Given the set of L linear simultaneous equations in unknowns uj subject to the linear constraint equation (input on CE command)
 | (14–198) |
where:
| Kkj = stiffness term relating the force at degrees of freedom k to the displacement at degrees of freedom j |
| uj = nodal displacement of degrees of freedom j |
| Fk = nodal force of degrees of freedom k |
| k = equation (row) number |
| j = column number |
| L = number of equations |
 | (14–199) |
normalize Equation 14–199 with respect to the prime degrees of freedom ui by dividing by Ci to get:
 | (14–200) |
where:
 |
 |
which is written to a file for backsubstitution. Equation 14–200 is expanded (recall
= 1) as:
 | (14–201) |
Equation 14–198 may be similarly expanded as:
 | (14–202) |
Multiply Equation 14–201 by Kki and subtract from Equation 14–202 to get:
 | (14–203) |
Specializing Equation 14–203 for k = i allows it to be written as:
 | (14–204) |
This may be considered to be a revised form of the constraint equation. Introducing a Lagrange multiplier λk, Equation 14–203 and Equation 14–204 may be combined as:
 | (14–205) |
By the standard Lagrange multiplier procedure (see Denn([8])):
 | (14–206) |
Solving Equation 14–201 for ui,
 | (14–207) |
so that
 | (14–208) |
Substituting Equation 14–208 into Equation 14–205 and rearranging terms,
 | (14–209) |
or
 | (14–210) |
where:
|
|
The constraint equation described by Equation 14–199 can also be written in the following matrix form:
 | (14–211) |
where [C] can be further composed into the dependent and independent degrees of freedom so that the direct-elimination method can be used. (In the direct elimination method, Equation 14–209 is used to solve equation systems Equation 14–198 and Equation 14–199 together.)
Equation 14–211 can be rewritten as:
 | (14–212) |
where {UD} is a displacement of the dependent degrees of freedom, and {UI} is the independent degree of freedom.
If external CE or CP commands are issued, the user must choose which degrees of freedom are dependent and which are independent (CE/CP). In many applications, Mechanical APDL automatically creates constraint equations and invokes the method of automatic selection.
When solving the equation with the direct elimination method, the {UD} variables can be removed from the system by applying the following transformation:
 | (14–213) |
Because the choice of {UD} is not unique, the program selects {UD} dependent degrees of freedom that ensure that Equation 14–199 has the best possible matrix conditioning (to avoid an ill-conditioned matrix) and fewer fill-ins when the sparse direct solver is used.
If a value for {UD} cannot be selected that makes Equation 14–198 and Equation 14–199 solvable together, the equation system is overconstrained. Typically, overconstraint is caused by contradictory constraint equations or an insufficient number of dependent degrees of freedom.
To calculate the dependent degrees of freedom and detect overconstrained degrees of freedom, Mechanical APDL uses two different methods automatically:
- The Topological Method
This simple method requires little computational time to check for overconstraints and determine unique dependent degrees of freedom. This method starts by traversing the topology (sparsity) pattern of the [C] matrix (Equation 14–211) row-by-row. During this process, the first possible degrees of freedom visited are considered dependent degrees of freedom. If the dependent degree of freedom has been used (or appeared) before, then it selects the next possible degree of freedom visited as a dependent degree of freedom. Although this method traverses the entire matrix, it is computationally efficient because no numerical computations are involved.
This method does not guarantee that overconstraint or unique dependent degrees of freedom can be determined for the entire equation system. If this method fails to detect dependent degrees of freedom, it swtches to the algebreic method automatically.
- The Algebraic Method
This is a mathematical method that requires more computational time than the topological method.
The dependent degrees of freedom are chosen using a QR factorization with full pivoting applied to the [C] matrix. If one pivot is too small (for example, lower than 1.e-08 proportionally), the associated constraint equation is considered redundant and removed from the set of constraint equations that must be satisfied.
If redundant constraint equations are detected, the program prints information regarding their removal.
Note that in cases where a model uses the CE/CP command and u-P elements involving Lagrange multipliers (for example, mixed u-P formulations), overconstraint could also come from the Lagrange multipliers (such as pressure degrees of freedom from the mixed u-P formulations), in addition to CE/CP overconstraint.
If CE/CP values are invalid or redundant, they are removed from equation (Equation 14–198) automatically, and the invalid or redundant CE/CP will not be used in the entire analysis process, which includes the Newton-Raphson nonlinear convergence loop. Additionally, if the invalid or redundant CE/CP are user defined, the application stops with an error message.