How to apply non zero Dirichlet boundary condition in finite elements?

Question

I am writing a code for steady state heat transfer on a rectangular domain. I am specifying temperature on the edges - nonzero Dirichlet boundary condition. The equations can be written in form of

$$KT=Q$$ $K$ is conductivity matrix, $T$ is unknown nodal temperature vector, and $Q$ is thermal load vector consisting of internal heat generation. For example, the system can look like

$$\begin {bmatrix} K_{11} & K_{12} & K_{13} & \dots & K_{1N}\\ K_{21} & K_{22} & K_{23} & \dots & K_{2N}\\ \vdots\\ K_{N1} & K_{N2} & K_{N3} & \dots & K_{NN} \end{bmatrix} \begin {bmatrix} 100\\ 200\\ \vdots\\ T_N \end{bmatrix} = \begin {bmatrix} Q_1\\ Q_2\\ \vdots\\ Q_N \end{bmatrix} $$

Non zero temperatures are applied on the edges of the rectangular domain (non-homogeneous Dirichlet boundary condition). How can I handle computationally non-zero Dirichlet boundary condition to solve for unknown temperature vector?

Answer 1

Consider the fact that your linear system can be broken up into two parts: the equations that correspond to the known Dirichlet dofs, and the equations that correspond to the unknown dofs. For the sake of convenience, let's say that your unknown vector is broken up so the first $k$ values of $T$ are known, and the remainder are unknown. Let a "$u$" subscript denote the unknown indices of $T$, and let "$d$" denote the known (Dirichlet) indices in $T$. Then, the linear system can be written in the following way:

$$ \left[\begin{matrix} K_{dd} & K_{du} \\ K_{ud} & K_{uu} \end{matrix}\right] \left[ \begin{matrix} T_d \\ T_u \end{matrix}\right] = \left[\begin{matrix} Q_d \\ Q_u \end{matrix}\right] $$

Since the values of $T_d$ are known, we can immediately discard the equations in those rows, leading to this system:

$$ \left[\begin{matrix} K_{ud} & K_{uu} \end{matrix}\right] \left[ \begin{matrix} T_d \\ T_u \end{matrix}\right] = \left[\begin{matrix} Q_u \end{matrix}\right] $$

And then move your known values over to the right-hand side of the linear system:

$$ K_{uu}T_u = Q_u - K_{ud}T_d $$

Solve this reduced set of linear equations, and you've got your unknowns. Another technique to avoid mucking around with the original $K$ matrix too much is to zero out $K_{du}$ and $K_{ud}$, set $K_{dd}$ to the Identity matrix, and set $Q_d = T_d$. You still need to modify the entries of $Q_u$ in the same way as described above, but this removes the need to actually re-create the sparse matrix to only include the $K_{uu}$ block while still preserving the symmetry of the linear system.

It's worth mentioning that these techniques are still appropriate, even when the index sets "$d$" and "$u$" are non-contiguous. There's a slightly higher bookkeeping burden for you, but it's conceptually identical.