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.

Since it is more convenient to write, 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 for the dynamical problem is as follows:

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

where the right-hand side vector

$$\begin{bmatrix} \underline{F}^u \\ \underline{F}^{\theta} \end{bmatrix}$$

denotes 'forces' stemming from the mechanical volume force & Neumann boundary condition and from the thermal heat source & Neumann boundary condition.

Dirichlet boundary conditions