5. Assembly of system equation (cont'd)

The system boundary conditions now consist of prescribed values for some of the \(w_i\). One way to handle these constrained \(w_i\) is to remove them from the vector of unknowns – with appropriate node numbering, \(\mathbf w\) can be partitioned into two parts, the unknown displacements \(\mathbf w _1\) and the prescribed values \(\mathbf w _2\). Partitioning \(\mathbf S\) and \(\mathbf f\) correspondingly, we obtain

$\left(\begin{array}{cc}{\text{S}}_{11}& {\text{S}}_{12}\\ {\text{S}}_{21}& {\text{S}}_{22}\end{array}\right)\left(\begin{array}{c}{\text{w}}_1\\ {\text{w}}_2\end{array}\right)=\left(\begin{array}{c}{\text{f}}_1\\ {\text{f}}_2\end{array}\right).$

(Eqn. 13)

This then gives

${\text{S}}_{11}{\text{w}}_1={\text{f}}_1-{\text{S}}_{12}{\text{w}}_2\text{.}$

(Eqn. 14)

The system to be solved is now smaller.

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