Given the set of L linear simultaneous equations in unknowns uj subject to the linear constraint equation (input on CE command)
 | (14–182) |
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 |
 | (14–183) |
normalize Equation 14–183 with respect to the prime degrees of freedom ui by dividing by Ci to get:
 | (14–184) |
where:
 |
 |
which is written to a file for backsubstitution. Equation 14–184 is expanded (recall
= 1) as:
 | (14–185) |
Equation 14–182 may be similarly expanded as:
 | (14–186) |
Multiply Equation 14–185 by Kki and subtract from Equation 14–186 to get:
 | (14–187) |
Specializing Equation 14–187 for k = i allows it to be written as:
 | (14–188) |
This may be considered to be a revised form of the constraint equation. Introducing a Lagrange multiplier λk, Equation 14–187 and Equation 14–188 may be combined as:
 | (14–189) |
By the standard Lagrange multiplier procedure (see Denn([8])):
 | (14–190) |
Solving Equation 14–185 for ui,
 | (14–191) |
so that
 | (14–192) |
Substituting Equation 14–192 into Equation 14–189 and rearranging terms,
 | (14–193) |
or
 | (14–194) |
where:
|
|
The constraint equation described by Equation 14–183 can also be written in the following matrix form:
 | (14–195) |
where [C] can be further composed into the slave and master DOFs so that the direct elimination method can be used. (In the direct elimination method, Equation 14–193 is used to solve equation systems Equation 14–182 and Equation 14–183 together.)
Equation 14–195 can be re-written as:
 | (14–196) |
where {Us} is a displacement of the slave DOFs, and {Um} is the master DOF.
If external CE or CP commands are issued, the user must choose which DOFs are slave and which are master (see the CE/CP command descriptions). In many applications, the ANSYS program automatically uses the CE command and invokes the method of automatic selection.
When solving the equation with the direct elimination method, the {Us} variables can be removed from the system by applying the following transformation:
 | (14–197) |
Because the choice of {Us} is not unique, the program selects {Us} slave DOFs that ensure that Equation 14–183 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 {Us} cannot be selected that makes Equation 14–182 and Equation 14–183 solvable together, the equation system is overconstrained. Typically, overconstraint is caused by contradictory constraint equations or an insufficient number of slave DOFs.
The ANSYS program has two automatic methods (which can be selected via the OVCHECK command) to compute the slave DOFs and detect overconstrained DOFs:
- The Topological Method (OVCHECK,TOPO)
This method is used by default. This simple method requires little computational time to check for overconstraints and determine unique slave DOFs. This method starts by traversing the topology (sparsity) pattern of the [C] matrix (Equation 14–195) row-by-row. During this process, the first possible DOFs visited are considered slave DOFs. If the slave DOF has been used (or appeared) before, then it selects the next possible DOF visited as a slave DOF. Although this method traverses the entire matrix, it is computationally inexpensive because no numerical computations are involved.
This method does not guarantee that overconstraint or unique slave DOFs can be determined for the entire equation system. If this method fails to detect slave DOFs, warning or error messages display in the program output files.
This method only traverses the constraint equation system and does not visit matrix coefficients associated with the P variable (i.e., Lagrangian multipliers). As a result, the topological method cannot detect overconstraints associated with u-P formulation elements.
- The Algebraic Method (OVCHECK,ALGEBRAIC)
This is a mathematical method that requires more computational time than the topological method.
The slave DOFs are chosen using 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.
In cases where a model uses the CE/CP command and u-P formulation elements, overconstraint could also come from the P DOFs, in addition to CE/CP overconstraint. Detection of overconstraint in this case is similar to the process described above; however, only the algebraic method supports this detection. The algebraic method packs CE/CP and matrix coefficients from the Lagrange multiplier equations into one assembled [C] matrix (Equation 14–195), then it executes the algebraic method process described above into the [C] matrix.
If CE/CP values are invalid or redundant, they are removed from equation (Equation 14–182) 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.
If some P constraint equations are invalid or redundant, the application stops and the user is notified which elements (e.g., MPC184, CONTA174, etc.) are causing overconstraint (i.e., the redundant equations).