Convective Operator

 We now consider the two-dimensional linear convective equation for u(x,y,t):

$$\frac{\partial u(x,y;t)}{\partial t} + Lu \equiv \frac{\partial u}{\partial t} + a(x,y)\diff{u}{x} + b(x,y)\diff{u}{y} = 0.$$

This has been formulated for triangular elements in [20], and we note that using quadrilateral elements requires very few changes to this method. We consider the weak form of this equation in $\Omega^k$, that is:

Find $u \in H^1(\Omega)$ such that for all $w \in H^1(\Omega)$

$$\left(\frac{\partial u}{\partial t} - a\frac{\partial u}{\pa... ...tial y},w\right) = 0, \hspace{12pt} \forall w \in H^1(\Omega).$$

Following a Galerkin formulation so that the trial and test spaces are spanned by the same basis we obtain:

$$\left( \phi_n, \phi_m \right) \frac{d \hat{u}_m}{dt} = \lef... ... \frac{\partial \phi_m}{\partial y} \right) \right] \hat{u}_m.$$

We now define two local operators B_k and $L_k(\theta)$:

where we have normalized the convection velocity appropriately so that its direction $\theta$ is defined by the scalars a,b. Based on these definitions we now construct two new operators for the entire domain:

$$L(\theta) = \left[\begin{array} {cccc} L_1 & 0 & ... & 0 \... ..... & ... & ... & ... \ 0 & 0 & ... & L_K \end{array} \right]$$

and similarly:

$$B = \left[\begin{array} {cccc} B_1 & 0 & ... & 0 \ 0 & B_... ... & ... & ... & ... \ 0 & 0 & ... & B_K \end{array} \right].$$

We can now assemble these elemental operators into a global operator, by means of the Z operator [40] that assembles the local coefficients into the global coefficients and ensures C⁰ continuity.

  

Figure 4.6: Illustration of local and global numbering for a domain containing one quadrilateral and one triangular elements. Here the expansion order is N=3, and we only show the boundary modes.

To illustrate this global assembly procedure, we consider a global domain made up of two elements as shown in figure 4.6. The expansion order shown here is N=3 which means there are six boundary modes on the triangle, and eight boundary modes on the quadrilateral.

The total number of local degrees of freedom is therefore N_local = 14. Since three modes meet along the connecting edge the number of global degrees of freedom is eleven and so for this case Z is a $14 \times 11$ matrix as shown in figure 4.7.

  

Figure 4.7: Z-matrix map from global to local degrees of freedom

The superscripts denote the local or global nodal number and the subscripts denote the element number. The absolute column sum gives the multiplicity of a mode and we see that columns 5,6 and 7 all have a multiplicity of 2. We also note that the absolute row sum is always 1 since there is only one value of each local mode. It is also possible to have a (-1) entry if we have two elements where: (1) edge 1 meets edge 1, (2) edge 1 meets edge 2, (3) edge 2 meets edge 2, (4) edge 3 meets edge 3, (5) edge 3 meets edge 4, or (6) edge 4 meets edge 4, then the local co-ordinates at the edges are in opposite directions. If we are using a hierarchical basis then we need to negate odd modes along one of the edges.

Having defined the assembly operation, we can now write the global system as:

$$\frac{d \vect{\hat{u}}^{g}}{dt} = G(\theta) \vect{\hat{u}}^{g},$$

where $G(\theta) = (Z^t B Z)^{-1}(Z^t L(\theta) Z).$

The behaviour of this operator is very important with regards to determining the maximum time step we will be able to use for any problem which involves an explicit treatment of convective contributions.