:
n
∆ t +∆t
γ =
2
n+
1
having
posed:
γ =∆t
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Finding the Lagrange Multipliers (3)
• Summarizing, the Lagrange multipliers λ are obtained by
solving the linear algebraic system:
We obtain one
*
B λ = w
multiplier for each
imposed constraint
where the known terms are given by:
* − 1 T
1
n+ 1/2 −1 e i
B ≡Cm C and w ≡ ( b −Cv ) −Cm ( f − f )
γ
γ =∆t
for constant ∆t
n n+
1
γ = ( ∆ t +∆t ) 2 for variable ∆t
in time
We obtain one
• Finally we compute the reactions:
reaction for each
T
r = C λ
constrained dof
and add them to the other known external forces.
• This is the only implicit part of the whole method.
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