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Essential Boundary Conditions
Essential conditions are imposed via Lagrange multipliers.
• Assume a linear set of constraints on the velocities:
Cv = b
• Both C and b are known, and may be function of time.
• The equilibrium equations for the subset of d.o.f.s
concerned become, introducing unknown reactions r :
e i
ma = f − f + r
• Without loss of generality, the unknown reactions can be
expressed via a vector λ of Lagrange multipliers:
r =
T
C λ
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Finding the Lagrange Multipliers
• Replacing into the equilibrium equations yields:
e i T
ma = f − f + C λ
• Multiplying both members by Cm −1 gives:
−1 e i −1
T
Ca = Cm ( f − f ) + Cm
C λ
B
*
• The Lagrange multipliers are obtained symbolically from:
* −1 e i
B λ = Ca−Cm ( f − f )
Matrix of
connections
• To obtain λ , we must first express the term Ca as a
function of known quantities, by using the constraint and
the time integration scheme.
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