Constraints are generally implemented using the Lagrange Multiplier Method (See Belytschko([345])). This formulation has been implemented in MPC184 as described in the Element Reference. In this method, the internal energy term given by Equation 3–90 is augmented by a set of constraints, imposed by the use of Lagrange multipliers and integrated over the volume leading to an augmented form of the virtual work equation:

(3–119)

where:

and

(3–120)

is the set of constraints to be imposed.

The variation of the augmented potential is zero provided (and, hence ) and, simultaneously:

(3–121)

The equation for augmented potential (Equation 3–119) is a system of n_tot equations, where:

(3–122)

where:

The solution vector consists of the displacement degrees of freedom and the Lagrange multipliers.

The stiffness matrix is of the form:

(3–123)

where:

= increments in displacements and Lagrange multiplier, respectively.