## 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
 $\left(\begin{array}{cc}{\text{S}}_{11}& {\text{S}}_{12}\\ {\text{S}}_{21}& {\text{S}}_{22}\end{array}\right)\left(\begin{array}{c}{\text{w}}_{1}\\ {\text{w}}_{2}\end{array}\right)=\left(\begin{array}{c}{\text{f}}_{1}\\ {\text{f}}_{2}\end{array}\right)\phantom{\rule[-0.5ex]{1ex}{0.5ex}}.$ (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.

R. Funnell