Page 100 - Process Modelling and Simulation With Finite Element Methods

Partial Differential Equations and the Finite Element Method 87

(2.47)

Now for the clever trick. The stiffness matrix equation is augmented by a vector
of unknowns A, called the Lagrange multipliers, which multiply NT, where the
superscript T means transpose:

K (uo (u - u, ) + N (u, >’ A = L (uo (2.48)
)
)
Why is this clever? Well, if the constraint (2.45) is satisfied, then there is a
unique set of Lagrange multipliers satisfying (2.48). (2.46) and (2.48) permit the
simultaneous solution of more than just boundary conditions, however. Any
constraint - internal pointwise, subdomain integral, edge or boundary that can be
expressed in weak form can be treated by the Lagrange multiplier method.

Lagrange Multipliers
So how do Lagrange multipliers ensure that M(U)=O is satisfied? By a variational
principle. With A=0 (2.48) is equivalent to the minimum principle for

(2.49)

If we wish to ensure the constraint (2.45) is satisfied simultaneously, then we add
a weighted penalty to (2.49) for the extent to which M(U)=O is not satisfied. The
weights are called Lagrange multiplers A.

-U~L+A.M (2.50)

Now we use the linearization of M(U)=M(Uo)+N(U,)(U-Uo) to simplify (2.50).
Note that constant terms do not contribute to the minimization.

(2.51)

The minimization (2.51) is then equivalent to (2.48) , i.e. the solution to (2.48)
minimizes (2.51). In the parlance of FEM, (2.51) is the “minimization in the
energy”, i.e. weighted by the stiffness matrix K. It should not be confused with
the least squares minimization, which by analogy with (2.50) is

(2.52)