Constraints are generally implemented using the Lagrange Multiplier Method (See Belytschko([349])). 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:
| W' = augmented potential |
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 ntot equations, where:
 | (3–122) |
where:
| ndof = number of degrees of freedom in the model |
| nc = number of Lagrange multipliers |
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. |