2.3 Constrained nodes.



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2.3 Constrained nodes.

Each irregular edge contains a middle node that does not contain an autonomous degree of freedom, as, in order to preserve continuity, the solution at this node must be equal to the average of values at edge's endpoints (Figure 3). Such nodes (and degrees of freedom) are called constrained, as opposed to unconstrained, ``real'' nodes and degrees of freedom. For each constraint, the equation is used in the following form:

where - vector of constrained degrees of freedom (actual vector will appear in - case, only one degree of freedom is constrained here)

- vector of unconstrained, actual degrees of freedom (although, due to the propagation of constraints explained below, any actual degree of freedom may be constrained further along the other edge)

- constraint matrix, in our simple case:

The constraint matrix will grow bigger and more complex later in our presentation, but the above form will always be used.

As it is customary in FEM, the transformation matrices may change size when they are applied to a different number of nodes (or degrees of freedom). In our case, constraint matrices are a special case of transformation, so we can write the above constraint for all five nodes of the element shown in the Figure 3. (Due to this constraint, the node nr 5 also ``belongs'' to this element as the solution in the element depends on this node).

One can see our previous simple matrix , as the fourth row of the above equation.

  
Figure 3: Solution at the irregular node before and after applying the constraint.



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