Finite Element discretization and Dirichlet boundary conditions

Discretized coupled governing equations

For now, we assume the reader knows how to discretize the weak form of the equation, to obtain the element matrices and then the global finite element matrices. A more detailed description might be added later.

In this page, the global vector of unknown degrees of freedom will be written $\ \begin{bmatrix} \underline{U} \\ \underline{\theta} \end{bmatrix} $, assuming the degrees of freedom have been organized with the displacement DOFs first and then the temperature DOFs.

The discretized system of equations is as follows:

$$\begin{bmatrix} [M^{uu}] & [0] \\ [0] & [0] \end{bmatrix} \; \begin{bmatrix} \underline{\ddot{U}} \\ \underline{\ddot{\theta}} \end{bmatrix} + \begin{bmatrix} [D^{uu}] & [0] \\ [D^{\theta u}] & [D^{\theta \theta}] \end{bmatrix} \; \begin{bmatrix} \underline{\dot{U}} \\ \underline{\dot{\theta}} \end{bmatrix} + \begin{bmatrix} [K^{uu}] & [K^{u \theta}] \\ [0] & [K^{\theta \theta}] \end{bmatrix} \; \begin{bmatrix} \underline{U} \\ \underline{\theta} \end{bmatrix} = \begin{bmatrix} \underline{F}^u \\ \underline{F}^{\theta} \end{bmatrix}$$

Dirichlet boundary conditions