Finite Element discretization and Dirichlet boundary conditions

Getting rid of the test function

In this section, we will adopt a few notations for simplicity. First, the global vector of inknown degrees of freedom will be written Since it is more convenient to write, in this page, the global vector of unknown degrees of freedom will be written

$$\begin{bmatrix} \underline{U}^{f} \\ \underline{\theta}^{f} \\ \underline{U}^{d} \\ \underline{\theta}^{d} \end{bmatrix} = \begin{bmatrix} \underline{\delta}^{f} \\ \underline{\delta}^{d} \end{bmatrix}$$

assuming the degrees of freedom have been organized with the free, unconstrained DOFs (superscript $\ f$) of the displacement and temperature given first and then the DOFs constrained by Dirichlet conditions second (superscript $\ d$).

The test function is assumed to follow a similar ordering, denoted

$$\begin{bmatrix} \underline{U}^{*\textrm{f}} \\ \underline{\theta}^{*\textrm{f}} \\ \underline{U}^{*\textrm{d}} \\ \underline{\theta}^{*\textrm{d}} \end{bmatrix} = \begin{bmatrix} \underline{\delta}^{*f} \\ \underline{\delta}^{*d} \end{bmatrix}$$

The global force vector $\ \underline{F}$ follows the same arrangement. The global finite element matrices this adopt a block decomposition according to these two groups of DOFs as well.

We denote by $\ \left< \cdot \; , \; \cdot \right>$ the scalar product of two vectors.

The discretized weak form gives the following equation:

$$\left< [M^g] \; \begin{bmatrix} \underline{\ddot{\delta}}^{f} \\ \underline{\ddot{\delta}}^{d} \end{bmatrix} \; , \begin{bmatrix} \underline{\delta}^{*f} \\ \underline{\delta}^{*d} \end{bmatrix} \right> + \left< [D^g] \; \begin{bmatrix} \underline{\dot{\delta}}^{f} \\ \underline{\dot{\delta}}^{d} \end{bmatrix} \; , \begin{bmatrix} \underline{\delta}^{*f} \\ \underline{\delta}^{*d} \end{bmatrix} \right> + \left< [K^g] \; \begin{bmatrix} \underline{\delta}^{f} \\ \underline{\delta}^{d} \end{bmatrix} \; , \begin{bmatrix} \underline{\delta}^{*f} \\ \underline{\delta}^{*d} \end{bmatrix} \right> = \left< \begin{bmatrix} \underline{F}^{f} \\ \underline{F}^{d} \end{bmatrix} \; , \; \begin{bmatrix} \underline{\delta}^{*f} \\ \underline{\delta}^{*d} \end{bmatrix} \right>$$

with $\ [M^g] \; , \; [D^g] \; , \; [K^g]$ the global mass, damping and stiffness matrices.

Since by definition of the chosen test function, it is null for degrees of freedom belonging to parts of the boundary where a Dirichlet condition is applied, the test function has a zero value for all these constrained DOFs, giving us:

$$\left< [M^g] \; \begin{bmatrix} \underline{\ddot{\delta}}^{f} \\ \underline{\ddot{\delta}}^{d} \end{bmatrix} \; , \begin{bmatrix} \underline{\delta}^{*f} \\ \underline{0} \end{bmatrix} \right> + \left< [D^g] \; \begin{bmatrix} \underline{\dot{\delta}}^{f} \\ \underline{\dot{\delta}}^{d} \end{bmatrix} \; , \begin{bmatrix} \underline{\delta}^{*f} \\ \underline{0} \end{bmatrix} \right> + \left< [K^g] \; \begin{bmatrix} \underline{\delta}^{f} \\ \underline{\delta}^{d} \end{bmatrix} \; , \begin{bmatrix} \underline{\delta}^{*f} \\ \underline{0} \end{bmatrix} \right> = \left< \begin{bmatrix} \underline{F}^{f} \\ \underline{F}^{d} \end{bmatrix} \; , \; \begin{bmatrix} \underline{\delta}^{*f} \\ \underline{0} \end{bmatrix} \right>$$

We can develop these scalar products, remove the zero-valued terms stemming from the test function on the constrained DOFs, and use the fact that the equation holds for every test function (as long as it is zero-valued on the constrained part of the boundary) to get the following general discretized system with a particular block decomposition for the matrices:$

$$\begin{bmatrix} [M^{ff}] & [M^{f d_u}] & [M^{f d_\theta}] \end{bmatrix} \begin{bmatrix} \underline{\ddot{U}}^{f} \\ \underline{\ddot{\theta}}^{f} \\ \underline{\ddot{U}}^{d} \\ \underline{\ddot{\theta}}^{d} \end{bmatrix} + \begin{bmatrix} [D^{ff}] & [D^{f d_u}] & [D^{f d_\theta}] \end{bmatrix} \begin{bmatrix} \underline{\dot{U}}^{f} \\ \underline{\dot{\theta}}^{f} \\ \underline{\dot{U}}^{d} \\ \underline{\dot{\theta}}^{d} \end{bmatrix} + \begin{bmatrix} [K^{ff}] & [K^{f d_u}] & [K^{f d_\theta}] \end{bmatrix} \begin{bmatrix} \underline{U}^{f} \\ \underline{\theta}^{f} \\ \underline{U}^{d} \\ \underline{\theta}^{d} \end{bmatrix} = \underline{F}^{f}$$

where, in the matrices, superscript $\ f$ relates to unconstrained, free DOFs, and where $\ d_{u}$ and $\ d_{\theta}$ superscripts relate to DOFs where displacement and temperature Dirichlet conditions are applied respectively.

Vectors $\ \underline{U}^d$ and $\ \underline{\theta}^d$ are assumed known and constant, thus zero-ing their time derivatives. We can develop their matrix products and put them on the right-hand side, yielding:

$$[M^{ff}] \underline{\ddot{\delta}}^f + [D^{ff}] \underline{\dot{\delta}}^f + [K^{ff}] \underline{\delta}^f = \underline{F}^{f} - [K^{f d_u}] \underline{U}^d - [K^{f d_\theta}] \underline{\theta}^d$$

For simplicity, from then on, we crite the previous system equation as such:

$$[M] \underline{\ddot{\delta}} + [D] \underline{\dot{\delta}} + [K] \underline{\delta} = \underline{F}$$

where $\ [M] \; , \; [D] \; , \; [K]$ being blocks of the global mass, damping and stiffness matrices corresponding to free DOFs, respectively, and where $\ \underline{F}$ is the force vector on these DOFs, incorporating the force terms stemming from Dirichlet conditions, as per the previous equations.

Solvable coupled governing equations

For now, we assume the reader knows how to discretize the weak form of the equation to obtain the element matrices' equations 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 shows a block decomposition for the matrices 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, the thermal heat source as well as Neumann and Dirichlet boundary conditions.

All matrix blocks have an expression at the element level (which for now, we assume the reader knows how to get), except $\ [D^{uu}]$ which is an added Rayleigh structural damping term whose expression is

$$[D^{uu}] = \alpha_M \; [M^{uu}] + \alpha_K \; [K^{uu}]$$

where $\ \alpha_M$ and $\ \alpha_K$ are damping ratios to be determined.